GEOMETRY

GEOMETRY* is the science which treats of extension, or form; that is extent of distance, extent of surface, and extent of capacity, or solid content.

DEFINITIONS
I

A SOLID is a magnitude which has length, breadth, and thickness.

II

A SURFACE, or SUPERFICIES, is the boundary of a solid, and has length and breadth only.

III

A LINE is the boundary of a surface, and has length only.

IV

A POINT is that which has no dimensions of any kind-neither length, nor breadth, nor thickness; it has position, but not magnitude, being the extremity of a line.

V

A

B

A STRAIGHT LINE, or RIGHT LINE, is that which lies evenly between its extreme points, without changing its direction, and is the nearest distance between the two points that terminate it, as A B.

NOTE. When the word "line" is used, it is understood to be a straight line, and is generally expressed by two letters at its extremes.

VI

A CURVED LINE is that which is continually changing its direction, as C D.

VII

PARALLEL LINES are such as extend in the same direction, being in every part at the same distance from each other, and which, if infinitely produced, would never meet; as the lines A B and C D.

VIII

A PLANE SURFACE, or PLANE, is that with which a straight line will wholly coincide when drawn between any two points on it.

Geometry (derived from two Greek words signifying land and to measure) originally meant nothing more than the measurement of land. The Egyptians are credited with its invention, as they had recourse to it in order to ascertain the effaced boundaries of their land after the annual inundations of the Nile; but the Science, in its present extended sense, constitutes the foundation of Mathematics.

IX

A CIRCLE is a plane figure bounded by a curved line, called the CIRCUMFERENCE, as A B D, which is in every part equally distant from a point within it, called the CENTRE, as C; it is formed by the revolution of a line about one of its extremities, which remains fixed.

The circumference is often called the circle; but properly the circle is the space contained within the cir- B cumference.

The circumference of a circle is divided into 360 equal parts, called DEGREES, which are subdivided into minutes and seconds (see p. 3). It is also divided into 32 equal parts of 11° 15′ each, which are called the POINTS OF THE COMPASS.

X

An ARC of a circle is any part of the circumference, as D F E.

A CHORD is a line joining the ends of an Arc, as DE; thus the line D E is the chord of the arc D F E.

A SEGMENT is a portion of a circle cut off by a chord ; and in this case divides the circle into two unequal parts, as D F E and D G E.

ΧΙ

A DIAMETER is a straight line drawn through the centre of a circle, and terminated at both ends by the circumference, as A C B; it divides the circle into two equal parts, called SEMICIRCLES, as A G B and A F B; A a semicircle contains 180°.

A QUADRANT is half a semicircle, or the fourth part of the whole circle, as A C G or G C B; a quadrant contains 90°.

B

A RADIUS, or SEMIDIAMETER, is a straight line drawn from the centre to any part of the circumference, and is the extent taken in the compasses to describe a circle, as C A, C G, or C B.

XII

A SECTOR of a circle is any part of a circle comprehended between two radii and their included arc, as A C B.

XIII

An ANGLE is the inclination or opening of two straight lines meeting in a point: the point where they meet is called the ANGULAR POINT, as A; and the lines that include the angle are the SIDES or LEGS, as A B, or A C.

B

An angle is sometimes expressed by three letters, the middle one always denoting the angular point, and the other two the legs that include it; but generally by the letter at the angular point only; as the angle B A C, or the angle A.

XIV

An angle is measured by an arc of a circle contained between its legs, making the angular point the centre of the circle; thus the arc A B is the measure of the angle A C B.

B

Increase or decrease in the length of the legs does not alter the angle, as the length of the lines is in no way connected with their direction towards each other.

XV

Angles are said to be equal to each other when the arcs that measure them are equal, thus the angle D E F and the angle A C B (in Def. XIV.) are equal, since the arcs D F and A B are equal.

XVI

One angle is greater or less than another angle, according as the arc between its legs is greater or less; thus the angle G H I is greater than the angles A C B or D E F (in Def. XIV. and XV.).

XVII

F

As all circles are divided into 360 equal parts, called degrees, etc., a certain number of these divisions will be contained between the two legs of the angle ; therefore an angle is said to measure as many degrees, minutes, etc., as are contained in the arc between the legs.

The arc which measures an angle may be described with any radius; for, since the whole circumference of every circle is supposed to be divided into the same number of parts, it hence follows that the divisions will be greater or less in the same proportion as the whole circumference.

XVIII

A RIGHT ANGLE.-When one line falls upon another, so as to make the angles on each side of it equal, it is called a PERPENDICULAR; and the angles formed by these lines, as the angles A C D, D C B, are called RIGHT ANGLES. Now as the semicircle A D B contains 180 degrees (the half of 360), all right angles will contain an arc of 90 degrees, equal to the fourth part of the whole circle.

C

XIX

An ACUTE ANGLE is that which contains less than a right angle, or 90 degrees, as the angle C A B.

XX

An OBTUSE ANGLE is that which contains more than

a right angle, or 90 degrees, as the angle F D E.

Acute and obtuse angles are called OBLIQUE ANGLES.

XXI

A PLANE TRIANGLE is a figure bounded by three right lines, and contains three angles, of which there are several kinds, both with respect to their sides and angles.

XXH

An EQUILATERAL TRIANGLE is that which has its three sides equal to one another, as A B C.

XXIII

An ISOSCELES TRIANGLE is that which has only two of its sides equal, as D E F.

XXIV

A SCALENE TRIANGLE is that which has all its sides unequal, as G H I.

XXV

A RIGHT ANGLED TRIANGLE is that which has one of its angles a right angle, or containing 90 degrees, as the angle A: the side opposite the right angle is called the HYPOTHENUSE, as BC; and of the other two sides or LEGS, that which stands upright is the PERPENDICULAR, as A C; and the other is the Base, as B A.

XXVI

An ACUTE ANGLED TRIANGLE is that which has all

its angles acute, as D E F.

D

XXVII

An OBTUSE ANGLED TRIANGLE is that which has one of its angles obtuse, as the angle H in the triangle G H I.

B