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The collections of ten hundreds of millions, or of thousands of millions, are called billions, and are distinguished, like the collections of millions.t

†The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we successively remount from individuals to species, from species to genera, and from genera to orders. The savage is lost in his attempts at numeration, and significantly expresses his inability to proceed by holding up his expanded fingers, or pointing to the hairs of his head.

Nature has furnished the great and universal standard for computation in the fingers of the hand. All nations have accordingly reckoned by fives; and some barbarous tribes have scarcely advanced any further. After the fingers of one hand had been counted once it was a second and perhaps a distant step to proceed to those of the other. The primitive words, expressing numbers, did not probably exceed five. To denote six, seven, eight, and nine, the North American Indians repeat the five with the successive addition of one, two, three, and four; could we safely trace the descent and affinity of the abbreviated terms denoting the numbers from five to ten, it seems highly probable, that we should discover a similar process to have taken place in the formation of the most refined languages.

The ten digits of both hands being reckoned up, it then became necessary to repeat the operation. Such is the foundation of our decimal scale of arithmetic. Language still betrays by its structure the original mode of proceeding. To express the numbers beyond ten, the Laplanders combine an ordinal with a cardinal digit. Thus, eleven, twelve, &c. they denominate second ten and one, second ten and two, &c. and in like manner they call twenty one, twenty two, &c. third ten and one, third ten and two, &c. Our term eleven is supposed to be derived from ein or one, and liben, to remain, and to signify one, leave or set aside ten. Twelve is of the like derivation, and means two, laying aside the ten. The same idea is suggested by our termination ty in the words twenty, thirty, &c. This syllable, altogether distinct from ten, is derived from ziehen, to draw, and the meaning of twenty is, strickly speaking, two drawings, that is, the hands have been twice closed and the fingers counted over.

After ten was firmly established, as the standard of numeration, it seemed the most easy and consistent to proceed by the same repeated

Each of the names just mentioned is considered as forming a unit of an order more elevated according as it is removed from simple unit. The names ten and hundred are continually repeated, and we have no occasion for new names, such as thousand, million, billion, except at every fourth order. The same law being observed, to billions succeed trillions, quadrillions, quintillions, &c. each, like billions, having its tens and hundreds.

Numbers expressed in this manner, when more than one word enters into the enunciation of them, are separated into their respective orders of units, mentioned above; for instance, the number expressed by five hundred thousand three hundred and two, is separated into three parts, viz. five hundreds of thousands, three hundreds of simple units, and two of these units.

5. The length of the expression, written in words, when the numbers were large, occasioned the invention of characters, exclusively adapted to a shorter representation, and hence originated the art of expressing numbers in writing by these charac.. ters, called figures, or written numeration.

The laws of the written numeration, now used, are very analogous to those of the spoken numeration. In it the nine first numbers are each represented by a particular character, viz.

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one, two, three, four, five, six, seven, eight, nine. When a number consists of tens and units, the characters representing the number of each are written in order from left to right, beginning with the tens. The number forty-seven, for instance, is written 47; the first figure on the left, 4, denotes the four tens, and consequently a value ten times greater than it would have standing alone; while the figure 7, placed on the right, expressing seven units, possesses only its original value.

composition. Both hands being closed ten times would carry the reckoning up to a hundred. This word, originally hund, is of uncertain derivation; but the term thousand, which occurs at the next stage of the progress, or the hundred added ten times, is clearly traced out, being only a contraction of duis hund, or twice hundred, that is, the repetition, or collection of hundreds. See Edinburgh Review, vol.

XVIII. art. VII.

In the number thirty-three, which is written 33, we see the figure 3 repeated, but each time with a different value; the value of the 3 on the left is ten times greater than the value of that on the right.

This is the fundamental law of our written numeration, that a removal, of one place, towards the left increases the value of a figure ten times.

