Square

“ 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.
Fig. 1. From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. ”

“ What would real wealthy people be willing to give for a perfect picture only one-half inch square, in which the artist had clearly defined every field and tree, the rivers, houses, roads, railways and all the beautiful landscape contained in a vast area of many square miles? ”

“ A square is less beautiful than a circle, because it is less simple: a circle has parts as well as a square; but its parts not being distinct like those of a square, it makes one entire impression; whereas, the attention is divided among the sides and angles of a square.... A square, though not more regular than a hexagon or octagon, is more beautiful than either, because a square is more simple, and the attention less divided. ”

“ 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. ”

“ A square unit is the square described on a line whose length is the linear unit. Thus a square inch is the square described on a line whose length is an inch; a square foot is the square described on a line whose length is a foot, &c.
3. If we take a linear foot, describe a square on it, divide two adjacent sides each into twelve equal parts, and draw parallels to the sides, we evidently divide the square foot into square inches; and as there will manifestly be 12 rectangular parallelograms, each containing 12 square inches, the square foot contains 144 square inches. ”

“ A square is a quadrilateral having all its sides equal, and all its angles right angles. The diagonal of a square is equal to the square root of twice the square of its sides: and the side of the square is equal to the square root of half the square of its diagonal. ”

“ From this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. For example, a square whose side is 13 inches in length contains 13 × 13 or 169 square inches. ”

“ If we look at it in this way we shall never mistake the square for the circle, but we shall see that as the circle grows, the corresponding square will grow with it. It is this dependence of the square on the circle that makes all the difference, and makes it a living, growing square. ”

“ 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. ”

“ If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. ”

“ The diagonals of a square, or rectangle, always divide each other into two equal parts. Using the point where the diagonals cross each other (intersect) as a center, draw a circle that will just touch the center of each side of the square. What is the diameter of this circle? How does this diameter compare with the length of the sides of the square? You have drawn what is known as an inscribed circle; that is, a circle whose circumference touches all sides of the containing figure but does not pass beyond the sides. What is the area of this square? ”

“ It is equal-sided, as long as it is broad. In its way, the square is as perfect as the circle. Four straight lines, each forming a right angle with its neighbors, are the boundaries of a perfect enclosure. ”

“ 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. ”

“ All lines that cross a square through its center divide the square evenly, as can be seen here where lines FG and HI cross each other at the center E. Lines FE and GE are equal, and lines HE and IE are equal as well, as are all lines that cross the square in this manner. ”

“ And universally, if a straight line contain a unit of length an exact number of times, the area of the square described on the straight line is denoted by the square of the number which denotes the length of the straight line. ”

“ In each case we make the square K the same size and at the same angle, its near corner being at A. It must be seen that by whichever method we work out this perspective, the result is the same, so that both are correct: the great advantage of the first or geometrical system being, that we can place the square at any angle, as it is drawn without reference to vanishing points. ”

“ If the reader will imagine this series of squares to represent a portable piece of open trellis-work, which might be set up at a window or in the open field, between the spectator and any object of interest at a distance—each square representing a number corresponding with a code in universal use—it will be obvious, that by noticing the squares which the outline of a hill would cover, and telegraphing the numbers of the squares, something in the way of form and outline may be quickly communicated from the other side of the world. ”

“ A square that was quite popular some sixteen or eighteen years ago known as “The Crenalated Square,” an illustration of which is shown in Fig. 16, is still preferred by many workmen. The peculiarity of this square is that the inner edge of the tongue is notched or crenalated, as shown in the illustration, the notches being intended as “gauge-points,” where a sharpened pencil may be inserted, then the square may be drawn along the timber or board, with the blade held snug against the edge, as shown, and mortises or tenons can be laid out at will. ”

“ Section 49 The area of a plane is how many square cubits, square handbreadths, or square finger-lengths are contained inside, whether the plane is square, circle, triangle, or any other shape imaginable. ”

“ A square place ten feet long and ten feet wide gives an area of one hundred feet. Now if it is required to double the square, and to make one of two hundred feet, we must ask how long will be the side of that square so as to get from this the two hundred feet corresponding to the doubling of the area. ”

