CALCULATING MACHINES. Machines devised for facilitating arith metical computations. From a remote period of antiquity, various mechanical devices were resorted to for this purpose. Amongst the Greeks and Romans, an instrument called " abacus' was employed, consisting of a number of parallel threads, upon which were strung a number of beads for counters, to represent units, tens, hundreds, &c. The Chinese at the present day use in their computations an instrument called a swanpan, nearly resembling the Greek abacus, and which may now be met with in the toy-shops of London. There are large and small swanpans ; those for mercantile purposes consist of many rows of small balls strung on wires, containing fifteen balls on each, with space for their being moved up and down with ease. These rows of balls are divided by a cross bar of wood extending from aide to side, leaving five balls above and ten below. Each ball of the upper row is of the value of the ten balls on the lower row, and by moving down to the bar one of the upper balls, the ten lower balls which had been moved up to the bar, are at liberty to repre sent any further sum until they have all again reached the bar, when a second ball is brought down from the upper row, and so on until the five balls are engaged, when their value is represented by one ball on the adjoining wire on the left hand, and so on to any amount. These instruments are universally employed throughout China for the purposes of computation; and so expert are the Chinese in the use of them, that few Europeans, with the assistance of pen and ink, can keep pace with them. The celebrated Napier, the inventor of logarithms, contrived an instrument by which the operation of multiplication is much facilitated, the product of any single figure with the multiplicand being represented at once by a very simple mechanical operation. This instrument, which consisted of a number of detached rods, each bearing at the top some one unit with the products of the same, multiplied by the nine units ranged in a line beneath it, is commonly known by the name of Napier's rods or bones. But by far the most useful contrivance of this kind is the Gunter's scale, so named after the inventor, Mr. Edmund Gunter, an eminent English mathematician, who likewise was the author of several other very useful inventions. In the extent to which the Gunter's scale has been adopted, it rivals the Chinese swanpan, whilst its powers far exceed those of the latter instrument. It consists of a fiat ruler of box wood, 2 feet long, having various lines laid down upon it, by means of which all the various problems relating to arithmetical trigonometry, and their depending sciences, may be performed by the extent of the compasses only. This will be best explained by a description of the line, called the line of number, or Gunter's line, which is adapted to the solution of arithmetical questions, and exhibiting a few practical examples. The line of numbers is
the logarithmic scale of proportionals, which, being graduated upon the ruler, serves to solve problems in the same manner as logarithms do arithmetically. It is usually divided into 100 parts, every tenth of which is numbered, beginning with 1 and ending with 10 ; so that if the first great division stand for the / of an integer, the next great division will stand for A, and the intermediate divisions will represent hundredths of an integer, whilst the large divisions beyond 10 will represent units ; and if the first set of large divisions represent units, the subdivisions will represent tenths, whilst the second set of large divisions will represent tens, and the subdivisions units, and so on. The general rule for using this instrument is as follows : since all questions are reducible to propor tions, if the compasses be extended from the first term to the third, the same extent will reach from the second to the fourth term. The following are a few examples of some of the uses of this line. 1. To find the product of any two numbers, as 4 and 8: extend the compasses from 1 to the multiplier 4, and the same extent applied the same way from 8, the multiplicand will reach the product 32. 2. To divide one number by another, as 36 by 4: extend the compasses from 4 to 1, and the same extent will reach from 9 to 36. 3. To find a fourth proportional to three given numbers, as 6, 8, 9 : extend the com from 6 to 8, and this extent laid the same way from 9, will reach 12, the fourth proportional required. 4. To extract the square root of a number, say 25 : bisect the distance between 1 on the scale and the point representing 25, then half this set off from 1 will give the point 5 = the root required. In the same manner the cube root, or the root of any higher power, may be found by dividing the distance on the line between 1 and the given number, into as many equal parts as the index of the power expresses, then one of those parts set from 1, will extend to the number representing the root required. A great improvement has been made in this instrument, by means of which compasses are rendered unnecessary. It consists in having two lines of numbers placed one over the other, one of which lines is engraved upon a slider moving in a groove, and is applied as follows : the first term of the proportional upon the slider being set against the third term upon the fixed scale, the second term of the proportion upon the slider will stand opposite the fourth term on the fixed scale. Instruments of this description, with scales suited to almost every branch of art in which calculation is required, are now common amongst in telligent workmen, and the scale of chemical equivalents, by which the labours of the chemist are so much facilitated, is constructed upon the same principle.