gressional, will be the geometrical progressional answering to the square root, cube root, &c. of the arithmetical progressional over it; and from hence arises the following common, though imperfect definition of logarithms; viz.

That they are so many arithmetical progressionals, answering to the same number of geometrical ones. Whereas, if any one looks into the tables of logarithms, he will find, that these do not all run on in an arithmetical progression, nor the numbers they answer to in a geometrical one; these last being themselves arithmetical progressionals. Dr. Wallis, in his history of algebra, calls logarithms the indexes of the ratios of numbers to one another. Dr. Halley, in the Philosophical Transactions, Number 216, says, they are the exponents of the ratios of unity to numbers. So, also, Mr. Cotes, in his "Harmonia Mensurarum," says, they are the numercial measures of ratios: but all these definitions convey but a very confused notion of logarithms. Mr. Maclaurin, in his "Treatise of Fluxions," has explained the natural and genesis of logarithms, agreeably to the notion of their first inventor Lord Neper. Logarithms then, and the quantities to which they correspond, may be supposed to be generated by the motion of a point : and if this point moves over equal spaces in equal times, the line described by it increases equally.

Again, a line decreases proportionably when the point that moves over it describes such parts in equal times as are always in the same constant ratio to the lines from which they are subducted, or to the distances of that point, at the beginning of those lines from a given term in that line. In like manner, a line may increase proportionably, if in equal times the moving point describes spaces proportional to its distances from a certain term at the beginning of each time. Thus, in the first case, let a c (Plate IX. Miscel. fig. 1 and 2.) be to a o, c d to co, de, to do, e fto e o, fg to fo, always in the same ratio of QR to QS: and suppose the point P sets out from a, describing a c, cd, de, e f, fg, in equal parts of the time; and let the space described by P in any given time, be always in the same ratio to the distance of P from o at the beginning of that time, then will the right line a o decrease proportionally.

In like manner, the line o a (fig. 3.) increases proportionally, if the point p, in equal times, describes spaces ac, cd, de, ef, fg, &c. so that a c is to a o, c d to co, de to do, &c. in a constant ratio. If we

now suppose a point P describing the line A G (fig. 4.) with an uniform motion, while the point p describes a line increas ing or decreasing proportionally, the line A P, described by P, with this uniform motion, in the same time that o a, by increasing or decreasing proportionally, becomes equal to o p, is the logarithm of op. Thus A C, A D, A E, &c. are the logarithms of o c, o d, o e, &c. respective. ly; ando a is the quantity whose logarithm is supposed equal to nothing.

We have here abstracted from numbers, that the doctrine may be the more general; but it is plain, that if A C, A D, A E, &c. be supposed 1, 2, 3, &c. in arithmetic progression; c, o d, o e, &c. will be in geometric progression; and that the logarithm of o a, which may be taken for unity, is nothing.

Lord Neper, in his first scheme of loga. rithms, supposes, that while op increases or decreases proportionally, the uniform motion of the point P, by which the logarithm of op is generated, is equal to the velocity of pat a; that is, at the term of time when the logarithms begin to be generated. Hence, logarithms, formed after this model, are called Neper's Logarithms, and sometimes Natural Logarithms.

When a ratio is given, the point p describes the difference of the terms of the ratio in the same time. When a ratio is duplicate of another ratio, the point p describes the difference of the terms in a double time. When a ratio is triplicate of another, it describes the difference of the terms in a triple time; and so on. Also, when a ratio is compounded of two or more ratios, the point p describes the difference of the terms of that ratio in a time equal to the sum of the times, in which it describes the difference of the terms of the simple ratios of which it is compounded. And what is here said of the times of the motion of p when op increases proportionally, is to be applied to the spaces described by P, in those times, with its uniform motion.

Hence the chief properties of loga rithms are deduced. They are the measures of ratios. The excess of the logarithm of the antecedent, above the logarithm of the consequent, measures the ratio of those terms. The measure of the ratio of a greater quantity to a lesser is positive; as this ratio, compounded with any other ratio, increases it. The ratio of equality, compounded with any other ratio, neither increases nor dimin ishes it; and its measure is nothing. The measure of the ratio of a lesser quantity

to a greater is negative; as this ratio, compounded with any other ratio, dimin ishes it. The ratio of any quantity A to unity, compounded with the ratio of unity to A, produces the ratio of A to A, or the ratio of equality; and the measures of those two ratios destroy each other when added together; so that when the one is considered as positive the other is to be considered as negative. By supposing the logarithms of quantities greater than oa (which is supposed to represent unity) to be positive, and the logarithms of quantities less than it to be negative, the same rules serve for the operations by logarithms, whether the quantities be greater or less than o a. When op increases proportionally, the motion of p is perpetually accelerated; for the spaces a c, ed, de, &c. that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as the lines o a, o c, od, &c. When the point p moves from a towards o, and op decreases proportion ally, the motion of p is perpetually retarded; for the spaces described by it in any equal times that continually succeed after each other, decrease in this case in the same proportion as op decreases.

