15. An isosceles triangle is that which has

two of its sides equal.

16. An equilateral triangle is that which has all its sides equal.

17. A triangle whose sides are unequal, is named scalene.

It will be shown (I. 9. cor.) that every triangle has at least two acute angles. The third angle may therefore, by its character, serve to discriminate a triangle.

18. A right-angled triangle is that which has a right angle.

19. An obtuse angled triangle is that

which has an obtuse angle.

20. An acute angled triangle is that

which has all its angles acute.

21. Any side of a triangle may be called its base, and the opposite angular point its vertex.

22. A quadrilateral figure is contained by four straight

lines.

23. Of quadrilateral figures, a trapezoid (1) has two parallel sides:

24. A trapezium (2) has two of its sides parallel, and the other two equal, though not parallel, to each other:

25. A rhomboid (3) has its opposite sides equal :

26. A rhombus (4) has all its sides equal:

27. An oblong, or rectangle, (5) has a right angle, and its opposite sides equal:

28. A square (6) has a right angle, and all

its sides equal.

29. A quadrilateral figure, of which the opposite sides are parallel, is called a parallelogram.

30. The straight line which joins obliquely the opposite angular points of a quadrilateral figure, is named a diagonal.

31. If an angle of a rectilineal figure be less than two right angles, it protrudes, and is called salient; if it be greater than two right angles, it makes a sinuosity, and is termed re-entrant.

Thus the angle ABC is re-entrant, and the rest of the angles of the polygon ABCDEF are salient at A, C, D, E and

F.

C

D

B

A

32. A rectilineal figure having more than four sides, bears the general name of a polygon.

33. A circle is a figure described by the revolution of a straight line about one of its extremities:

34. The fixed point is called the centre of the circle, the describing line its radius, and the boundary traced by the remote end of that line its circumference.

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

It is obvious that all radii of the same circle are equal to each other and to a semidiameter. It likewise appears, from the slightest inspection, that a circle can only have one centre, and that circles are equal which have equal diameters.

36. Figures are said to be equal, when, applied to each other, they wholly coincide; they are equivalent, if, without coinciding, they yet contain the same space.

A PROPOSITION is a distinct portion of abstract science. It is either a problem or a theorem.

A PROBLEM proposes to effect some combination.

A THEOREM advances some truth, which is to be established.

A problem requires solution, a theorem wants demonstration; the former implies an operation, and the latter generally needs a previous construction.

A direct demonstration proceeds from the premises, by a regular deduction.

An indirect demonstration attains its object, by showing that any other hypothesis than the one advanced would involve a contradiction, or lead to an absurd conclusion.

A subordinate property, included in a demonstration, is sometimes, for the sake of unity, detached, and then it forms a LEMMA.

A COROLLARY is an obvious consequence that results from a proposition.

A SCHOLIUM is an excursive remark on the nature and application of a train of reasoning.

The operations in Geometry suppose the drawing of straight lines and the description of circles, or they require in practice the use of the rule and compasses.

PROPOSITION I. PROBLEM.

To construct a triangle, of which the three sides are given.

Let AB represent the base, and G, H two sides of the triangle which it is required to construct.

From the centre A, with the distance G, describe a circle; and, from the centre B, with the distance H, describe another circle, meeting the former in the point C: ACB is the triangle required.

Because all the radii of the same circle are equal, AC is equal to G; and, for the same reason, BC is equal to H.

G

H

B

Consequently the triangle ACB answers the conditions of the problem. The limiting circles, after mutually intersecting, must obviously diverge from each other, till, crossing the extension of the base AB, they return again and meet below it; thus marking two positions for the required triangle.

Corollary. If the radii G and H be equal to each other, the triangle will evidently be isosceles; and if those lines be likewise equal to the base AB, the triangle must be equilateral.