SQUARE

, in Geometry, a quadrilateral figure, whose angles are right, and sides equal. Or it is an equilateral rectangle. Or an equilateral rectangular parallelogram.

A Square, and indeed any other parallelogram, is bisected by its diagonal. And the side of a Square is incommensurable to its diagonal, being in the ratio of 1 to √2.

To find the Area of a Square. Multiply the side by itself, and the product is the area. So, if the side be 10, the area is 100; and if the side be 12, the area is 144.

Square Foot, is a Square each side of which is equal to a foot, or 12 inches; and the area, or Square foot is equal to 144 square inches.

Geometrical Square, a compartment often added on the face of a quadrant, called also Line of Shadows, and Quadrant.

Gunner's Square. See Quadrant.

Magic Square. See Magic Square.

Square Measures, the Squares of the lineal measures; as in the following Table of Square Measures:

Squa. Inches.	Sq. Feet.	Sq. Yards.	Sq. Poles.	S. Chs.	Acres.	S. Miles.
144	1
1296	9	1
39204	272 1/4	30 1/4	1
627264	4356	484	16	1
6272640	43560	4840	160	10	1
4014489600	27878400	3097600.	102400	6400	640	1

Normal Square, is an instrument, made of wood or metal, serving to describe and measure right angles; such is ABC. It consists of two rulers or branches fastened together perpendicularly. When the two legs are moveable on a joint, it is called a bevel.

To examine whether the Square is exact or not. Describe a semicircle DBE, with any radius at pleasure; in the circumference of which apply the angle of the Square to any point as B, and the edge of one leg to one end of the diameter as D, then if the other leg pass just by the other extremity at E, the Square is true; otherwise not.

Square Number, is the product arising from a number multiplied by itself. Thus, 4 is the Square of 2, and 16 the Square of 4. The series of Square integers, is 1, 4, 9, 16, 25, 36, &c; which are the Squares of 1, 2, 3, 4, 5, 6, &c. Or the Square fractions 1/4, 4/9, 9/16, 16/25, 25/36, 36/49, &c, which are the Squares of 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, &c.

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 consists of 9 little squares, the side being equal to 3; 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 follow: 1. Of the

Natural series of Squares,	12, 22, 32,  42, &c,
which are equal to	1 , 4 , 9 , 16 , &c;

The mean proportional mn between any two of these Squares m² and n², is equal to the less square plus its root multiplied by the difference of the roots; or also equal to the greater square minus its root multiplied by the said difference of the roots. That is, ; where is the difference of their roots.

2. An arithmetical mean between any two Squares m² and n², exceeds their geometrical mean, by half the Square of the difference of their roots. That is .

3. Of three equidistant Squares in 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 difference of their roots. That is, ; where m, n, p, are in arithmetical progression, the common difference being d.

4. The difference between the two adjacent

Squares m2, and n2, is	;
in like manner,	, the differ-

ence between the next two adjacent Squares n² and p²; and so on, for the next following Squares. Hence the difference of these differences, or the second difference of the Squares, is only, because ; that is, the second differences of the Squares are 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 continual addition of the first differences; and thus the whole series of Squares is constructed by addition only, as here below:

2d Diff.		2	2	2	2	2	2	&c.
1st Diff.	1	3	5	7	9	11	13	&c.
Squares.	1	4	9	16	25	36	49	&c.

And this method of constructing the table of Square numbers I sind first noticed by Peletarius, in his Algebra.

5. Another curious property, also noted by the same author, is, that the sum of any number of the cubes of the natural series 1, 2, 3, 4, &c, taken from the beginning, 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, &c,
the diffs. of which are	1, 2, 3,  4,  5,  6, &c,

viz, ; where n is the number of the terms or cubes.

Square Root, a number considered as the root of a second power or Square number: or a number which multiplied by itself, produces the given number. See Extraction of Roots, and also the article Root, where tables of Squares and roots are inserted.

T. Square, or Tee Square, an instrument used in drawing, so called from its resemblance to the capital letter T.

This instrument consists of two straight rulers AB and CD, fixed at right angles to each other. To which is sometimes added a third EF, moveable about the pin C, to set it to make any angle with CD.—It is very useful for drawing parallel and perpendicular lines, on the face of a smooth drawing-board.

SQUARED - square, Squared-cube, &c. See Power.

SQUARING. See QUADRATURE.

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 yet been able to resolve this problem accurately, 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 s; .

See Circle, Diameter, and Quadrature. |