Table III. may be constructed from Table I. thus, Add 1.05 the amount of 11. at compound interest. 3.1525 the third term of Table III. Add 1 157625 the third of Table I. 4.310125 the fourth term of Table III. and so proceeding, add to each new term in Table III. the same term in Table I. For it is manifest that by this process we get the sums of this Progression, 1.1.05, 11025, 1.157625. But these sums are still the amounts of an annuity of 11. It is manifest that Table IV. may be constructed in like manner from Table II. The Use. The use of these Tables is very easy, being only to multiply any given sum, by the tabular number, in the same row with the given number of years. Example. Suppose it be required to find the amount of 136l. 15s. 6d. in 20 years, at 6 cent. annum, compound interest? £.136 15 6 136 7525 which multiply by the tabular number under 6 cont and in the same row with 20, viz. 3.207135, and the product will be l. 438.5834. i. c. 4381. 11s. 8d. and so of any other. Let the sundry Examples of this Chapter be done by the Tables. L CHAP. VI. OF LOGARITHMS. OGARITHMS are Numbers so contrived and adapted to other Numbers that the Sums and Differences of the former former correspond to, and shew the Products and Quotes of the latter, and also their Powers and Roots. The Logarithms of these numbers in a decuple progression from 1, (to which progression the Logarithms now in use are applied) are called Characteristicks, because they denote how many places the corresponding natural number consists of, which is easily apprehended. Num. Logarithms. 1 0.0000000 10 1 0.00000 100 2.0000000 3.0000000 10000 4 0000000 100000 5.0000000 For the Logarithm of 1 is 0. for 1 is not distant from itself; of 10,1.0000000 wherefore the Logarithm of every number between 1 and 10 must be a decimal fraction: Likewise since the Logarithm of 10 is .1, and of 100,.2, the Logarithm of every number between 10 and 100 must be greater than 1, and less than 2, i. e. 1 and a decimal, and between 100 and 1000 the Log, will be 2 and some decimal; so on the contrary, if a Logarithm be a decimal fraction the natural number must be between 1 and 10, if it consists of 1 and a decimal, it is between 10 and 100, if of 2 and a decimal between 100 and 1000, &c. That is, if the characteristick of a Logarithm be O, the natural number is a single figure; if the characteristick be 1, the natural number consists of 2 figures, if 2 of 3, if 3 of 4, &c. 2. The Logarithms of all numbers in a decople proportion differ only in their characteristicks, as if the Logarithm of 6.748 be 0.8291751, then, the Logarithm of its decuples will be as under: For 6.748X10 produces 67.48 wherefore the Log. of 67.48, viz. 0.8291751+1 the Log. of 10=1.8291751, the Log. of 67.48, and so of the rest. A geometrical progression may in fractions be continued downward below unity, infinitely, in the same proportion as it ascends in whole numbers above it, viz. 1 Or Decimally, .001, .01, .1, 1, 10, 100, 1000, &c. And since the distance of from unity is equal to the distance of 10; ofo that of 100 the Log. of To will be equal to that of 10 of equal to that of 100, &c. But then since the Logarithm of unity is 0, the Logarithms of fractions are negative or descending below 0, for they go on the contrary way to whole numbers, and are there to be marked with the sign -3 -2 -1 0 1 I 2 10, 100, 3 -,as, 1000 And as fractions in their multiplication and division have contrary effects to whole numbers, so have their Logarithms, viz. a fraction multiplying a whole number diminishes the whole number and the contrary. So a nega tive Logarithm must be subtracted from a positive Logarithm when addition is implied, and the contrary; but negative Logarithms are to be added or subtracted amongst themselves, as Addition or Subtraction is implied, as will be easily apprehended from an example or two of the foregoing progression, viz. X 10—1—1+1. i, e. 1 subtracted from 10 the Log. of 1. TX 1000 10 and −2+3, i. c. 2 subtracted from 3=1. Again, if the fraction ᄒ divide 10 the quotient will be 100. And so Subtraction being implied, add their Logarithms +1=2 the Logarithm of 100. PROBLEMS, shewing more particularly the use of Tables of Logarithms. To find the