2. An isosceles triangle is one which has only two of its sides equal.

3. A scalene triangle is one which has its three sides unequal.

4. An acute-angled triangle is one which has its three angles acute.

5. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse, and the other two sides, the base and perpendicular.

6. An obtuse-angled triangle is one which has an obtuse angle.

24. There are three kinds of QUADRILATERALS:

1. The trapezium, which has none of

its sides parallel.

2. The trapezoid, which has only two of its sides parallel.

3. The parallelogram, which has its

25. There are four kinds of PARALLELOGRAMS:

1. The rhomboid, which has no right

angle.

2. The rhombus, or lozenge, which is an equilateral rhomboid.

3. The rectangle, which is an equiangular parallelogram, but not equilateral.

4. The square, which is both equilateral and equiangular.

A DIAGONAL of a figure is a line which joins the vertices of two angles not adjacent.

EXPLANATION OF SIGNS.

26. The sign is the sign of equality; thus, the expression A = B, signifies that A is equal to B.

27. To signify that A is smaller than B, the expression AB is used.

28. To signify that A is greater than B, the expression A> B is used; the smaller quantity being always at the vertex of the angle.

30. The sign is called minus; it indicates subtraction:

Thus, AB, represents the sum of the quantities A and B; A-B represents their difference, or what remains after B is taken from A; and A- B+C, or A+ C− B, signifies that A and C are to be added together, and that B is to be subtracted from their sum.

31. The sign X indicates multiplication: thus A× B represents the product of A and B.

The expression A× (B+ C − D) represents the product of A by the quantity B+C-D. If A+D were to be multiplied by A- B+C, the product would be indicated thus;

whatever is enclosed within the curved lines, being considered as a single quantity. The same thing may also be

indicated by a bar: thus,

A+B+CXD,

denotes that the sum of A, B and C, is to be multiplied by D.

32. A figure placed before a line, or quantity, serves as a multiplier to that line or quantity; thus, 3AB signifies that the line AB is taken three times; A signifies the half of the angle A.

33. The square of the line AB is designated by AB ; its cube by AB3. What is meant by the square and cube of a line is fully explained in Geometry.

34. The sign indicates a root to be extracted; thus, √2 means the square-root of 2; √AX B means the square

GEOMETRICAL CONSTRUCTIONS.

35. Before explaining the method of constructing geometrical problems, we shall describe some of the simpler instruments and their uses.

DIVIDERS.

36. The dividers is the most simple and useful of the instruments used for drawing. It consists of two legs ba, bc, which may be easily turned around a joint at b.

One of the principal uses of this instrument is to lay off on a line, a distance equal to a given line.

For example, to lay off on CD a distance equal to AB. For this purpose, place the forefin

ger on the joint of the dividers, and A

set one foot at A: then extend, with

the thumb and other fingers, the CL

E

B

D

other leg of the dividers, until its foot reaches the point B. Then raise the dividers, place one foot at C, and mark with the other the distance CE: this will evidently be equal to AB.

RULER AND TRIANGLE.

37. A Ruler of convenient size, is about twenty inches

ness. It should be made of a hard material, perfectly straight and smooth.

The hypothenuse of the right-angled triangle, which is used in connection with it, should be about ten inches in length, and it is most convenient to have one of the sides considerably longer than the other. We can solve, with the ruler and triangle, the two following problems.

I. To draw through a given point a line which shall be parallel to a given line.

C

38. Let C be the given point, and AB the given line. Place the hypothenuse of the triangle against the edge of the ruler, and then place the ruler and triangle on the paper, so that one of the sides of the triangle shall coincide exactly with AB: the triangle being below the line.

A

B

Then placing the thumb and fingers of the left hand firmly on the ruler, slide the triangle with the other hand along the ruler until the side which coincided with AB reaches the point C. Leaving the thumb of the left hand on the ruler, extend the fingers upon the triangle and hold it firmly, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB.

II. To draw through a given point a line which shall be perpendicular to a given line.

A

D

B

39. Let AB be the given line, and D the given point. Place the hypothenuse of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that one of the sides of the triangle shall coincide exactly with the line AB. Then slide the triangle along the ruler until the other side reaches the point D: draw through D a right line,