necessary to calculate the logarithms of the prime numbers only, and that those of their various multiples may then be found by addition. When part of a table has thus been constructed, one portion of it may be used to verify another: thus when we have found the logarithms of 3, 5 and 6, we should have 1 = log 10 = log 30 3 = log 30 – log 3 = log 5 + log 6 – log 3: and by means such as these, a check may be applied at any stage of the process in order to ascertain the correctness of the previous computations. 199. For the reader's exercise we put down here the logarithms of the prime numbers less than 100 without their characteristics; and he will thus be enabled to construct for himself a table of the logarithms of all other numbers up to 100. These logarithms are extracted from Mr Babbage's Tables which every practical Student should have in his possession. Examples for Practice. (1) Required the logarithms of 5 and 168. Answers: .6989700 and 2.2253093. (2) Determine the logarithms of 1.04 and 3690. (3) What are the logarithms of 12 and? (4) Express the logarithm of 225 by means of the logarithms of 2 and 3, and verify it. Answer: 2-2 log 2 + 2 log 3. (5) Given the logarithms of 3 and 7, find the logarithm of 14700, and verify it. Answer: 2+ log 3 + 2 log 7. (6) Given the logarithms of 2 and 3, deduce the logarithm of .0072, and prove the converse. Answer: 3 log 2+2 log 3-4. (7) Find the logarithm of 50000 in terms of the logarithms of 216 and .081. (8) Express the logarithms of 8 and 9 in terms of those of 6 and 15. 3 3 Answers: 1.5+log 6-log 15 and log 6+ log 15-1. 2 (9) Find the logarithm of 83349 from the logarithms of 3 and .21 Answer: 6+2 log 3 + 3 log .21. (10) Given the logarithms of 15 and 16, find those of 27 and 420. 3 3 Answers: 3log 15+ log 16-3 and 4log 15+ log 16-5. 4 (11) Find the logarithms of 15.625 and .00475. (12) Required the logarithms of of the logarithms of 2, 3 and 5. 9 2 and in terms 16 375 2 log 2-2, Answers 2 log 5 + 2 log 3 (13) Determine the logarithms of 1.625, by means of those of 2, 3, 5 and 13. Answers: 2 log 2-log 3 + 2log 5 − 1, 3 4 (14) Express the logarithm of 7 in terms of the logarithms of 2 and .714285. Answer: 1-log 2 – log .714285. (15) Given the logarithm of 10424 4.0180353: find the fifth root of 1%. Answer: 1.0424. (16) Determine the value of the expression by means of logarithms. Answer: 6.25. 26 × 25o 43 × 102 (17) Find a fourth proportional to the quantities 1.3, .0104 and 2.375 by logarithms. Answer: .019. (18) Determine by logarithms a mean proportional between the magnitudes .004 and 72250. Answer: 17. (19) Given .200686= log 1.58740 = 2 log 1.25992: find the value of 4-2. Answer: 32748. 5 (20) Given 2.2309306 = log 170.188: it is required to find the value of 8 × √7 √2 × 3√3. Answer: 13.61504. (21) Required the number of figures in the product of 324 and 126, by means of logarithms. Answer: 5. (22) Find the numbers of digits in the results of the involutions of 210 and 312, by means of logarithms. Answers: 4 and 6. (23) Required by a table of logarithms, the index of 5 which shall give a result equal to 20. (24) Find the logarithm of 180 in a system whose base is 12, by means of a table of common logarithms. 1 + log 2 + 2 log 3 Answer: 2 log 2 + log 3 (25) Shew that the Mantissa of a logarithm depends upon the figures and not upon the pointing off: and that the Characteristic depends upon the pointing off and not upon the figures. The invention of Logarithms is due to the celebrated JOHN NAPIER or NEPER, Baron of Merchiston in Scotland, who was born in the year 1550 and died in the 68th year of his age. The base of the Napierian System of Logarithms is the mixed magnitude 2.71828 &c.; but, for the great improvement in the subject hinted at in Article (191), we are indebted to Mr. HENRY BRIGGS, Professor of Geometry at Oxford, by whom a Table of Logarithms was published in the year 1624. The reader who may be desirous of further information upon this portion of science is referred to Dr. HUTTON'S Mathematical Tables which contain an account of the discoveries of the most celebrated writers connected with it but he will not be able to appreciate their ingenuity and merits without a much more extensive knowledge of numerical calculations than can be acquired from this or any other treatise on Arithmetic. H. A. 9 CHAPTER IX. THE APPLICATION OF ARITHMETIC TO and 200. DEF. 1. In some of the preceding chapters the symbols and signs of Pure Arithmetic have been transferred from abstract magnitudes so as to represent concrete magnitudes and their relations to each other; it is on the same principle that the objects of Geometry or Geometrical Magnitudes, as Lines or Distances, Superficies or Areas, and Solid Contents or Volumes, are valued and compared by means of the numbers representing their respective Dimensions. A line having length only has one dimension: a superficies having length and breadth comprises two dimensions; and a solid has three dimensions, inasmuch as it is defined by three magnitudes, length, breadth and depth or thickness. 201. DEF. 2. A Measure in Geometry is a magnitude assumed as an Unit with which other magnitudes of the same kind may be compared: and though one magnitude neither contains another nor is contained in it an exact number of times, there may still be a third and smaller magnitude which is capable of measuring them both. A measure has therefore the same relation to quantity as unit has to number; and all quantities and numbers are said to be equal to the aggregates or sums of their measures and units respectively. When the magnitudes of lines are numerically expressed, the Principles of Geometry must furnish_the means of valuing or comparing with each other, those of both superficies and solids of which lines naturally form the dimensions: and on this account we shall first establish the Theory of Lineal Measure and then deduce those of Superficial and Solid Measure from it. |