THESE Tables, which are executed with a new and elegant type on good paper, form a fmall octavo volume. There is every probability in favors of their correctnefs. They are copied from the London edition of Gardiner printed in 1742, which is in the higheft eftimation for that quality. Meffieurs Callet. Leveque and Prud'homme, three good mathematicians, revifed the proof fheets, as did alfo the editor M. Jombert three feveral times. M. Didot fenr. the printer formed the models of the types and founded them on purpose, and the editor avers that, during the courfe of the impreffion, none of the figures came out of their place; a precious advantage which he imputes to the juftness of the principles that M. Didot has established in his foundery.

THERE is an additional improvement, which I am furprised none of the editors of our common logarithms has thought of making. What I allude to is the uniting, to the tables of the logarithms of the natural numbers and of the fines and cofines, the logarithms of their reciprocals (their arithmetical complements, as they are called). By this means, all the common operations by logarithms might be performed by addition only, without any trouble. The logarithms of the natural numbers might be difpofed on the left hand, and thofe of their reciprocals on the right hand pages. The characteristics of the latter, being equal to the difference between 10 and the number of integral figures in the natural numbers, would be as cafily found as thofe of the former. The logarithms of the reciprocals of the fines and cofines might, in each page, be put in the fame line with the logarithms of the fines and cofines, having

* The arithmetical complements of the logarithms were firft thought of by John Speedell, who, in his new logarithms fift published in 1619, and feveral times afterwards, avoided the inconvenienceof the figns in Napiers logarithms by that contrivance.

having their common differences between them, as the logarithms of the tangents and cotangents, which are reciprocals of each other, have theirs. It is very likely that the prefent edition of the Tables portatives will foon be exhaufted. If, in a fecond edition, M. Jombert adopts the propofed amelioration, he will do an effential fervice to the community. 1. The computation might be accomplished, by a good arithmetician, in little more than three hours labour every day for half a year. 2. The type and length of the page being the fame, the book would be little more than a fourth part thicker, and would ftill be of a convenient fize.

IN the month of May, 1784, there were published propofals for publishing, by fubfcription, A Table of Logarithmic fines and tangents, taken at fight to every fecond of the quadrant, accurately computed to feven places of fi gures befides the index: to which will be prefixed a table of the logarithms of numbers from 1 to 100000, inferibed, by permiffion, to the right honourable and bonourable the Commiffioners of the Board of Longitude, by Michael Taylor, one of the computers of the Nautical Ephemeris, and author of a Sexagefimal Table, published by order of the Commiffioners of the Board of Longitude. The plan of this work was fubmitted to the Board of Longitude, who came to a refolution to give Mr Taylor a gratuity of three hundred pounds fterling towards defraying the expence of printing and publishing it. This circumftance ought to be a fufficient recommendation of Mr Taylor, and it is to be hoped, that his laborious and useful undertaking will meet with the encouragement and recompence from the public which it fo juflly deferves. In the fpecimen annexed to the propofals, the degrees being as ufual at the top and bottom of the page, the feconds occupy

the

the first column: the minutes are difpofed along the tops and bottoms of the other columns: immediately below the minutes at top stand the characteristics, and below them the three next common figures of the logarithms; the other four figures filling the columns. It is to be regretted, that an improvement, fimilar to M. Callet's, has not been adopted in this work, the printing of which was begun before the date of the proposals.

THE tables of logarithms which, with thofe that have been mentioned, are most in estimation, are thofe of the edition of Sherwin, which was corrected and published by Gardiner in the fame year (1742) with his own tables-Those by Deparcieux*, and those of the small editions of Ulacq published at Lyons in 1670, and 1760 †.

THE London edition of Gardiner, which has been defervedly esteemed as containing the most accurate fet of tables, is not entirely free from errors. There is, at the end of Dr Hutton's tables, a list of about fifty errors in the logarithms of the natural numbers, fines and tangents; twenty of which he himself discovered in collating the proofs of his book with the like parts of Gardiner's; all of thefe, however, that gentleman obferves, are not in all the copies of this edition. In the Avignon edition of Gardiner (1770), the errors pointed out by Dr Hutton are above feventy. All the errors of the London edition are corrected in the Tables portatives, excepting that of the logarithm of the natural numbers 64445.

BEFORE

Z

* Montucla.

+ Hutton.

BEFORE Concluding this fection, we fhall fay a few words of the logarithms called logistic. The logistic logarithm of a number of feconds is the excefs of the logarithm of 3600" above the logarithm of that number of feconds. A table of these logarithms was first given by Strut in his Aftronomia Carolma published in 1661 *. A fimilar one is given in feveral of the common logarithmic tables.

Tab. portatives.

SECTION

SECTION VII.

THE USE OF THE LOGARITHMS.

THE general ufe of the logarithms, as was before obferved, is to convert every species of multiplication and divifion into addition and subtraction, and to raife quantities to any given power, and to extract their roots by easy multiplications and divifions. Examples of these operations, particularly in trigonometry, are prefixed to almoft all the most confiderable tables of logarithms. We beg leave to refer the reader to Gardiner, Callet, Sherwin, and Hutton, where he will find the theory, construction, and application of thefe numbers.

THE theory of the logarithms has put it in our power to folve, with great eafe, an equation in algebra, which before could not be folved but with difficulty and tatonnement. In the equation ab, if b is the unknown quantity, its value is found by multiplying a by itself as often as there are units in x-1: Again, if a is the unknown quantity, its value may be found by extracting the xth root of b. But if x is the unknown quantity, algebra, without the logarithms, can furnish no direct rule for finding its value. This, however, is eafily accomplished

by