SQUARE. We believe that the old English meaning of this word had reference only to the corners of a figure, or at most to right-angled corners. The old word for any oblong, or rectangle, is a four-square figure ; the carpenter's rule for drawing a right angle is called a T-square to this day. The French word eqUerr (anciently esquerre, originally derived, like the Italian squadra, from quadratum) is the immediate origin ; and this (in French) means also an instrument for drawing a right angle. In Recorde's ` Ground of Arts,' the earliest English geometry extant, he calls what is now a square by the name of square quadrate (square, right-angled, quadrate, four-sided figure); and it is not until he is considerably advanced in his work that he seems to find out that he may drop the second word and retain the first only. There was still an incorrectness, for a square figure should have meant one having all its angles right angles ; that is, what we now call a rect angle, whether its sides were equal or not. To complete the proof of connection between the square and the right-angled corner, we may mention that before now a right-angled triangle has been called a square. In or about 1613, Thomas Bedwell published a work, of which the title was' Trigonum Architectonicum, the Carpenter's Squire.' In geometry, a square means a four-sided plane figure with all its sides equal, and all its angles right angles. In algebra, it signifies the number produced by multiplying a number by itself. The reason of the double meaning is obvious enough. [RECTANGLE.] A square of 7 units long contains 7 x 7 square units ; so that the operation 7 x 7 is the arithmetic of finding the content of a square of 7 units in length and breadth. We have spoken, in the article just cited, of the confusion which is caused by this double use of the word square ; and proposed to correct it by speaking of the square on a line in geometry, and the square of a number in algebra. It has been the fashion of late years to publish what are called symbolical editions of Euclid, in which A n= is made to stand for the square on the line A H, because al stands for the square of the number a. The learner who uses this species of symbol will not, without great care, avoid false reasoning in making the connection between geometry and alFebra.
A square is divided by its diagonal into two isosceles right-angled triangles : its diagonals are equal, and bisect each other at right angles. The easiest way of drawing a square is to describe two circles on the side a u, and bisecting A C in n, to make c x and c equal to a )3; the figure a it a is then a square.
Of all similar figures, squares are those to the areas of which reasoning is most easily applied. If three similar figures be described too small, we go on adding 100 to the square root, until no more hun Ir+ds can be added ; all the while forming the square% by the rule, each square from the preceding. Wo then begin to add tens, forming the squares also, until the addition of one more ten would bring the square past 104713; we then add units, until either the square is exactly 104713, or the nearest to it. Or, instead of continual additions, we might subtract every number, as we get It, from 104713 until no more subtractions can be made. Both modes are exhibited in the following :— on the, sides of a right-angled triangle, the sum of those on the sides is equal to that on the hypothenuse. This proposition is learned with reference to squares (II YPOTIIENUSE] long before it can be proved with reference to similar figures in general : the consequence is, that the general proposition is almost overlooked by the previous occurrence of the particular case. We lime noted in the article just cited the Hindu proof of this celebrated case ; the simplest properties of the square may be made to give a more easy proof, founded on the same prin ciple ; it being remembered that the first four propositions of the second book of Euclid do not require the last two of the first book. The proof is as follows :—Let a B, n c, be the sides of a right-angled triangle, and on their sum describe a square ; make c E, F II, X L, each equal to AB. It is easily proved that LBELI is the square on the hypothenuse of the triangle ; and it is made by subtracting four times the triangle from the whole square A F. But if four times the triangle In the first column we feel our way, so to speak, by hundreds, by tens, and by units, up to the result that is too small, and too great ; so that we see that 104713 has no square root. In the second column we go down from 104713, and subtracting the squares already formed in the first column, we come to the result that 104713 is 384 more than 323', but less than The results of tho second column are— be subtracted by taking away the rectangles ass and a D, we have (Enc., ii. 4) the sum of the squares on the sides, which is therefore the same as the square on the diagonaL