LOGARITHMS.

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0.954251, which is excedingly near the true logarithm of 9.3dly. If in like manner you find mean proportionals between1.00000000 and 3.1022777, and assign the proper logarithmsto each, you will, at length, have the logarithm of the num-ber 2, and so of the rest.

The preceding was the method employed by the earlycalculators of logarithms ; and although they had in particu-lar cases certain means of abridging the operation, yet itis evident that the computations were not effected withoutimmense labour. An interesting and definite account ofwhich, with an explanation of the several degrees of modi-fication of the above method made use of by differentauthors, may be seeu in Dr. Huttons Mathematical Tables.

It may be unnecessary to observe, that these computationswere only required for prime numbers; but these beingonce obtained, the logarithms of all other numbers wereprocured by simple addition.

Having only spoken of logarithms as applicable to nu-merical computation: they are of infinitely more extensiveuse in the higher branches of geometry, and particularly intrigonometry; in the doctrine of fluxions, of which thevery eminent Mac Laurin, in his Treatise on Fluxions, pub-lished ir, answer to Bishop Berkeley s Charge against Mathe-maticians, has rendered a most satisfactory account, w T hoproceeded upon the doctrine of Napier, as delivered in hisMirif. Logar. Canon. Descript. He there supposes loga-rithms, and the quantities to which they correspond, to begenerated by the motion of a point. If this point movesover equal spaces in equal times, the line described by itincreaseth equally.

Again, a line decreases proportionally when the pointwhich moves over it describes such parts in equal times asare always in the same constant ratio to the lines from whichthey are subducted, or to the distances of that point at thebeginning of those times from a given term in the line. Inthe like manner, a line may increase proportionally, if inequal time the moving point describes spaces proportionalto its distances from a certain term at the beginning of eachline. Conceiving this mode to be applicable to the solutionof questions in spherical trigonometry, for the sake of illus-tration we will give a diagram of the solar system, 1 andsuppose the point S to represent the sun, and the otherastronomical characters to stand for the other planets.Here the orbit of Mercury ( $ ) is removed from the mean

1 See Frontispieoe.

Bb 2