“ With a square and a compass it is a comparatively easy matter accurately to step off 45 degrees, as it is the line C, midway between A and B, and the other degrees may be calculated from the line C and the cardinal lines A or B.
Degrees Without a Compass.—But in the absence of a compass and when you do not wish to step off a circle, you will in such case lay down the square, and mark off at the outer margin of the limbs two equal dimensions. Suppose we take 2 inches on each limb of the square. The angle thus formed by the angle square blade is 45 degrees. ”

“ Section 75 In considering a right triangle, the area contained within a square constructed from the length of its hypotenuse is equal the area of the squares constructed from the lengths of its vertical leg and base respectively. The proof of this is found when a square is created by doubling the vertical leg and the base. ”

“ In every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. ”

“ 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. ”

“ Indeed, the use of square and angular forms is one of the characteristics of the style; we have square panels, square foliage, square crockets and finials, square forms in the windows,—caused by the introduction of so many transoms,—and an approach to squareness in the depressed and low pitch of the roofs in late examples.{209}
Doorway, Fotheringhay, Northants, A.D. 1435.
It is frequently said that it is not easy to distinguish the moldings of the Perpendicular style from those of the earlier styles, but this is in general because people do not pay attention to them. ”

“ 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 square is excluded from the conception of a circle, therefore the predicate "square" must not be given to a circle. For the same reason a straight line must not be crooked, and a lie must not be true. ”

“ If the cities were each to be imagined as a single point, a square could be drawn around the point so that, at nearest, each side of the square would be one thousand cubits from the city, while each side would be two thousand cubits long. ”

“ THE SQUARE PUZZLE.
261. A man has a square of land, out of which he reserves one-fourth (as shown in the diagram) for himself. The remainder he wishes to divide among his four sons so that each will have an equal share and in similar shape with his brother. ”

“ An infinite variety of puzzles may be made introducing new conditions into the magic square. In The Canterbury Puzzles I have given examples of such squares with coins, with postage stamps, with cutting-out conditions, and other tricks. I will now give a few variants involving further novel conditions.
399.—THE TROUBLESOME EIGHT.
Nearly everybody knows that a "magic square" is an arrangement of numbers in the form of a square so that every row, every column, and each of the two long diagonals adds up alike. ”

“ As Biceps could not tell how to extract a circle from a square, his friend gave him the following solution: “Let the given square be Sloane Square; find the Inner Circle, and take its lines to any point, at any distance from that square, paying the proper fare. That’s the ticket!”
Return to description
252
Every Cretan is said to lie,
And steeds that canter pant.
The gods drank nectar, old and dry,
And all men may recant. ”

“ 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. ”

“ Solution
[8]
No. VIII.—A HARDY ANNUAL
A magic square can be formed with the 81 numbers from 172 to 252 inclusive, which in all its rows, columns, and diagonals will total 1908. It may interest our solvers to complete the square.
Image
We have filled in, as a solid start, 45 of the 81 cells. ”

“ If from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to 2n times the square of the radius. ”

“ I do not think I have mentioned the bright form of quadrature in which a square is made equal to a circle by making each side equal to a quarter of the circumference. The last squarer of this kind whom I have seen figures in the last number of the Athenæum for 1855: he says the thing is no longer a problem, but an axiom. He does not know that the area of the circle is greater than that of any other figure of the same circuit. ”

“ 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. ”

“ First draw a square with its diagonals (Fig. 178) , and from its centre O inscribe a circle; in this circle inscribe a square, and in this again inscribe a second circle, and so on. Through their intersections with the diagonals draw lines to base, and 155 number them 1, 2, 3, 4, &c.; transfer these measurements to the base of the perspective square (Fig. 179) , and proceed to construct the circles as before, drawing lines from each point on the base to the point of sight, and drawing the curves through the inter-sections of these lines with the diagonals. ”

“ The same would be true of the man who would claim that he could give, in exact figures, the diagonal of a square of which the side is exactly one foot or one yard, and it is also true of the man who claims that he can give the exact area of a circle of which either the circumference or the diameter is known with precision. ”