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. Two triangles which are both of them right angled, or obtuse, or acute, are said to have the same affection. 22. Any side of a triangle may be called its base, and the opposite angular point its vertex.

23. A quadrilateral figure is contained by four straight lines.

24. Of quadrilateral figures, a square has one right angle, and all its sides equal.

25. An oblong has one right angle, and its opposite sides equal.

26. A rhombus has all its sides equal.

27. A rhomboid has its opposite sides equal.

28. A trapezium has two of its sides parallel and the other two equal to each other

29. A trapezoid has two parallel sides.

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

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

32. If an angle of a polygon 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.

D

Thus the angle ABC is re-entrant, and the rest of the angles of the polygon ABCDEF

B

are salient at A, C, D, E and F.

F

33. A circle is a plane figure described by the revolution

of a straight line about one of its extremi

ties.

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.

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

A PROPOSITION is a distinct undivided 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 involves a contradiction, or leads to an absurd conclusion.

A subordinate property, involved 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 tri

angle 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. Con

H

sequently the triangle ACB answers the conditions of the problem.

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.

PROP. II. THEOREM.

Two triangles are equal, which have all the sides of the one equal to the corresponding sides of the other.

Let the two triangles ABC and DFE have the side AB equal to DF, AC to DE, and BC to FE: These triangles are equal.

For let the triangle ACB be applied to DEF, in the same position. The point A being laid on D, and the side AC on DE, their other extremities C and E must coincide, since AC is equal to DE. And because AB is equal to DF, the point B must be found in the circumference of a circle described from D, with the distance DF; and for the same reason, B must also be found in the circumference of a circle described from E, with

K

the distance EF: The vertex of the triangle ACB must, therefore, occur in a point which is common to both those circles, or in F the vertex of the triangle DFE. Consequently those two triangles, being rectilineal, must entirely coincide. The angle CAB is equal to EDF, ACB to DEF, and CBA to EFD; the equal angles being thus always opposite to the equal sides.

PROP. III. THEOR.

Two triangles are equal, if two sides and the angle contained by these in the one be respectively equal to two sides and the contained angle in the other.

Let ABC and DEF be two triangles, of which the side AB is equal to DE, the side BC to EF, and the angle ABC contained by the former equal to DEF which is contained by the latter: These triangles are equal.

For let the triangle ABC be applied to DEF: The vertex B being placed on E, and the side BA on ED, the extremity A must fall upon D, since "AB is equal to DE. And because the angle or divergence ABC is equal to DEF, and the side AB coin

cides with DE, the other side BC must lie in the same direction with EF, and being of the same length, must en