Recorde uses nothing more than millions repeated ; so that it seems the billions and higher denominations were never anything but a fancy of arithmetical writers, conceived after the time when elementary works ceased to be written in Latin. The probability of this is increased by their meaning different things in different countries : with us the billion is a million of millions, a trillion is a million of billions, and each denomination is a million of times the one pre ceding. With the French and the other Continental nations (except some of the older writers, at least, among the Italians), the billion is a thousand millions, and each denomination is a thousand times the preceding. According to English writers, the number 1,234567,891234,567891 is one trillion, 234567 billions, 891234 mil lions, and 567891 • according to the French writers, it is one quin tillion, 234 quadrillions, 567 trillions, 891 billions, 234 millions, 567 thousands, and 891. For common purposes, the denominations higher than a million may be abandoned, it being remembered that all the figures on the left, after six are taken off ou the right, are so many millions, and all above twelve figures so many millions of millions. In writing, round numbers of millions should be written as such ; thus, 638 millions, not 638,000,000: in computation it is of course a different thing. Some authors seem to think it very scientific to parade ciphers, sometimes by the dozen ; and so it is, no doubt, since it shows they know how many ciphers go to a million or a million of millions; but no reader likes to stop and examine 000,000,000,000, when the words "million of millions " would have done equally well.
The decimal system, made complete, supposes a point always to be placed at the end of the units, to separate them from the fractions which may follow. When there are no fractions, the point is useless, as in or which is 675. The numbers on the right of the point, successively denoting tenths, hundredths, thousandths, &e. of a unit [FnAcTtotcsj, are in denominations which have not received distinct names. The modern French call them dechnes, centimes, &e.; and the attempt has before now been made (see Wybard's Tacto metria' 1650) to introduce centesms, millesms, &e. into English, but with no success.
The principle of local value which distinguishes our system of numeration from that of the Greeks and Roiltans, is applicable to any system, whether decimal or not. If I0 stands for tcn, that is, if its units in the second column are ten times in value those of the first column, nine numeral symbols besides the cipher are requisite ; but if 10 had signified fifteen, it would have been necessary to have fourteen distinct symbols of number besides the cipher, since 10, II, &e. now stand for sixteen, seventeen, &c. In such an explana
tion, the frame-work of our numerical language (being decimal) is not well calculated to give an easy comprehension of the change : we should rather invent a word for fifteen, or five and ten, say A; whence A-one A-two, &c. would be the spoken sounds answering to what we now Call sixteen, seventeen, &e. ; while ten, eleven, twelve, thirteen, and fourteen would require new names not connected in etymology with ten.
The method of reducing a number, decimally expressed, to another iu which the radix or base of the system (as ten is that of the common one), is a, is as follows : divide the number successively by a, expressed in the decimal system ; the remainders give the units, as, aas, &c. of the new expression. Thus if 12376 is to be expressed in the pinery system, whose base is 5, we should have the following process :- This exhibits both the reduction to the quinary system and the restitution of the decimal expression ; but if the number had been given in the quinary system, it might have been reduced to the decimal system by the same rule, the new base ten being, in the old or quinary system, represented by 20, and the rule of division being performed by the use of five in the same manner as ten is used in the decimal system.
The quinary being supposed the old system, as soon as we come to the remainder II, we have to invent a new symbol (say 6), since II, in the new system, is to stand for eleven. For further examples, see the Library of Useful Knowledge : Treatise on the study of Mathematics.' In teaching the elements of numeration by the abacus [Am&eus], it is desirable that exercise should be given in several different systems, were it only to prevent the formation of that impression which so many students long retain, that the decimal system is natural and necessary. The want of words for the denominations will be the only difficulty ; this may be got over by using the letters A, B, C, &e., to represent them. Thus if the system be quinary, A counts as one ball on the second row or five on the first, n as one ball on the third row, five on the second, or twenty-five on the first, and so on. All the balls on the seeond.row may be marked A, those on the third B, &c.