Logarithm of any number in general. It is either found by inspection, being placed to the right-hand of the number, or if otherwise placed, suitable directions are prefixed or annexed to the tables for the use of them. Problem II. To find the Logarithm of an integral number exceeding the limits of the Table of Logarithms: for example exceeding 10,000.. Rule. Take as many figures to the left hand of the given numbers as there are in the Table, (viz. 4 of them if the Table goes only to 10,000, or 5 if to 100,000) and in the place place of the figures not taken, aumex O's: Again, to the number expressed by the figures taken, add 1, and annex the same number of O's: Then take the difference of these two numbers; also the difference between the given number and the first of these, and make this proportion. As the difference of the first two is to the difference of their Logarithms: So is the difference of the last: to the difference of their Logarithms, which added to the Logarithm of the number less than the given number, gives the Logarithm of the number proposed. Application. To find the Logarithm of 123459 from a table carried only to 10,000. The two numbers less and greater than 123459 taken according to the rule, are 123400 and 123500 whose Logarithms are 5.0913152 and 5.0916670 for the Logarithm of 1234 is 3.0913152, to which add 2 the Logarithm of 100, (because 1234001234 × 10) the sum 50913152 is the Log. of 123400; also the Logarithm of 1235 is 3.0916670, and so that of 123500 is 5.0946670, and the proportion is From 123500 5.0916670 123459 Take 123400 5.0913152 123400 As 100 is .0003518 so is 59 to .00020756. &c. which added to 5.0913152 the Logarithm of 1234000 the sum is 50915276, &c. the Logarithm of 123459 nearly. Problem III. To find a number corresponding to any Log, which being the result of an operation with Logarithms, found in the table, is not itself found exactly in the table. 1. If the characteristick, and first 4 of 5 decimal figures are found in the table, that's near enough for common use; and the number against that Logarithm in the table, which is nearest the resulting Logarithm, may be taken as the number sought. But if greater exactness is desired, or the characteristick is beyond the limits of the table. 2. Take the two Logarithms, in the table, whose decimal figures are next less, and greater, and also their corresponding numbers, and make this proportion: As the difference of the greater and lesser Logarithms is to the difference of their corresponding numbers. So is the difference of the given and next lesser Logarithms to the diff. of their corresponding numbers. Which difference added to the number corresponding to that lesser Logarithm, makes the number corresponding to the given Logarithm nearly. Application. Let the given Logarithm be 4669347; the next lesser and greater are .3010300 the Log. of 2, and .4771213 the Log. of 3: so the proportion is thus formed. From .4771213 3 .4669347 .176 913 : 1 ::.1659047, ; .94215, which added to 2, makes 2.94215, &c. number sought. Again if it be required to find the number answering to the Logarithm 5.0915121 from a table not extending beyond 10,000, I seek for the highest Logarithms in the table, viz. those which have 3 for their characteristick, and the decimal figures next less and greater, than those of the given Logarithm, and find them to be 3.0913152 the Log. of 1234 and 3.0916670; from which we form the following analogy. From 5.0916670 Log. of 123500 given L. 5.0915121 5.0913152 which added to the lesser makes 123456. Now it will be proper to shew the use of Logarithms in calculation, and then conclude. 1. Of their Use in MULTIPLICATION. It is manifest that the sum of the Logarithms of the factors is the Log, of the product, wherefore To multiply one number by another, add their Logarithms together, and in the table find the natural number corresponding to their sum: that number is their product. Examples. Multiply 144 Log.-2.1583625 | 1728 Log-3.2375537 } Add 2. Mult. 1385 by 185 4. Mult. .1385 by .0185 7589 by 6757 2. In DIVISION. To divide one number by another, subtract the Logarithm of the divisor from the Log. of the dividend, and the remainder is the Log. of the quotient. Examples. |