BOOK I.

THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS.

DEFINITIONS.

THE POINT.

1. A point is that which has position but not dimensions.

A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions-that is, it has neither length, breadth, nor thickness.

THE LINE.

II. A line is length without breadth.

A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness.

III. The intersections of lines and their extremities are points.

IV. A line which lies evenly between its extreme points is called a straight or right

line, such as AB.

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B

If a point move without changing its direction it will describe a right line. The direction in which a point moves is called its sense. If the moving point continually changes its direction it will describe a curve; hence it follows that only one right line can be drawn between two points. The following illustration is due to Professor Henrici :-" If we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape. If we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line."

THE PLANE.

v. A surface is that which has length and breadth.

A surface is space of two dimensions. It has no thickness, for if it had any, however small, it would be space of three dimensions.

VI. When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.

A plane is perfectly flat and even, like the surface of still water, or of a smooth floor.-NEWCOMB.

FIGURES.

VII. Any combination of points, of lines, or of points and lines in a plane, is called a plane figure. If a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure.

VIII. Points which lie on the same right line are called collinear points. A figure formed of collinear points is called a row of points.

THE ANGLE.

IX. The inclination of two right lines extending out from one point in different directions is called a rectilineal angle.

x. The two lines are called the legs, and the point the vertex of the angle.

A right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater. Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point.

Thus if AB, AC be the legs, a line may turn from the position AB to the position AC in the two ways indicated by the arrows. The smaller of the angles thus formed is to be understood as the angle contained by the lines. The larger, called a re-entrant angle, seldom occurs in the "Elements."

A

B

XI. Designation of Angles.-A particular angle in a figure is denoted by three letters, as BAC, of which the middle one, A, is at the vertex, and the other two along the legs. The angle is then read BAC.

XII. The angle formed by joining two or more angles together is called their sum. Thus the sum of the two

angles ABC, PQR is the angle AB'R, formed by applying the side QP to the side BC, so that the vertex Q shall

fall on the vertex B, and the side QR on the opposite side of BC from BA.

XIII. When the sum of two angles BAC, CAD is such that

the legs BA, AD form one right

D

/C

A

B

line, they are called supplements of each other.

Hence, when one line stands on another, the two angles which it makes on the same side of that on which it stands are supplements of each other.

XIV. When one line stands on another, and makes

the adjacent angles at both

IC

A

Hence a right angle is equal to its supplement.

XV. An acute angle is one which is less than a right angle, as A.

A

A

B

XVI. An obtuse angle is one which is greater than a right angle, as BAC.

The supplement of an acute angle is obtuse, and conversely, the supplement of an obtuse angle is acute.

XVII. When the sum of two angles is a right angle, each is called the complement of the other. Thus, if the angle BAC be right, the angles BAD, DAC are complements of each other.

A

B

CONCURRENT LINES.

XVIII. Three or more right lines passing through the same point are called concurrent lines.

XIX. A system of more than three concurrent lines is called a pencil of lines. Each line of a pencil is called a ray, and the common point through which the rays pass is called the vertex.

THE TRIANGLE.

xx. A triangle is a figure formed by three right lines joined end to end. The three lines are called its sides.

XXI. A triangle whose three sides are unequal is said to be scalene, as A; a triangle having two sides equal, to be isosceles, as B; and and having all its sides equal, to be equilateral, as C.

XXII. A right-angled triangle is one that has one of its angles a right angle, as D. The side which subtends the right angle is called the hypotenuse.

XXIII. An obtuse-angled triangle is one that has one of its angles obtuse, as E.

XXIV. An acute-angled triangle is one that has its three angles acute, as F.

XXV. An exterior angle of a triangle is one that is formed by any side and the continuation of another side.

Hence a triangle has six exterior angles; and also each exterior angle is the supplement of the adjacent interior angle.

THE POLYGON.

XXVI. A rectilineal figure bounded by more than three right lines is usually called a polygon.

XXVII. A polygon is said to be convex when it has no re-entrant angle.