If the velocity of the point be always as the distance op, then will this line increase or decrease in the manner supposed by Lord Neper; and the velocity of the point p being the fluxion of the line op, will always vary in the same ratio as this quantity itself. This, we presume, will give a clear idea of the genesis, or nature of logarithms; but for more of this doctrine, see Maclaurin's Fluxions.

LOGARITHMS, construction of. The first makers of logarithms had in this a very laborious and difficult task to perform; they first made choice of their scale or system of logarithms, that is, what set of arithmetical progressionals should answer to such a set of geometrical ones, for this is entirely arbitrary; and they chose the decuple geometrical progressionals, 1, 10, 100, 1000, 10000, &c. and the arithmetical ones, 0, 1, 2, 3, 4, &c. or 0.000000; 1.000000; 2.000000; 3.000000; 4.000000, &c. as the most convenient. After this they were to get the logarithms of all the intermediate numbers between 1 and 10, 10 and 100, 100 and 1000, 1000 and 10000, &c. But first of all they were to get the logarithms of the prime numbers, 3, 5, 7, 11, 13, 17, 19, 23, &c. and when these were once had, it was easy to get those of the compound numbers made up of the prime ones, by the addition or subtraction of their logarithms.

and

1.0000000000000001278191493200323442 so near to 1 as not to differ from it so much as 1000ōō o ō ó ó ó ó ó ōōōō part, and found its logarithm to be 0.000000000000000055515123125782702 00000000000000012781914932003235 to be the difference whereby 1 exceeds the number of roots or mean proportionals found by extraction; and then, by means of these numbers, they found the logarithms of any other numbers whatsoever; and that after the following manner: beis wanted, and 1, they found a mean profween a given number, whose logarithm ber (mixed) be found, such a small matportional as above, until at length a numter above 1, as to have 1 and 15 cyphers after it, which are followed by the same number of significant figures; then they said, as the last number mentioned above is to the mean proportional thus found, so is the logarithm above, viz. 0.00000000000000005551115123125782702 to the logarithm of the mean proportional number, such a small matter exceeding 1, being as often doubled as the number of as but now mentioned; and this logarithm mean proportionals, (formed to get that number. And this was the method Mr. number) will be the logarithm of the given Briggs took to make the logarithms. But if they are to be made to only seven places of figures, which are enough for find 25 mean proportionals, or, which is the common use, they had only occasion to same thing, to extract the 33331172th root of 10. Now having the logarithms of 3, 5, 7, they easily got those of 2,4, 6, 8, and 9; for since 2, the logarithm of 2 will be the difference of the logarithm of 10, and 5 the logarithm of 4 will be two times the logarithm of 2; the logarithm of 6 will be the sum of the logarithm of 2 and 3; and the logarithm of 9 double the logarithm of 3. So, also, having found the logarithm of 13, 17, and 19, and also of 23 and 29, they did easily

=

get those of all the numbers between 10 and 30, by addition and subtraction only; and so having found the logarithms of other prime numbers, they got those of other numbers compounded of them.

But since the way above hinted at, for finding the logarithms of the prime numbers, is so intolerably laborious and troublesome, the more skilful mathematicians that came after the first inventors, employing their thoughts about abbreviating this method, had a vastly more easy and short way offered to them, from the contemplation and mensuration of hyperbolic spaces contained between the portions of an asymptote, right lines perpendicular to it, and the curve of the hyperbola: for if E C N (Plate IX. fig. 5.) be an hyperbola, and A D, A Q, the asymptotes, and A B, A P, A Q, &c. taken upon one of them, be represented by numbers, and the ordinates B C, PM, Q N, &c. be drawn from the several points B, P, Q, &c. to the curve, then will the quadrilinear spaces BC MP, PMN Q, &c. viz. their numerical measures, be the logarithms of the quotients of the division of A B by A P, A P by A Q, &c. since when A B, A P, A Q, &c. are continual proportionals, the said spaces are equal, as is demonstrated by several writers concerning conic sections. See HYPERBOLA.

Having said that these hyperbolic spaces, numerically expressed, may be taken for logarithms, we shall next give a specimen, from the great Sir Isaac Newton, of the method how to measure these spaces, and consequently of the construction of logarithms.

=

1

Let C A (fig. 6.) A F be = 1, and AB = Ab - be: =x; then will. 1+x

BD and

-x

bd; and putting these

Now if this difference of the areas be added to, and subtracted from, their sum before found, half the aggregate, viz 0.1053605156578263, will be the greater area A d, and half the remainder, viz 0.0953101798043249, will be the lesser area A D.

By the same tables, these areas, A D and A d, will be obtained also when AB + A dare supposed to be or C B = 1.01, and C b 0.99, if the numbers are but duly transferred to lower places, as

=

1000 x 0.998

2

=

7 X 11

= 13, and

= 499; it is plain that the

area A F G H may be found by the composition of the areas found before, when C G 100, 1000, or any other of the numbers above mentioned; and all these areas are the hyperbolic logarithms of those several numbers.

