Again, suppose that in reducing our observation have to multiply the uncorrected result by correcting factor occurs in reducing observed barom heights to o° C., and other examples of its use are g in Chap. III.) By ordinary algebraical division we We have already seen that if both a and B are s quantities compared with unity, their product ma neglected; and the same is true for a2 and ẞ2. Ex. 6. Find correctly to three decimal places value of The answer, correctly to three decimal places 15.244. The student can easily verify for himself the follow results, which are approximately correct when quantities a and B are small compared with unity that their squares and higher powers may be glected). es the es, is >wing the y (so ne 12. Use of Logarithms.—It is proved in treatises on algebra that different powers of any fixed number can be multiplied by adding together the indices of those powers. We may assume that a list can be drawn up, giving the indices of the powers of some fixed number which are equal to every whole number, say from 1 to 10,000. Such a list is called a table of logarithms, and the fixed number is called the base of the system of logarithms: we may therefore define the logarithm of a number to a given base as being the index of that power of the base which is equal to the given number. Thus if an, then x is called the loga rithm of n to the base a. From motives of convenience the number 10 is chosen as the base of the common system of logarithms, and a table will be found at the end of this book giving the decimal parts (to four places) of the common logarithms of numbers from 1 to 9999. To find the Logarithm of a Number from the Table.—The first two figures of the number are to be found in the left-hand column, and the third in the first series of figures (0 to 9) in the top column; the number opposite the first two figures, and below the third, is the decimal part of the logarithm. When the number whose logarithm is required contains four figures, the fourth figure is to be looked for in the second series of figures (1 to 9) in the top column; the proportional part, which is found opposite the first two figures and below the fourth, is to be added to the part of the logarithm already found, the righthand figure of the proportional part being added to the right hand Again, since 100=1, and 10-1=0.1, it follows that the rithm of any number between o and o.1 is a negative d fraction, and may therefore be written in the form (2.) The characteristic of the logarithm of a number les unity is negative, and is one more than the number of immediately following the decimal point. Thus the logarithm of 0.314 is - 1+0.4969, which is viated thus: I-4969; the logarithm of 0.0314 is 2.496 so on. The operation of multiplication is performed by addi gether the logarithms of the numbers which are to be multi the sum is the logarithm of their product. Division is perf by subtracting the logarithm of the divisor from that dividend: the remainder is the logarithm of the quotient. racteristic; may be de manner in which these operations are carried out, and the method of finding a number when its logarithm is given, will be best explained by an example. he logadecimal 1 0.01 ile: s than iphers bbre and log dividend = 4.1407 = 4.9431 log quotient = I.1976 =2 x 0.4048=0.8096 log divisor = 4.9431 0.1976 is not one of the logs given in the table: the next lower one is o.1959, which is the log of 1.57. Now o.1976 -0.1959 0.0017. Looking along the row (in which the log is given) for 17, we find that it stands in a column headed by the figure 6, and this is the fourth figure of the number. Lastly, by rule (2), we see that Ĩ∙1976=log 0.1576. The value of the fraction is therefore o.1576. A full account of the methods of logarithmic calculation will be found in Chambers's Mathematical Tables, and these may be used for more accurate work; but the table of four-place logarithms at the end of this book will be found sufficient for working out most of the problems given. The student is advised to practise the methods of approximation indicated in § 11, and to make himself thoroughly familiar with the use of logarithms, as a large amount of arithmetical calculation will thus be avoided. In working out examples he should aim at something more than merely getting a correct numerical answer: diagrams or rough sketches should be given wherever they render the solution more intelligible, and formulæ should not be quoted without explanation unless the rela tions which they 1. |