now proceed to explain, by means of the principles of the common algebraic analyfis, in the following manner.

(39.) Let r denote any pofitive number what ever, different from unity. Then, by affuming proper exponents, the powers of r may become equal to all pofitive numbers whatever, whether thofe numbers be whole or fractional. Thus if 2, we have 20—1, 21—2, 224.23—8, 24=16, &c. As to the intermediate numbers 3, 5, 6, 7, &c. they may be expreffed, at leaft nearly, by fractional powers of 2, Thus,

21.5850=3; 22.3319=5, 22.550=6, 12.80737. &c. So alfo the powers of 10 may become, either exactly, or nearly, equal to all pofitive numbers whatever. Thus,

10o =I 10.30102

10-781

8451

ΙΟ 10-9931

169542 101

6

78

10.0990 (40.) In general, if a denote any positive number, it is fufficiently evident, that it is poffible to conceive a correfponding number A, fuch, that r^a; and A, that is the exponent of r, which gives a power equal to a, is called the logarithm

of a.

(41.) From this manner of defining logarithms, we readily derive all their properties. For a and 6 denoting any two numbers; alfo A and B their logarithms; we have r =a, and r3b, there-·

A

B

B*

r tr = ab; but r X r =r therefore A+B is the logarithm of ab; that is, the fum of the logarithms of any two numbers is equal to the logarithm of the product. Again,

(42.) There may be various systems of logarithms, according to the different values which may be given to the number r, which is called the radical number of the fyftem. In the common fyftem of the logarithms, r is 10; but in the fyftem of Napier it is 27182818. It is evident from the definition given in § 40, that the loga rithm of the radical number in every fyftem muft be unity. In different fyftems, the logarithms of the fame number are always to one another in a

depends only

Thus it appears that the fraction on r, and r; and therefore must be the fame

whatever be the value of the number a.

(43.) Hence it follows, that if the logarithms of numbers, according to any one fyftem, be given, the logarithms of the fame numbers, according to any other propofed fyftem, may be readily found. Thus, if the given fyftem be the common logarithms, the radical number of which is 10, and it be required to find the logarithm of any number a, according to Napier's fyftem, of which the radical number is 2'7182818; let A denote the logarithm of a, according to the former system; and the logarithm of a, according to the latter. Then, by fubftituting 10 and 2'7182818 for r and 'r'; alfox for A',in the laft equation of § 42, we have 1027182818, and, from the nature of logarithms, log. 10= log. 2'7182818, $41: Hence

x=

log. 1o XA = XA log. 2 7182818 4342945 =2'3025851 X A. Thus it appears, that Napier's logarithm of any number is equal to the common logarithm of the fame number, multiplied by 2 3025851; or divided by 4342945.

= y.

(44.) Let us now denote any number whatever by y and its logarithm by x; then, r reprefenting as. before the radical number of the fyftem, the relation between a number and its logarithm is reprefented by the algebraic equation r* = This equation fuggefts two fubjects of enquiry, both capable of being refolved by means of the algebraic method of analyfis. Thefe are: First, To determine y when x is given; or to determine the number which correfponds to a given logathat is, to determine the logarithm corresponding rithm. Secondly, To determine x when y is given; to a given number.

(45.) We proceed to the firft fubject of enquiry, namely, to find an algebraic expreffion for y, in terms of x and r; or to exprefs, generally, any number by means of its logarithm, and the bafe or radical numbers of the fyftem: for this purpose, let us affume

r=A+Bx + C x2 + D x3 + Ex1 + &c? Here A, B, C, D, &c. are fuppofed to be coefficients independent of x. Let z denote any other quantity; then, in like manner, we have r2=A+B+C≈2 + D≈3 + Ex++ &c. Taking now the difference between the affumed equations, and dividing both fides of the refult by *-*, we have

(47.) Thus we have obtained a series for y, which will always converge, whatever be the values of the quantities m and x. Before, however, the feries can be applied to practice, it will be neceffa ry to compute the value of m, which, by putting r-1 for its value a, is equal to this other feries r1_(r−1)2+(r—1)3. (r—1)2 + (x1)$ -&c.

I

2

But as this laft feries will not converge, unless I be less than unity, it will be neceffary to have recourse to another method of obtaining its fum than by the mere addition of its terms. (48.) Refuming, therefore, the equation m x m2x2 m3x-3 m*x*

I

+-+-+

let us fuppofe =

I' 2 1'2°3 I'2'3'4

and we have

m m2 m3

r=1+2+ ·+ +

+ &c.

1°2 1°2°3 1*2*3°4

This feries gives the value of r, when m is known. Therefore, if we suppose a system of logarithms to be fuch, that m=1, and put e for the radical number of that fyftem, we have

<=1+

and by reducing a sufficient number of the terms of this feries to decimal fractions, and taking their fum, we readily find e=27182818, nearly. Now, as e is the radical number of a fyftem, in which m1, for the fame reafon that

mx

I

I

and as this must be true, whatever be the value of x, we may suppose x=m; hence we have

; or,

I

log.e

Let this value of

log.rlog.e

(r=1)+.

+ &c.

