areas of the squares constructed upon the sides AB, ab, or as the areas of the squares upon the sides BC, bc, &c. For, by drawing in the polygon ABCDEF, the diagonals AC, AD, AE, and in the polygon abcdef, the corresponding diagonals, ac, ad, ae, the triangle ACB, is similar to the triangle abc, the triangle ACD, to the triangle acd, &c.; because, if the whole polygons ABCDEF, abcdef, are similar, their similarly disposed parts must also be similar; and the same proportion which exists between their parts, must necessarily exist also between the whole polygons; consequently, as the areas of the triangles ABC, abc, ACD, acd, &c., are in the ratio of the areas of the squares constructed upon their corresponding sides, the whole polygons must be in the same ratio, which may be expressed thus:

Polygon ABCDEF : polygon abcdef

= = AB × AB : ab × ab.

*

RECAPITULATION OF THE TRUTHS IN THE THIRD SECTION.

Ques. 1. How do you determine the length of a line? 2. How do you find out which of two lines is the greater?

3. How can you measure a surface?

4. What do you call the area of a surface?

5. If you take one of the sides of a triangle for the basis, how do you determine the height of the triangle? 6. How is the height of a parallelogram determined? How that of a rectangle? A rhombus ? A square? 7. When do you call a triangle equal to a square? to a parallelogram? to a rectangle, &c. ?

8. When can you call two geometrical figures equal to one another, though these figures do not coincide with each other?

9. Can you repeat the different principles respecting the areas of geometrical figures, which you have learned in this section?

Ans. 1. The area of a rectangle is found by multiplying its basis, given in miles, rods, feet, inches, &c., by its height expressed in units of the same kind.

2. The area of a square is found by multiplying one of its sides by itself.

3. If a parallelogram stands on the same basis as a rectangle, and has its height equal to the height of that rectangle, the area of the parallelogram is equal to the area of the rectangle?

4. The areas of all parallelograms, which have equal bases and heights, are equal to one another.

5. Parallelograms upon equal bases, and between the same parallels, are equal to one another.

6. The area of a parallelogram is found by multiplying the basis given in rods, feet, inches, &c., by the height, expressed in units of the same kind.

7. The area of a rhombus or lozenge is found like that of a parallelogram.

8. The areas of parallelograms are to each other, as the products obtained by multiplying the bases of the parallelograms by their heights.

9. Rectangles, or parallelograms which have equal bases, are to each other as their heights.

10. Rectangles, or parallelograms which have equal heights, are to each other as their bases.

11. If two triangles stand on the same basis, and have equal heights, their areas are equal to one another.

12. Every triangle is half of a parallelogram upon equal basis and of the same height.

13. The area of a triangle is half of the area of a parallelogram upon equal basis and of the same height; and, therefore, the area of a triangle is found by multiplying the length of its basis by its height, and dividing the product by 2.

14. The areas of triangles upon the same basis, and between the same parallels, are equal.

15. The areas of triangles are to each other, as the products of their bases by their heights.

16. The areas of triangles upon equal bases are to each other, as the heights of the triangles.

17. The areas of triangles, which have equal heights, are to each other, as their bases.

18. The area of a trapezoid is found by multiplying half the sum of the two parallel sides, by their distance.

19. The area of any rectilinear figure, terminated by any number of sides, is found by dividing that figure, either by diagonals or by any other means, into triangles, and then adding the areas of these triangles together.

20. If, upon each of the three sides of a right-angled triangle, a square is constructed, the square upon the hypothenuse equals, in area, the two squares constructed upon the two sides, which include the right angle.

21. The bases of similar triangles are to each other, as the heights of the triangles.

22. The areas of similar triangles are to each other, as the areas of the squares upon the corresponding sides. 23. The areas of similar polygons are to each other, as the squares constructed upon the corresponding sides.

SECTION IV.

OF THE PROPERTIES OF THE CIRCLE.*

QUERY I.

In how many points can a straight line, CD, meet the circumference of a circle?

there is but one point in the line CD, on each side of the perpendicular, such, that a line, drawn from it to the point O of the perpendicular, has the length of the radius ON. (Page 46, 6thly.)

In what cases do the circumferences of two circles cut each other?

A. When the distance,

OP, between their centres, O and P, is less

QUERY II.

than the sum of their radii, OM, PM.

M

* Before entering on this section, the teacher ought to recapitulate with his pupils the definitions of a circle, of an arc, of a chord, a segment, &c.