Square (geometry)

“ The problem of finding a square equal in area to a given circle, like all problems, may be increased in difficulty by the imposition of restrictions; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the “squaring” of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the Euclidean geometry. ”

“ And so the square of the side shall be 8. and the same square shall be 9, to wit, the square of 3. And so even shall be odde, which is most absurd.
Hither may be added that at the 42 p x. That the segments of a right line diversly cut; the more unequall they are the greater is their power.
13. If the base of a right angled triangle be cut by a [158] perpendicular from the right angle in a doubled reason, the power of it shall be halfe as much more, as is the power of the greater shanke: But thrise so much as is the power of the lesser. ”

“ The possibility of a right line is a square H. A right line is said posse quadratum, to be in power a square; because being multiplied in it selfe, it doth make a square.
5 If two conterminall perpendicular equall right lines be closed with parallells, they shall make a quadrate. ”

“ The Gothic window also offers some interesting possibilities in connection with the study of the square. For example, the illustration given on page 223 shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles. [72]
The properties of the square, a figure now easily constructed by the pupils, are not numerous. ”

“ And this figurate number is but the shadow of a Geometricall figure, and doth not indeede expresse it fully, neither is such a quadrate rationall: Yet notwithstanding the numerall side of the greatest square in such like number may be found: As in 148. The greatest quadrate continued is 144 and the side is 12. And there doe remaine 4. Therefore of such kinde of number, which is not a quadrate, there is no true or exact side: Neither shall there ever be found any so neare unto the true one; but there may still be one found more neare the truth. ”

“ Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. ”

“ For I doubt not, but if it had been a thing contrary to any man's right of dominion, or to the interest of men that have dominion, that the three angles of a triangle should be equal to two angles of a square, that doctrine should have been, if not disputed, yet by the burning of all books of geometry suppressed, as far as he whom it concerned was able. ”

“ You need only consider squares of one colour, for whatever can be done in the case of the white squares can always be repeated on the black, and they are here quite independent of one another. This equality, of course, is in consequence of the fact that the number of squares on an ordinary chessboard, sixty-four, is an even number. If a square chequered board has an odd number of squares, then there will always be one more square of one colour than of the other. ”

“ The definition of a square assumes more than is necessary. For if a four-sided figure have all its sides equal and one angle a right angle, it may be shewn that all its angles are right angles; or if a four-sided figure have all its angles equal, it may be shewn that they are all right angles.
Postulates. The postulates state what processes we assume that we can effect, namely, that we can draw a straight line between two given points, that we can produce a straight line to any length, and that we can describe a circle from a given centre with a given distance as radius. ”

“ MAGIC SQUARE, a square divided into equal squares, like a chess-board, in each of which is placed one of a series of consecutive numbers from 1 up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant. ”

“ Let the square be represented by a quadrangle, the sides of which are unknown to us, though they are equal among themselves, as also the angles. This is the quadrate A D, which comprises the three roots and the four of numbers mentioned in this instance. In every quadrate one of its sides, multiplied by a unit, is its root. We now cut off the quadrangle H D from the quadrate A D, and take one of its sides H C for three, which is the number of the roots. ”

“ The numbers here given to the squares are the same as in draughts. The point to be borne in mind in playing the game is to try to unite a perpendicular with a horizontal line of squares, or with a diagonal line. By careful play three men can be got on one of these lines, and three on a communicating line, each having a vacant square. The player of the opposite colour can fill only one of these squares, and consequently cannot prevent the formation of the quaternion on the other line. ”

“ It will be seen that the constant of the adding square is n times that of the subtracting square derived from it, where n is the number of cells in the side of square. And the manner of derivation here is simply to reverse the two diagonals. Both squares are "associated"—a term I have explained in the introductory article to this department.
The third square is a multiplying magic. The constant, 216, is obtained by multiplying together the three numbers in any line. It is "associated" by multiplication, instead of by addition. ”

“ At this point, square ABCD is the square made from side CD, which is the large side of the triangle, and square DEHI is made from side DH, which is the short side of the triangle. Square CHJL is made from side CH, which is the hypotenuse, and line DK is the line from the vertex that meets line JL forming a right angle. ”

