23. Plane figures are equal, when, by supposing them to be applied to each other, they would coincide throughout; and they are said to be equivalent, when they enclose equal portions of space, and at the same time are incapable of such coincidence.

Definition of Terms employed.

An ariom is a self-evident proposition.

A theorem is a proposition which requires demonstration.

A problem proposes an operation to be performed.

A corollary is a consequence obviously resulting from a demonstration.

An hypothesis is a supposition.

A scholium is a remark subjoined to a demonstration.

Axioms.

1. Things which are equal to the same thing, are equal to each other.

2. Things which are double, triple, &c. of the same or equal things, are equal to each other.

3. Things which are the one half, one third, &c. of the same or equal things, are equal to each other.

4. If equals be added to, or taken from equals, the results will be equal.

5. If equals be added to, or taken from unequals, the results will be unequal.

6. The whole is greater than a part.

Explanation of Signs.

1. The sign=signifies equality; thus A=B signifies that the quantity represented by A is equal to that represented by B; and is read, A equal to B.

2. To represent that A is less than B, the expression A<B is used, and is read, A less than B.

3. The sign is called plus, and signifies addition. 4. The sign is called minus, and signifies subtraction. Thus, A+B signifies the sum of the two quantities A and B, and is read, A plus B.

A-B signifies the difference of these quantities, and is read, A minus B.

5. A dot. signifies that the quantities between which it is placed are multiplied together; thus A.B signifies that A is multiplied by B.

6. The expressions A (B+C), A(B+C+D), signify that, in the first, A is multiplied into the sum of the quantities B and C ; and in the second, that A is multiplied into the sum of the quantities B, C, and D.

7. The figure 2 placed above and at the right hand of a quantity, signifies that the quantity is multiplied by itself; thus, A2 represents that the quantity A is multiplied by itself; it is read, A square.

PROPOSITION I. THEOREM.

When one straight line meets another straight line, the sum of the adjacent angles is equal to two right angles. Let the straight line DC meet the line AB at C;

meet, the line CE perpendicular to AB; the angles ACE, ECB, will be right angles (Def. 6), and their sum, consequently, two right angles; but the opening between the lines AC and CD, which is the angle ACD, added to the opening between the lines DC and CB, which is the angle DCB, evidently makes the same amount of opening as that between the lines AC and CE, added to that between EC and CB, being in both cases the whole amount of opening between the parts AC and CB of the line AB; now the angle ACE+ECB we have seen is equal to two right angles, hence ACD+DCB must be equal to two right angles.

equal to two right angles. For it is evident that the sum of all the openings between the lines drawn in the figure, and indeed between any number of lines drawn from the point C, on the same side of AB, is the same as the opening between any one of the lines, as CE, for exam ple, and the parts AC and CB of the line AB; bu the sum of these two angles, we learned above, is equal

to two right angles; hence the sum of all the angles must also be equal to two right angles.

Cor. 2. The sum of all the angles formed on both sides the line A B, by lines meeting at C, is equal to four right angles, since those formed on each side are equal to two right angles.

Cor. 3. If one of the angles, formed by one straight line meeting another, is a right angle, the other angle must be also a right angle.

PROP. II. THEOREM.

If two straight lines meet, and, at the point of their meeting, a third be drawn so as to make with them adjacent angles, which, taken together, are equal to two right angles, the two first lines will form but one continued straight line.

Let the two straight lines, AB, CB, meet in the point B, and let a third line DB be drawn, so that the sum of the adjacent angles ABD, DBC, shall be A

equal to two right an

B

Fig. 3.

D

F

gles; then we have to prove that AB, BC, form one continued straight line, that is, that BC is the continuation of AB. If it is not, let BF be the continuation of AB, so that ABF shall form one continued straight line; then (by Prop. 1) the sum of the angles ABD, DBF, is equal to two right angles; but by the supposition ABD+DBC is equal to two right angles;

therefore ABD+DBF is equal to ABD+DBC; now take away ABD from each of these equals, and DBF will be equal to DBC (Ax. 4); but DBF is only a part of DBC, and cannot be equal to it (Ax. 6), therefore BF cannot be a continuation of AB; and as the same absurdity would arise from taking any line except BC as the continuation of AB, it follows that BC must be that continuation.

PROP. III. THEOREM.

If two straight lines intersect each other, the opposite angles, formed at their intersection, will be equal.

If the lines AB, CD, intersect at E, we have to prove that the opposite angles CEA, BED, are equal.

On account of the straight line CE meeting AB, the angles AEC+CEB are e

Fig. 4.

E

B

D

qual to two right angles (Prop. 1); and on account of the straight line BE meeting CD, the angles CEB +BED are equal to two right angles; taking CEB from each of the equals, there remains CEA=BED (Ax. 4.) In a similar manner it may be shown that CEB-AED.

PROP. IV. THEOREM.

If two sides and the included angle, in one triangle, be respectively equal to the two sides and included angle in another triangle, these two triangles will be equal. (Def. 23.)