For

49

3.89182029798

48×6×8=49. Consequently the half of this, viz. 1.94591014899, is the hyperbolic logarithm of 7; for 7 ×7=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 2.94443897994125 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.

2

14. It appears from articles 1, 3, and 7, that if all the logarithms of the geometrical progression 1, 1+a,)', 1+d2, 1+a3, 1+al+, 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:

I

1, 1+a, 1+a2, 1+al3, 1+a+, 1+als, 1+a, &c. 1, 1+all, 1+1+1 1+ 1+ası, 1+1, &c. 1, 1+ alm, 10m, 1+alm 1+alm, 1+alm, 1+am, &c.

1

2

4

5

6

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 1+a+ be equal to 7, then 1+at, is equal

6

to 7, as is also 1+am. If 1+a be equal to 10, then 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

m2 m2 m2

&c. may be derived from them; for the hy

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

4

4

1

ratio. Thus 4l: ::1: 4lm=m; also 6l:

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 methods for calculating hyperbolic logarithms explained in the foregoing articles*, had been known to Mr. Briggs, his trouble would bave 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 byperbolic logarithm of 10. If, therefore, for the sake of illustration, as in article 14, we suppose 1+1=10, and allow, in addition to the hypothesis

6

1

234

there formed, that &c. denote common lo

m'

, mmm

6

garithms, then 61 = 2.30258509324, and -, =1; and

m

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 3.

17. As the constant ratio, for the reduction of hyperbolic to common logarithms, is that of 2.30258509324 to 1, it is evident that the reduction may be made by multiplying the hyperbolic logarithm, of the number whose com

mon logarithm is sought, by

818.

1 2.30258509324

= .4342944

Thus 1.94591014899, the hyperbolic logarithm of 7, being multiplied by .4342944818, 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

€

dition or subtraction. For addition of logarithms, in any set or kind, answers to the multiplication of the natural numbers to which they belong, and consequently subtraction of logarithms to the division of the natural nuinbers. Hyperbolic logarithms are not only useful as a medium through which common logarithms may be obtained: they are absolutely necessary for finding the fluents of many fluxional expressions of the highest importance.

It is deemed unnecessary, in this place, to show the utility of logarithms by examples. Being once calculated and arranged in tables, not only for common numbers, but also for natural sines, tangents, and secants, it is manifest that a computor may save himself much time, and a great deal of labour, by means of their assistance; as otherwise multiplications and divisions of high numbers, or of decimals to a considerable number of places, would enter into his inquiries.

The writer of the foregoing articles now considers the design with which he set out as completed. He has endeavoured to explain, with perspicuity, the first principles of logarithms, and their relations to one another when of different sets or kinds; and he has laid before the young mathematical student the most improved and expeditious methods by which they may be calculated. If the reader should be desirous of further information on the subject, he may meet with full gratification by a perusal of the history of discoveries and writings relating to logarithms, prefixed to Dr. Hutton's Mathematical Tables. He will also find the Tables of Logarithms, contained in that volume, the most useful for calculation, if in his computations he does not go beyond degrees and minutes: if he aims at a higher degree of accuracy, he will have recourse to Taylor's Tables, in which the Logarithmic Sines and Tangents are calculated to every second of the Quadrant.

THE pole of a circle of the sphere is a point in the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal.

II.

A great circle of the sphere is any whose plane passes through the centre of the sphere, and, whose centre therefore is the same with that of the sphere.

III..

A spherical triangle is a figure upon the superficies of a sphere comprehended by three arches of three great circles, each of which is less than a semicircle.

IV.

A spherical angle is that on which the superficies of a sphere is contained by two arches of great circles, and is the same with the inclination of the planes of these great circles.

PROP. I.

GREAT circles bisect one another.

1

As they have a common centre, their common section will be a diameter of each which will bisect them.

: PROP. II. FIG. 1.

THE arch of a great circle betwixt the pole and the circumference of another is a quadrant.

Let ABC be a great circle, and D its pole; if a great circle DC pass through D, and meet ABC in C, the arch DC will be a quadrant.

Let the great circle CD meet ABC again in A, and let AC be the common section of the great circles, which will pass through E, the centre of the sphere: Join DE, DA, DC: By def. 1. DA, DC are equal, and AE, EC are also equal, and DE is common; therefore (8.1), the angles DEA, DEC are equal; wherefore the arches DA, DC are equal, and consequently each of them is a qua drant. Q.E. D.