Diagonal

“ In the limiting case, when the diagonal is exactly equal to the tension of the corresponding surface, the system may often be determined to be unstable by the application of the principle enunciated to an adjacent point of the line in which the surfaces of discontinuity meet. ”

“ A set of four perpendicular, equal, and equidistant straight lines are cut by a diagonal line, which runs from right to left; that is to say, from the higher end of the fourth line to the lower extremity of the first line. This diagonal then represents number 5, and completes the scale or tally of 5.
2. A similar set of four lines are cut by another diagonal, which passes from left to right, or from the higher extremity of number one, to the lower extremity of number four. The diagonal thus completes the second score or tally for number 5. ”

“ As the men never leave the color upon which they are first placed, all moves must be diagonal. A man can move only one square at a time, and only to a square which is in front of him diagonally and is not occupied. ”

“ With the exception of the top squares, every square is divided into two parts by a diagonal, the units being written on one side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal. Fig. 1 shows the slips corresponding to the numbers 2, 0, 8, 5 placed side by side in contact with one another, and next to them is placed another slip containing, in squares without diagonals, the first nine digits. The slips thus placed in contact give the multiples of the number 2085, the digits in each parallelogram being added together ”

“ 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. ”

“ The line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle—
divides each pair of opposite sides into inversely proportional segments; is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides; is divided by both pairs of opposite sides into segments which, measured from either diagonal, have the same ratio to each other. ”

“ It is obvious that if a scale of feet showing inches diagonally be required, twelve equidistant parallel lines must be drawn instead of eight as in the foregoing example where furlongs are required. The diagonal scale possesses the important advantages of accuracy and distinctness of division which render it very suitable as a scale of construction. Another practical advantage is that it is less rapidly defaced by use than the other kinds, in consequence of the measurements being taken on so many different lines. ”

“ With respect to the formation of new surfaces of discontinuity in the vicinity of the point especially considered, the system is stable, if every diagonal of the polygon is less, and practically unstable, if any diagonal is greater, than the tension which would belong to the surface of discontinuity between the corresponding masses. ”

“ The diagonal is a right line drawn across a quadrilateral figure, from one angle to another. The sum of the squares of the two diagonals of every parallelogram is equal to the sum of the squares of the four sides.
A parallelogram is a quadrilateral, whose opposite sides are parallel.
A rectangle is a parallelogram having four right angles.
A rhomboid is an oblique-angled parallelogram.
A rhombus, or lozenge, is a quadrilateral, whose sides are all equal but its angles oblique.
A trapezium is a quadrilateral, which has none of its sides parallel to each other. ”

“ A is a hexagon, and if one of its sides be measured, it will [138] be found that it measures the same as along line 1 from O B to the diagonal line 45 degrees, which distance is shown by a thickened line.
At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened horizontal line) , to the diagonal marked 49 degrees, be measured, it will be found exactly equal to the length of a side on the polygon. ”

“ CUTTING DIAGONALS. Diagonals should not be cut with two bias edges meeting as the lines will come together at entirely different angles. (See illustration 68.) This difficulty can be overcome if you have wide enough material. A diagonal rarely has a perceptible nap and the lengthwise and crosswise of the material are so alike that there is no dissimilarity either in appearance or in wearing qualities between them. So if you want the diagonal to run the same way in both sides of your garment cut one-half lengthwise of the material and the other half crosswise. ”

“ We have shown our axioms sufficient to establish the existence of a complete quadrangle; are they sufficient to prove the ordinary properties of such a figure? They are not. Axioms 1-6 do not decide whether the three diagonal points, 2, 3, 5, are or are not collinear. In the ordinary geometry, those points are non-collinear and form what is called the diagonal triangle. If, however, we suppose that they are collinear (one may assist one's imagination by means of the dotted line) then on rereading our six postulates they will all be found verified. ”

“ On AC the diagonal of a square ABCD, a rhombus AEFC is described of the same area as the square, and having its acute angle at A: if AF be joined, shew that the angle BAC is, divided into three equal angles. ”

“ Scales of Construction. —Scales of construction are intended to afford means of measuring more minute quantities than scales of distances. Of the former there are two kinds, known respectively as the Diagonal and the Vernier scale. The diagonal is the more frequently employed. Its construction involves no peculiar difficulty, as it consists simply of an ordinary scale of distances, with the addition of a number of parallel lines crossed by other parallel lines drawn diagonally from the smaller points of division. An example will best show the construction and mode of using this scale. ”

“ Measuring from the centre lines draw other lines every three inches until you reach to within three inches of the outline of the space, thus dividing the surface into three-inch squares, as dotted lines, Figure 113A. Rule diagonal lines, as A, B, C, to cut through the intersections of the dotted lines, and make these diagonal lines sufficiently heavy to be legible when tufting, as the smaller squares enclosed by the diagonal lines outline the size of the tufts and the intersection of the diagonals mark the points for the buttons. ”

“ By taking various points on the diagonal, panels of any height or width but still of the proper proportions may be obtained (see diagram 210) . Diagram 211 illustrates a variation of the previous method of enlarging the proportions of a panel, in which, by the use of two diagonals, both perpendicular and horizontal center lines are retained. ”

