form motion.

Hence the chief properties of logarithms 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 diminishes 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, diminishes 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 ac, cd, 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, oc, od, &c. When the point p moves from a towards o, and op decreases proportionally, 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 p 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 sucu a

arbitrary; and they chose the decuple geometrical progressionals, 1, 10, 100, 1000, 10000, &c. and the arithmetical one, 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, 25, &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.

In order to this, they found a mean proportion between 1 and 10, and its logarithm will be one-half that of 10; and so given, then they found a mean proportional between the number first found and unity, which mean will be nearer to 1 than that before, and its logarithm will be one-half of the former logarithm, of one-fourth of that of 10; and having in this manner continually found a mean proportional between 1 and the last mean, and bisected the logarithms, they at length, after finding 54 such means, came to a number 1.0000000000000001278191495200323442, so near to 1 as not to differ from it so much as 10000000000000600 part, and found its logarithan to be

0.00000000000000005551115123125782702

and

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: between a given number, whose logarithm is wanted, and 1, they found a mean proportional, as above, until at length a number (mixed) be found, such a small matter 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, as but now mentioned; and this logarithm being as often doubled as the number of mean proportionals, (formed to get that number) will be the logarithm of the given

number. And this was the method Mr.

=

Briggs took to make the logarithms. But if they are to be made to only seven places of figures which are enough for comnion use, they had only occasion to find 25 mean proportionals, or, which is the same thing, to extract the 35th root of 10. Now having the logarithms of 3, 5, and 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 logarithms of 10 and 5; the logarithm of 4 will be two times the logarithin 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 logarithms 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.

+ } x2 + } x2 + 2°, &c.

Now if A B, or ab, bex, Cb being = 0.9, and CB 1.1, by putting this value of x in the equations above, we shall have the area b d D B 0.2006706954621511

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 ECN (Plate IX. fig. 5.) be an hyperbola, and A D, AQ, the asymptotes, and AB, AP, AQ, &c. taken upon one of them, be represented by numbers, and the ordinates BC, PM, QN, &c. be drawn from the several points B, P, Q, &c. to the curve, then will the quadrilinear spaces BCMP, PMN Q, &c. riz. their numerical measures be the loga rithms of the quotients of the division of AB by AP, AP by AQ, &c. since when AB, AP, AQ, &c. are continual propor tionals, 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 said great Sir Isaac Newton, of the method how to measure these spaces, and consequently of the construction of logarithms.

Let CA (fig. 6)=AF be = 1, and AB =Ab=x; then will be BD, and

1

1

1+

=bd; and putting these expressions

Term of the scries.

see in this table,

=

=

ing A d=0.2231, &c. A D = 0.1823, &c. and Ad 0.1053, &c. together, their sum is 0.5108, &c. and this added to 1.0986, &c. the area of A F G H, when CG= 3. You will have 1.6093379124341004 AFG H, when CG is 5; and adding that of 2 to this, gives 2.3025850929940457 = AFGH, when CG is equal to 10: and since 10 X 10=100; and 10 x 100 = 1000; and √ 5 × 10 × 0.98 =7, and 10 × 1.1=11, 1000 X 1.091 1000 X 0.998 7 X 11 499; it is plain that the area AFGH may be found by the composition of the arcas found before, when CG= 100, 1000, or any other of the numbers above-mentioned; and all these areas are the hyperbolic logarithms of those several numbers.

and

= 13, and

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 and subtraction 84 X 1020 8 × 9963 =2; 9945 981

1.2 0.9

only: for

10

1001

=3;

7 X 11

added thus, } 0.28768, &c.

=23;

0.40546, &c.

Total 0.69314, &c.

the area of

AFHG, when CG is 2.

