Square formation

“ 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? ”

“ 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 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. ”

“ 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 a first thought it might appear as if these eight movements had nothing in common with trigonometry, but a very fundamental connection may be seen as follows: If the vertex of the angle A is the center of a square and the initial line of A coincides with a line of symmetry of the square, the operations of taking the complement and the supplement of A correspond to movements transforming the square into itself. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ You can draw an oblong with your test square in the same way that you drew a square, measuring the sides to get the length you wish. In Sketch A the semi-diameters are bisected and the points connected, forming a diamond-shaped space, something like a square on its diagonals. In making the unit used in the upper half of this space, the lines of the triangle are changed very slightly, but this change makes an interesting decoration. In Sketch B the sides are quadrisected, and the space is divided by connecting some of the opposite points, making an oblong on its diameters for the middle space. ”

“ 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. ”

“ It will be seen that these squares adapt themselves to other work as well as to framing, a quality very few of the combination squares possess, and while combination squares have their special uses and should be in the tool chest of 20 every expert workman, the square pure and simple, like this of Nicholls or similar ones, should never be absent from the “kit” of the ordinary workman, for with it, if he thoroughly understands it, he can accomplish all that is possible even with a combination square. ”

“ Square each part, and multiply all that come before by twice that part. Hence a reverse rule for extracting the square root presents itself with more than usual simplicity. It is: To extract the square root of a number N, choose a number A, and see if N will bear the subtraction of the square of A; if so, take the remainder, choose a second number B, and see if the remainder will bear the subtraction of the square of B, and twice B multiplied by the preceding part A: if it will, there is a second remainder. ”

“ 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. ”

“ As Lethaby reminds us, modern people take squareness very much for granted as being a self-evident form, but the discovery of the square was a great step in geometry. [4] It opened a new era in the story of the builders. Early inventions must have seemed like revelations, as indeed they were; and it is not strange that skilled craftsmen were looked upon as magicians. If man knows as much as he does, the discovery of the Square was a great event to the primitive mystics of the Nile. ”

“ Each quarter will have two sides at right angles, and by inserting the apex into the opening of the angle drawn by the pupil, any incorrectness will be detected. A small brass or iron square will serve the same purpose, but does not satisfactorily show that a right angle is equal to a quarter of a circle, which is also called a quadrant.
25. Cross a right line with a perpendicular. (fig. 10.)
The right line should be drawn in various directions, to show the pupil that a perpendicular may be raised on any right line, whether horizontal or oblique. ”

“ In like manner we may solve the problem, to produce a given straight line so that the square on the whole straight line made up of the given straight line and the part produced, may exceed the square on the part produced by a given square, which is not less than the square on the given straight line. ”

“ If it were possible so to multiply squares thus inscribed that a figure coincident with the circumference of a circle might be constructed, such a figure would not be a square but a circle. The manifested universe is like that—the process of inscribing squares within a circle. It is ever becoming, evolving, developing, but never quite attains. ”

“ 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. ”

“ Now if we divide this quadrant into halves, thus establishing the point E, and if from E we draw a line to the center C, which is 12 of the blade, it will be found that it cuts also 12 of the tongue. If we complete the figure by erecting a perpendicular line from the point X, and intersecting it with a horizontal line from G, thus establishing the point O, it becomes very evident that CE is the miter line of a square. ”

“ 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. ”

“ 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. ”

“ However, given that he had the experience of a cube being pushed through his plane, he would know there was some figure, whereof his square was part, which was bounded by his square on one side, and by a White square on another side. [108] We have supposed him to make models of these boundaries, a Black square and a White square. The Black square, which is his solid matter, is only one boundary of a figure in Higher Space. ”

“ 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. ”

“ Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define "rhombus" in such a manner as to include a square. We use "parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square." For "oblong" we use "rectangle," letting it include square. Euclid's definition of "square" [Pg 149] illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear; for he might have said that it "is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. ”

“ 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. ”

“ 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. ”

“ And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. ”