you had put the heights CM, DO, instead of the bases AB, AE? *

A. The proportion would have been changed

into

Triangle ABC : triangle AED = AM × AM : DO X DO;

that is, the areas of similar triangles are to each other, as the areas of the squares upon the heights of the triangles.

QUERY VIII.

From the ratio which you have proved to exist between the areas of similar triangles, can you now find out the ratio which exists between the areas of similar polygons? (See Definitions.)

A. Yes. The areas of similar polygons are to each other, as the areas of the squares constructed upon the corresponding sides. The areas of the two similar polygons ABCDEF,

* As the heights of similar triangles are in the same ratio as the bases, you can at pleasure, use one ratio for the other.

abcdef, for instance, are to each other, as the areas of the squares constructed upon the sides AB, a b, are to the areas of 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 a c, a d, ae, the triangle ABC, will be similar to the triangle abc, the triangle ACD, similar to the triangle a cd, the triangle ADE similar to the triangle a de, &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 between the whole polygons; and as the areas of the triangles are in the ratio of the areas of the squares constructed upon the corresponding sides, the whole polygons must be in the same ratio, which may be expressed thus:

Polygon ABCDEF: polygon a b c d e f = AB X AB: a b x ab.

*

*

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RECAPITULATION OF THE TRUTHS IN THE THIRD SECTION.

Question 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 do 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, in general, call two geometrical figures equal to one another, though these figures do not coincide with each other?

9. Will you now 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 the length of the basis, given in miles, rods, feet, inches, &c. by the 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 areas of parallelograms are to each other, as the products obtained by multiplying the length of the bases of the parallelograms by their heights.

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

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

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

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

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

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

14. The areas of triangles are to each other, as the products obtained by multiplying the length of their bases by their heights.

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

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

17. The area of a trapesoid is found by multiplying the sum of the two parallel sides by their distance.

18. 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.

19. If upon each of the three sides of a rightangular 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.

20. The areas of similar triangles are to each other, as the squares constructed upon the sides opposite to the equal angles, and also as the squares upon the heights of the triangles.

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