MCC XX IIII . . . . . . . . 1224 The Roman required the learning of no addition facts; the units (ones, fives, tens, fifties . . . ) were seen at once, and the only difficulty was that of "carrying." In pointing to the figures, it is easier to "add up" than to add downwards, and therefore certain writers (e.g., Maximus Planudes, c. 1340) suggested writ ing the sum at the top.
Subtraction has been the subject of various experiments. On the abacus or with counters (see ABACUS) the simplest plan of taking 46 from 423 was to change 423 to 300+110+13, after which there was no difficulty. This is essentially the plan of bor rowing used by most people to-day. With the advent of the Indo Arabic numerals other devices were suggested. One of these, the complementary plan, was known in India in the 12th century. It is based upon the identity a — b = a+ ( i o — b) —1 o ; that is, 12- 7 = '2+0o-7)-10=12+3-10=5. It found a worthy use when the modern calculating machine became common, sub traction being performed by adding the complement of a number, 10-7 being the complement of 7. The plan of borrowing 1 from the tens of the minuend and repaying it by adding 1 to the tens of the subtrahend appears in Borghi's arithmetic (1484), but was already old in Europe. It seems to have been of Arabic origin, for Fibonacci (q.v., 1202), who was much indebted to the Muslim writers, used it. The "addition method," seen in "making change" and sometimes called the Austrian method, was suggested at least as early as 1559, but did not become widely known till the i9th century. No one of these methods has shown such points of superiority in actual practice as to make it generally accepted as the best, and with the coming of calculating machines (q.v.) it is not important that it should.
The Indian writer, Bhaskara (c. 1150), gave five methods of multiplication, and Pacioli (q.v., 1494) gave eight. To these may be added the ancient one of repeated addition as developed with considerable skill by the Egyptians before 1700 B.C., numerous special methods developed by the Arabs, and the method of quarter squares (q.v.).
Although the primitive method of dividing may have been that of repeated subtraction, the earliest one of which we have definite record is that of duplation and mediation—finding (by doubling and halving) the number of times the divisor must be used to make the dividend. The only method that was for any length of time a rival to our present plan is the Galley Method here shown.
As with all initial steps in any branch of learning, the early work in arithmetic requires the tacit assumption of certain laws, and the mechanism of the operations is accepted with a minimum of explanation. The prime necessity is that the work should be come mechanical as soon and as completely as possible. The child—who is quite unable to grasp the significance of radix, scale, place value, associative law and commutative law involved in the operation—should tacitly assume all this knowledge. (See