Page:Proceedings of the Royal Society of London Vol 1.djvu/260

Rh by copies: nor is the uncertainty of copies being accurately taken greater in any case than in that of copied numbers. It is conse- quently useful to contrive new and easy methods for computing new tables, or for examining those we already have; and it is particularly useful to contrive methods by which any part of a table may be ve- riﬁed, independently of the rest; for by examining parts taken at random, we may, in some cases, acquire a moral certainty respecting the accuracy of the whole.

Among the various methods of computing logarithms, none, o_ur author says, possesses the advantage of forming them with tolerable ease, independently of each other, by means of a few easy bases. This desideratum, he trusts, the method described by him will supply; being very easy of application, as it requires no division, multiplica- tion, or extraction of roots, and has its relative advantages highly in- creased by increasing the number of decimal places to 'which the computation is carried.

The chief part of the work, according to the method laid down by Mr. Manning, consists in merely setting down a number, under itself, removed one or more places to the right, and subtracting; then re- peatiog this operation. This method, consequently, is very little liable to error; and may be performed, after a few minutes instruc- tion, by any one who is able to subtract. Besides, from the com- modious situation of the ﬁgures, the work may be revised with great rapidity. It is as easy for large numbers as for small ones; and, on an average, about 27 subtractions will furnish a logarithm accurately to ten places of decimals.

A similar method, by addition only, may, Mr. Manning says, in some cases, be used with advantage; and various artiﬁces may be occasionally employed to shorten the computation, both in the me- thod by subtraction and in that by addition: the two methods may also he sometimes advantageously combined together. It must, how- ever, be observed, that the method by subtraction has many advan- tages over that by addition; particularly as from its being more simple, and more completely mechanical, it may be conﬁded to the most unskilful persons without much danger of error.

Various examples of our author’s mode of computation and rules for conducting it are given, for which we must necessarily refer to the paper itself.

The basin, which is here described by Mr. Martin, is delineated in a map annexed to the paper; it is formed of limestone, and contains all the strata of coal and iron ore in South Wales : it is upwards of 100 miles in length; and its average breadth in the counties of Monmouth, Glamorgan, Carmarthen, and part of Brecon, is