ered 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.

7. 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.

3

8. 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, will be explained in its proper place.

9. The sign ✔ indicates a root to be extracted; thus √2 means the square-root of 2; √AXB means the square-root of the product of A and B.

AXIOMS.

1. Things which are equal to the same thing, are equal to one another.

2. If equals be added to equals, the wholes will be equal. Untrue in the scomitus siuse of Equal

3. If equals be taken from equals, the remainders will be equal.

4. If equals be added to unequals, the wholes will be unequal.

5. If equals be taken from unequals, the remainders will be unequal.

6. Things which are doubles of equal things, are equal to each other.

7. Things which are halves of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. The whole is equal to the sum of all its parts.

10 All right angles are equal to each other.

11. From one point to another only one straight line can be drawn.

12. A straight line is the shortest distance between two points.

13. Through the same point, only one straight line can be drawn which shall be parallel to a given line.

14. Magnitudes, which being applied the one to the other, coincide throughout their whole extent, are equal

POSTULATES.

1. Let it be granted, that a straight line may be drawn from one point to another point.

2. That a terminated straight line may be prolonged, in a straight line, to any length.

3. That if two straight lines are unequal, the length of the less may always be laid off on the greater.

4. That a given straight line may be bisected: that is, divided into two equal parts.

5. That a straight line may bisect a given angle.

6. That a perpendicular may be drawn to a given straight line, either from a point without the line, or at a point of a line.

7. That a straight line may be drawn, making with a given straight line, an angle equal to a given angle.

+

PROPOSITION I. THEOREM.

If one straight line meet another straight line, the sum of the two adjacent angles will be equal to two right angles.

Let the straight line DC meet the straight line AB at C; then will the angle ACD plus the angle DCB, be equal to two right angles.

=

At the point C suppose CE to be drawn perpendicular to AB: then, ACE + ECB two right angles (D. 13).* But ECB is equal to ECD + DCB (A. 9): hence, ACE + ECD + DCB = two right angles. But ACE + ECD fore, ACD + DCB = two right angles.

=

E

D

-B

ACD (A. 9): there

Cor. 1. If one of the angles ACD or DCB, is a right angle, the other will also be a right angle.

right angle (D. 13). But since AC meets DE at the point C, making one angle ACD a right angle, the adjacent angle ACE will also be a right angle (c. 1). Therefore, AB is perpendicular to DE (D. 13).

Cor. 3. The sum of the successive angles BAC, CAD, DAE, EAF, formed on the same side of the line BF, is equal to two right angles; for, their sum is equal to that of the two adjacent angles BAC and CAF.

B

D

E

-F

A

*In the references, A. stands for Axiom-D. for Definition-B. for Book--P. for Proposition-C. for Corollary-S. for Scholium, and Prob. for Problem.

PROPOSITION II. THEOREM.

Two straight lines, which have two points common, coincide the one with the other, throughout their whole extent, and form one and the same straight line.

Let A and B be the two common points of two straigh lines.

In the first place, the two lines will coincide between the points A and B; for, otherwise there would be two straight lines between A and AB, which is impossible (A. 11).

B

F

-D

Suppose, however, that in being prolonged, these lines begin to separate at some point, as C, the one becoming CD, the other, CE. At the point C, suppose CF to be drawn, making with AC, the right angle ACF.

Now, since ACD is a straight line, the angle FCD will be a right angle (P. I., C. 1): and since ACE is a straight line, the angle FCE will also be a right angle. Hence, the angle FCD is equal to the angle FCE (A. 10): that is, a whole is equal to one of its parts, which is impossible (A. 8): therefore the two straight lines which have two points, A and B, in common, cannot separate at any point, when prolonged; hence, they form one and the same straight line.*

PROPOSITION III. THEOREM.

If, when a straight line meets two other straight lines at a common point, the sum of the two adjacent angles which it makes with them, is equal to two right angles, the two straight lines which are met, form one and the same straight line.

Let the straight line CD meet the two lines AC, CB, at their common point C, and let the sum of the two adjacent angles, DCA, DCB, be equal to two right angles: then

* See Note A. It is earnestly recommended to every pupil to read and understand this Note. Also, see Logic and Utility of Mathematics, § 262.

will CB be the prolongation of AC; or, AC and CB will form one and the same straight line.

For, if CB is not the prolongation of AC, let CE be that prolongation. Then the line ACE being straight, the sum of the angles ACD, DCE, will be equal to two right angles (P. 1). But by hypothesis, the sum of the angles ACD, DCB, is also equal to two right angles: therefore (A. 1),

Ꭰ

A

B

E

ACD+DCE must be equal to ACD+DCB.

Taking away the angle ACD from each, there remains the angle DCE equal to the angle DCB: that is, a whole equal to a part, which is impossible (A. 8): therefore, A and CB form one and the same straight line.

PROPOSITION IV. THEOREM.

When two straight lines intersect each other, the opposite or vertical angles, which they form, are equal.

Let AB and DE be two straight lines, intersecting each other at C; then will the angle ECB be equal to the angle ACD, and the angle ACE to the angle DCB.

For, since the straight line DE A

is met by the straight line AC, the sum of the angles ACE, ACD,

E

is equal to two right angles (P. I.); D

B

and since the straight line AB is

met by the straight line EC, the sum of the angles ACE, and ECB, is equal to two right angles: hence (a. 1),

ACE+ACD is equal to ACE+ECB.

Take away from both, the common angle ACE, there remains (A. 3) the angle ACD, equal to its opposite or vertical angle ECB. In a similar manner it may be proved that ACE is equal to DCB.

Scholium. The four angles formed about a point by two straight lines, which intersect each other, are together equal