370

LOGARITHMS'.

theless, Kepler , the great .astronomical and philosophicalprecursor,of Newton, published his Chilias Logarithmorumad lotidiam numeros rotundas, at Marpnrg, in 17-24.

Spidell, in his New Logarithms, published in 1619, andsoon reprinted with considerable additions, and a sixthimpression in 1624; Benjamin Ursinius, in his Table ofLogarithms , printed at Cologne, in 1625, with others inEngland, and other parts of Europe , laboured at the com-putation of logarithms, and constructed various smalltables, agreeable to the designs of Baron Merchislon: how-ever, of all these, Professor Briggs was the most successful;he so much improved upon the learned Baron’s idea,—anidea, we are told, he had himself entertained twenty yearsprevious to his having heard of Napier’s design,—that itis now upon his system most logarithmic tables are con-structed.

In the present state of our knowledge upon this subject,we can perceive the many easy methods which may now beemployed for this purpose, at the time the labour of earlywriters, to effect similar productions may be appreciated.The toils of the first computations were exceedingly great,some idea of which may be collected from the followingillustration of Briggs’s method, who has given this rule :To find the logarithms of any number according to hismethod. 1st. Because 1, 10, 100, 1000, 10,000, &c. con-stitute a geometrical progression, their logarithms may betaken at pleasure; but to be able then to express thelogarithms of the intermediate numbers by decimal fractions:take 0.00000000, 1.00000000, 2.0000000, 3.00000000,4.00000000, &c. 2dly. It is manifest, that for thosenumbers which cannot be contained in this scale of geo-metrical progression, the just logarithms cannot be had;yet they may be had so near the truth, that as to mattersof utility, they shall be altogether just. To render thisapparent, suppose the logarithm of 9 be required, between1.0000000 and 10.0000000, find a mean proportional, andbetween their logarithms 0.00000000 and 1.00000000, anequi-diffei’ent mean will be the logarithm thereof, that is,of a number exceeding 3 by ; which is therefore

far remote from 9. Between 3 and 10, therefore, find someother mean proportional which may come somewhat nearer0 ; and between 10 and this mean, another still; and so onbetween the.number next greater and next less than 9, tillyou arrive at 9 vg aaiaaaa ; which, without any sensibleerror, may be taken for 9 itself. Seeking then, in such case,for the logarithms of the mean proportionals, you will have