Rectangle

“ 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 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. ”

“ We do not teach all the names relative to the geometric figures, giving only those of the most familiar forms, such as square, circle, rectangle, triangle, oval. We now call attention to the fact that there are rectangles which are narrow and long, and others which are broad and short, while the squares are equal on all sides and can be only big and little. These things are most easily shown with the insets, for, though we turn the square about, it still enters its frame, while the rectangle, if placed across the opening, will not enter. ”

“ One curious result is that even the very simplest objects can produce an esthetic effect. You would scarcely suppose, for example, that a mere rectangle could produce any esthetic effect, or that it would make any difference what exact proportions the rectangle possessed; and yet it is found that some rectangles are preferred to others, and that the popular choice falls upon what the art theorists have long known as the "golden section", a rectangle with a width about sixty-two per cent, of its length. ”

“ If the base and the difference of the base angles of a triangle be given, the rectangle contained by the perpendiculars from the vertex on two lines through the middle point of the base, parallel to the internal and external bisectors of the vertical angle, is constant. ”

“ Figs 12-13 Figure 13, though rectangular, is a scalene also.
28. Draw a rectangle. (fig. 14.)
A rectangle is properly a figure with four sides, of which each two opposite sides are equal and parallel, and of which all the angles are right angles. ”

“ The seventeenth proposition of the sixth book is the problem “To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.” ”

“ Let the base CD (= x) increase to 10 inches (the diameter) ; then the altitude will decrease to zero and the area will become zero. Now let the base decrease to zero; then the altitude will increase to 10 inches and the area will again become zero. It is therefore intuitionally evident that there exists a greatest rectangle. By a careful study of the figure we might suspect that when the rectangle becomes a square its area would be the greatest, but this would at best be mere guesswork. A better way would evidently be to plot the graph of the function (1) and note its behavior. ”

“ Second. A rectangle, Fig. 121. On the base E F draw the perpendiculars E H, F G, equal to the height of the rectangle, and join G H.
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Fig. 122.
Ex. 17.—To construct a parallelogram of which the sides and one of the angles are given, Fig. 122. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, forming the given angle C. From E with D F as radius, describe an arc, and from F, with the radius D E cut the arc at G. Draw F G, E G. Or, the remaining sides may be drawn as parallels to D E, D F.
Fig. ”

“ This variation of what is termed their value gives us a guide to their proper placing in the hue rectangle.
Values and Horizontal Value Lines
Value is that quality by which we may distinguish a dark hue from a light one. For design purposes we will imagine the hue rectangle to grade from white at the top to black at the bottom. We will draw horizontal lines or steps across the rectangle, marking nine even value steps from white to black ”

“ The confining edge of the design is usually a rectangle, on the printed page. The location of a point of [Pg 25] balance within this rectangle tends to divide it. How shall it be divided in the most interesting way? By applying the ratio of good proportion. So the point of balance may be located usually on a line which divides the page into parts of 2 and 3.
When equal masses are to be balanced it is obvious that they will be equidistant from the point of balance. (Fig. 12.)
Fig. 12. Equal masses balanced at equal distance from the center point. ”

“ Gradually the frame is completely filled with figures; first, with very dissimilar figures, as, for example, a square, a very narrow rectangle, a triangle, a circle, an 48 ellipse and a hexagon, or with other figures in combination. ”

“ The Surface of every Round Solid is equal to a Rectangle, whose Base is the Circumference of the Figure, by the Rotation whereof the Solid is begot, and the Height equal to the Circumference, which the Center of Gravity of the Perimeter of the Figure describeth. ”

“ In the middle rectangle, measure off and cut out a rectangle 2-1/4×3 inches in size. Fold rectangle No. 3 up and back of rectangle No. 2. Holding the two firmly together, punch two holes, one-fourth inch apart, on each side, and one-fourth inch from the outer edges (see diagram) . Draw a piece of raffia or ribbon through these holes and tie in a bow. Fold back rectangle No. 1 for support.
PICTURE FRAME No. I— (For description see page 37.)
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PICTURE FRAME No. II— (For description see pages 37 and 39.) 10 Portfolio
Material—Heavy manila paper, 7-1/2×12 inches. ”

“ In the early dawn of human knowledge the arrangement of points in squares and triangles, and the further conception of areas by their subdivision into triangles, undoubtedly arose from the inability of the human mind to retain distinct images of figures more complicated than rectangles or squares. ”

“ 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. ”

“ They may form the covering of any rectilinear figure at all approaching to regularity of form; as the triangle, the square, the canted square, polygons, either regular or elongated, oblongs or parallelograms of any kind; but the usual form is the octagon or other polygon, and for our general purpose it may be best to limit them to figures capable of being inscribed in a circle or an ellipse.
I have introduced this variety of dome as occurring in those which are raised high above their pendentives or corbels. They occur, however, in numerous positions. ”