Horace speaks of boys going to school with the table and counters suspended on their left arm. * There is an imperfect abacus sometimes used in our schools for teaching the multiplication table. It consists of 12 rods, on each of which are twelve moveable balls. A still simpler abacus is employed to teach the numeration table, which has only 6 wires, upon which a number of balls circulate. An abacus, but much less simple and efficient than the Chinese, is described by Dr. Peacock.† Counters were used for the pur * Quo pueri magnis e centurionibus orti, † Pp. 409-10. pose of carrying on this process; and as the whole partakes of a decimal character, the explanatory extract is here given at length. "They commenced by drawing seven lines with a piece of chalk, or other substance, on a table, board, or slate, or by a pen on paper; the counters (which were usually of brass) on the lowest line represented units, on the next tens, and so on as far as a million on the last and uppermost line; a counter placed between two lines was equivalent to five counters on the lower line of the two. Thus the disposition NOTATION. of counters in the annexed example represents the number 3,629,638; and it is clearly very easy to increase the number of lines, so as to comprehend any number that might be required to be expressed. "ADDITION.-Suppose it was required to add together 788 and 383, The sum express the numbers to be added in the two first columns. of the counters on the lowest line is 6; write, therefore, 1 on that line in the third column; carry one to the first space, which, added to the 1 already there, is equal to 1 on the second line; place a counter there, and add all the counters on that line together, the sum is 7; leave, therefore, two counters on that line together, and pass one to the next space; add the counters on that space together, which are three; leave one there, and place one also on the next line; add all the counters on that line together, the sum is 6. Leave one counter, and pass another to the next space; add all the counters on that space together, which are 2; leave no counter in the space, but pass one to the next, or fourth line; we thus represent the sum, which is 1,171. "The principle of this operation is extremely simple; and the process itself, after a little practice, would clearly admit of being performed with great rapidity. In giving a scheme of this operation, we have made use of three columns; but in practice no more would be required than are sufficient to represent the sums to be added, the counters on each line being removed as the addition proceeds, and being replaced by the counters which are requisite to denote the sum. "SUBTRACTION.-We shall now proceed to a second examplenamely, to subtract 682 from 1,375. Write the numbers in the first and second columns. The two counters on the last line have none to correspond to them in the minuend; bring down the counter in the first space, and suppose it replaced by 5 counters; take away two, and three remain on the lowest line of the remainder. Again the three counters on the second line must be subtracted from 7, (bringing down 5); and therefore leaving 4 on the second line of the remainder. "The counter on the second space has now no counter corresponding with it in the minuend; remove one counter from the next line, and replace it by two counters in the next inferior space; there will remain, therefore, one counter for that space in the remainder. "There is now one counter on the third line to subtract from two in the minuend, and there remains one for the remainder. The counter in the next space has nothing corresponding to it; and we must, therefore, bring down the counter on the highest line and replace it by two counters in the space below it; if one counter be subtracted from them, there will remain one, and the whole remainder will be 693. "Recorde writes the smaller number in the first column, and commences the subtraction with the highest counters; and very little consideration will show in what manner the operation must be performed, with such a change in the process. "MULTIPLICATION.-We shall now give an example of multiplication; and let it be proposed to multiply 2,457 by 43. "Write the multiplicand in the first column, and the multiplier in the second; multiply first by three, and write the product in the third column, and then by 4 in a superior place, and write the result in the fourth column; add the numbers in these two columns together, and the sum is the product required. "DIVISION. We shall conclude with an example of division; and let it be required to divide 12,832 by 608. "Write the dividend in the first column, and the divisor in the second, reserving the third for the quotient; then, since 6 is contained twice in 12, in the line above that in which 6 is written, we may put down 2, in the last line but one in the column for the quotient; multiply 6 by 2, and subtract; there is no remainder; multiply 8 by 2, and subtract 16 from the number expressed by the counters remaining in the dividend in the line above the last; first take one counter from the three in the third line, and two remain; next take 6, which is done by taking 1 from the second line from the bottom, and bringing 1 from the third line, replacing it by 2 in the space below, and then subtracting one of them, thus leaving 67 in the remainder, to be denoted in the second and third lines and the spaces above them; the remaining two counters in the dividend are transferred to the corresponding line in the column for the first remainder; the operation is now repeated, the next figure in the quotient, as 1, being written on the lowest line; it is now merely necessary to subtract the divisor from the first remainder, and we get 64 to the second and last remainder. It is evident that the same process may be repeated to any extent that may be required; and that the complication of the process as exhibited in a scheme is much greater than in practice, when the dividend is replaced by the first remainder, and so on successively until the remainder is zero, or less than the divisor." There is an instrument frequently seen in the houses and buildings of China, called the Lines of Fohi, to which mysterious virtues are attached. The broken lines represent zero, entire line units; and it will be seen, that taking the top line to represent digits, the second line ten, and the third hundreds, the result will be that the following numbers will be represented: The Jesuits lauded this instrument as containing the elements of all knowledge, and as a part of revelation from the greatest of sovereigns and philosophers, by which all the revolutions of the celestial orbs were to be calculated, and all the mysteries of nature developed and explained. But they do not give us the key to the mystery, or interpret the teachings of the oracle. |