Example.

Required the logarithmic rising answering to 45045!?

4:50 45:72:41:15, the natural versed sine of which is 702417; the common log. of this is 5. 846595, which, therefore, is the logarithmic rising required.

TABLE XXVIII.

Logarithms of Numbers.

Logarithms are a series of numbers invented, and first published in 1614, by Lord Napier, Baron of Merchiston in Scotland, for the purpose of facilitating troublesome calculations in plane and spherical trigonometry. These numbers are so contrived, and adapted to other numbers, that the sums and differences of the former shall correspond to, and show, the products and quotients of the latter.

Logarithms may be defined to be the numerical exponents of ratios, or a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression; as,

5.

6.

7.

0. 1. 2. 3. 4. 8. ind. or log. 1. 10. 100. 1000. 10000. 100000. 1000000. 10000000. 100000000 ge. pro.

Whence it is evident, that the same indices serve equally for any geometrical series; and, consequently, there may be an endless variety of systems of logarithms to the same common number, by only changing the second term 2. 3. or 10. &c. of the geometrical series of whole num

bers.

In these series it is obvious, that if any two indices be added together,

their sum will be the index of that number which is equal to the product of the two terms, in the geometrical progression to which those indices belong thus, the indices 2. and 6. being added together, make 8; and the corresponding terms 4. and 64. to those indices (in the first series), being multiplied together, produce 256, which is the number corresponding to the index 8.

It is also obvious, that if any one index be subtracted from another, the difference will be the index of that number which is equal to the quotient of the two corresponding terms: thus, the index 8. minus the index 3 = 5; and the terms corresponding to these indices are 256 and 8, the quotient of which, viz., 32, is the number corresponding to the index 5, in the first series.

And, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power; thus, the index, or logarithm of 16, in the first series, is 4; now, if this be multiplied by 2, the product will be 8, which is the logarithm of 256, or the square

of 16.

Again,-if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root: thus, the index or logarithm of 256 is 8; now, 8 divided by 2 gives 4; which is the logarithm of 16, or the square root of 256, according to the first series.

The logarithms most convenient for practice are such as are adapted to a geometrical series increasing in a tenfold ratio, as in the last of the foregoing series; being those which are generally found in most mathematical works, and which are usually called common logarithms, in order to distinguish them from other species of logarithms.

In this system of logarithms, the index or logarithm of 1, is 0; that of 10, is 1; that of 100, is 2; that of 1000, is 3; that of 10000, is 4, &c. &c.; whence it is manifest, that the logarithms of the intermediate numbers between 1 and 10, must be 0, and some fractional parts; that of a number between 10 and 100, must be 1, and some fractional parts; and so on for any other number: those fractional parts may be computed by the following

Rule. To the geometrical series 1. 10. 100. 1000. 10000. &c., apply the arithmetical series 0. 1. 2. 3. 4. &c., as logarithms. Find a geometrical mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series between which the proposed number lies. Between the mean thus found and the nearest extreme, find another geometrical mean in the same manner, and so on till you arrive at the number whose logarithm is sought. Find as many arithmetical means, according to the order in which the geometrical ones were found, and they will be the logarithms

of the said geometrical means; the last of which will be the logarithm of

the proposed number.

Example.

To compute the Lóg. of 2 to eight Places of Decimals :--

Here the proposed number lies between 1 and 10.

First,

The log. of 1 is 0, and the log. of 10 is 1;

therefore 0+ 1+ 2.5 is the arithmetical mean,

and ✓ 1 x 10 = 3.1622777 is the geometrical mean :
hence the log. of 3. 1622777 is .5.

Second, The log. of 1 is 0, and the log. of 3. 1622777 is .5;
therefore 0+5+2= 25 is the arithmetical mean,

and 1 x 3.1622777 = 1.7782794 the geometrical mean : hence the log. of 1.7782794 is. 25.

Third, The log. of 1.7782794 is. 25, and the log. of 3. 1622777 is .5; therefore. 25+.5 ÷ 2 = .375 is the arithmetical mean,

and 1.7782794 × 3.1622777 = 2.3713741 the geo. mean: hence the log. of 2. 3713741 is .375.

Fourth, The log. of 1.7782794 is. 25, and the log. of 2.3713741 is. 375; therefore. 25+ .375 ÷ 2 = .3125 is the arithmetical mean,

Fifth,

and 1.7782794 x 2.3713741 = 2.0535252 the geo. mean : hence the log. of 2.0535252 is .3125.

