1. It is an interesting subject for reflection that from the earliest times mechanical assistance has been required in mental operations. The word calculation at once reminds us of the employment of pebbles for marking units, and it is asserted that the word α ' ρiϴμòc is also derived from the like notion of a pebble or material sign. Even in the time of Aristotle the wide extension of the decimal system of numeration had been remarked and referred to the use of the fingers in reckoning; and there can be no doubt that the form of the most available arithmetical instrument, the human hand, has reacted upon the mind and moulded our numerical system into a form which we should not otherwise have selected as the best. 2. From early times, too, distinct mechanical instruments were devised to facilitate computation. The Greeks and Romans habitually employed the abacus or arithmetical board, consisting, in its most convenient form, of an oblong frame with a series of cross wires, each bearing ten sliding beads. The abacus thus supplied, as it were, an unlimited series of fingers, which furnished marks for successive higher units and allowed of the representation of any number. The Russians employ the abacus at the present day under the name of the shtshob , and the Chinese have from time immemorial made use of an almost exactly similar instrument called the schwanpan .