OF LOGARITHMS*.

LOGARITHMS

MOGARITHMS are made to facilitate troublesome calculations in numbers. This they do, because they perform multiplication by only addition, and division by only subtraction, and raising of powers by multiplying the logarithm by the index of the power, and extracting of roots by dividing the logarithm of the number by the index of the root. For, logarithms are numbers 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, &c.

Or, more generally, logarithms are the numerical exponents of ratios; or they are a series of numbers in arithmetical

The invention of Logarithms is due to Lord Napier, Baron of Merchiston, in Scotland, and is properly considered as one of the most useful inventions of modern times. A table of these numbers was first published by the inventor at Edinburgh, in the year 1614, in a treatise entitled Canon Mirificum Logarithmorum; which was eagerly received by all the learned throughout Europe. Mr. Henry Briggs, then professor of geometry at Gresham College, soon after the dis. covery, went to visit the noble inventor; after which, they jointly undertook the arduous task of computing new tables on this subject, and reducing them to a more convenient form than that which was at first thought of. But Lord. Napier dying soon after, the whole burden fell upon Mr. Briggs, who, with prodigious labour and great skill, made an entire Canon, according to the new form, for all numbers from 1 to 20000, and from 90000 to 10100, to 14 places of figures, and published it at London, in the year 1624, in a treatise entitled Arithmetica Logarithmica, with directions for supplying the intermediate parts."

metical progression, answering to another series of numbers in geometrical progression.

Thus,

0, 1, 2, 3,
1, 2, 4, 8, 16, 32,

4, 5, 6, Indices, or logarithms. 64, Geometric progression.

Or 0, 1, 1, 3,

2, 3, 4, 5,

ཉ་

0,

Or

9, 27, 81, 243, 729, Geometric progression.

1, 2, Indices, or logs. 1, 10, 100, 1000, 10000, 100000, Geom. progress.

Where it is evident, that the same indices serve equally for any geometric series; and consequently there may be an

6, Indices, or logarithms.

4,

This Canon was again published in Holland by Adrian Vlacq, in the year 1628, together with the Logarithms of all the numbers which Mr. Briggs had omitted; but he contracted them down to 10 places of decimals. Mr. Briggs also computed the Logarithms of the sines, tangents, and secants, to every degree, and centesm, or 100th part of a degree, of the whole quadrant; and annexed them to the natural sines, tangents, and secants, which he had before computed, to fifteen places of figures. These Tables, with their construction and use, were first published in the year 1633, after Mr. Brigg's death, by Mr. Henry Gellibrand, under the title of Trigonometria Britannica.

Benjamin Ursinus also gave a Table of Napier's Logs. and of sines, to every 10 seconds. And Chr. Wolf, in his Mathematical Lexicon, says that one Van Loser had computed them to every single second, but his untimely death prevented their publication. Many other authors have treated on this subject; but as their numbers are frequently inaccurate and incommodiously disposed, they are now generally neglected. The Tables in most repute at present, are those of Gardiner in 4to, first published in the year 1742; and my own Tables in 8vo, first printed in the year 1785, where the Logarithms of all numbers may be easily found from 1 to 10000000; and those of the sines, tangents, and secants, to any degree of accuracy required.

Also, Mr. Michael Taylor's Tables in large 4to, containing the common logarithms, and the logarithmic sines and tangents to every second of the quadrant. And, in France, the new book of logarithms by Cllet; the 2d edition of which, in 1795, has the tables still farther extended, and are printed with what are called stereotypes, the types in each page being soldered together into a solid mass or block.

Dodson's Ant logarithmic Canon is likewise a very elaborate work, and used for finding the numbers answering to any given logarithm.

endless

endless variety of systems of logarithms, to the same common numbers, by only changing the second term, 2, 3, or 10, &c. of the geometrical series of whole numbers; and by interpolation the whole system of numbers may be made to enter the geometric series, and receive their proportional logarithms, whether integers or decimals.

It is also apparent, from the nature of these series, 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 geometric progression, to which those indices belong. Thus, the indices 2 and 3, being added together, make 5; and the numbers 4 and 8, or the terms corresponding to those indices, being multiplied together, make 32, which is the number answering to the index 5.

In like manner, 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 terms to which those indices belong. Thus, the index 6, minus the index 4, is = 2; and the terms corresponding to those indices are 64 and 16, whose quotient is = = 4, which is the number answering to the

index 2.