The log. of 1.7782794 is. 25, and the log. of 2. 0535252 is.3125; therefore. 25+.31252 = . 28125 is the arith. mean,

and 1.7782794 x 2.0535252 = 1.9109530 the geo. mean: - hence the log. of 1.9109530 is . 28125.

Sixth, The log. of 1.9109530 is. 28125, & the log. of 2. 0535252 is.3125; therefore. 28125 + .3125 + 2 = .296875 is the arith. mean,

and 1.9109530 × 2.0535252 = 1.9809568 the geo. mean: hence the log. of 1. 9809568 is. 296875.

Seventh, The log.of 1.9809568 is. 296875, & the log. of 2.0535252 is.3125; therefore. 296875 +.3125 ÷ 2 =.3046875 is the arith. mean, and 1.9809568 x 2.0535252 = 2.0169146 the geo. mean: hence the log. of 2. 0169146 is. 3046875.

Eighth, The log.of 2. 0169146 is. 3046875, & log. of 1.9809568 is 296875; therefore. 3046875 +.296875 +2.30078125 is the ar. mean, and 2.0169146 x 1.9809568 = 1.9988548 the geo, mean: hence the log. of 1.9988548 is. 30078125.

Proceeding in this manner, it will be found, after 25 extractions, that the log. of 1.9999999 is .30103000; and since 1.9999999 may be considered as being essentially equal to 2 in all the practical purposes to which it can be applied, therefore the log. of 2 is. 30103000.

If the log. of 3 be determined, in the same manner, it will be found that the twenty-fifth arithmetical mean will be .47712125, and the geometrical mean 2.9999999; and since this may be considered as being in every respect equal to 3, therefore the log. of 3 is . 47712125.

Now, from the logs. of 2 and 3, thus found, and the log. of 10, which is given=1, a great many other logarithms may be readily raised; because the sum of the logs. of any two numbers gives the log. of their product; and the difference of their logs. the log. of the quotient; the log. of any number, being multiplied by 2, will give the log. of the square of that number; or, multiplied by 3, will give the log. of its cube; as in the following examples:

Since the odd numbers 7. 11. 13. 17. 19. 23. 29. &c. cannot be exactly deduced from the multiplication or division of any two numbers, the logs. of those must be computed agreeably to the rule by which the logs. of 2 and 3 were obtained; after which, the labour attending the construction of a table of logarithins will be greatly diminished, because the principal part of the numbers may then be very readily found by addition, subtraction, and composition.

Of the Table.

This Table, which is particularly adapted to the reduction of the apparent to the true central distance, by certain concise methods of computation, to be treated of in the Lunar Observations, is divided into two parts: the first of which contains the decimal parts of the logs., to six places of figures, of all the natural numbers from unity, or 1, to 999999; and the second, the logs. to the same extent, of all the natural numbers from 1000000 to 1839999;-and although the logs. apparently commence at the natural number 100, yet the logs. of all the natural numbers under that are also given thus, the log. of 1, or 10, is the same as that of 100; the log. of 2, or 20, is the same as that of 200; the log. of 3, or 30, is equal to that of 300; that of 11, to 110; that of 17, to 170; that of 99, to 990; and so on: using, however, a different index. And as the indices are not affixed to the logs., they must therefore be supplied by the computer: these indices are always to be considered as being one less than the number of integer figures in the corresponding natural number. Hence the index to the log. of any natural number, from 1 to 9 inclusive, is 0; the index to the log. of any number from 10 to 99 inclusive, is 1; that to the log. of any number from 100 to 999, is 2; that to the log. of any number from 1000 to 9999, is 3; &c. &c. &c. The second part of the Table will be found very useful in computing the lunar observations, by certain methods to be given hereafter, when the apparent distance exceeds 90 degrees, or when it becomes necessary to take out the log. of a natural number consisting of seven places of figures, and conversely.

In the left-hand column of the Table, and in the upper or lower horizontal row, are given the natural numbers, proceeding in regular succession; and, in the ten adjacent vertical columns, their corresponding logarithms.

As the size of the paper would not admit of the ample insertion of the logs., except in the first column, therefore only the four last figures of each log. are given in the nine following columns; the two preceding figures belonging to which will be found in the first column under 0 at top, or over 0 at bottom; and where these two preceding figures change, in the body of the Table, large dots are introduced instead of O's, to catch the