which answers to the second part of the expression in article 12. Consequently the hyperbolic logarithm of the number 2 is 0.57536414488+0.1177830356=0.69514718054.

The hyperbolic logarithm of 2 being thus found, that of 4, 8, 16, and all the other powers of 2, may be obtained by multiplying the logarithm of 2, by 2, 3, 4, &c. respectively, as is evident from the properties of logarithms stated in article 6. Thus, by multiplication, the hyperbolic logarithm of 4 1.38629436103 =

of 8=2.07944154162 &c.

From the above the logarithm of 3 may easily be obtained.

4

3

For 44x =4x-3; and therefore as the logarithm of

3

4

was determined above, and also the logarithm of 4, From the logarithm of 4, viz.

1.38629436108,

viz. 0.28768207244,

4

Subtract the logarithm of

And the logarithm of 3 is

4

3

Having found the logarithms of 2 and 3, we can find, by addition only, the logarithms of all the powers of 2 and 3, and also the logarithms of all the numbers which can be produced by multiplication from 2 and 3. Thus,

And the sum is the logarithm of 6 1.79175946918. To this last found add the logarithm of 2, and the sum 2.48490664972 is the logarithm of 12.

The hyperbolic logarithms of other prime numbers may be more readily calculated by attending to the following article.

13. Let a, b, c, be three numbers in arithmetical progression, whose common difference is 1. Let b be the prime number, whose logarithm is sought, and a and c even numbers whose logarithms are known, or easily obtained from others already computed. Then, a being the least of the three, and the common difference being 1, a b-1, and c=b+1. Consequently axc-b-1xb+1 b ac+1 b, and ac+16'; and therefore This is a general expression for the fraction which it will be.

ac

ac

ac+1

ac

-X

ac+acx=ac+1-acx-x, and therefore 2 acx

25

Consequently,

x=0.0204081632

=0.0000028332

=0.0000000007

0.020410997 1

2

0.0408219942

But×8x3=25, and the addition of Logarithms an

24

swers to the multiplication of the natural numbers to which they belong. Consequently,

And the sum is the log. of 25 3.2188758254

The half of this, viz. 1.6094379127, is the hyperbolic loga

49

48

For x6x8=49 Consequently the half of this, viz. 1.94591014899, is the hyperbolic logarithm of 7; for 7 x7=49.

If the reader perfectly understand the investigations and examples already given, he will find no difficulty in calculating the hyperbolic logarithms of higher prime numbers. It will only be necessary for him, in order to guard against any embarrassment, to compute them as they advance in succession above those already mentioned. Thus, after what has been done, it would be proper, first of all, to calculate the hyperbolic logarithm of 11, then that of 13, &c.

Proceeding according to the method already explained, it will be found that

The hyperbolic logarithm of 11 is 2.397895273016 of 13 is 2.564999357538 of 17 is 2.833213344878 of 19 is 2.944438979941

Logarithms were invented by Lord Neper, Baron of Merchiston, in Scotland. In the year 1614, he published at Edinburgh a small quarto, containing tables of them, of the hyperbolic kind, and an account of their construction and use. The discovery afforded the highest pleasure to mathematicians, as they were fully sensible of the very great utility of logarithms; but it was soon suggested by Mr. Briggs, afterwards Savilian Professor of Geometry in Oxford, that another kind of logarithms would be more

convenient, for general purposes, than the hyperbolic. That one set of logarithms may be obtained from another will readily appear from the following article.

14. It appears from articles 1, 3, and 7, that if all the logarithms of the geometrical progression 1, 1+a',l+al2, 1+a3, l+a, 1+as, &c. be multiplied or divided by any given number, the products and also the quotients will likewise be logarithms, for their addition or subtraction will answer to the multiplication or division of the terms in the geometrical progression to which they belong. The same terms in the geometrical progression may therefore be represented with different sets or kinds of logarithms in the following manner:

1, ì+a1, l+@2, 1+a3, 1+a4, 1+a 5, 1+al¤; &c. 1, 1+a2, 1+a2,1+a8, 1+aa2, 1+ a' 3⁄41, 1+a'¤¿, &c.

1, 1+a”, l+a", i+a”, l+a", i+a", l+am, &c. In these expressions land m denote any numbers, whole or fractional; and the positive value of the term in the geometrical progression, under the same number in the index, is understood to be the same in each of the three series. Thus if a* be equal to 7, then 1+a42, is equal to 7, as is also i+a". If 1+a6 be equal to 10, then

6

1+a is equal to 10, as is also 1+am, &c. If therefore 1, 21, 31, &c. be hyperbolic logarithms, calculated by the methods already explained, the logarithms expressed by &c. may be derived from them; for the hy

1 2 3

m' m' m

perbolic logarithm of any given number is to the logarithm in the last-mentioned set, of the same number, in a given

15. Mr. Briggs's suggestion, above alluded to, was that 1 should be put for the logarithm of 10, and consequently 2 for the logarithm of 100, 3 for the logarithm of 1000, &c. This proposed alteration appears to have met with the full approbation of Lord Neper; and Mr. Briggs afterwards, with incredible labour and perseverance, calculated extensive tables of logarithms of this new kind, which are

now called common logarithms. If the expeditious me thods for calculating hyperbolic logarithms explained in the foregoing articles*, had been known to Mr. Briggs, his trouble would have been comparatively trivial with that which he must have experienced in his operations.

16. It has been already determined that the hyperbolic logarithm of 5 is 1.6094379127, and that of 2 is 0.69314718054, and therefore the sum of these logarithms, viz. 2.30258509324 is the hyperbolic logarithm of 10. If, therefore, for the sake of illustration, as in article 14, we suppose 1+ a = 10, and allow, in addition to the hypothesis 1 2 3 4 &c. denote common lo

there formed, that,

m' m m m

6

garithms, then 61 = 2.30258509324, and = 1; and the ratio for reducing the hyperbolic logarithm of any number to the common logarithm of the same number, is that of 2.30258509324 to 1. Thus in order to find the common logarithm of 2, 2.30258509324 : 1 :: 0.693147 18054 0.3010299956, the common logarithm of 2. The common logarithms of 10 and 2 being known, we obtain the common logarithm of 5, by subtracting the common logarithm of 2 from 1, the common logarithm of 10; for 10 being divided by 2, the quotient is 5. Hence the common logarithm of 5 is 0.6989700044. Again, to find the common logarithm of 3, 2.30258509324 : 1 :: 1.0986122 8864: 0.4771212546 the common logarithm of S.

17. As the constant ratio, for the reduction of hyperbolic to common logarithms, is that of 2.30258509824 to 1, it is evident that the reduction may be made by multiplying the hyperbolic logarithm, of the number whose common logarithm is sought, by

818.

1 2.50258509324

.4342944

Thus 1.94591014899, the hyperbolic logarithm of 7, being multiplied by .4842944818, the product, viz. .8450980378, &c. is the common logarithm of 7.

The common logarithms of prime numbers being derived from the hyperbolic, the common logarithms of other numbers may be obtained from those so derived, merely by ad

* Some of the principal particulars of the foregoing methods were discovered by the celebrated Thomas Simpson. See also Mr. Hellins' Mathematical Essays, published in 1788.