the first four operations by machinery alone was that of Blaise Pascal (1623-1662), when a lad of eighteen. The close application to this work undermined a not over strong constitution, and he died at the early age of 39. The Pascal machine, which is here illustrated, was constructed on the principle of a wheel upon the circumference of which were marked the first 9 numerals. One turn of this wheel caused the next wheel, similarly marked, to pass through a tenth of a revolution, and so forth. Pascal's machine was not built, however, strictly on a decimal scale, as it was designed for monetary work. A similar attempt was made by Leibnitz, the German mathematician. The most elaborate calculating engine ever attempted was designed by Charles Babbage (1791-1871), on which he expended a private fortune of over $100,000, and toward which the British Government contributed $80,000 and a fireproof building for its construction. While the machine was never completed, the work on it left an indelible stamp on British artizanship. The most successful machine was constructed by George and Edward Scheutz, who were inspired by the attempt of Babbage. This machine, which computes and prints logarithmic and other tables, finally came into the possession of the Dudley Observatory at Albany, N. Y. The last few years have seen a great advance in the art of constructing computing machines for purely commercial purposes. The inverse process of involution is evolution, the problem of which is to determine one of a given number of equal factors when their product alone is given. The factors so found are called square root, cube root, fourth root, etc., depending upon the number of factors involved. The square root of 4 is 2, the cube root of 27 is 3. The simplest method of extracting a root is to divide the number by its lowest prime factor and continue the process. It may be illustrated in finding the cube root of 216. Since there are three factors 2, and three factors 3, there are three factors 2 X 3, or 6; or the cube root of 216 is 6. 2 ) 216 2) 108 2) 54 3)27 3)9 3)3 Σ The symbol of evolution is V an abbreviation, r, for root, followed by the vinculum; a figure is placed above the √ to indicate the root taken, except in the case of square root, when it is usually omitted. The ordinary algorithm or scheme for finding square root is given in a paraphrase of the work of Theon, of Smyrna, who flourished about 139 A.D.: "We learn the process from Euclid, II, 4, where it is stated, 'If a straight line be divided by any point, the square on the whole line is equal to the squares of both parts, together with twice the oblong which may be found from those segments.' S with a number like 144, we take a lesser square, say 100, of which the root is 10. We multiply 10 by 2, because in the remaining gnomon, ABCDEF, there are two ob |