Having thus obtained the hyperbolic logarithms of the numbers 10, 0.98, 0.99, 1.01, 1.02; if the logarithms of the four last of them be divided by the hyperbolic logarithm 2.3025850, &c. of 10, and the index 2 be added: or, which is the same thing, if it be multiplied by its reciprocal 0.4342944819032518, the value of the subtangent of the logarithmic curve, to which Briggs's logarithms are adapted, we shall have the true tabular logarithms of 98, 99, 100, 101, 102. These are to be interpolated by ten intervals, and then we shall have the logarithms of all the numbers between 980 and 1020; and all between 980 and 1000, being again interpolated by ten intervals, the table will be as it were constructed. Then from these we are to get the logarithms of all the prime numbers, and their multiples less than 100, which may be done by addition

10

10

84×1020 9945

= 3; 2 = 5;

/8 x 9963

✔98

= 2;

2

984

99

Sum = 0.28767,

&c.

=7; 9

= 11;

= 13;

7 X 11

6

added thus, $0.40546, &c.

998

9936

20.28768, &c.

Total = 0.69314, &c. = the area of AF H G, when C G is - 2. Also since

17; 4 X 13

= 19;

=23;

16 x 27

986

992

999

= 29;

37;

2 X 17

32

27

987

41:

47;

23

21

0.8

9911

9971

53;

11 x 17

13 x 13

9949

994

=

61;

67;

3 X 49 9954

14

996

79;

83;

59;

9882 2 x 81 9928 8×17

71;

9968

21

7 X 16

of the lesser number: for if the numbers

are represented by Cp, CG, CP, (fig. 16.) and the ordinates ps, PQ, be raised; if n be wrote for C G and x for C P, or 2r xa x3 Gp, the area p8 Q P, or + + 2n2 n S n' &c. will be to the area ps H G, as the difference between the logarithms of the extreme numbers, or 2 d, is to the difference between the logarithms of the lesser, and of the middle one; which, therefore, dx dx' d x3

+ + -,&c. 2n 3n

Ꮖ x3

x5

+ + &c. ኪ 3n 5n'

&c.

d.x 2n

The two first terms d+ of this se

ries being sufficient for the construction of a canon of logarithms, even to 14 places of figures, provided the number, whose logarithm is to be found be less than 1000; which cannot be very troublesome, because x is either 1 or 2; yet it is not necessary to interpolate all the places by help of this rule, since the logarithms of numbers, which are produced by the multiplication or division of the number last found, may be obtained by the numbers whose logarithms were had before, by the addition or subtraction of their logarithms. Moreover, by the difference of their logarithms, and by their second and third differences, if necessary, the void places may be supplied more expeditiously, the rule beforegoing being to be applied only where the continuation of some full places is wanted, in order to obtain these differences.

By the same method rules may be found for the intercalation of logarithms, when of three numbers the logarithm of the lesser and of the middle number are given, or of the middle number and the

greater; and this although the numbers should not be in arithmetical progression. Also by pursuing the steps of this me thod, rules may be easily discovered for the construction of artificial sines and

tangents, without the help of the natural tables. Thus far the great Newton, who says, in one of his letters to M. Leibnitz, construction of logarithms, at his first that he was so much delighted with the setting out in those studies, that he was ashamed to tell to how many places of figures he had carried them at that time: and this was before the year 1666; because, he says, the plague made him lay aside those studies, and think of other things.

Dr. Keil, in his treatise of logarithms, at the end of his Commandine's Euclid, gives a series, by means of which may be found easily and expeditiously the logarithms of large numbers. Thus let z be an odd number, whose logarithm is sought: then shall the numbers z-1 and logarithms, and the difference of the lo +1 be even, and accordingly their garithms will be had, which let be called y. Therefore, also the logarithm of a number, which is a geometrical mean between z- - 1 and + 1, will be given, viz. equal to half the sum of the loga 1 rithms. Now the series y + + 4: 242 &c. shall be equal

181

+15120 =7

13 + 25200 29'

to the logarithm of the ratio, which the geometrical mean between the numbers z-1 and z + 1, has to the arithmetical mean, viz. to the number z. If the number exceeds 1000, the first term of the y

series, viz. is sufficient for producing 4=

the logarithm to 13 or 14 places of figures, and the second term will give the loga rithm to 20 places of figures. But if = be greater than 10000, the first term will exhibit the logarithm to 18 places of figures: and so this series is of great use in filling up the chiliads omitted by Mr. Briggs logarithm of 20001: the logarithm of For example, it is required to find the 20000 is the same as the logarithm of 2, with the index 4 prefixed to it; and the difference of the logarithms of 20000 and 20001, is the same as the difference of the logarithms of the numbers 10000 and 10001, viz. 0.0000434272, &c. And if this difference be divided by 4 z, or 80004, the quotient shall be

4 z