“ Now may it appeare, to your eye, that the great square made of the whole line A.B, and of one of his partes that is A.C, [f.ii.v] (which is equall with D.G.) is equal to two partiall squares, whereof the one is made of the saide greatter portion A.C, in as muche as not only D.G, beynge one of his sides, but also D.H. beinge the other side, are eche of them equall to A.C. The second square is H.E.F.K, in which the one side H.E, is equal to C.B, being the lesser parte of the line, A.B, and E.F. is equall to A.C. which is the greater parte of the same line. ”

“ If you will look at the lettered square you will understand that there are only ten really differently placed squares on a chessboard—those enclosed by a dark line—all the others are mere reversals or reflections. For example, every A is a corner square, and every J a central square. Consequently, as the solution shown has a turning-point at the enclosed D square, we can obtain a solution starting from and ending at any square marked D—by just turning the board about. ”

“ The difference between the square on one of the equal sides of an isosceles triangle, and the square on any line drawn from the vertex to a point in the base, is equal to the rectangle contained by the segments of the base. ”

“ All my readers know what a magic square is. The numbers 1 to 9 can be arranged in a square of nine cells, so that all the columns and rows and each of the diagonals will add up 15. It is quite easy; and there is only one way of doing it, for we do not count as different the arrangements obtained by merely turning round the square and reflecting it in a mirror. Now if we wish to make a magic square of the 16 numbers, 1 to 16, there are just 880 different ways of doing it, again not counting reversals and reflections. ”

“ The given number must not itself be a square, because a square multiplied by a square produces a square, and no square plus 1 can be a square. My remarks throughout must be understood to apply to whole numbers, because fractional soldiers are not of much use in war. ”

“ If the combined area of squares constructed from the two sides that are not opposite the angle under consideration is greater than the area of a square constructed from the side opposite the angle under consideration then this angle will be acute. The reason is that the larger an angle is the larger the side opposite it is. It is further explained in section 75 that a square constructed from the hypotenuse of a right triangle contains an area equal to squares constructed from the horizontal and vertical sides together. ”

“ The solution to the puzzle is shown in Figs. 25 and 26. The direction of the cut dividing A and C in the first diagram is very obvious, and the second cut is made at right angles to it. That the four pieces should fit together and form a square will surprise the novice, who will do well to study the puzzle with some care, as it is most instructive.
I will now explain the beautiful rule by which we determine the size of a square that shall have the same area as a Greek cross, for it is applicable, and necessary, to the solution of almost every dissection puzzle that we meet with. ”

“ This we will Pg 175do in Diagram 4. The circle K has one-quarter the area of the circle containing Yin and Yan, because its diameter is just one-half the length. Also L in Diagram 3 is, we know, one-quarter the area. It is therefore evident that G is exactly equal to H, and therefore half G is equal to half H. So that what F loses from L it gains from K, and F must be half of Yin or Yan.
159.—THE SQUARE OF VENEER.—solution
Any square number may be expressed as the sum of two squares in an infinite number of different ways. ”

“ With the hundred a square is connected. If you subtract from this hundred and square the two squares negative connected with fifty, then one square disappears by reason of the other, and the remainder is a hundred and fifty, minus a square, and minus ten roots. ”

“ This gives for inside measurements blade, 221⁄2 inches and tongue 16 inches.
I have described the square as the embodiment of a right angle. If the square is not a 36 right angle, or to use common terms, if the tool is “out of square,” that is, if it is in the least inaccurate, its usefulness is destroyed. When the square is inaccurate instead of solving intricate geometrical problems correctly it becomes a snare and a delusion, leading to false results and misfits in general. It is somewhat remarkable how few workmen test their squares. ”

“ Seat yourself comfortably and don't work hurriedly.
The first thing necessary in this system of squares and circles is to know [132]
How to Cut a Circle Quickly
easily, and accurately, and always without a pattern. Here is a method which never fails:
Cut a square the size you wish to make your circle. That is, if you want a circle with a diameter of four inches, cut a four-inch square (Fig. 226) . Fold the square diagonally through the centre according to the dotted line on Fig. 226, and you have a triangle (Fig. 227) . ”