“ Here is a problem that has never yet been solved, nor has its impossibility been demonstrated. Play the knight once to every square of the chessboard in a complete tour, numbering the squares in the order visited, so that when completed the square shall be "magic," adding up to 260 in every column, every row, and each of the two long diagonals. I shall give the best answer that I have been able to obtain, in which there is a slight error in the diagonals alone. Can a perfect solution be found? ”

“ I may not know much math, but I did check it."
"Even after the Columbia boys said it was nonsense? Well, does it come out right?"
"No, it doesn't! You multiply the inverted matrix by the original and you not only don't get zeros for all elements outside the diagonal, you get a haphazard assortment of ones and twos. Worse still, every element in the diagonal comes out equal to zero. The product of the two matrices is about as different from the identity matrix as anything could be. ”

“ For instance, a true opinion that the diagonal is commensurable with the side of a square would be absurd, but since the diagonal concerning which the opinions are held remains the same whether the opinion about it be right or wrong, the object of the two kinds of opinions is one and the same, while according to their essential nature and definition those opinions are different. ”

“ If a straight line be drawn through A one of the angular points of a square, cutting one of the opposite sides, and meeting the other produced at F, shew that AF is greater than the diagonal of the square. ”

“ Again, a body solicited by two forces in directions making an angle with one another, moves in the diagonal; and it seems a paradox to say that motion in the diagonal is the sum of two motions in two other lines. ”

“ Thus in the history of geometry the recognised principles of the science were temporarily disorganised by the admission of incommensurable magnitudes which was forced upon the early Greek mathematicians by the discovery that the side and diagonal of a square have no common measure, and the discrepancy was only removed when it became possible to revise the principles of the theory of numbers itself. ”

“ If one of the sides cuts α′ or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The diagonals may also serve to check the drawing, for two corresponding diagonals must meet in the trace α′.
Having got the plan we easily find the elevation. The elevation of G is above G1 in α′′, and that of F is at F2 in the axis. This gives the elevation F2G2 of FG and in it we get A2B2 in the verticals through A1 and B1. ”

“ In very few cases is the vista behind an attractive or noticeable part of the picture, the fact showing that it acts in opposition to the latter, leading the eye away from it, and thus serving at once the variety and richness of the picture, and its unity. A complete diagonal would have line and vista both working at the extreme outer edge of the picture, and thus too strongly,-- unless, indeed, balanced by very striking elements near the outer edge. ”

“ By cutting the same parallelogram in two, along a diagonal A C, it may be easily shown that the two triangles can be placed exactly one upon the other, and that, consequently, they have equal areas. These lessons constitute a series of classical theorems in geometry which the child can try with his fingers and learn without even giving them the form of theorems. I might show the same as to the area of the trapeze and with many other theorems, but my purpose is only to present as many examples as will make my idea understood, without going into details. ”

“ In cross-hatching it is better that the diagonal lines do not quite meet the edges of the piece, than that they should in the least overrun, as is shown in Figure 89, where in the top half the diagonals slightly overrun, while in the lower half they do not quite meet the outlines of the piece. ”

“ Our own view was formerly in favor of a simple ratio between the sides; but experiments have convinced us that persons of good taste, and who have never been prejudiced by reading Hay's ingenious speculations, do nevertheless agree in preferring rectangles and ellipses which fulfil his law of simple ratio between the angles made by the diagonal. ”

“ If two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals, this perpendicular will bisect the angle between the diagonals. ”

“ Returning to Fig. 62, we see the reason why the rectangle e d c f should have one or both of its diagonals introduced. A load placed, for example, at d would tend to depress the piece d e, and thus deform the rectangle, but when the diagonals are introduced this deformation is impossible.
444. Hence one of these frames is almost as strong as a beam supported at the points c and d, and therefore, from the principles of Art. 388, its strength is three times as great as that of an unsupported beam. ”

“ There are really three cases, if one wishes to be overparticular, corresponding to the three pairs of equal sides. But if we are allowed to take the longest side for the common base, only one case need be considered.
Of the applications of the proposition one of the most important relates to making a figure rigid by means of diagonals. For example, how many diagonals must be drawn in order to make a quadrilateral rigid? to make a pentagon rigid? a hexagon? a polygon of n sides. In particular, the following questions may be asked of a class:
1. ”

“ The diagonals of cells which follow each other passing once only through each zone and section, form similar and equal closed curves passing once quite round the circular axis of the ring and once quite round the centre of the ring. The position of each number is regarded as the intersection of two diagonals of its cell. The numbers are most easily seen if the smallest circle on the surface of the ring, which circle is concentric with the axis, be one of the zonal circles. ”

“ First make a geometrical plan A at the angle required. Then draw its perspective K. Divide line 4b into four equal parts, and continue these measurements all along the base: from each division draw lines to V, and draw the hexagon K. Having this one to start with we produce its sides right and left, but first to the left to find point G, the vanishing point of the 182 diagonals. Those to the right, if produced far enough, would meet at a distant vanishing point not in the picture. But the student should study this figure for himself, and refer back to Figs. ”