NUMERATION is a term generally applied to the art of repre senting numbers by distinct names and symbols, a sense in which the word is used by the oldest writers. Every treatise on arithmetic must necessarily begin with something on this art of counting and repre senting the results of counting, on the goodness of which, slight and easy as any method may appear to which we have been habituated from childhood, the progress of the arts of life, to say nothing of the mathematical sciences, is in no slight degree dependent. The time is' gone by for a formal eulogy upon the benefits of any fundamental method4f expression; we will therefore content ourselves with quoting a part of that which is found in the first English work on arithmetic), Robert Recorde'a Grounde of Artes ' (1540). We quote this also because it is an instance (the only one we over met with in a mathe matical work) of the species of doggrel comic verse afterwards in use on the stage (see the 'Comedy of Errors '), which has a sort of measure and rhyme, though printed in the form of simple prose in the work from which we cite (we put the syllables which are meant to rhyme in italics) :—" Muster. Wherefore in all great workea are clerks so much desired? Wherefore are auditors so richly fed I What causeth geo metricians so highly to be inhaunsed I Why are astronomers so greatly advanced ? Because that by number such things they find, which else would far excel man's mind Master. Exclude number, and answer to this question : How many years old are you r Scholar. Muni. Master. So that if number want, you answer all by mamma ? How many miles to London I Scholar. A peak full of plums Master. If number be. lacking, it maketh men dumb, so that to moat questions they must answer mum. Scholar. This is the cause, sir, that I judged it so rile, because it is so common in talking every while ; for plenty is not dainty, as the common saying is. Master. No; nor store is no sore, perceive you this! The more common that the thing is being needfully required, the better is the thing and the more to be desired. But in numbring, as some of it is light and plain, so the most
part is difficult and not redo to attain." The earliest method of signifying a large number must have been such a one as the scholar uses above, when he designates a large number of miles as a " peak full of plums," namely, the similitude of some visible or well-known collection of units. The fingers of the baud, or of both hande, or the united number of fingers and toes furnished natural collections of reference on which the various quinary, decimal, and vicenary scales in existence have proceeded. The transition from counting by tens to counting by dozens might have been caused by the facility of subdi'ision which the number twelve possesses, though we rather doubt this explanation, at least unless we assume that the division of the Roman As into twelve unite is to be explained on the time principle. From this we think came the method of reckoning by dozens to be introduced throughout Europe, as would that by thirteens, if the Roman coin or weight had been so divided.
Our present numerative system is stated by writers to employ the words unit, ten, hundred, thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, &o. But the greater part of this is pure statement ; for the terms billion, trillion, &c., though defined by arithmetical writers, have never found their way into common use, the want of such large numbers having never been experienced. The French Indeed have naturalised the term milliard, moaning one thousand millions, in matters of public debt and revenue, which only shows how little the term billion has been used among them, since, according to their writers, the milliard and billion are the same things. Tonstal expressly says, that in his time (Ilenry VIII.) the common reckoning from millions was made by millions of millions, Ac., and the word millie is noted as a vulgarism by (neither is it among the recognised barbarisms of Ducange).