the end of the table the last place thus obtained cannot always be depended upon within a unit, that is, provided the mantissa of log. is greater than 9388. Thus, for instance, the log. 3'7575 corresponds to the no. 5721 and the log. 3'7576 to 5722, nearly. This remark should be kept in mind, because it is mere waste of time to employ more figures than are required to insure a certain degree of precision in the result. MISCELLANEOUS. We here insert a collection of numbers, the logarithms of which are to be taken out of the tables. Required the natural numbers of the following logarithms :— I. 2'309630 13. 1565000 25. 1188591 37. 5'654243 49. 0763947 2. 3'676968 14. 2.621754 4. 5. 6. 7. 8. 9. 000000.0 2.845098 26. 3*020887 38. 0434294 50. 883030 39. 5.606389 51, 3625343 40. 2.397050 52. 1725364 41. 5'000000 53. Z627407 42. 2.881955 54. 3686216 167317 55. 0'400573 56. 1567343 57. 0927632 5002559 15. 3786942 27. 2954243 16. 0'565021 28. 3 959041 17. 0778441 29. 4705864 18. 2769504 30. 1415674 19. 5'774152 31. O'415974 43. 20. 5'421604 32. I'000000 44. 7875061 21. 3.000000 33. 3'954243 45. 0'000186 2.056905 22. 6'394452 34. 2.716003 46. 6'947385 2*564494 23. 1'415674 35. 4'000000 47. 2953031 12. 3'563362 24. 2*640978 36. 0230449 48. 2564772 IO. II. 2 000000 1944483 MULTIPLICATION BY LOGARITHMS. In multiplication we proceed by RULE XIII. 58. 1°. Find the logarithms of the numbers, the product of which is required. (For the method of taking out the log. of a number, see pages 19 to 22.) Add these together, the sum will be the logarithm of the product. 2°. 3. Find from the tables the corresponding number. (For the method of finding the corresponding number to a log., see pages 23 to 25.) This will be the required product. (a) When the characteristics of the logs. to be added are both positive, it is evident that their sum will be positive. When they are both negative, their sum (diminished by what is to be carried from the sum of the positive decimal parts) will be negative, When one is negative and the other positive, subtract the less from the greater, and prefix to the difference the sign belonging to the greater. We have here added the logs, of the given factors together, and having sought for the given mantissa 672935, which is not to be exactly found in the tables, we obtain the next less mantissa 672929, which we subtract from the given mantissa, the difference is 6, to which two cyphers are annexed, and then we divide by the tabular difference 92, whence we obtain 07 nearly; the remainder, 48, being more than half the divisor, I is added to the last figure in the quotient (6); attaching these to the four figures obtained previously, we have 470907; the index 4 determines five of these to be integral; hence the product is 47090'7 (Rule IX, page 23). 2. Multiply 476 by 50. 476 log. 2.677607 50 log. 1'698970 23800 log. 4376577 The mantissa of log., viz. 376577, is found exactly in the table in a line with 238 and under 0; but as the characteristic 4 requires 5 digits in the integer part, we therefore add a o (cypher), which gives 23800 as the nat. no. corresponding to the proposed log. This is according to Rule XI, (a). 4. Multiply 0567 by 00339. 0567 log. 8753583 00339 log. 7530200 oco1922 log. 6.283783 Or thus, using negative indices: 0567 log. 2753583 00339 log. 3530200 000192 log. 4'283783 In adding, when we come to the places of tenths, the process is 5 and 7 are 12, 2 to put down and 1 to carry, and since the indices are both negative 2 and 3, we diminish their sum (5) by the number carried (1), which leaves 4 for the index (see Rule XIII, (a). We prefix 3 cyphers because the index being 4 the first significant figure of product must stand in the fourth place from the decimal point. 15. Required the product of 17*25, o'82, and o'c65. 