whence, by the method of indeterminate coefficients, qa = p, or a

and the indices of v ascend regularly.

Thirdly. Let the index be any negative number, integer or fractional; that is, let nr.

Then

and it follows, therefore, that the coefficient of the second term is, in all cases, the index of the power to which the binomial is to be raised.

II. To determine the law of the coefficients of v in the several terms in the expansion of (1 + v)".

(1 + y)" + n (1 + y)”~1 2 + B (1 + y)"−2 z2 + c (1 + y)”—3 23 + . . . or again (arranging the expanded expressions according to powers of z),

(I.)

(II.)

(putting B C1 1, n

n

2,

VOL. I.

.)

B2, C2, ... for the coefficients when the index n is changed to

Now the series (II.) and (III.) being but expansions of the same expression, the coefficients of the powers of z must be such, as of themselves fulfil the condition of the equation. Hence equating the corresponding ones in the two series, we obtain

and the process may obviously be continued as far as n such steps when n is a positive whole number; or as far as we please, when n is a negative integer, or a fraction either positive or negative*.

Hence the law of the indices, and the law of the coefficients of the formula in its second form, has been established; and the former is obtained, multiplying all the terms by a", as is evident on inspection.

P

THE EXPONENTIAL THEOREM.

WHETHER x be a known quantity, such as we have assumed n to be, or not, still the developement of (1 + b)* is

the developement depending upon b and a being specific symbols of number, and not upon the knowledge what those numbers are in any individual case, or of the manner in which the expression containing them was actually formed.

results will be composed of positive integral powers of x, as follows †,

b3

or arranging the expression according to the powers of x, and attaching to them their collected coefficients, it takes the following form

(1 + b)* = 1 + Ax + вx2 + cx3 + . . in which A, B, C, &c. can be computed from the above expression.

*For in the former case, one of the factors, (the n + 1th) that comes in will be n-n, or 0, and hence the term which involves it, (which that, and all subsequent to that, do,) will be 0; and in the latter case, obviously, none of the factors can become 0, since all the numbers subtracted are positive integers, and the values of n by hypothesis are either negative integers or fractions.

+ The readiest mode of computing the numeral coefficients of a in these expressions is that explained at page 228.

Put 1+b=a: then, whatever x or y may denote,
a = 1 + AX + вx2 + сx3 +
a2 = 1 + Ay + BY2 + cy3 +

And hence by subtraction

a — a2 = ▲ (x − y) + в (x2 - y2) + c (x3 — y3) + . . . a*

B

= (x − y) {A + B (x + y) + c (x2 + xy + y2) + . . . . } .

....

·y) + c (x − y)2 +
x

(1)

(2)

By comparing equations (1) and (2), and making a = y, there follows (since y has previously disappeared as a common factor

a' is then equal to a*, and x
in the two values of a*. a.)
A+ 2Bx + 3cx2 + 4Dа3 +.

Now, since this is true whatever x may signify, the coefficients of the several powers of a must be equal on each side: that is

Now we have only to find a the coefficient of the first power of x, to be able to express the complete developement. But a very slight attention to the formation of the coefficients of x in the first or second equations at the opening of the section will show that they are 1, - 1.2, 1. 2. 3, - 1. 2. 3. 4, &c. in the cases where the denominators the quantity b in the second equation are 1, 1, 1.2, .; or in other words, taking in the powers of b, that the collected terms form the expression

b2 b3
+

2

1)

3 4

(a

b4
+

These values and expressions will find their use in the next chapter.

* It will be a useful exercise for the student to demonstrate this theorem by the method of indeterminate coefficients.

And by actually multiplying a + Bx + Cx2. . . by itself, to six or eight terms, and equating the two values of (a* )2, the same result will be obtained as in the text.

LOGARITHMS.

IN the equation a* = N, a is called the base, N the number, or value of the expression, and x is called the index of the power of a, or, the logarithm of the number N, to the base a.

