EXAMPLES FOR PRACTICE. 1. Divide 6391 by 77; 21636 by 36; 6384 by 76; and 31250 by 250. 2. Divide 4752000 by 49; 4500 by 9; 3654 by 38; and 58469 by 981. 3. Divide 1755 by 39; 646 by 34; 2160 by 30; and 365:55 by 5.5. 4. Divide 36808 by 127; 147392 by 440; 70009 by 630; and 16882 by 734. 5. Divide 248-25 by 364·87; 32:08 by 8·8; 7. 5·6949 by 53·058; and 3876000 by 1200. 87641 by 368; and 147000 by 1470. 6. Divide 2020 by 202; 7156 by 2.68878; Divide 135056 by 734; 8746'9 by 36′4; 674.80 by '0763; and 3372 36 by 5'37. 8. Divide 06314 by 0007241; 0472 by 3'12; 03755 by 025; and 048869 by 0071698. 9. 1972; 1972; 1972; and 190072. 10. II. 12. 001237 108.46; 287642834'56; 47232′2; and 1001'1÷99'3. When it is proposed to find the value of an expression in which both multiplication and division are signified, the sum of the logarithms of the factors of the dividend, diminished by the sum of the logarithms of the factors of the divisor will be the logarithm of the value required. It is very often expedient to transform the logarithm of a divisor into that of a multiplier, and it is customary, in such calculations, to avoid not only negative logarithms, but negative indices also, by substituting for a subtraction logarithm its arithmetical complement (See page 46). This makes the operation consist of a single addition; only we must diminish the result by subtracting 10 for every arithmetical complement that has been used. To apply this method to the example above:---Having found in the Table the log. of the divisor 287, we may at once transform it into the addition logarithm 7.542118, and similarly for the log. of 2101 we may write 6.677574, and then the calculation will proceed continuously as follows: 14. Divide 06314 X 7438 X 102367 by 007241 X 12'9476 × 496523, and compare the result with the product of 8.71979 X 057447 X 0206168. TABLES OF NATURAL SINES, ETC. Trigonometrical magnitudes are numbers capable of being calculated from geometrical principles, and accordingly, Tables, called "Tables of Natural Sines," have been computed, in which the value of the Sines, Cosines, &c., of every degree and minute of the quadrant are registered. The statement of the method by which such Tables are constructed is unsuited to the present treatise. The mode of using them in computation we shall now proceed to explain. In using these Tables we have either to find the sine, cosine, &c., of an angle whose value is given in degrees, minutes, and seconds; or being given the value of the sine, cosine, &c.; to find the corresponding angle in degrees, minutes, and seconds. Referring to the Tables (Table XXVI, Norie), it will be seen that the degrees are given at the top of the page, and the minutes down the left hand side of the page for the sines. And, for the cosines, the degrees are given at the bottom of the page, and the minutes up the right hand side of the page. If the value of the angle be given in degrees and minutes, the sine, cosine, &c., is found directly from the Tables, in which are registered the values of trigonometrical quantities. All the numbers must be If the angle contains seconds, we must proceed by the method of proportional parts, as in the following examples : RULE XXIV. 1°. Find from the Table the nat. sine, cosine, &c., which corresponds to the degrees and minutes. (Norie, Table XXVI.) 20. Multiply the tabular difference by the seconds, and divide by 100. 3°. If the required quantity be a nat. sine, tangent, or secant, add the result to the last figures obtained in 1°; if it be a cosine, cotangents, or cosecant, subtract. The result will be the required sine, cosine, &c. The reason of this rule is founded on the principle that for a small interval, such as one minute, the increase of the sine is proportional to the increase of the angle, If the value of the sine, cosine, &c., be given, and it is required to find the angle, we use the following rule : : RULE XXV. 1o. Find in the Tables the next lower nat. sine, nat. cosine, &c., and note the corresponding degrees and minutes. 2°. Subtract this from the given sine, cosine, &c., multiplying the difference by 100, divide by the tabular difference and consider the result as seconds. 3. If the given value be that of a sine, tangent, or secant, add these seconds to the degrees and minutes found in 1°, if it be that of a cosine, cotangent, &c., subtract. The result will be the required angle. Tab. diff. 327 327)900(3" nearly (additional seconds for nat. sine). The log. 732156 is sought for in Table XXVI, Norie, but as it cannot be found exactly, the next less is taken which corresponds to 47° 4′. The difference of the logs. is then found, two cyphers added (which is equivalent to multiplying by 100), and the product divided by the tabular difference, the quotient is the additional seconds. 2. Given the natural cosine 853267: find the angle. In order to apply logarithmic calculations to trigonometrical quantities, it is necessary to construct tables of logarithms of the natural sines, cosines, &c., and the real logarithmic sines, tangents, &c., are just the logarithms of those numbers which are the natural sines, tangents, &c.* In practice the logarithmic are generally far more useful than the natural sines, &c., though the latter are often necessary, or, in some simple kinds of calculation, preferable. As all the sines and cosines, all the tangents from o° to 45°, and all the cotangents from 45° to 90°, are less than radius or unity; the logarithms of the values of these quantities are decimal fractions and have There are independent methods of calculating logarithmic tables, but any investigation of these methods would be out of place in these pages, |