The angle opposite the base of a triangle is called the vertex.

[Ill.u.s.tration: Fig. 67.]

[Ill.u.s.tration: Fig. 68.]

A figure that is bounded by four straight lines is termed a quadrangle, quadrilateral or tetragon. When opposite sides of the figure are parallel to each other it is termed a parallelogram, no matter what the angle of the adjoining lines in the figure may be. When all the angles are right angles, as in Figure 66, the figure is called a rectangle. If the sides of a rectangle are of equal length, as in Figure 67, the figure is called a square. If two of the parallel sides of a rectangle are longer than the other two sides, as in Figure 66, it is called an oblong. If the length of the sides of a parallelogram are all equal and the angles are not right angles, as in Figure 68, it is called a rhomb, rhombus or diamond. If two of the parallel sides of a parallelogram are longer than the other two, and the angles are not right angles, as in Figure 69, it is called a rhomboid. If two of the parallel sides of a quadrilateral are of unequal lengths and the angles of the other two sides are not equal, as in Figure 70, it is termed a trapezoid.

[Ill.u.s.tration: Fig. 69.]

[Ill.u.s.tration: Fig. 70.]

[Ill.u.s.tration: Fig. 71.]

If none of the sides of a quadrangle are parallel, as in Figure 71, it is termed a trapezium.

THE CONSTRUCTION OF POLYGONS.

[Ill.u.s.tration: Fig. 71_a_.]

[Ill.u.s.tration: Fig. 72.]

The term polygon is applied to figures having flat sides equidistant from a common centre. From this centre a circle may be struck that will touch all the corners of the sides of the polygon, or the point of each side that is central in the length of the side. In drawing a polygon, one of these circles is used upon which to divide the figure into the requisite number of divisions for the sides. When the dimension of the polygon across its corners is given, the circle drawn to that dimension circ.u.mscribes the polygon, because the circle is without or outside of the polygon and touches it at its corners only. When the dimension across the flats of the polygon is given, or when the dimension given is that of a circle that can be inscribed or marked within the polygon, touching its sides but not pa.s.sing through them, then the polygon circ.u.mscribes or envelops the circle, and the circle is inscribed or marked within the polygon. Thus, in Figure 71 _a_, the circle is inscribed within the polygon, while in Figure 72 the polygon is circ.u.mscribed by the circle; the first is therefore a circ.u.mscribed and the second an inscribed polygon. A regular polygon is one the sides of which are all of an equal length.

NAMES OF REGULAR POLYGONS.

A figure of 3 sides is called a Trigon.

" 4 " " Tetragon.

polygon 5 " " Pentagon.

" 6 " " Hexagon.

" 7 " " Heptagon.

" 8 " " Octagon.

" 9 " " Enneagon or Nonagon.

[Ill.u.s.tration: Fig. 73.]

[Ill.u.s.tration: Fig. 74.]

The angles of regular polygons are designated by their degrees of angle, "at the centre" and "at the circ.u.mference." By the angle at the centre is meant the angle of a side to a radial line; thus in Figure 73 is a hexagon, and at C is a radial line; thus the angle of the side D to C is 60 degrees. Or if at the two ends of a side, as A, two radial lines be drawn, as B, C, then the angles of these two lines, one to the other, will be the "angle at the centre." The angle at the circ.u.mference is the angle of one side to its next neighbor; thus the angle at the circ.u.mference in a hexagon is 120 degrees, as shown in the figure for the sides E, F. It is obvious that as all the sides are of equal length, they are all at the same angle both to the centre and to one another. In Figure 74 is a trigon, the angles at its centre being 120, and the angle at the circ.u.mference being 60, as marked.

The angles of regular polygons:

Trigon, at the centre, 120, at the circ.u.mference, 60.

Tetragon, " 90, " " 90.

Pentagon, " 72, " " 108.

Hexagon, " 60, " " 120.

Octagon, " 45, " " 135.

Enneagon, " 40, " " 140.

Decagon, " 36, " " 144.

Dodecagon, " 30, " " 150.

THE ELLIPSE.

An ellipse is a figure bounded by a continuous curve, whose nature will be shown presently.

The dimensions of an ellipse are taken at its extreme length and narrowest width, and they are designated in three ways, as by the length and breadth, by the major and minor axis (the major axis meaning the length, and the minor the breadth of the figure), and the conjugate and transverse diameters, the transverse meaning the shortest, and the conjugate the longest diameter of the figure.

In this book the terms major and minor axis will be used to designate the dimensions.

The minor and major axes are at a right angle one to the other, and their point of intersection is termed the axis of the ellipse.

In an ellipse there are two points situated upon the line representing the major axis, and which are termed the foci when both are spoken of, and a focus when one only is referred to, foci simply being the plural of focus. These foci are equidistant from the centre of the ellipse, which is formed as follows: Two pins are driven in on the major axis to represent the foci A and B, Figure 75, and around these pins a loop of fine twine is pa.s.sed; a pencil point, C, is then placed in the loop and pulled outwards, to take up the slack of the twine. The pencil is held vertical and moved around, tracing an ellipse as shown.

[Ill.u.s.tration: Fig. 75.]