Also since

dx

=

dr

214

of this series,

The two first terms d+ 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, becanse r 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 method, 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, that he was so much delighted with the construction of logarithms, at his first 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 loga rithms of large numbers. Thus, let z be an odd number, whose logarithm is sought: then shall the numbers z -1 and z+1 be even, and accordingly their logarithms, and the difference of the logarithms will be had, which let be called y. Therefore, also the logarithm of a number, which is a geometrical mean between 2- 1 and 1, will be given, riz. equal to half the sum of the logarithms. Now the series y x+

1

42'

+

181 13 15120 z7 25200 =

1

4 z &c. shall be

+ 24 z3 equal to the logarithm of the ratio, which the geometrical mean between the numbers %-1 and 2+1, has to the arithmetical mean, viz. to the number z. If the number exceeds 1000, the first term of the series, viz. y is sufficient for producing the logarithm to 13 or 14 places of figures, and the second term will give the logarithm to 20 places of figures. But if z 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. For example, it is required to find the logarithm of 20001: the loga rithm of 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

บ

Wherefore it is manifest that to have the logarithm to 14 places of figures, there is no necessity of continuing out the quotient beyond 6 places of figures. But if you have a mind to have the logarithm to 10 places of figures only, the two first figures are enough. And if the logarithms of the num-, bers above 20000 are to be found by this way, the labour of doing them will mostly consist in setting down the numbers. This series is easily deduced from the consideration of the hyperbolic spaces aforesaid. The first figure of every logarithm towards the left hand, which is separated from the rest by a point, is called the index of that logarithm ; because it points out the highest or remotest place of that number from the place of unity in the infinite scale of proportionals towards the left hand: thus, if the index of the logarithm be 1, it shows that its highest place towards the left hand is the tenth place from unity; and therefore all logarithms which have 1 for their index, will be found between the tenth and hundredth place, in the order of numbers. And for the same reason all logarithms which have 2 for their index,will be found between the hundredth and thousandth place in the order of numbers, &c. Whence universally the index or characteristic of any logarithm is always less by one than the number of figures in whole numbers, which answer to the given logarithm; and, in decimals, the index is negative.

As all systems of logarithms whatever are composed of similar quantities, it will be easy to form, from any system of logarithms, another system in any given ratio; and consequently to reduce one table of logarithms into another of any given form. For as any one logarithm in the given form is to its corresponden: logarithm in another form, so is any other logarithm in the given form to its correspondent logarithm in the required form; and hence we may reduce the logarithms of Lord Neper into the form of Briggs's, and contrary wise. For as 2.302585092, &c. Lord Neper's logarithm of 10, is to 1.0000000000, Mr. Briggs's logarithm of 10; so is any other logarithm in Lord Neper's form to the correspondent tabular logarithm in Mr. Briggs's form: and because the two first numbers constantly remain the same; if Lord Neper's logarithm

of any one number be divided by 2.302585, &c. or multiplied by .4342944, &c. the ratio of 1.0000, &c. to 2.50258, &c. as is found by dividing 1.00000, &c. by 2.30258, &c. the quotient in the former, and the product in the latter, will give the correspondent logarithm in Briggs's form, and the contrary. And, after the same manner, the ratio of natural logarithms to that of Briggs's will be found 868538963806.

The use and application of LOGARITHMS. It is evident, from what has been said of the construction of logarithms, that addition of logarithms must be the same thing as multiplication in common arithmetic; and subtraction in logarithms the same as division: therefore, in multiplication by logarithms, add the logarithms of the multiplicand and multiplier together, their sum is the logarithm of the product.

num. logarithms. Example. Multiplicand.. 8.5 0.9294189 1.0000000 Multiplier..... 10 Product....... 85 1.9294189

LOGARITHM, to find the complement of a. Begin at the left hand, and write down what each figure wants of 9, only what the last significant figure wants of 10; so the complement of the logarithm of 456, viz. 2.6589648, is 7.3410352.

In the rule of three. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first, the remainder is the logarithm of the fourth. Or, instead of subtracting a logarithm, add its complement, and the result will be the same,

LOGARITHMS, to raise powers by. Multiply the logarithm of the number given by the index of the power required, the product will be the logarithm of the power sought.

Example. Let the cube of 32 be required