If it were required to express fifty, or five tens, as there are no units in this number, there would be nothing to write but the figure 5, and consequently it would be necessary to show, by some particular mark, that in the expression of this number, the figure ought to occupy the first place on the left. To do this we place on the right the character 0, cipher or nought, which of itself has no value, and serves only to fill the place of the units, which are wanting in the enunciation of the proposed number.

6. Thus with ten characters, by means of the rule before laid down concerning the value which figures assume, according to the places they occupy, we can express all possible numbers.

With two figures only, we can write all, as far as to nine tens and nine units, making 99, or ninety nine. After this comes the hundred, which is expressed by the figure 1, put one place farther towards the left, than it would be, if used to express tens only; and to denote this place, two ciphers are placed on the right, making 100.

The units and tens, afterwards added to form numbers greater than 100, take their proper places; thus a hundred and one will be written in figures 101; a hundred and eleven, 111. Here the same figure is three times repeated, and with a different value each time; in the first place on the right it expresses an unit, in the second, a ten, in the third, a hundred. It is the same with the number 222, 333, 444, &c. Thus, in consequence of the rule laid down before when speaking of units and tens, the same figure expresses units ten times greater, in proportion as it is removed from right to left, and by a simple change of place, acquires the power of representing, successively, all the different collections of units, which can enter into the expression of a number.

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7. A number dictated, or enunciated, is written then, by placing one after the other, beginning at the left, the figures which express the number of units of each collection; but it is necessary to keep in mind the order in which the collections succeed each other, that no one may be omitted, and to put ciphers in the room of those, which are wanting in the enunciation of the number to be written. If, for example, the number were three hun-· dred and twenty-four thousand, nine hundred and four, we should put 3 for the hundreds of thousands, 2 for the twenty thousand, or the two tens of thousands, 4 for the thousands, 9 for the hundreds; and as the tens come immediately after the hundreds, and are wanting in the given number, we should put a cipher in the room of them, and then write the figure 4 for the units; we should thus have 324904.

In the same way, writing ciphers in the place of tens of thousands, thousands and tens, which are wanting in the number five hundred thousand three hundred and two, we should have 500302.

8. When a number is written in figures, in enunciating it, or expressing it in language, it is necessary to substitute for each of the figures the word which it represents, and then to mention the collection of units, to which it belongs according to the place it occupies. The following example will illustrate this;

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The figures of this number are divided by commas, into portions

of three figures each, beginning at the right; but the last division on the left, which in the present instance has but two figures, may sometimes have but one. Each of these divisions corresponds to the collections designated by the words unit, thousand,

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∞ Hundreds of Billions

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million, billion, trillion, and their figures express successively the units, tens, and hundreds of each. Consequently, the expression of the whole number given is made in words, by reading each division of figures as if it stood alone, and adding, after its units, the name of their place.

The above example is read, twenty four trillions, eight hundred and ninety seven billions, three hundred and twenty one millions, five hundred and eighty thousand, three hundred and forty six units.

9. Numbers admit of being considered in two ways; one is, when no particular denomination is mentioned, to which their units belong, and they are then called abstract numbers; the other when the denomination of their units is specified, as when we say, two men, five years, three hours, &c. these are called concrete numbers.

It is evident, that the formation of numbers, by the successive union of units, is independent of the nature of these units, and that this must also be the case with the properties resulting from. this formation; by which properties we are enabled to compound and decompound numbers, which is called calculation. We shall now explain the principal rules for the calculation of numbers, without regard to the nature of their units.

ADDITION.

10. THIS operation, which has for its object the uniting of several numbers in one, is only an abbreviation of the formation of numbers by the successive union of units. If, for instance, it were required to add five to seven, it would be necessary, in the series of the names of numbers, one, two, three, four, five, six, seven, &c. to ascend five places above seven, and we should then come to the word twelve, which is consequently the amount of seven units added to five. It is upon this process that the addition of all small numbers depends, the results of which are committed to memory; its immediate application to larger numbers would be impossible, but in this case, we suppose these numbers divided into the different collections of units contained in them, and we may add together those of the same name. For instance, to add 27 to 52, we add the 7 units of the first number

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