One of the most important of the mathematical sciences; as it relates to the form, extension, and magnitude of bodies, and is consequently the foundation of Mensuration. The most generally useful portion of this science has been selected for our pages, which we here annex under the usual distinctive appellation of Practical Geometry.

Geometry Problems

A point is that which hath position, but not magnitude.

A line hath length, but not breadth or thickness, and may therefore be conceived to be generated by the motion of a point.

A right line is what is commonly called a straight line, or that tends every where the same way.

A curve is a fine which continually changes its direction between its extreme points.

Parallel lines are such as are equally distant from each other, and which, if prolonged ever so far, would never meet: such are the lines A B and C D, Fig. 1.

An angle is the space or corner between two lines meeting in a point, as A B C, B denoting the vertex, or angular point.

A right angle is represented when one line stands upon another, so as not to lean more upon one side than upon the other, as in the angles A B C in Fig. 3; all right angles are equal to each other, being all equal to 90°; and the line A B is said to be perpendicular to C D.

Fig. 1.

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Fie. 2.

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Fig. 3.

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Beginners are very apt to confound the terms perpendicular, and plumb or vertical line. A line is vertical when it is at right angles to the plane of the horizon, or level surface of the earth, or to the surface of water, which is always level. The sides of a house are vertical. But a line may be perpendicular to another, whether it stand upright or incline to the ground, or even if it lie flat upon it, provided only that it make the two angles formed by meeting with the other line equal to each other; as, for instance, if the angles ABC and A B D be equal, the line A B is perpendicular to C D, whatever may be its position in other respects.

When one line B E, Fig. 3, stands upon another, C D, so as to incline, the angle EBC, which is greater than a right-angle, is called an obtuse angle; and that which is less than a right angle, is called an acute angle, as the angle E B D.

Two angles which have one leg in common, as the angles ABC and ABE, are called contiguous angles, or adjoining angles; those which are produced by the crossing of two lines, as the angles E B D and C B F, formed by C D and E F crossing each other, are called opposite or vertical angles.

A figure is a bounded space, and is either a surface or a solid.

A superficies, or surface, has length and breadth only. The extremities of a superficies are lines.

A surface may be bounded either by straight lines, curved lines, or both these. Every surface, bounded by straight lines only, is called a polygon. If the sides be all equal, it is called a regular polygon. If they be unequal, it is called an irregular polygon. Every polygon, whether equal or unequal, has the same number of sides as angles, and they are denominated sometimes according to the number of sides, and sometimes from the number of angles they contain. Thus, a figure of three sides is called a triangle, and a figure or four sides, a quadrangle.

A pentagon is a polygon of five sides.

A hexagon has six sides.

A heptagon has seven sides.

An octagon has eight sides.

A nonagon, nine sides.

A decagon, ten sides.

An undecagon, eleven sides.

A duodecagon, twelve sides.

When they have a greater number of sides, it is usual to call them polygons of 13 sides, 14 sides, and so on.

Triangles are of different kinds, according to the length of their sides.

A right-angled triangle is that which has one right angle, as A B C, Fig. 1.

An acute-angled triangle, is a triangle which has all its sides acute, as Figs. A and B.

An obtuse-angled triangle, is a triangle having one obtuse angle, as Fig. C.

An equilateral triangle, is a triangle having all its side3 equal, as Fig. A.

Fig. 1.

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An isoceles triangle, is a triangle having two equal sides, as Fig. B. A scalene triangle, is a triangle having no two of its sides equal, as Fig. C. Quadrangles or quadrilateral figures are of various denominations, as their sides are equal or unequal, or as all their angles are right angles or not.

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Every four-sided figure, whose opposite sides are parallel, is called a parallelogram. Provided that the sides opposite to each other be parallel, it is immaterial whether the angles be right or not. Figs. D E F and G, are all parallelograms.

When the angles of a parallelogram are all right angles, it is called a rectangular parallelogram, or a rectangle, as Figs. E and D.

A rectangle may have all its sides equal, or only the opposite sides equal. When all its sides are equal, it is called a square, as Fig. D.

When the opposite sides are parallel, and all the sides equal to each other, but the angles not right angles, the parallelogram is called a rhombus, as Fig. G.

A parallelogram, having all its angles oblique, and only its opposite equal, is called a rhomboid, as Fig F.

When a quadrilateral, or four-sided figure, has none of its sides parallel, it is called a trapezium, as Fig. H, consequently, every quadrangle or quadrilateral, which is not a parallelogram, is a trapezium.

A trapeziod has only one pair of its sides parallel, as Fig. 1.

A circle is a plane figure, bounded by a curve line returning into itself, called its circumference, b de, Fig. O, every where equally distant from a point within the circle, which is called the centre, c.

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The radius of a circle is a straight line drawn from the centre to the circum ference, as c d, Fig. O. The radius is the opening of the compass when a circle is described; and consequently, all the radii of a circle must be equal to each other.

A diameter of a circle is a straight line drawn from one side of the circumference to the other through the centre, as a b, Fig. P. Every diameter divides the circle into two equal parts.

A segment of a circle is a part of a circle cut off by a straight line drawn across it. This straight line f h, Fig. R, is called the chord. A segment may be either equal to, greater, or less, than a semi-circle, which is a segment formed by the diameter of the circle, and is equal to half the circle, as in Fig. P.

A tangent is a straight line, drawn so as just to touch a circle without cutting it, as d e, Fig. T. The point where it touches the circle, is called the point of contact; and a tangent cannot touch a circle in more points than one.

A sector of a circle is a space comprehended between two radii and an arc, as a c b, Fig. T.

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The circumference of every circle, whether great or small, is supposed to be divided into 360 equal parts, called degrees; and every degree into 60 parts, called minutes; and every minute into 60 seconds. To measure the inclination of lines to each other, or angles, a circle is described round the angular point, as a centre, as a b, Fig. T, and according to the number of degrees minutes, and seconds, cut off by the sides of the angle, so many degrees, seconds, and minutes, it is said to contain. Degrees are marked by °, minutes by ', and seconds by "; thus an angle of 48 degrees, 15 minutes, and 7 seconds is written in this manner, 48° 15' 7".