Now, this equation muft hold

true, whatever be the value of r; we may therefore fubftitute 1+ y for r, and confequently y for r-1; where y denotes any number whatever; hence we find

log.(1 +3)= log.e(3—2++&c.)

Thus we have found a feries, which expreffes the logarithm of any number whatever, by means of the number itself, and the logarithm of another given number e.

(so.) The feries which we have just now found will be of little or no ufe in the actual conftruction of logarithms, unless y be a small fraction; we may however derive from it another feries, much better adapted to that purpose, in the following

manner.

I 100000000

8th term does not exceed ;; by redu cing, therefore, the firft 7 terms to decimals, and taking their fum, find log. 23465736 X 2 M='6931472 X log e; but we cannot find the abfolute value of the logarithm of 2, without previously affigning the particular fyftem to which the logarithm required is to belong. The moft fimple hypothefis we can affume is, that log. e=1; hence we have log, 26931472. This fyftem, which corresponds

to the equation y=e*, (where y denotes a num

ber, its logarithm, and e the base of the fyftem, 2'7182818) is the fame as that firft adopted by NAPIER; the logarithms of which are also called Hyperbolic, for the reason already affigned in § 21.

(52.) It appears therefore, from § 49 and § 50, that NAPIER'S, or the Hyperbolic logarithm of a ny numbers z, is equal to either of these two series (2-1) (2 - 1)}^

m

2

+

3

(1

3

}

and that the logarithm of z, accord

ing to any other fyftem, is equal to the hyperbolic logarithm of the fame number, multiplied by a certain conftant quantity M= log. e; which, being peculiar to that fyftem, has been called by writers on logarithms the Modulus of the fyftem.

Now it appears, by recurring to § 47. that or m, is equal to this series

-

(~ — 1 )2 (r — 1)3

&c.

log.

which is evidently the hyperbolic logarithm of r, the radical number of the fyftem: hence it follows, that the modulus of any fyftem of logarithms is the reciprocal of the hyperbolic logarithm of the radical number of that fyftem.

(53.) In the common system of logarithms, the radical

But, as this feries converges too flowly to be of any ufe, we may find another that converges fafter, by confidering, that fince 10 = 2 X 5, therefore, log. 1olog. 2 + log. 5: Now we have already found hyp. log. 2; it is only neceffary, therefore, to compute hyp. log, 51

5;

(54.) But, in feeking hyp. log. 1o, it will be convenient to begin with even a lefs number than and for this purpose, we shall inveftigate a rule, by which, the logarithm of any number being fup pofed given, the logarithm of another number, a little greater than the given one, may be found with great facility.

Let ʼn be a number, whose logarithm is given, and » + v another number, whofe logarithm is it

required ; then, fince n + w➡n (x + 7 ),

of the common system of logarithms.

(56.) We may now find the common logarithm of 2, for nothing more is neceffary than to mul tiply its hyp, log. already found, 51, by the m dulus, 4342945; or divide it by 2 3025851, the reciprocal of the modulus, and, in either way, we find the common logarithm of 2 to be *3010300. The common logarithm of 5 may now alfo be found; for fince 5 = we have log. 5 = log.

10

Let thefe values of x and

inferted in the feries of $50, and we presently have 2 log. log. (n—v) + log. (n + v) + 2 M x

โ.

I

kc.}

(58.) We fhall conclude this fection, by fhewing the application of the three feries which have been inveftigated in 51. § 54. and $ 57. to the calculation of the common logarithms of the numbers between 1 and 10; and as it will be moft convenient in rectly, inftead of firft finding the hyperbolic logapractice, to compute the common logarithm dirithm, and afterwards multiply by the modulus of the common fyftem; we shall put down each feries in a form fuited to that purpose. The value of in the two feries laft found will generally be unity, we fhall therefore put them down under that form. Let N = 2 M = ·8685889638 + &c. Let n − 1, n, n + 1 denote three numbers.

+ &c.)

This feries converges very faft, fo that the three firft terms are fufficient to give a refult true to the 7th decimal; and accordingly their fum, when multiplied by the coefficient 2, will be found = 2231435. Now we formerly found hyp. log. 2 6931472, 51, therefore, to the number juft

II. Log. (n+1)= log. n. + NB+

+ &c.

NB3 NB5

III. 2 Log. n= log. (n − 1) + log. (n + 1) NC3 NCS

NC + + + &c.

3

5

N8685890, NC '0089545

NC3

log. 67781513

•8450980 *845098

(59.) EXAMPLE I. It is required to find the common logarithm of 2, to 6 places of figures, by the firft feries.

log. of 7

log. 14 = log. 2 + log. 7.

2 log. 13 = : log. 12 + log. 14+N{

I

+ &c. }

3°3373

I

337

+

and in this way might a table of logarithms be calculated; it would however be neceffary to compute the logarithms of the numbers at the beginning of the table, to many more figures than were intended to be retained; because, that at each operation, the laft figure of the logarithm is fet down only to the nearest figure. But in conftructing a table, there are many expedients by which the calculations may be abridged; thefe we cannot here find room to explain; and muft therefore refer the curious reader to Dr HUTTON'S Mathematical Tables, where he will meet with ample information on the subject.

SECT. IV. Of certain CURVES related to LOGA