i. A point is that which has position but not dimensions.
ii. A line is length without breadth.
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.
v. A surface is that which has length and breadth.
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.
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.
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.
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.
xiii. When the sum of two angles BAC, CAD is such that the legs BA, AD form one right line, they are called supplements of each other.
xiv. When one line stands on another, and makes the adjacent angles at both sides of itself equal, each of the angles is called a right angle, and the line which stands on the other is called a perpendicular to it. Hence a right angle is equal to its supplement.
xv. An acute angle is one which is less than a right angle.
xvi. An obtuse angle is one which is greater than a right angle. 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.
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.
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; a triangle having two sides equal, to be isosceles; and and having all its sides equal, to be equilateral.
xxii. A right-angled triangle is one that has one of its angles a right angle. 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.
xxiv. An acute-angled triangle is one that has its three angles acute.
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.
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.
xxviii. A polygon of four sides is called a quadrilateral.
xxix. A quadrilateral whose four sides are equal is called a lozenge.
xxx. A lozenge which has a right angle is called a square.
xxxi. A polygon which has five sides is called a pentagon; one which has six sides, a hexagon, and so on.
xxxii. A circle is a plane figure formed by a curved line called the circumference, and is such that all right lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre.
xxxiii. A radius of a circle is any right line drawn from the centre to the circumference.
xxxiv. A diameter of a circle is a right line drawn through the centre and terminated both ways by the circumference.
Let it be granted that -
i. A right line may be drawn from any one point to any other point. When we consider a straight line contained between two fixed points which are its ends, such a portion is called a finite straight line.
ii. A terminated right line may be produced to any length in a right line. Every right line may extend without limit in either direction or in both. It is in these cases called an indefinite line. By this postulate a finite right line may be supposed to be produced, whenever we please, into an indefinite right line.
iii. A circle may be described from any centre, and with any distance from that centre as radius.
i. Things which are equal to the same, or to equals, are equal to each other.
ii. If equals be added to equals the sums will be equal.
iii. If equals be taken from equals the remainders will be equal.
iv. If equals be added to unequals the sums will be unequal.
v. If equals be taken from unequals the remainders will be unequal.
vi. The doubles of equal magnitudes are equal.
vii. The halves of equal magnitudes are equal.
viii. Magnitudes that can be made to coincide are equal.
ix. The whole is greater than its part.
ix0. The whole is equal to the sum of all its parts.
x. Two right lines cannot enclose a space.
xi. All right angles are equal to one another.
xii. If two right lines (AB; CD) meet a third line (AC), so as to make the sum of the two interior angles (BAC; ACD) on the same side less than two right angles, these lines being produced shall meet at some finite distance.
Prop 1: On a given finite right line (AB) to construct an equilateral triangle.
Sol.-With A as centre, and AB as radius, describe the circle BCD (Post. iii.).
With B as centre, and BA as radius, describe the circle ACE, cutting the former circle in C.
Join CA, CB (Post. i.).
Then ABC is the equilateral triangle required.
Dem.-Because A is the centre of the circle BCD, AC is equal to AB (Def. xxxii.).
Again, because B is the centre of the circle ACE, BC is equal to BA.
Hence we have proved.
AC = AB;
and BC = AB
But things which are equal to the same are equal to one another (Axiom i.); therefore AC is equal to BC;
therefore the three lines AB, BC, CA are equal to one another. Hence the triangle ABC is equilateral (Def. xxi.); and it is described on the given line AB, which was required to be done.
Prop 2: From a given point (A) to draw a right line equal to a given finite right line (BC).
Sol.-Join AB (Post. i.);
on AB describe the equilateral triangle ABD [i.].
With B as centre, and BC as radius, describe the circle ECH (Post iii.).
Produce DB to meet the circle ECH in E (Post. ii.).
With D as centre, and DE as radius, describe the circle EFG (Post. iii.).
Produce DA to meet this circle in F.
AF is equal to BC.
Dem.-Because D is the centre of the circle EFG, DF is equal to DE (Def. xxxii.).
And because DAB is an equilateral triangle, DA is equal to DB (Def. xxi.).
Hence we have
DF = DE;
and DA = DB;
and taking the latter from the former, the remainder AF is equal to the remainder BE (Axiom iii.).
Again, because B is the centre of the circle ECH, BC is equal to BE;
and we have proved that AF is equal to BE; and things which are equal to the same thing are equal to one another (Axiom i.).
Hence AF is equal to BC.
Therefore from the given point A the line AF has been drawn equal to BC.
Prop 3: From the greater (AB) of two given right lines to cut off a part equal to (C) the less.
Sol.- From A, one of the extremities of AB, draw the right line AD equal to C [ii.];
and with A as centre, and AD as radius, describe the circle EDF (Post. iii.) cutting AB in E.
AE shall be equal to C.
Dem. Because A is the centre of the circle EDF, AE is equal to AD (Def. xxxii.), and C is equal to AD (const.); and things which are equal to the same are equal to one another (Axiom i.);
therefore AE is equal to C.
Wherefore from AB, the greater of the two given lines, a part, AE, has been out off equal to C, the less.