Chap. VIII.

GEOMETRY.

A Trapezium is any other figure whose opposite four sidesare not parallel, as ABC I).

A Scalene Trapezium has two of its sides parallel, but itsfour sides unequal, as E F G II.

A Rectangular Trapezium has two of its sides parallel, andtwo right angles, as IKLM.

An Irregular Trapezium has none of its sides parallel, asPQNO.

"When one pair of opposite sides, as in the figure ABCD,are parallel, it is called a Trapezoid , and among the remarkableelementary properties of this trapezium are the followingThe sum of any three sides is greater than the fourth side :the sum of the squares of the diagonals is equal to the sum ofthe squares of the sides, and four times the square of the linejoining the middle points of the diagonals. The lines joiningthe middle points of the sides form a parallelogram ; and if thefigure can be inscribed within a circle, the sum of each pair ofopposite angles in two right angles, and the sum of the rectan-gles of each pair of opposite sides, are equal to the rectangle ofthe diagonals.

When a diagonal is drawn in a parallelogram, and two otherright lines parallel with the sides are made to cut it, the twoparallelograms which the diagonal does not cut are called thesupplements or complements. In the figure 11STZ the pa-rallelograms ItX Y and X VT are such.

Land surveying requires that every plane figure should beresolved into some of the forms we have described, or con-sidered as composed of a certain number of triangles; for com-puting the area of which it is necessary that we should havethe length of at least one side; and when this is ascertained,together with any two of its other parts, those remaining andthe area may be computed by the rules of trigonometry.

Polygons which are regular have their angles equal each toeach, because they are contained the same number of times inthe same number of right angles, and their sides about theequal angles are to one another in the same ratio.

If the circumference of a circle be divided into any numberof equal parts, the chords joining the points of divisioninclude a regular polygon inscribed in the circle, and thetangents drawn through those points include a regular poly-gon of the same number of sides circumscribed about thecircle : therefore when we have a regular polygon inscribedin a circle, by drawing tangents through the angular points, wecan readily construct another on the outside.

Pentagon is bounded by five sides, and having within it asmany angles; it is called regular when they are all equal, as thefigure A.

In the regular pentagon, if we inscribe within it a triangle,whose base corresponds with one side, and its point that wheretwo opposite sides meet, we shall have an isosceles triangle:in such a triangle we have the angles of the greater doublethat of the less.

An Irregular Pentagon is where the sides and angles vary,as B.

Hexagon is a rectilinear figure of six sides and as manyequal angles; this is called also irregular when the sides areunequal, as in the figure D.

The side of a regular hexagon is equal to the radius of thecircle in which it is inscribed, and it will also be foundthat the side of a regular decagon is equal to thegreater segment of the radius divided medially,and the side square of a regular pentagon is greaterthan the square of the radius by the side square of aregular decagon inscribed in the same circle. Thehexagon is composed of six equilateral triangles, andits figure was much adopted formerly by architects,from the facility which it affords for subdivision.

3 b 4

l'ig. CCD.

Fig. 070.

Fig. 671.

Fig. 672

Fig. i

Fig. 674.

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