an assemblage of several units, or things of the same kind. See Arithmetic, and Metaphysics, n° 205—208.

Number, says Malcolm, is either abstract or applicable: Abstract, when referred to things in general, without attending to their particular properties; and applicable, when considered as the number of a particular sort of things, as yards, trees, or the like.

When particular things are mentioned, there is always something more considered than barely their numbers; so that what is true of numbers in the abstract, or when nothing but the number of things is considered, will not be true when the question is limited to particular things; for instance, the number two is less than three; yet two yards is a greater quantity than three inches: and the reason is, because regard must be had to their different natures as well as number, whenever things of a different species are considered; for though we can compare the number of such things abstractedly, yet we cannot compare them in any applicable sense. And this difference is necessary to be considered, because upon it the true sense, and the possibility or impossibility, of some questions depend.

Number is unlimited in respect of increase; because we can never conceive a number so great but still there is a greater. However, in respect of decrease, it is limited; unity being the first and least number, below which therefore it cannot descend.

Kinds and definitions of Numbers. Mathematicians, considering number under a great many relations, have established the following definitions.

Broken numbers are the same with fractions.

Cardinal numbers are those which express the quantity of units, as 1, 2, 3, 4, &c., whereas ordinal numbers are those which express order, as 1st, 2d, 3d, &c.

Compound number, one divisible by some other number besides unity; as 12, which is divisible by 2, 3, 4, and 6. Numbers, as 12 and 15, which have some common measure besides unity, are said to be compound numbers among themselves.

Cubic number is the product of a square number by its root: such is 27, as being the product of the square number 9 by its root 3. All cubic numbers, whose root is less than 6, being divided by 6, the remainder is the root itself; thus 27 ÷ 6 leaves the remainder 3, its root; 215, the cube of 6, being divided by 6, leaves no remainder; 343, the cube of 7, leaves a remainder 1, which added to 6, is the cube root; and 512, the cube of 8, divided by 6, leaves a remainder 2, which added to 6, is the cube root. Hence the remainders of the divisions of the cubes above 216, divided by 6, being added to 6, always gives the root of the cube so divided till that remainder be 5, and consequently 11, the cube-root of the number divided. But the cubic numbers above this being divided by 6, there remains nothing, the cube root being 12. Thus the remainders of the higher cubes are to be added to 12 and not to 6, till you come to 18, when the remainder of the division must be added to 18; and so on ad infinitum.

Determinate number is that referred to some given unit, as a ternary or three; whereas an indeterminate one is that referred to unity in general, and is called quantity.

Homogeneous numbers are those referred to the same unit; as those referred to different units are termed heterogeneous.

Whole numbers are otherwise called integers.

Rational number is one commensurable with unity; as a number, incommensurable with unity, is termed irrational, or a surd.

In the same manner, a rational whole number is that whereof unity is an aliquot part; a rational broken number, that equal to some aliquot part of unity; and a rational mixed number, that consisting of a whole number and a broken one.

Even number, that which may be divided into two equal parts without any fraction, as 6, 12, &c. The sum, difference, and product, of any number of even numbers, is always an even number.

An evenly even number, is that which may be measured, or divided, without any remainder, by another even number, as 4 by 2.

An unevenly even number, when a number may be equally divided by an uneven number, as 20 by 5.

Uneven number, that which exceeds an even number, at least by unity, or which cannot be divided into two equal parts, as 3, 5, &c.

The sum or difference of two uneven numbers makes an even number; but the factum of two uneven ones makes an uneven number.

If an even number be added to an uneven one, or if the one be subtracted from the other, in the former case the sum, in the latter the difference, is an uneven number; but the factum of an even and uneven number is even.

The sum of any even number of uneven numbers is an even number; and the sum of any uneven number of uneven numbers is an uneven number.

Primitive or prime numbers are those divisible only by unity, as 5, 7, &c. And prime numbers among themselves, are those which have no common measure besides unity, as 12 and 19.

Perfect number, that whose aliquot parts added together make the whole number, as 6, 28; the aliquot parts of 6 being 3, 2, and 1 = 6; and those of 28, being 14, 7, 4, 2, 1, = 28.

Imperfect numbers, those whose aliquot parts added together make either more or less than the whole. And these are distinguished into abundant and defective: an instance in the former case is 12, whose aliquot parts 6, 4, 3, 2, 1, make 16; and in the latter case 16, whose aliquot parts 8, 4, 2, and 1, make but 15.

Plain number, that arising from the multiplication of two numbers, as 6, which is the product of 3 by 2; and these numbers are called the sides of the plane.

Square number is the product of any number multiplied by itself; thus 4, which is the factum of 2 by 2, is a square number.

Even square number added to its root makes an even number.

Polygonal or polygonous numbers, the sums of arithmetical progressions beginning with unity: these, where the common difference is 1, are called triangular numbers; where 2, square numbers; where 3, pentagonal numbers; where 4, hexagonal numbers; where 5, heptagonal numbers, &c.

Pyramidal numbers, the sums of polygonous numbers, collected after the same manner as the polygons themselves, and not gathered out of arithmetical progressions, are called first pyramidal numbers; the sums of the first pyramids are called second pyramids, &c.

If they arise out of triangular numbers, they are called triangular pyramidal numbers; if out of pentagons, first pentagonal pyramidal.

From the manner of summing up polygonal numbers, it is easy to conceive how the prime pyramidal numbers are found, viz., \(\frac{(a-2)n^3+3n^2-(a-5)n}{6}\) expresses all the prime pyramids.

The number nine has a very curious property, its products always composing either 9 or some lesser product of it. We have already given an account of this, with the examples from Hume, under the article Nine; and we need not repeat them. Did our limits permit us, we could instance in a variety of other properties numbers both curious and surprising. Such speculations are indeed by some men considered as trifling and useless: but perhaps they judge too hastily; for few employments are more innocent, none more ingenious, nor, to those who have a taste for them, more amusing.

Numbers were by the Jews, as well as the ancient Greeks and Romans, expressed by letters of the alphabet: hence we may conceive how imperfect and limited their arithmetic was, because the letters could not be arranged in a series, or in different lines conveniently enough for the purposes of ready calculation. The invention of the cypher, or arithmetical figures, which we now make use of, has given us a very great advantage over the ancients in this respect.

Mankind, we may reasonably suppose, first reckoned by their fingers, which they might indeed do in a variety of ways. From this digital arithmetic, very probably, is owing the number 10, which constitutes the whole set of arithmetical figures.

The letters chiefly employed by the Romans to express numbers were, M, for 1000; D, for 500; C, for 100; L, for 50; V, for 5; X, for 10; and I, for 1.—M, probably signified 1000, because it is the initial of mille; D stands for 500, because it is dimidium mille; C signifies 100, as being the first letter of the word centum; L stands for 50, because it is the half of C, having formerly been wrote thus C; V signifies 5, because V is the fifth vowel; X stands for 10, because it contains twice V or V in a double form; I stands for one, because it is the first letter of initium. These however are fanciful derivations. See Numerical Letters.

The Jewish cabbalists, the Grecian conjurors, and the Roman augurs, had a great veneration for particular numbers, and the result of particular combinations of them. Thus three, four, five, seven, nine, ten, are full of divine mysteries, and of great efficacy.

Golden Number. See Chronology, no. 27.