If x and N be given, the unknown quantity is a, and the rules for finding it have already been given under the head of extracting roots; and if a and x are given, N is the unknown, and the rule for finding it has been given under the title of raising powers: but when a and N are given, the process becomes much more complicated, and the actual numerical solution of the question which involved it is extremely laborious. As, however, the cases which occur in the actual practice of mathematics, are those in which a = = 10, and a = 2.718281828.. tables have been computed which express the various values of x in the equations 10* = 1, 10* = 2, . . . 10* = 1000, &c.

but the former logarithms are most frequently required, and therefore often given in collections which do not contain the latter. It will, however, be presently made to appear that whenever we have calculated the logarithms, or values of x, to any one base, as that of a = 10, the logarithm of the same number N to any other base, as a 2.718. or 5, or 100,

can be obtained by simply multiplying all the logarithms in the given table by a certain constant number adapted to each particular base. On this account the logarithms to other bases than 10, and 2.718 . . . have not been calculated at all.

The logarithms calculated to the base of our common decimal scale, 10, are called the common logarithms, or Briggs's Logarithms, from Mr. Henry Briggs, the mathematician who first computed them to that scale: and those to the base 2-718... are called the Hyperbolic Logarithms, from an analogy which will be pointed out at the end of the Conic Sections, or Napierean Logarithms, from Lord Napier, who not only first computed those particular logarithms, but who also invented the system itself, as well as conferred several other advantages on the practical processes of mathematical calculation *.

*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 treatiseentitled 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 discovery, went to visit the noble inventor; after which, they jointly undertook the arduous task of computing new tables on the 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 101000, 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.

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 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

But before proceeding either to explain the arrangement of the tables, or the calculations of the common logarithms, it will be necessary to announce one or two of the simpler properties of the logarithms themselves, whatever the given base may be.

1. The sum of the logarithms of two numbers is equal to the logarithm of their product, and the difference of their logarithms to the logarithm of their quotient.

Let a be the base, and let b, c, be the numbers, and x, y their logarithms. Then a* = b; a" = c; and by multiplication and division a*+

b

=

b

bc, and a-y

== that is the logarithm of bc is x + y, and the logarithm of- is x y.

с

C

Hence, also, the sum of the logarithms of any number of quantities is equal to the logarithm of their continued product: and x times the logarithm of any number is equal to the logarithm of the ath power of that number.

2. The nth part of the logarithm of any number is equal to the logarithm of its nth root.

3. If there be a series of numbers in geometrical progression, their logarithms will be in arithmetical progression; and, on the contrary, if the logarithms be in arithmetical progression, the numbers themselves will be in geometrical progression..

For let k, ak, a2k, a3k,

be the numbers, then if ka", the several terms will be a", a"+1a”÷1, the logarithms or indices of which are in arithmetical progression. The converse is obvious.

4. To every base the logarithm of 1 is 0. may be, and in no other case

For generally ao 1, whatever a

published in the year 1633, after Mr. Briggs'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 10800000; and those of the sines, tangents, and secants, to any degree of accuracy required: also the Tables of Professor Babbage, and a very useful collection has been recently published by Mr. Galbraith, of Edinburgh. Also a duodecimo volume, by Professor Young, of Belfast, the tables in which appear to be very accurate, and are certainly very well arranged.

Mr. Michael Taylor's Tables in large 4to, containing the common logarithms, and the logarithmic sines and tangents to every second of the quadrant, are very valuable. And, in France, the new book of logarithms by Callet; the 2d edition of which, in 1795, has the tables still further extended, and are printed with what are called stereotypes, the types in each page being cast in a mould in one piece.

Dodson's Antilogarithmic Canon is likewise a very elaborate work, and used for finding the numbers answering to any given logarithm, each to Il places.

Neat portable tables have been published by Barker and Whiting; the smallest, and at the same time very accurate, are the tables of Mr. Woollgar, published in vol. xiii. of the Mechanics' Magazine.

* It is true that we have also 1± = 1, or generally that while a is an integer 1* = 1; yet as such a base is excluded by other considerations from logarithmic systems, the conclusion above-stated is perfectly valid in respect to our present object.