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Geometry expressed as Rise & Run

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The two straight lines are called the sides of the angle, and the common point of intersection the vertex.
An acute angle is less than a right angle.
An obtuse angle is greater than a right angle.
A polygon, or rectilinear figure, is a portion of a plane terminating on all sides by straight lines. 1, 2, 3, 4 and 5 in Fig. 1.

The polygon of three sides is called a triangle ; that of four sides, a quadrilateral; that of five sides, a pentagon; that of six, a hexagon; that of seven, a heptagon; that of eight, an octagon; that of nine, a nonagon; that of ten, a decagon; that of twelve, a dodecagon.

On Fig. 1 is shown the equilateral parallelogram or perfect square, 5, whose diagonal line, A, rises at an angle of 45 degrees, the vertex, 6, being an angle of 90 degrees. The angle of the vertex is always double the number of degrees which the diagonal line rises from the horizon, or given line.

The parallelogram 5, Fig. 1, is an equilateral figure, or perfect square, described within a circle ; the four vertices, 9, 10, 11, 12, being the four cardinal or perfect points in the circle.
At 1 on the same figure will be seen the nonagon, or polygon of nine sides, whose diagonal line rises at an angle of 20 degrees from the horizon and whose vertex is an angle of 40 degrees.

At 2 is the octagon of eight sides, whose diagonal line rises at an angle of 22 degrees 30 minutes, and whose vertex is an angle of 45 degrees.

At 3 the hexagon of six sides, whose diagonal line rises at an angle of 30 degrees and whose vertex is an angle of 60 degrees.

At 4 the pentagon, or figure of five sides. whose diagonal line rises from the horizon at an angle of 30 degrees and whose vertex is an angle of 72 degrees.

To find the number of degrees the diagonal rises from the horizon, divide the number of sides in the polygon into 360 degrees and divide by 2.

On Fig. 1 we find all the lines radiating from the center. 6, represented on the circumference of the circle by degrees, minutes and seconds, meaning the number of degrees, minutes, etc., which the line rises from the horizon.

In explaining the steel square and its uses, and in all other illustrations, the terms "rise" and run" will be used in the place of degrees, minutes, etc., meaning the number of inches a line rises in every foot on the horizon, that being a common term among mechanics.

The Standard Steel Square, Figs. 2 and 3, has a blade 24 inches long and 2 inches wide, and a tongue 16 or 18 inches long and 1i inches wide ; the blade is exactly at right' angles with the tongue, forming the base and perpendicular of a right angled triangle. The angle at the vertex being an angle of 90 degrees and one of the four cardinal points shown on the circle. The square shown at Figs. 2 and 3 is the Standard Steel Square, containing the diagonal scale of hundredths, brace measure and lumber measure.

On the lumber scale will be found nine parallel lines running parallel with the sides of the blade, divided at intervals of one inch into sections or spaces by cross lines. On each side of the cross lines, and often spaced over the lines, are figures denoting the lengths and widths of lumber.

Suppose we had a board 12 feet long and 6 inches wide; looking on the outer edge of the blade find 12. Between the fifth and sixth lines will be found 12 again ; this is the length of the board. Now follow the space toward the tongue till you come to the cross line under 6 (on the outer edge of the blade ), six being the width of the board ; in this space will be found the figure 6 again which is the answer in board measure, viz : 6 feet. A careful examination of the scale will soon make anyone familiar with it.

On the same side of the square near the vertex will be found the diagonal scale of hundredths, or centesimal scale, as by it a unit may be divided into one hundred equal parts. The scale shown on the Standard Square is 1 inch square; then if it be required to take off .43 of an inch, set one foot of the dividers in the third parallel line from J. extend the other foot to the fourth diagonal on the same parallel ; always start with the parallel line whose number corresponds with the unit figure of the distance required.

In the center of the tongue on the same side will be found the brace measure ; between two parallel lines a half inch apart, near the extreme end, will be found 24-24 33-94. The 24-24 indicates the rise and run of the brace, or the base and perpendicular of a right angle triangle; the 33.94 represents the hypothenuse or length of the brace which would be 33.94-100 inches, but in heavy frame work would practically be 34 inches, allowing the 6-100 for crowding of the wood by draw-bore and key.

On the opposite side of the tongue from the brace measure will be found the octagonal scale, Fig. 3, situated between two parallel lines, divided into intervals and numbered thus: 10, 20, 30, 40, 50, 60. Now let the square, Fig. 4, represent the end of a stick of timber 6 inches square, bisect the lines at aaaa; then with the compass take off as many spaces from the scale as the stick is inches square and with aaaa as centers point off the distances each way and connect the lines, which will be the octagon required.

On Fig. 2 will be seen the method of getting the angles for cutting the joints for any polygon whose sides are of equal length or for cutting the rise and run cuts for any roof. First, let the outer edge of the blade of the square represent the horizon of a circle at 180 degrees, and 12 the center of the circle and base of all operations on the square, for it always represents the run, the rise being shown on the tongue. On the Quadrant of the circle we find a line running from the horizon at an angle of 45 degrees. Starting at the center, 12, and at the intersection on the tongue, we find 12 again ; then with the square being laid upon the edge of the board at 12 and 12 the angle upon either side would be an angle of 45 degrees, and as 45 degrees is one-half of the whole quadrant of 90 degrees, we say " half pitch," hence the pitch of a roof or line is according to the number of degrees it rises from the horizon in the whole quadrant.

Many carpenters in cutting a quarter pitch roof use the figures 6 and 12 on the principle that if 12 and 12 will cut the rise on a half pitch, 6, being the half of that, will cut the quarter pitch, but that is not correct as will be seen. The reasons for taking the figures 6 and 12 for quarter pitch, 8 and 12 for third pitch, etc., is because they represent one-fourth or one-third of the span ofthe building, and has become an established rule, yet, the same rule will not apply to any polygon, except the perfect square. 22 degrees 30 minutes being the half of 45 degrees, or one-fourth of 90 degrees, would be the line upon which the roof rises and the diagonal line intersects the tongue of the square at 5, that being the cut for a quarter pitch roof ; also the diagonal cut of the octagon as shown in Fig. 4.

In the practical application of the square, it is necessary to study thoroughly the relation which one object or piece of work bears to the other. In determining what a square is, we first have to resort to the circle, naturally beginning with its center.

PLATE II.On this plate is given a number of geometrical figures and the method of constructing them.

Fig. 1 shows an easy method of describing an octagon from a given square. From the corners A, B, C, D, as centers, and one-half the diagonal line as a radius, describe the arcs shown, connect the intersecting points and the figure is complete.

Fig. 2 shows the method of erecting a perpendicular from a given line with the compass. Let A, B represent the edge of a board and C the point from which to square it; with one point of the dividers at C step off any equal distance each way on the edge of the board, E, F; then with E and F as centers describe the arcs G, then the points G and C will be the points from which to draw the perpendicular line.

Figs. 3, 4 and 5 show the method of describing a polygon of any number of equal sides when only one side of the polygon is given. Fig. 5 shows the first operation. A, B is the given side of a hexagon; at B erect the perpendicular B, C indefinitely; then with A, B as a radius describe the quadrant of a circle A, C, divide the quadrant into as man y parts as there are sides in the polygon; through the second division from C, draw the diagonal line B, D, bisect A, B and erect the perpendicular E. and the point of intersection with the diagonal line will be the center from which to describe the circle intersecting all