26. A quadrilateral, in which two sides are equal and parallel, is a parallelogram.

27. If from one of the vertices of a rectilinear figure, diagonals are drawn to all the other vertices, the figure is divided into as many triangles as it has sides less two.

28. The sum of all the angles in a rectilinear figure, is equal to as many times two right angles as the figure has sides less two.

RECAPITULATION OF THE TRUTHS CONTAINED IN PART II.

1. On Proportions.

Ques. 1. How is a geometrical ratio determined? Q. 2. What is the ratio of a line 3 inches in length, to a line of 12 inches? What, the ratio of a line 2 inches in length, to one of 10 inches, &c.?

Q. 3. When two geometrical ratios are equal to one another, what do they form?

Q. 4. What is a geometrical proportion?

Q. 5. What signs are used to express a geometrical proportion?

Q. 6. What sign is put between the two terms of a ratio?

Q. 7. What sign is put between the two ratios of a proportion?

Q. 8. What are the first and fourth terms of a geometrical proportion called?

Q. 9. What are the second and third terms of a geometrical proportion called?

Q. 10. What are the most remarkable properties of geometrical proportions?

Ans. a. In every geometrical proportion the two ratios may be inverted,

b. In every geometrical proportion the order of the means or extremes may be inverted.

c. If two geometrical proportions have a ratio common, the two remaining ratios make again a proportion.

d. If you have several geometrical proportions, of which the second has a ratio common with the first, the third a ratio common with the second, the fourth a ratio common with the third, &c., the sum of all the first terms will be in the same ratio to the sum of all the second terms, as the sum of all the third terms is to the sum of all the fourth terms; that is, the sums make again a proportion.

e. The second term of a proportion being added once, or any number of times, to the first term, and the fourth term the same number of times to the third term, they will still be in proportion; and in the same manner can the first term be added a number of times to the second term, and the third the same number of times to the fourth term, without destroying the proportion.

f. The second term may also be once, or any number of times, subtracted from the first term, and the fourth from the third term, without destroying the proportion; or the first term may also be subtracted from the second, and the third from the fourth-and the result will still be a geometrical proportion.

g. If all the terms of a geometrical proportion are multiplied or divided by the same number, the proportion remains unaltered.

h. From three terms of a geometrical proportion the fourth term can be found.

i. If four lines are together in a geometrical propor

tion, their lengths, expressed in numbers of rods, feet, inches, &c., are in the same proportion.

k. In every geometrical proportion, the product of the two mean terms is equal to that of the two extremes.

7. When the two mean terms of a geometrical proport tion are equal to each other, either of them is called a mean proportional between the two extremes.

QUESTIONS ON SIMILARITY OF TRIANGLES.

Ques. What other principles do you recollect in the second part of the second section?

Ans. 1. If one side of a triangle is divided into any number of equal parts, and then, from the points of division, lines are drawn parallel to one of the two other sides, the side opposite to the one that has been divided will, by these parallels, be divided into as many equal parts as the first side.

2. If, in a triangle, a line is drawn parallel to one of the sides, that parallel divides the two other sides into such parts as are in proportion to each other and to the whole of the two sides themselves; and the reverse of this principle is also true; namely, a line must be parallel to one of the sides of a triangle, if it divides the two other sides proportionally.

3. If, in a triangle, a line is drawn parallel to one of the sides, the triangle which is cut off by it is similar to the whole triangle.

4. If the three angles in one triangle are equal to the three angles in another triangle, each to each, the two triangles are similar to one another; and the same is the case if only two angles in one triangle are equal to two angles in another, each to each.

5. If an angle in one triangle is equal to an angle in another, and the two sides which include that angle in the one triangle are in proportion to the two sides which include the equal angle in the other, these two triangles are similar to each other.

6. If the three sides of one triangle are in proportion to the three sides of another, the two triangles are similar to each other.*

7. If, in a right-angled triangle, a perpendicular is let fall from the vertex of the right angle upon the hypothenuse, that perpendicular divides the whole of the triangle into two parts, which are similar to the whole triangle, and to each other.

8. The perpendicular let fall from the vertex of a rightangled triangle upon the hypothenuse, is a mean proportional between the parts into which it divides the hypothenuse.

9. In every right-angled triangle, each of the sides which include the right angle is a mean proportional between the hypothenuse and that part of it, which lies between the extremity of that side and the foot of the perpendicular let fall from the vertex of the right angle upon the hypothenuse.

* The teacher will do well to let the pupil repeat the different cases where two triangles are similar to each other. (Page 73.)

SECTION III.

OF THE MEASUREMENT OF SURFACES.

Preliminary Remarks. We determine the length of a line, by finding how many times another line, which we take for the measure, is contained in it. The line which we take for the measure is chosen at pleasure; it may be an inch, a foot, a fathom, a mile, &c. If we have a line upon which we can take the length of an inch 3 times, we say that line measures 3 inches, or is 3 inches long. In like manner, if we have a line upon which we can take the length of a fathom 3 times, we call that line 3 fathoms, &c. To find out which of two lines is the greater, we must measure them. If we take an inch for our measure, that line is the greater, which contains the greater number of inches. If we take a foot for our measure, that line is the greater, which contains the greater number of feet, &c.

To measure the extension of a surface, we make use of another surface, commonly a square (□), and see how many times it can be applied to it; or, in other words, how many of those squares it takes to cover the whole surface. The length of the square side is arbitrary. If it is an inch, the square of it is called a square inch; if it is a foot, a square foot; if it is a mile, a square mile, &c. The extension of a surface, expressed in numbers of square miles, rods, feet, inches, &c., is called its area.

Remark 2. If we take one of the sides of a triangle for the basis, the perpendicular let fall from the vertex of the opposite angle, upon that side, is called the altitude or height of the triangle.

If, in the triangle ABC, (Fig. I.) for instance, we call AC the basis, the perpendicular BD will be its height. If the perpendicular BD should fall without the triangle ABC (as in Fig. II.),

Fig. I.

B

A

D C

Fig. II.

B

A

D