6. Required the sum of 40 terms of the series (1×2) +(3 × 4) + (5 x 6) + (7×8), &c.

X

Ans. 86884.
2x-1
2x

7. Required the sum of n terms of the series

2x-3 2x-5 2x -7
+
2x
2x
2x

+

,

&c.

8. Required the sum of the infinite series

1 1 + + 3.4.5.6 4.5.6.7'

&c.

9. Required the sum of the series

&c., continued ad infinitum.

+

Ans. n(2).

1 1 + 1.2.3.4 2.3.4.5 2.3.4. 1

Ans.

18°

1 35'

1

2

10. It is required to find the sum of n terms of the series 1+8x+27x2+ 64x2 + 125x*, &c., continued ad infinitum.

3

قسم

11. Required the sum of n terms of the series ++

8.12

+
2n (1+2n)

6

6

6

14. Required the sum of the series 27+7.12+12.17

Σ

1

3n (4 + 2n)

1

,

1

n

1

n

Ans. 2 = 24' 8 = 2(3+6) 4(6+6)

16. Required the sum of the series

2

3.5

(1+2n).(3 + 2n)

1

3 4 + 5.7 7.9

1
-==-

Ans. 2 = 12,8 =

17. Required the sum of the series

7 8 + 3.4.5 4.5.6'

12

.1 4(3+4n)

5 6 + + 1.2.3 2.3.4

LOGARITHMS are a set of numbers that have been computed and formed into tables, for the purpose of facilitating many difficult arithmetical calculations; being so contrived, that the addition and subtraction of them answers to the multiplication

* The series here treated of are such as are usually called algebraical, which, of course, embrace only a small part of the whole doctrine. Those, therefore, who may wish for farther information on this abstruse but highly curious subject, are referred to the Miscellanea Analytica of of Demoivre, Sterling's Method Differ., James Bernouilli's de Seri. Infin., Simpson's Math. Dissert., Waring's Medii Analyt., Clark's translation of Lorgna's Series, the various works of Euler, and Lacroix Traite du Calcul. Diff. et Int., where they will find nearly all the materials that have been hitherto collected respecting this branch of analysis.

and division of natural numbers with which they are made to correspond.*

Or, when taken in a similar but more general sense, logarithms may be considered as the exponents of the powers to which a given or invariable number must be raised, in order to produce all the common, or natural numbers. Thus, if

ax = y, ax' = y', ax" = y", &c. then will the indices x, x', x", &c., of the several powers of a, be the logarithms of the numbers y, y', y", &c., in the scale, or system, of which a is the base.

So that from either of these formulæ it appears, that the - logarithm of any number, taken separately, is the index of that power of some other number, which, when involved in the usual way, is equal to the given number.

And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural number.

It is likewise farther evident, from the first of these equations, that when y = 1, x will be = 0, whatever may be the value of a; and consequently, the logarithm of 1 is alvrays 0, in every system of logarithms.

And if x = 1, it is manifest from the same equation, that the base a will be = y; which base is therefore the number whose proper logarithm, in the system to which it belongs, is 1.

* This mode of computation, which is one of the happiest and most useful discoveries of modern times, is due to Lord Napier, Baron of Merchiston, in Scotland, who first published a table of these numbers in the year 1614, under the title of Canon Mirificum Logarithmorum ; which performance was eagerly received by the learned throughout Europe, whose efforts were immediately directed to the improvement and extensions of the new calculus that had so unexpectedly presented itself.

Mr. Henry Briggs, in particular, who was, at that time, professor of geometry in Gresham College, greatly contributed to the advancement of this doctrine, not only by the very advantageous alteration which he first introduced into the system of these numbers, by making 1 the logarithm of 10, instead of 2.3025852, has had been done by Napier; but also by the publication, in 1624 and 1633, of his two great works, the Arithmetica Logarithmica and the Trigonometrica Britanica, both of which were formed upon the principle abovementioned; as are, likewise, all our common fogarithmic tables at present in use.

See, for farther details on this part of the subject, the Introduction to my Treatise of Plane and Spherical Trigonometry, 8vo. 2d edit., 1813; and for the construction and use of the tables, consult those of Sherwin, Hutton, Taylor, Callet, and Borda, where every necessary information of this kind may be readily obtained,

Also, because a = y, and ax = y', it follows from the multiplication of powers, that a × a', or ax + x = yy'; and consequently, by the definition of logarithms, given above, x + x' = log. yy', or,

log. yy' = log. y + log. y'.

And, for a like reason, if any number of the equations ax = y, xx = y', a" = y", &c., be multiplied together, we shall have ax + x + x/", ', &c., yyy", &c.; and consequently, x + x' + x", &c., = log. yy'y," &c.; or,

=

log. yy'y", &c.; = log. y + log. y' + log. y", &c. From which it is evident, that the logarithm of the product of any number of factors is equal to the sum of the logarithms of those factors.

Hence, if all the factors of a given number, in any case of this kind, be supposed equal to each other, and the sum of them be denoted by m, the preceding property will then become

log. ym = m log. y.

From which it appears, that the logarithm of the mth power of any number is equal to m times the logarithm of that number.

ax

ora

In like manner, if the equation at = y be divided by ax' = y', we shall have, from the nature of powers, as before, = 2; and, by the definition of logarithms laid y y down in the first part of this article, a - x' = log. or y"

Hence, the logarithm of a fraction, or of the quotient arising from dividing one number by another, is equal to the logarithm of the numerator minus the logarithm of the denominator.

And if each member of the common equation a y be raised to the fractional power denoted by, we shall have,

m

in that case, an

m

13

= y";

m

And consequently, by taking the logarithms, as before,

m

m

- x = log. y", or log. yn =

n

Where it appears that the logarithm of a mixed root, or power, of any number, is found by multiplying the logarithm

of the given number by the numerator of the index of that power, and dividing the result by the denominator.

And if the numerator m, of the fractional index, be in this

case taken equal to 1, the above formula will then become

From which it follows, that the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number.

Hence, besides the use of logarithms, in abridging the operations of multiplication and division, they are equally applicable to the raising of powers and extracting of roots; which are performed by simply multiplying the given logarithm by the index of the power, or dividing it by the number denoting the root.

But although the properties here mentioned are common to every system of logarithms, it was necessary, for practical purposes, to select some one of them from the rest, and to adapt the logarithms of all the natural numbers to that particular scale.

And, as 10 is the base of our present system of arithmetic, the same number has accordingly been chosen for the base of the logarithmic system, now generally used.

So that, according to this scale, which is that of the common logarithmic tables, the numbers

10-4, 10, 102, 101, 10°, 101, 102, 103, 104, &c.

1

1 1 1

10000' 1000' 100' 10'

Or,

1, 10,100, 1000, 10000, &c.,

have for their logarithms

. -4, -3, -2, -1, 0, 1, 2, 3, 4, &c.

Which are evidently a set of numbers in arithmetical progression, answering to another set in geometrical progression; as is the case in every system of logarithms.

And therefore, since the common or tabular logarithm of any number (n) is the index of that power of 10, which, when involved, is equal to the given number, it is plain, from the following equation,

that the logarithms of all the intermediate numbers, in the above series, may be assigned by approximation, and made to occupy their proper places in the general scale.

It is also evident, that the logarithms of 1, 10, 100, 1000,