17.25 log. 1236789 82 log. 9913814 065 log. 8.812913 o'919425 log. 9·963516 Or thus: 1725 log. 1*236789 82 log. T913814 065 log. 2812913 0919425 log. I 963516 Here we have one to carry from the mantissa, which added to the positive index 1 (index of log. 17:25, see above) makes positive 2. Now the sum of the negative indices is 3 (negative 3), and, therefore, since where one is positive and the other is negative the difference is the index; we have + 2 from 3 leaves ī for the index, and the first significant figure of the quotient must occupy the first place to the right of the decimal point (Rule IX, page 24, top). EXAMPLES FOR PRACTICE. 444 by 22·2; 1. Multiply by logs. 85 by 70; 100 by 10; 39 by 27; 37 by 98; and 97 by 79. 2. Multiply 83 by 77; 100'3 by 12; 3. Multiply 38 by 174; 601 by 18; 4. Multiply 288 by 24; 517 by 659; 5. Multiply 127 by 304; 476 by 100; 6. Multiply 37 6 by 249; 7. Multiply 76 by 42; 82:33 by 15'3; 250 by 125; 9001 by 90; 98 by 194; 182·7 by 250; 47'6 by 6·82; 99'9 by 8.63; 387 by 3'2; 76'4 by 5'4; and 307 by 30.. 82 by 12; and 101 by 11. 5760 by 12; and 970 by 630. 34 by 6; and 3654 by 500. 3907 by 20; and 28 9 by 96'5 and 100co by 10. 12. Multiply 307 by 31; 37 by 6.70; 5900 by 00071; and 23.807 by 02. 13. Multiply 02865 by 2196; 334387 by 05454; and 03948 by 9·1959. 14. Multiply 1 by 1; 01 by 001; 10 X 0 X 001; and 1000 by 100, 15. Multiply 00146 by 039; 02895 by 2196; 4189 by 00071; and 37 by 670. 16. Multiply 673 14 by 35; 5900 by 00071; 1000 by 100; and ·00000275 by 0336. Required by means of logarithms the product of : 17. 002784 X 4 X 1600; 7'84 00083 X 000027; 52 X 734 X 6. 18. 37 X 426 X 004 X 275 X 326; 72 X 96 X 124 X 05; and 32 X 181 X 4. 19. 24 X 007 X 54 X 1; 6 X 4 X 12 X 32; and 18 X 48 X ú'2 X 4. 36 X 48 X 62 X 4; 2861 × 19 × 163 × 4; and ·772 = (12) × 3. 20. DIVISION BY LOGARITHMS. In division we proceed by 1o. 2°. RULE XIV. Find the logarithms of the numbers the quotient of which is required. Subtract the logarithm of the divisor from that of the dividend; (adding 10 to the index of this last, if required), the difference will be the logarithm of the quotient. 3°. Find from the tables the corresponding number. This will be the required quotient. In subtracting the logarithm of the divisor, if it is negative, change the sign of its characteristic, and then proceed as if this were to be added to the characteristic of the dividend; but before making the characteristic of the divisor positive, subtract what was borrowed (if anything), in subtracting its decimal part. For, since the decimal part of a logarithm is positive, what is borrowed, in order to make it possible to subtract the decimal part of the logarithm of the divisor from that of the dividend, must be so much taken away from what is positive, or added to what is negative in the remainder. We change the sign of the negative characteristic, and then add it; for, adding a positive is the same as taking away a negative quantity. 3. Divide 830772 by 982. The log. of 830772 is taken out by Rule VIII, page 21. We seek in the left hand column of the table (No.) for 830 (the first three digits), and also at the top of the page in one of the horizontal columns for the fourth figure 7, then in a line with the first and under the latter we hive 919444. In a line with the quantity in the right hand column marked Diff. stands 52, which multiplied by the remaining figures of the nat. number, viz., 72 produces 3744; then cutting off two digits from these (since we multiplied by two digits) it becomes 37, which being added to 919444, the mantissa of 8307, makes 919481, and with the characteristic 5, is 5'919481. The work will stand thus: Log, 8307 919444 tab, diff. 52 37 6. Divide 10000 by 10.0. 830772 log, 5'919481 1000'0 log, 3.000000 982 log. 2'992111 100 log. 1000000 |