practically used. We shall not enter here upon the method of forming the tables themselves. The following is a specimen of the way in which the logarithms of numbers are usually tabulated : No. 0 1 2 | 3 | 4 | 5 | 6 o "Thus, if the number consist of four figures only, we have simply to copy out the figures in the column headed 0, prefix a decimal point, and the proper characteristic. Ex. Log 7991 = 3.9026011, log 7.995 = .9028185. When we speak of a number consisting of four figures only, we include such numbers as .003654, 07682, &c., the number of zeros immediately following the decimal points not being counted. Thus, log •07997 = 2.9029271 log •007992 = 3.9026555. When the number contains five figures, as, for instance, 79936, we look along the line containing the first four figures viz., 7993_of the number until the eye rests upon the column headed 6, the fifth figure. We then take the first three figures of the column headed 0, and affix the four figures of the column headed 6 in the horizontal line of the first four figures of the number. Thus, log 79936 = 4.9027424 log •079927 = 2.9026935. It will be seen from the portion of the logarithmic table above extracted, that when the first three figures of the logarithm-viz., 902-have been once printed, they are not repeated, but must be understood to belong to every four. figures in each column, until they are superseded by higher figures, as 903. When, however, this change is intended to be made at any place not at the commencement of a horizontal row, the first of the four figures corresponding to the change is usually printed either in different type, or, as above, with a bar over it. Thus we have above 0031, indicating that from this point we must prefix 903 instead of 902. Thus, log 79.986 = 1.9030140, log •0079987 = 3.9030194. 26. To find the logarithm of a number not contained in the tables. Ex. Find the logarithm of 799.1635. Since * the mantissa of the number 79916:35 is the same as the mantissa of the given number, and that the first five figures are contained in the tables, we may proceed as follows (1.) Take out from the tables the mantissa corresponding to the number 79916. This is :9026337. (2.) Take out the mantissa of the next higher number in the tables-viz., 79917. This is •9026392 (3.) Find the difference between these mantissæ. This is called the tabular difference, being the difference of the mantissæ for a difference of unity in the numbers. We find tab. diff. = .0000055, which we call D. (4.) Then assuming that small differences in numbers are proportional to the differences of the corresponding logarithms, we find the difference for •35 = .35 x .0000055 = .0000019, retaining only 7 figures. This is often called d. (5.) Now adding this value of d to the mantissa for the number 79916, we get the mantissa corresponding to the number 79916:35. (6.) Lastly, prefix to this mantissa the proper characteristic. The whole operation may stand thus M. of log 79916 = .9026337.. . : M. of log 79917 = .9026392 Tabular difference or D = .0000055 * * Thus log 79916.35 = log (100 x 799·1635) = 2 + log 799·1635.. ........... (1) Hence, difference for •35 or d = .35 •0000055 = .0000019 .... ........... (2). Hence, adding (1.) and (2.)— M. of log 79916.35 = .9026356. :: log 799.1635 = 2.9026356. or better thus, omitting the useless ciphers M. of log 79916 = .9026337 .:. D = 55 ... M. of log 7991635 = .9026356, as before. In the next article we shall show how the required difference may be obtained by inspection from the tables. 27. Proportional parts. We saw in the example just worked that the tab. diff. (omitting the useless ciphers) is 55, and if we examine the table in Art. 25, we shall find the difference between the mantissæ of any two consecutive numbers there to be 54 or 55—generally 54. The number 54 is therefore placed in a separate column at the right of the table, and headed D. The student will understand that the tab. diff. changes from time to time, and is not always 54 or 55. Now assuming as in (4.) of the last article, we have“. Diff. for •1 = 54 x 1 = 5 Diff. for •6 = 54 x .6 •2 = 54 x .2 = 11 •7 = 54 x 7 = •3 = 54 x :3 = 16 •8 = 54 x 8 = , •4 = 54 x 4 = 22 , •9 = 54 x .9 = 49 , •5 = 54 x •5 = 27 We find therefore the numbers 5, 11, 16, 22, 27, 32, 38, 43, 49 placed in a horizontal row at the bottom marked P, in the columns respectively headed 1, 2, 3, 4, &c. Hence, if we require the difference for (say) •7, we take out the number 38 from the horizontal row marked P, instead of being at the trouble to find it by actual computation. The following example will illustrate how we proceed when we require the difference for a decimal containing more than one decimal figure. No explanation is needed. touching the line which meets the circle, the line which meets the circle shall touch it. Let any point D be taken without the circle ABC, and from it let two straight lines, DCA, DB, be drawn, of which DCA cuts the circle, and DB meets it; and let the rectangle AD, DC be equal to the square on DB. Then DB shall touch the circle. Draw DE CONSTRUCTION.-Draw the straight line DE, touching the the circle. circle ABC (III. 17); Find F the centre (III. 1' and join FB, FD, FE. PROOF.—Then FED is a right angle (III. 18). And because DE touches the circle ABC, B and DCA cuts it, the reetangle AD, DC is equal to the square on DE (III. 36). But the rectangle AD, DC is equal to the square on DB (Hyp.); Therefore the square on DE is equal to the square on DB (Ax. 1); Therefore the straight line DE is equal to the straight line DE = DB. DB. And EF is equal to BF (I. Def. 15); Therefore the two sides DE, EF are equal to the two sides DB, BF, each to each; And tri- And the base DF is common to the two triangles DEF, DBF and DEF are equal in Therefore the angle DEF is equal to the angle DBF (1.8). every re But DEF is a right angle (Const.); spect. Therefore also DBF is a right angle (Ax. 1). :: DBF is And BF, if produced, is a diameter; and the straight line angie; which is drawn at right angles to a diameter, from the ex tremity of it, touches the circle (III. 16, Cor.); and there Therefore DB touches the circle ABC. fore DB touches Therefore, if from a point, &c. Q.E.D. Then log .00123 = 7-3001031 -1230345 196 1230541 *1230541 = 4922164 = 2.5078867 M. of log N, M. of log 32202; 29. rical Tables. tables much in the same way carithms of numbers. natural sines, cosines, &c., and wes, &c. It is with the latter gh many of our remarks apply dural sines and cosines of all unde (Art. 11) less than unity, it logarithms are negative. To em in a negative form, and for add 10 to their real value, and must allow for this. The same pase of logarithmic tangents, coInts. true logarithmic sine by log sin, sine by L sin. A = L sin A - 10, = L cos A. - 10, &c. in using the tables that, although mund tangent of an angle increase as |