Hollow square

“ 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. ”

“ By a square in a town, one does not necessarily understand an area with four equal sides and as many right angles, but an open space that is left for air and beauty. There are air and beauty enough to satisfy any reasonable man. A square may be a parallelogram, or a triangle, or any other shape one pleases.”
“This, then, is Hanover Square!—a New York square, or a Nassau Hall square, Corny; but not a Yale College square, take my word for it. ”

“ 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. ”

“ There can be nothing in the world more charming than the way the Square receives its friends. Let it number amongst its guests a Duchess, that is no reason why it should scorn "Colonel Harry" or "Mouldy Jim," the singer of hymns. Scorn, indeed, cannot be found within its grey walls, soft grey, soft green, soft white and blue—in these colours is the Square's body clothed, no anger in its mild eyes, nor contempt anywhere at its heart.
The Square is proud, and is proud with reason, of its garden. It is not a large garden as London gardens go. It has in its centre a fountain. ”

“ Brookbine?" asked the principal. "I thought you gave me lists of all that would be wanted."
"When I did so I had no idea that the boys would ever have occasion to frame a building. The first things we want are a steel square and compasses for each student, for I intend to have the boys lay out the work as well as do it; that is, after the general plan for framing the house is arranged. Everyone must learn to use the square and compass."
"Square and compasses," added the captain, laughing. "We use a compass on board of a vessel." ”

“ It is like nothing that one has seen before, and its beauty equals its novelty. Imagine an immense marble quadrangle, open to the sky and enclosed within lofty walls, with, at each side, a vast recess framed in by a single arch. The quadrangle is more than 100 feet square, and the walls are more than 100 feet high. ”

“ 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. ”

“ 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. ”

“ Before placing the men upon the board, it must be decided whether the white or the black squares are to be played on, as the whole must be put on one color only. If the white squares are selected, there must be a black square in the right-hand corner ”

“ Instances of this kind are shown by the record in connection with squares 667 and east of 667, squares 665 and 666, and squares 662 and 709. And in the case of squares s. s. 667, lying to the south of the street, which consisted of considerable fast land, a sale was made of a lot in that square with the privilege east of the same, being an unnumbered square lying in the water. ”

“ Thus a community of 3,200 families live and work in each section. For further benefit to the people, each two triangles of a quadrant is combined [287] under the title of ‘square.’ Each square, therefore, has its own diagonal street, meeting the other diagonals of the section in the center, or place of occupation. Again, as each square is a smaller community of the section, it has its own shops, stores, etc. All of these places of business are located midway on the diagonals, and are styled the ‘bazar.’ There is allowed in each bazar, only one store for each particular trade ”

“ For all straight lines which meet at any point on the horizon are perspectively parallel to each other, just as two geometrical parallels crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) that all squares formed between the two vanishing lines AS, BS, and by the aid of these diagonals, are also equal, and further, that any number of squares such as are shown in this figure (Fig. 50) , formed in the same way and having equal bases, are also equal; and the nine squares contained in the square abcd being equal, they divide each side of the larger square into three equal parts. ”

“ If run is measured from the point B, which will be the outer edge of the wall plate, it will be necessary to run a gauge line through B parallel to the edge of the rafter, and subtract a distance from the height of the ridge to give us the correct rise. The square must then be applied to the line L. A rafter of any desired rise and run may be laid off in this manner by selecting proportional parts of the rise and run for the blade and tongue of the square. For a half-pitch roof use 12 in. ”

“ 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. ”

“ A space, perhaps forty feet square, has been cut down fifteen or twenty feet, perhaps originally twenty-five or thirty feet, and the whole of the rock removed, leaving the sides regular and smooth, like the walls of a house. Parallel to the south side, and at the distance of eight or ten feet, a graduated road has been made fifteen or twenty feet wide, on a moderate descent, which brings it down near the south-east corner, to nearly the same level with the floor within. A noble archway is hewn through this wall, wide enough to admit three or four men abreast. ”

“ 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. ”

“ So when the great square was filled with people, come to see the race, insomuch that the men swarmed like ants, and the windows and roofs were all as full as an egg, Lightning came out and took his station at the top of the square, waiting for the signal. ”

“ Then, when examining the preliminary sketch-plan, he had put his finger on a square space and asked what room that was. “That isn’t a room; that’s the hall,” said Mr Orgreave. “But it’s square!” Edwin exclaimed. He thought that in houses (houses to be lived in) the hall or lobby must necessarily be long and narrow. Now suddenly he saw no reason why a hall should not be square. ”

“ They are unlike our college buildings, being constructed with an imposing façade, through whose centre is an arch, under the second story, opening into an enclosed quadrangle. These quadrangles vary in dimensions, from one hundred to one hundred and fifty feet square, and the students' rooms open into them. Out of the quadrangle nearest the street, similar arches may often lead into other quadrangles in the rear of the first, or at its sides; so that the establishment may be extended, in a series of buildings, without losing its primal characteristics. ”

“ The space to be occupied by the stairs is divided into nine equal squares, of which those in the corners represent the landings, while the intermediate ones are occupied by the steps; the middle square is the "well-hole." ”

“ 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. ”

“ From a perfect magic ring of n2 cells containing one number each, n2 distinct magic squares can be read off; as the four numbers round each intersection of a zonal circle and a transverse circle constitute corner numbers of a magic square. The shape of a magic ring gives it the function of an indefinite extension in all directions of each of the aforesaid n2 magic squares. ”

“ 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. ”

“ The square of the diagonal line between the two towers is equal to the sum of the square of the horizontal distance between the two towers and the square of the remainder of the subtraction of the tower heights. This is the calculation for heights. For depth calculations on may subtract the two heights [that are above the low point] or the two depths [if one is at the highest point] . If one point is higher than the observer and the other is lower he should add the height to the depth and then square the summation. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ A later stage was to repeat the squares with varieties of colour; and also to introduce details into the squares, and so make them compound patterns, as in the XVIIth dynasty at El Kab, where the sequence of the blue, green, and red lines makes a brilliant effect from these simple elements. Not only a square, but also a hexagon, was worked into the same design. This, from the nature of it, suggests a rush-work screen, and probably it was plaited with rushes in three directions, and hence the production of this particular angle. The previous zigzag patterns all suggest weaving ”

“ 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. ”

“ The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will assume that it is not. ”