Now it is obvious, from this method of construction, that there will be at every point in the pencil's path a length of twine from the final point to each of the foci, and a length from one foci to the other, and the length of twine in the loop remaining constant, it is demonstrated that if in a true ellipse we take any number of points in its curve, and for each point add together its distance to each focus, and to this add the distance apart of the foci, the total sum obtained will be the same for each point taken.

[Ill.u.s.tration: Fig. 76.]

[Ill.u.s.tration: Fig. 77.]

In Figures 76 and 77 are a series of ellipses marked with pins and a piece of twine, as already described. The corresponding ellipses, as A in both figures, were marked with the same loop, the difference in the two forms being due to the difference in distance apart of the foci.

Again, the same loop was used for ellipses B in both figures, as also for C and D. From these figures we perceive that--

1st. With a given width or distance apart of foci, the larger the dimensions are the nearer the form of the figure will approach to that of a circle.

2d. The nearer the foci are together in an ellipse, having any given dimensions, the nearer the form of the figure will approach that of a circle.

3d. That the proportion of length to width in an ellipse is determined by the distance apart of the foci.

4th. That the area enclosed within an ellipse of a given circ.u.mference is greater in proportion as the distance apart of the foci is diminished; and,

5th. That an ellipse may be given any required proportion of width to length by locating the foci at the requisite distance apart.

The form of a true ellipse may be very nearly approached by means of the arcs of circles, if the centres from which those arcs are struck are located in the most desirable positions for the form of ellipse to be drawn.

[Ill.u.s.tration: Fig. 78.]

Thus in Figure 78 are three ellipses whose forms were pencilled in by means of pins and a loop of twine, as already described, but which were inked in by finding four arcs of circles of a radius that would most closely approach the pencilled line; _a b_ are the foci of all three ellipses A, B, and C; the centre for the end curves of _a_ are at _c_ and _d_, and those for its side arcs are at _e_ and _f_. For B the end centres are at _g_ and _h_, and the side centres at _i_ and _j_. For C the end centres are at _k_, _l_, and the side centres at _m_ and _n_.

It will be noted that, first, all the centres for the end curves fall on the line of the length or major axis, while all those for the sides fall on the line of width or the minor axis; and, second, that as the dimensions of the ellipses increase, the centres for the arcs fall nearer to the axis of the ellipse. Now in proportion as a greater number of arcs of circles are employed to form the figure, the nearer it will approach the form of a true ellipse; but in practice it is not usual to employ more than eight, while it is obvious that not less than four can be used. When four are used they will always fall somewhere on the lines on the major and minor axis; but if eight are used, two will fall on the line of the major axis, two on the line of the minor axis, and the remaining four elsewhere.

[Ill.u.s.tration: Fig. 79.]

In Figure 79 is a construction wherein four arcs are used. Draw the line _a b_, the major axis, and at a right angle to it the line _c d_, the minor axis of the figure. Now find the difference between the length of half the two axes as shown below the figure, the length of line _f_ (from _g_ to _i_) representing half the length of the figure (as from _a_ to _e_), and the length or radius from _g_ to _h_ equalling that from _e_ to _d_; hence from _h_ to _i_ is the difference between half the major and half the minor axis. With the radius (_h i_), mark from _e_ as a centre the arcs _j k_, and join _j k_ by line _l_. Take half the length of line _l_ and from _j_ as a centre mark a line on _a_ to the arc _m_. Now the radius of _m_ from _e_ will be the radius of all the centres from which to draw the figure; hence we may draw in the circle _m_ and draw line _s_, cutting the circle. Then draw line _o_, pa.s.sing through _m_, and giving the centre _p_. From _p_ we draw the line _q_, cutting the intersection of the circle with line _a_ and giving the centre _r_. From _r_ we draw line _s_, meeting the circle and the line _c, d_, giving us the centre _t_. From _t_ we draw line _u_, pa.s.sing through the centre _m_. These four lines _o_, _q_, _s_, _u_ are prolonged past the centres, because they define what part of the curve is to be drawn from each centre: thus from centre _m_ the curve from _v_ to _w_ is drawn, from centre _t_ the curve from _w_ to _x_ is drawn.

From centre _r_ the curve from _x_ to _y_ is drawn, and from centre _p_ the curve from _y_ to _v_ is drawn. It is to be noted, however, that after the point _m_ is found, the remaining lines may be drawn very quickly, because the line _o_ from _m_ to _p_ may be drawn with the triangle of 45 degrees resting on the square blade. The triangle may be turned over, set to point _p_ and line _q_ drawn, and by turning the triangle again the line _s_ may be drawn from point _r_; finally the triangle may be again turned over and line _u_ drawn, which renders the drawing of the circle _m_ unnecessary.

To draw an elliptical figure whose proportion of width to breadth shall remain the same, whatever the length of the major axis may be: Take any square figure and bisect it by the line A in Figure 80. Draw, in each half of the square, the diagonals E F, G H. From P as a centre with the radius P R draw the arc S E R. With the same radius draw from O as a centre the arc T D V. With radius L C draw arc R C V, and from K as a centre draw arc S B T.