SQUARE, in Feometry, a quadrilateral figure, both equilateral and equiangular. To find the area of a square, seek the length of one side ; multiply this by itself, and the product is the area of the square.
Sayeat number, the product of a num ber multiplied into itself. Thus 4, is the product of 2 multiplied by 2 ; or 16, the product of 4 multiplied by 4, are square numbers.
A square number is so called, either because it denotes the area of a square, whose side is expressed by the root of the square number ; as in the annexed square, which cons4ta of nine little squares, the side being equal to three ; or else, which is much the . same thing .because the points in the number may . be ranged in the form of a square, by making the root, . or factor, the side of the square.
Some properties of squares are as fol low : 1. Of the Natural series of squares 1', 2', 3', 4', &c. which are equal to . 1 , 4, 9 , 16, &c.
The mean proportional m n between any two of these squares m' and n', is equal to the less square plus, its root multiplied by the difference of the roots; or also equal to the greater square mi nus, its root multiplied by the said dif -ference of the roots. That is m n + dm =i a' — d n; where d = is — m is the difference of their roots.
2. An arithmetical mean between any two squares in and a', exceeds their geo metrical mean, by half the square of the differencc of their roots.
That is ins' ± fn. = m n id'.
S. Of three equidistant squares is the series, the geometrical mean between the extremes is less than the middle square, by the square of their common distance in the series, or of the common differ ence of their roots.
That is, in p a' — d'; where m, n, p, are in arithmetical progres sion, the common difference being d.
4. The difference between the two ad jacent squares in' and n., is n'—m' =2m -I-1; in like manner, p. --n' = 2n + 1, the difference between the next two ad jacent squares n' and p. ; and so on, for
the next following squares. Hence the difference of these differences, or the se cond difference of the squares, is 2n 2 x — m = 2 only, because n 1 ; that is, the second differences of the squares arc each the same constant number 2 ; therefore the first differences will be found by the continual addition of the number 2; and then the squares themselves will be found by the con tinual addition of the first difference ; and thus the whole series of squares is constructed by addition only, as here below : any number of the cubes of the natural series 1, 2, 3, 4, &c. taken from the be ginning, always makes a square number, and that the series of squares, so formed, have for their roots the numbers .. . 1, 3, 6, 10, 15, 21, Ei.c. the Jiffs. of which are 1, 2, 3, 4, 5, 6, &c. viz.
11 1', 13 + 23 = 3., 13 -I- 21 -I- 33 = 6', 13 + 23 ? 33 + 4 = l0'; and the ge neral 1' 23 + 33 ± =- (1 + 2 + 3 + n.n + 1; where n is the number of the terms or cubes.
Squaring the circle, is the making or finding a square whose area shall be equal to the area of a given circle. The best mathematicians have not vet been able to resolve this problem accu rately, and perhaps never will. But they can easily come to any proposed degree of approximation whatever ; for instance, so near as not to err so much in the area as a grain of sand would cover, in a circle whose diameter is equal to that of the orbit of Saturn. The following proportion is near enough the truth for any real use, viz. as 1 is to .88622692, so is the diameter of any circle to the side of the square of an equal area. Therefore, if the diameter of the circle be called d, and the side of the equal square a; then is 8 = .88622692d = • and d nearly.