For the same reason, 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 4, in the above series, is 2; and if this number be multiplied by 3, the product will be 6; which is the logarithm of 64, or the third power of 4.

And, 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 64 is 6; and if this number be divided by 2, the quotient will be = 3; which is the logarithm of 8, or the square root of 64.

The logarithms most convenient for practice, are such as are adapted to a geometric series increasing in a tenfold proportion, as in the last of the above forms; and are those which are to be found, at present, in most of the common tables on this subject. The distinguishing mark of this system of logarithms is, that the index or logarithm of 10 is 1; that of 100 is 2; that of 1000 is 3; &c. And, in decimals,

decimals, the logarithm of ⚫1 is

1; that of 01 is

2; that of .001 is -3; &c. The log. of 1 being 0 in every system. Whence it follows, that the logarithm of any number between 1 and 10, must be 0 and some fractional parts; and that of a number between 10 and 100, will be 1 and some fractional parts; and so on, for any other number whatever. And since the integral part of a logarithm, usually called the Index, or Characteristic, is always thus readily found, it is commonly omitted in the tables; being left to be supplied by the operator himself, as occasion requires.

=

Another Definition of Logarithms is, that the logarithm of any number is the index of that power of some other number, which is equal to the given number. So, if there be N=r", then n is the log. of N; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the different systems of logarithms. When n is = 0, then N is 1, whatever the value of r is; which shows, that the log. of 1 is always O, in every system of logarithms. When n is 1, then N is = r; so that the radix r is always that number whose log is 1, in every system. When the radix r is = 2-718281828459, &c. the indices n are the hyperbolic or Napier's log. of the numbers N ; so that n is always the hyp. log. of the number N or (2·718 &c.).

Thus

But when the radix r is = 10, then the index n becomes the common or Briggs's log. of the number N: so that the common log. of any number 10 or N, is n the index of that power of 10 which is equal to the said number. 100, being the second power of 10, will have 2 for its logarithm and 1000, being the third power of 10, will have 3 for its logarithm: hence also, if 50 be 101.69897, then is 1.69897 the common log. of 50. And, in general, the following decuple series of terms,

viz. 104, 103, 102, 101, 10°, 10-1,
or 10000, 1000, 100, 10,
2, 1,

3,

=

10-2, 10-3, 10-4, 1, •1, 01, 001, 0001, have 4, -4, -2, -3, 0, -1, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as above mentioned.

PROBLEM.

PROBLEM.

To compute the Logarithm to any of the Natural Numbers 1, 2, 3, 4, 5, &c.

RULE I*.

TAKE the geometric series. 1, 10, 100, 1000, 10000, &c. and apply to it the arithmetic series, 0, 1, 2, 3, 4, &c. as logarithms. Find a geometric mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series, between which the number proposed lies.-In like manner, between the mean, thus found, and the nearest extreme, find another geometrical mean; and so on, till you arrive, within the proposed limit of the number whose logarithm is sought.-Find also as many arithmetical means, in 'the same order as you found the geometrical ones, and these will be the logarithms answering to the said geometrical

means.

EXAMPLE.

Let it be required to find the logarithm of 9.

Here the proposed number lies between 1 and 10. First, then, the log. of 10 is 1, and the log. of 1 is 0; theref. 1025 is the arithmetical mean, and 10X1✔ 10 = 3.1622777 the geom. mean; hence the log. of 3.1622777 is 5.

Secondly, the log. of 10 is 1, and the log. of 31622777 is ·5; theref. 15 2·75 is the arithmetical mean,

and 10 X 3.16227775-6234132 is the geom. mean; hence the log of 5.6234132 is 75.

Thirdly, the log. of 10 is 1, and the log. of 5-623-4132 is 75; theref. 175 ÷ 2 = 875 is the arithmetical mean,

and 10 X 5.6235132 = 7.4989422 the geom. mean; hence the log. of 7.4989422 is ·875.

Fourthly, the log. of 10 is 1, and the log. of 7-4989422 is ·875; theref. 1·875 2 =9375 is the arithmetical mean, and 10 X 7.4989422 8.6596431 the geom. mean; hence the log. of 8-6596431 is .9375.

The reader who wishes to inform himself more particularly concerning the history, nature, and construction of Logarithms, may consult the Introduction to my Mathematical Tables, lately published, where he will find his curiosity amply gratified.

Fifthly,