NUMBER. The general considerations which this word would suggest cannot be treated independently of those required in treating the notion of ratio in general. As this will form a part of the sub ject of PROPORTION, we refer to that article as the continuation of the present one.
The notion of number is suggested by succession; and it is customary to call the actual things repeated, considered as a collection, a concrete number ; while the notion formed from com paring the collection with one of the things collected is called an abstract number. This abstract number arises from repetition of objects, in which the attention is directed to the repetitions as repetitions, and not to the objects as distinguished from any other objects. It is therefore a number of times, not a number of things. [MurirsriAscAmroN.] If we never, numbered any things capable of division iuto parts like themselves, our notion of number would rest in what is now called whole number. If the intellect were taught to count by the beating of a clock, and never came in contact with any other magnitude except that of the intervals between the beats, it is difficult to see how the idea of fractions would be obtained. But when we come to put together continuous magnitudes, which might increase or decrease without any alteration except that of magnitude, such as lines, sur faces, &c.. we then begin to see that the unit is.purely arbitrary, con sidered as a magnitude, so that the consideration of smaller or larger units, and the reduction of processes from one unit to another, become necessary. Hence the doctrine of fractions, and finally that of INCOMMENSURABLES.
The unit of magnitude and tho unit of repetition are as distinct as concrete and abstract number. A given magnitude being chosen, wo may fix our own ideas of other magnitudes and convey them to other persons by describing the repetitions of the given unit which will severally give the other magnitudes : but it is incorrect to say that in arithmetic we can perform all operations upon magnitudes repre sented by numbers ; the operations are performed by our minds upon notions of repetition, not of magnitude. Any question of numbers arising out of geometry might, so far as the pure arithmetical pro cesies are concerned, as well have the prototypes of its numbers in collections of beats of a clock or motions of the arm, as in repe titions of lengths or areas. It is not true that such simple successions would suggest as many problems as geometry or commercial business ; but that is a distinct consideration.
Discussion formerly took place upon the question whether I repro aa seats a number; it being asserted that number must be more than one. The settlement of such a question depends upon convention entirely, and is very easy. In the common sense of the word neither 1 nor 2 are numbers: a number of men, or of pebbles, would suggest the idea of more than two ; in fact the colloquial word number means indefinitely many ; more than the eye can decide on without count ing; serrral, that is to say, as many as require the sereriny which takes place in counting. With different persons this commencement of number, vulgarly speaking, may be different ; all persons discern dove without counting, and probably four ; but it is certain that fire must be severed by most persons, and six probably by all. Those who watch the progress of children can easily see that their scales of reckoning are successively one and more ; one two, and more; owe, two, three, and more In the common playing-cards we decide by forms, not by numbers ; and were not the nine distinguished from the seven by the different positions of the odd spot, there would be continual mistakes.
In mathematical language, every numerical symbol is called number, including 0, 1, fractions, whole numbers, and even infinity.
The talent of easily combining and remembering numbers, or of calculation, is a perfectly distinct thing from that of mathematical invention, reasoning, or application; though the two are frequently confounded. Taking mathematicians of the highest order, some have been singularly gifted in this respect, some distinguished in neither way, and some more than commonly deficient.
A very deceptive mode of speaking is common with regard to numbers, which divides them into cardinal and ordinal. Thus one, two, three, &a., are cardinal numbers, while first, second, third, fie., are ordinal. The real distinction is that of numeral nouns and numeral pronouns, to the latter of which the term ordinal might properly be applied. That first, second, third, fic., are really pronouns is obvious if we consider that, so far as they go, this, that, and the other would supply their places. The so-called cardinal numbers denotecollections ; the ordinal numbers point out only the places of the several units of which a collection is composed. Even one, when its force is simply selective or distinctive, is a pronoun, as in "one or another." [Atuniurric ; MAGNITUDE; PROPORTION ; QUANTITY; UNIT. •