17. A right angled triangle is one which has a right angle.

18. In a right angled triangle the side opposite the right angle is called the hypothenuse.

If, for example, the angle A is right, the A side BC is the hypothenuse.

B

Any side of a triangle may be considered as its base, but it is usual, in the case of the isosceles triangle, to confine this term

to that side which is not equal to either of the others.

19. A rhomboid or parallelogram is a quadrilateral whose opposite sides are parallel.

20. If only two of the opposite sides are parallel, the figure is a trapezium.

21. A rhombus is a rhomboid, two of whose adjacent, sides are equal.

22. A rectangle is a rhomboid having a right angle.

23. And a square is a rhombus having a right angle.

24. The straight line which joins the vertices of two opposite angles of a quadrilateral, is called a diagonal.

Thus the line AC joining the vertices of the opposite angles DAB,DCB of the quadrilateral ABCD, is a diagonal.

D

25. 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 are at the same time incapable of such coincidence.

AXIOMS-POSTULATES.

An Axiom is a self-evident truth.

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A Postulate requires us to admit the possibility of an ope

ration.

A Theorem is a truth, the evidence of which depends upon a train of reasoning.

The reasoning by which a truth is established, is called a demonstration. It is a direct demonstration when the truth is inferred directly from the premises as the conclusion of a regular series of inductions. The demonstration is indirect when the conclusion shows that the introduction of any supposition, contrary to the truth advanced, necessarily leads to an absurdity.

A Problem proposes an operation to be performed.

A Lemma is a subsidiary truth, the evidence of which must be established preparatory to the demonstration of a succeeding theorem.

A Proposition is a general term for either a theorem, a problem, or a lemma.

A Corollary is an 'obvious consequence, resulting from a demonstration.

An Hypothesis is a supposition, and may be either true or false.

A Scholium is a remark subjoined to a demonstration.

AXIOMS.

1. Magnitudes which are equal to the same, are equal to each other.

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

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

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

5. But if equals be either added to, or taken from, unequals, the results will be unequal.

6. The whole is greater than a part.

7. The whole is equal to the sum of the parts into which it is divided.

POSTULATES.

1. Grant that a straight line may be drawn from one point to another.

2. And that it may be either increased till it be equal to a greater straight line, or diminished till it be equal to a less. 3. Grant also, that an angle may be increased till it be equal to a greater angle, or diminished till it be equal to a less.

4. And lastly, that from a point either within or without a straight line, a perpendicular thereto may be drawn.

It is necessary to remark, that in the first eight books of these elements, the lines concerned in each proposition are supposed to be all situated in the same plane.

PROPOSITION I. THEOREM.

From the same point, in a given straight line, more than one perpendicular thereto cannot be drawn.

Let BD be perpendicular to the straight line AB, or AC, BC being the production of AB, and if the truth of the theorem be denied, let some other line, as BE drawn from the same point B, be also perpendicular to AC.

Then because the angles ABD, CBD are equal, (Def. 3.) the angle ABD must be greater than the angle EBC (Ax. 6.). But BE is perpendicular to AC, by hypothesis, therefore the A angle EBC must be equal to the angle ABE

B

K

(Def. 3.). It follows, therefore, that the angle ABD is greater than the angle ABE, a manifest absurdity; therefore BE cannot be perpendicular to AC.

PROPOSITION II. THEOREM.

All right angles are equal to each other.

Let ABC be a right angle, and DEF any other right angle, then, if it be denied that these two angles are equal to each other, one of them, as ABC, must be supposed greater than the other, so that DEF must be equal to some portion of ABC. Let ABƒ represent that portion, then, because ABƒ is a right angle, Bf is perpendicular to AB (Def. 4.); but ABC is also a right angle, therefore BC is likewise perpendicular to AB, that is from the

B D

same point B in the straight line AB, two perpendiculars thereto are drawn, which is impossible. (Prop. I.) Therefore ABC cannot be greater than DEF, and in a similar manner it may be proved that DEF cannot be greater than ABC; the two angles are, therefore, equal.

PROPOSITION III. THEOREM.

The adjacent angles which one straight line makes with another which it meets, are together equal to two right angles.

Let the straight line DB meet AC in B, making adjacent angles ABD, CBD; these angles shall together be equal to two right angles.

For, let BE be perpendicular to ABC, then the angle ABD is equal to the right angle ABE, together with the angle EBD, and this angle EBD, together with DBC, make up the other right angle EBC; consequently the sum

A

of the angles ABD, CBD, is equal to two right angles. Corollary 1. Hence, if either side of a right angle be produced through the vertex, the adjacent angle formed will be right.

Cor. 2. Therefore the sides of a right angle are mutually perpendicular.

Cor. 3. The sum of all the angles formed by straight lines drawn on the same side of another straight line from any point in it, is equal to two right angles; for, be these angles ever so numerous, they are evidently only subdivisions of the two right angles, which a perpendicular from the point forms with the adjacent portions of the line.

PROPOSITION IV. THEOREM. (Converse of Prop. III.) If, to the point where two straight lines meet, a third be drawn, making with them adjacent angles, which are together equal to two right angles; the two lines so meeting form but one continued straight line.

Let the two straight lines AB, CB, meet in the point B, to which let a third DB be drawn, so that the adjacent angles DBA, DBC, may together be equal to two right angles, then will ABC be one straight line.

For, if it be denied, let BF, and not BC, be the continuation of AB; then the angles ABD, FBD, are together equal to two right angles (Prop. III.). But the angles A ABD, CBD, are together also equal to two

B