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 lo-' garithms of the geometrical progression 1, 1+al', Ital2, 1+a3, 1+a+, 1+als, &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 fol lowing manner.

1, l+a'', i+a2, 1+@3, ita', i+as, i+ao, &c. 1, 1+a, 1+a21, 1+a31, 1+a 41, +5,

3

5

&c.

1, 1+am, Itam, itam, Itam, 1+am, 1+am, &c.

In these expressions 7 and m denote an 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+, is equal to 7, as is also 1+alm If 1+ be equal to 10, then 1+a is equal to 10,

6

6

as is also 1+am, &c. If therefore 1, 21, 31, &c. be hyperbolic logarithms, calculated by the methods already explained, the

1 2 3
m

logarithms expressed by m m &c. may be derived from them; for the hyperbolic logarithm of any given number is to the logarithm in the last-mentioned set, of the same

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 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, the for the sake of illustration, as in article 14, we suppose 1+a°=10, and allow, in addition to the hypothesis there formed, that

&c. denote common logarithms, then 6l=

6

m

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; 1:0.69314718054 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 :: 1.09861228864 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 common logarithm is sought, by

2.30258509324..

4342944818.

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 hyperbolie, the common logarithms of other num

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.

bers may be obtained from those so derived, merely by addition 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 numbers. 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 calculations.

A. ROBERTSON, Savilian Professor of Geometry, Oxford

DEFINITIONS.

I.

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 the 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 which on 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.

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 circle, which will

pass through Ethe 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 quadrant, Q. E. D.

IF a

PROP. III. FIG. 2.

a great circle be described meeting two great circles AB, AC passing through its pole A in B, C, the angle of the centre of the sphere upon the circumference BC, is the same with the spherical angle BAC, and the arch BC is called the measure of the spherical angle BAC.

Let the planes of the great circles AB, AC intersect one another in the straight line AD passing through D their common centre: join DB, DC.

Since A is the pole of BC, AB, AC will be quadrants, and the angles ADB, ADC right angles; therefore (6. def. 11.) the angle CDB is the inclination of the planes of the circles AB, AC; that is, (def. 4.) the spherical angle BAC. Q. E. D.

COR. If through the point A, two quadrants AB, AC, be drawn, the point A will be the pole of the great circle BC, passing through their extremities B, C.

Join AC, and draw AE, a straight line to any other point E, in BC; join DE: Since AC, AB are quadrants, the angles ADB, ADC are right angles, and AD will be perpendicular to the plane of BC: Therefore the angle ADE is a right angle, and AD, DC are equal to AD, DE, each to each; therefore AE, AC are equal, and A is the pole of BC, by def. 1. Q.E. D.

PROP. IV. FIG. 3.

IN isosceles spherical triangles, the angles at the base are equal.

Let ABC be an isosceles triangle, and AC, CB the equal sides; the angles BAC, ABC at the base AB, are equal.