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MENSURATION.

§ 375. MensuRATION consists in applying the principles of Geometry (§ 4) to the determination of the dimensions and contents of surfaces and solids.

GEOMETRICAL DEFINITIONS.

§ 376. A point is mere position, without length, breadth, or thickness.-A line is an extension in length, without breadth, or thickness.-A straight line is one which has the same direction throughout its whole extent.-A curved line is one which continually changes its direction.

§ 377. An Angle is the divergence from each other of two straight lines which have a common point. This point is called the vertex of the angle, and the lines themselves are the sides of the angle.

A right angle is one of the two equal angles which one straight line may make at a point in another straight line.

Thus if the two angles A B C and CBD are equal to each other, each of them is a right angle.

E

A

B

An acute angle is one which is less than a right angle, as the angle C BE; and an obtuse angle is one which is greater than a right angle, as the angle A BE.

§ 378. One straight line is perpendicular to another when they form a right angle with each other.

Thus the straight line B C is perpendicular to A D.

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§ 379. Two straight lines are parallel to each other when they are everywhere equidistant from each other.

FG and HI may represent two straight lines which are parallel to each other.

H

F

G

-I

Two parallel straight lines, it is evident, would never meet, how far soever they might be produced.

$380. A surface is an extension in length and breadth, without thickness; and a plane is a surface in which if any two points be taken, the straight line which joins them lies wholly in the surface.

Polygons.

§ 381. A plane figure is a bounded plane surface.—A rectilineal figure, or polygon, is a plane bounded by three or more straight lines which are called the sides of the polygon, and all the sides together make up the perimeter of the polygon.

§ 382. A triangle is a polygon of three sides. A triangle is equilateral when its three sides are equal-isosceles when two of its sides are equal-and scalene when no two of its sides are equal.

383. A right angled triangle is one which has a right angle. The side opposite to the right angle is called the hypothenuse: the other two are the perpendicular sides.

Thus D G F is a right angled triangle, having the right angle at G. The side D F is the hypothenuse.

§ 384. Any side of a triangle may be considered as its base, and the perpendicular distance from the base, or the base produced, to the vertex of the opposite angle, is the altitude of the triangle.

Thus G F is the altitude of the triangle D E F or HIF.

§ 385. A quadrilateral is a polygon of four sides. A paral lelogram is a quadrilateral whose opposite sides are parallel; a rectangle is a parallelogram whose four angles are right angles; and a square is a rectangle whose sides are all equal.

A trapezoid is a quadrilateral which has only two of its sides parallel.

ABCD is a parallelogram; and ABED is a trapezoid, having the two sides A B and D E parallel to each other.

§ 386. Any side of a parallelogram may be considered as its base, and the perpendicular distance from the base to the opposite side, is the altitude of the parallelogram.

The altitude of a trapezoid is the perpendicular distance between its two parallel sides.

Thus B E is the altitude of the parallelogram A C, or of the trapezoid A E.

§ 387. A diagonal is a straight line which joins the vortices of two angles of a polygon that are not adjacent to each other.

The diagonal of a parallelogram divides the parallelogram into two equal triangles.

Thus D B is a diagonal, and it divides the parallelogram A C into the two equal triangles A B D and C D B.

§ 388. A pentagon is a polygon of five sides; a hexagon has six sides,-a heptagon seven,-an octagon eight, a nonagon nine, a decagon ten,-an undecagon eleven,-a dodecagon twelve, &c.

§ 389. A regular polygon is one which has all its sides equal, and all its angles equal,-being equilateral and equiangular.

§ 390. Two polygons are similar when they have the angles of the one equal to the angles of the other, each to each, and the sides which contain equal angles proportional.

Circle and Ellipse.

§ 391. A circle is a plane figure bounded by a curve line called the circumference, all points of which are equidistant from a point within called the centre.

The radius of a circle is a straight line drawn from the centre to the circumference; and the diameter is a straight line drawn through the centre, and terminated both ways by the circumference.

§ 392. An arc of a circle is any part of the circumference; and the chord of an arc is a straight line joining the two extremities of the arc.

A segment of a circle is the space enclosed by an arc and its chord; and a sector of a circle is the space enclosed by an arc and two radii drawn to its extremities.

A semicircumference is half the circumference-a semicircle is a segment equal to half the circle-a quadrant is a fourth part of the circumference, or a sector equal to a fourth part of the circle.

ABDEFA is the circumference of the circle, C the centre, CA a radius, DF a diameter, A B D an arc, the straight line AD is the chord of the arc, ABDA is a segment, A C D BA

a sector.

393. An ellipse is a plane figure bounded by a curve line, from any point of which if two straight lines be drawn to two fixed points, their sum will be equal to a straight line drawn through the same two points, and terminated both ways by the

curve.

Thus ADBEG A may represent an ellipse, in which the two lines E F and E f, drawn from any point of the curve, as E, to the two points F and ƒ, are together equal to A B.