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Logarithms and Interpolation

The arithmetical operations of multiplication, division, the determination of any power of a number, and the extraction of roots, may be performed, to the required degree of approximation, by the use of tables of logarithms. The method of using these for the purposes mentioned is so well known that it is not necessary to enter into details here. A table of logarithms to four places of decimals is given in Lupton's book, and is sufficient for most of the calculations that we require. If greater accuracy is necessary, Chambers's tables may be used. Instead of tables of logarithms, a 'slide-rule' is sometimes employed. The most effective is probably 'Fuller's spiral slide rule,' which is made and sold by Stanley of Holborn. By this, two numbers of four figures can be multiplied or divided.

Besides tables of logarithms, tables of squares, cubes, square roots, cube roots, and reciprocals may be used. Short tables will be found in Lupton's book (pp. 1-4); for more accurate work Barlow's tables should be used. Besides these the student will require tables of the trigonometrical functions, which will also be found among Lupton's tables.

An arithmetical calculation can frequently be simplified on account of some special peculiarity. Thus, dividing by 5 is equivalent to multiplying by 2, and moving the decimal point one place to the left. Again, π2 = 9.87 = 10-0.13, and many other instances might be given; but the student can only make use of such advantages by a familiar acquaintance with cases in which they prove of service.

In some cases the variations of physical quantities are also tabulated, and the necessity of performing the arithmetic is thereby saved. Thus, No. 31 of Lupton's tables gives the logarithms of (1 + 0.00367t) for successive degrees of temperature, and saves calculation when the volume or pressure of a mass of gas at a given temperature is required. A table of the variation of the specific resistance of copper with variation of temperature, is given on p. 47 of the same work.

It should be noticed that all tables proceed by certain definite intervals of the varying element; for instance, for successive degrees of temperature, or successive units in the last digit in the case of logarithms; and it may happen that the observed value of the element lies between the values given in the table. In such cases the required value can generally be obtained by a process known as 'interpolation.' If the successive intervals, for which the table is formed, are small enough, the tabulated quantity may be assumed to vary uniformly between two successive steps 'of the varying element, and the increase in the tabulated quantity may be calculated as being proportional to the increase of the varying element We have not space here to go more into detail on this question, and must content ourselves with saying that the process is strictly analogous to the use of ' proportional parts' in logarithms. We may refer to 12, 19, 77 for examples of the application of a somewhat analogous method of physical interpolation.

Last Update: 2011-03-27