NUMBER, an assemblage of several units, or things of the same kind. Number is either abstract or applicate; abstract, when referred to things in general, without attending to their particular properties; and applicate, 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 although we can compare the number of such things abstractedly, yet we cannot compare them in any applicate 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 there is still a greater. However, in respect of decrease, it is limited; unity being the first and least number, below which, therefore, it cannot descend.

Mathematicians, considering number under a great many relations, have established the following distinctions. 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 is 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 amongst themselves. Cubic number is the product of a square number by its root; as 27, which is the product of the square number 9 by its root 3. All cubic numbers, the root of which is less than 6, being divided by 6, the remainder is the root itself. Thus 27 ÷6 leaves the remainder 3, its root; 216, 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 give 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 in 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. Homogeneal numbers are those referred to the same unit; as those referred to different units are termed heterogemeal. 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 of [7:16:300]which 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 is 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 evenlv 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 arc those divisible only by unity, as 5, 7, &c.; and prime numbers amongst themselves are those which have no common measure besides unity, as 12 and 19. Perfect number is that the aliquot parts of which 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 are those the aliquot parts of which 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, the aliquot parts of which, 6, 4, 3, 2, 1, make 16; and in the latter case 16, the aliquot parts of which 8, 4, 2, and 1, make but 15. Plane number is 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. An even square number added to its root makes an even number.

Figurate numbers are such as represent some geometrical figure, in relation to which they are always considered; as triangular, pentagonal, and pyramidal numbers. Figurate numbers are distinguished into orders, according to their place in the scale of their generation, being all produced one from another, viz. by adding continually the terms of any one, the successive sums are the terms of the next order, beginning from the first order, which is that of equal units, 1, 1, 1,1, &c.; then the second order consists of the successive sums of those of the first order, forming the arithmetical progression 1, 2, 3, 4, &c.; those of the third order are the successive sums of those of the second, and are the triangular numbers 1, 3, 6, 10, 15, &c.; those of the fourth order are the successive sums of those of the third, and are the pyramidal numbers 1, 4, 10, 20, 35, &c.; and so on, as below:
[table]

The above are all considered as different sorts of trian gular numbers, being formed from an arithmetical progression the common difference of which is 1. But if that common difference be 2, the successive sums will be the series of square numbers; if it be 3, the series will be pentagonal numbers, or pentagons; if it be 4, the series will be hexagonal numbers, or hexagons; and so on. Thus:
[table]

The reason of the names, triangles, squares, pentagons, hexagons, and the like, is, that those numbers may be placed in the form of these regular figures or polygons. But the figurate numbers of any order may also be found without computing those of the preceding orders; which is done by taking the successive products of as many of the terms of the arithmeticals, 1, 2, 3, 4, 5, &c. in their natural order, as there are units in the number which denominates the order of figurâtes required, and dividing these products always by the first product. Thus the triangular numbers are found by dividing the products 1 X 2, 2 X 3,3 X 4, 4 X 5, &c. each by the first product 1X2; the first pyramids by dividing the products 1 × 2 × 3,2 X 3 X 4, 3 × 4 × 5, &c. by the first 1 × 2 X 3. And, in general, the figurate numbers of any order n, are found by substituting successively 1, 2, 3, 4, 5, &c. instead of x . . . , . x ∙ x + 1 ∙ X + 2 ∙ x + 3 ∙ &c.

in this general expression —— ~ A; where the factors in the numerator and denominator are supposed to be muitiplied together, and to be continued till the number in each be less by 1 than that which expresses the order of the figurâtes required.^[1. .See Maclaurin s. l’ luxions, art. 351, in the notes; also Simpson’s Algebra, p. 213; or Malcolm’s Arithmetic, p∙ 396, where the subject of figurâtes is treated in a very extensive and perspicuous mauner. Hutton’s Mathematical Dictionary may also be con. suited. ]

Polygonal or polygonous numbers are the sums of arithmetical progressions beginning with unity, and these, where the common difference is 1, are called triangular numbers·, where it is 2, square numbers ; where it is 3, pentagonal numbers ; where it is 4, hexagonal numbers ; where it is 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 ; and the sums of the first pyramidals are called secondpyrami∙ dals, &c. If they arise out of triangular numbers, they are called triangular pyramidal numbers ; and if out of pentagons, first pentagonal pyramidals. From the manner of summing up polygonal numbers, it is easy to conceive how the prime pyramidal numbers are found. The formula (a — 2) ra 3 + 3 n 3 — (a — 5) η i —— — expresses all the prime pyramidals.

The number 9 has a very curious property, its products always composing either 9 or some lesser product thereof. If our limits permitted us, we could instance in a variety of other numbers properties 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 [7:16:301]and Romans, expressed by means of 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 ciphers, or arithmetical figures, which we now make use of, has given us in this respect a very great advantage over the ancients; to say nothing of that which is derived from these figures having a value in position. See Arithmetic.

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 1 for one. M probably signifies 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 written thus, L; V signifies 5, because V is the half of X, which stands for 10; 1 stands for one, because it is the first letter of initium. These, however, are fanciful derivations, and an explanation more accordant with philosophical principles has been given in the article Arithmetic. See also Numeral 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 numbers. Thus three, four, six, seven, nine, ten, were full of divine mysteries, and of great efficacy.

Numbers, in Poetry and Oratory, are certain measures, proportions, or cadences, which render a verse, period, or song agreeable to the ear.

Poetical numbers consist in a certain harmony in the order, quantities, and adjustment of the feet and syllables which render the piece musical to the ear, and capable of being sung, for which all the verses of the ancients were intended. It is of these numbers Virgil speaks in his ninth Eclogue, when he makes Lycidas say, Numeros memini, si verba tenerem ; meaning, that although he had forgotten the words of the verses, yet he remembered the feet and measure of which they were composed.

Rhetorical or prosaic numbers are a sort of simple unaffected harmony, less glaring than that of verse, but such as is perceived and affects the mind with pleasure.

Book of Numbers, the fourth book of the Pentateüch, taking its denomination from the numbering of the families of Israel. A part of this book is historical, relating to several remarkable passages in the march of the Israelites through the wilderness. It contains a distinct relation of their several movements from one place to another, or their forty-two stages through the wilderness, and many other things, by which we are instructed and confirmed in some of the most weighty truths which have immediate reference to God and his providence in the world. But the greater part of this book is occupied in enumerating those laws and ordinances, whether civil or ceremonial, which were given by God, but not mentioned before in the preceding books.