from one of these to the other, the distance can neither have been increasing nor diminishing, for that, had either been the case, since the lines are straight, the distance of their corresponding points would have continued to increase or to diminish. Should this, however, as we can easily imagine, fail to satisfy the student, we must refer him to measurement for such a degree of conviction as it can afford.

In fact, although it may be shown without difficulty, that certain straight lines will never meet one another; the converse, viz., that straight lines, which never meet one another, must have certain properties, has never been strictly demonstrated. It is agreed by Geometrical writers that some assumption is indispensable.*

The following is, perhaps, as simple as any that can be proposed, while it has the advantage also of not being many steps distant from the proposition in question.

"If from two points of one straight line to another, there fall two unequal perpendiculars, the straight lines will meet one another, if produced, upon the side of the lesser perpendicular."

Hence, if two straight lines be parallel, the perpendiculars drawn from the points of the one to the other, must all

either of them.

A modern geometer of great celebrity, M. Legendre, has, after more than one alteration, suggested, finally, an experimental proof not very different from that which is here adopted, as best suited to an Elementary Treatise. He is, notwithstanding, of opinion, that the grand truth with which it is so intimately connected, viz., that "the three angles of a triangle are together equal to two right angles," may be referred to the general principle of homogeneity. Of this it is unnecessary to say more in this place, than that it teaches us in the present instance, that the angles of a triangle depend, not upon the absolute magnitude of its sides, but upon their relative magnitude; so that a triangle whose sides are 3, 4, and 5 times some given line will have the same angles whether the given line be

an inch or a mile; a truth which, indeed, seems to be nearly related to the more simple truth, that an angle is not increased by producing the sides which contain it, and leads directly to the theory of parallel straight lines. But, though the principle be of extensive application, the reasoning by which it is established has been shown to be incomplete, and such as, if great circumspection be not used, may even lead to fallacies.

of them be equal; and hence, if a straight line be drawn at right angles to one of two parallels, it may easily be shown to cut the other at angles, which are equal to one another, that is, at right angles.

It is demonstrated in Prop. 16., that, if two straight lines be parallel, the perpendiculars drawn from the points of the one to the other must all of them be equal but that demonstration itself rests upon the converse part of Prop. 14., which is here in question. The reader must not imagine, therefore, that the above assumption is at all assisted by that demonstration.

PROP. 15. (EUc. i. 27, 28 and 29.) Straight lines which make equal angles with the same straight line, towards the same parts, are parallel: and, conversely, if two parallel straight lines be cut by the same straight line, they shall make equal angles with it towards the same parts.

Let the straight lines A B, CD make equal angles BEG, DFG, with the same straight line E F, towards the same parts: AB shall be parallel to CD..

TE

H

Bisect E F in H (Post. 3.), from H draw HK perpendicular to CD (12.), and produce KH to meet A B in L. Then because the angle HEL is equal. to BE G (3.), and that B E G is equal to DFG or HFK, the angle HEL is equal to the angle H FK (ax. 1.). The vertical angles EH L, FHK are likewise equal to one another (3.). Therefore, the triangles HEL and HFK having two angles of the one equal to two angles of the other, each to each, and their sides H E, H F, which lie between the equal angles, also equal to one another, are equal in every respect (5.). Therefore, the angle H LÊ is equal to the angle H K F, that is, to a right angle. But straight lines which are at right angles to the same straight line KL are parallel (14.). Therefore, A B is parallel to C D.

Next let A B be parallel to C D, and let them be cut by the same straight line GEF; the angles BEG, D FG, which are towards the same parts, shall be equal to one another.

Bisect E F, and draw the straight line I. K, as before. Then, because LK is at right angles to C D, and that A B is parallel to CD, L K is at right angles also to AB (14.). And, because, in the right-angled triangles H EL, H FK, the hypotenuse HE, and adjacent angle EHL of the one, are equal to the hypotenuse HF, and adjacent angle FHK (3.) of the other, they are equal in every respect (13.). Therefore, the angle HEL is equal to the angle HFK, and B E G, which is equal to HEL (3.), is equal to D F G.* Therefore, &c.

Cor. 1. When two straight lines A B, CD, are cut by a third, E F, the angle BE G, is called the exterior angle, and the angle D FG, the interior and opposite angle upon the same side of the line. Therefore, if one straight line fall upon two other straight lines, so as to make the exterior angle equal to the interior and opposite upon the same side, those two straight lines shall be parallel: and conversely.

Cor. 2. The angles A E F, E FD, are called alternate angles. And A E F is always equal to BEG (3.). Therefore, if one straight line fall upon two other straight lines so as to make the two alternate angles equal to one another, those two straight lines shall be parallel: and conversely.

Cor. 3. The angles B E F, and DFE, are called interior angles on the same side of the line. Now, when BEG is equal to DFG, or DFE, the angles BEF and DFE are together equal to two right angles, because BEF and BEG, are together equal to two right angles (2.). Therefore, if one straight line, falling upon two other straight lines, make the two interior angles, on the same side, together equal to two right angles, the two straight lines shall be parallel: and conversely.

Cor. 4. (Euc. i. ax. 12.) If one straight line, falling upon two other straight lines, make the two interior angles, on the same side, together less than two right angles,

* A shorter demonstration may be had, by considering that if the angles BEG, DFG, be equal

to one another, the two interior angles upon each side of E F will be together equal to two right angles; and, therefore, cannot be two angles of a triangle (8.), that is A B, CD cannot meet upon either side of EF; and hence the converse, because (14. Cor. 2.) only one parallel can be drawn through the same point to the same straight line. That which is given in the text, however, seems preferable, as pointing out the connexion of the proposition with Prop. 14., which immediately precedes it.

the two straight lines shall meet upon that side, if produced far enough.

PROP. 16.

Parallel straight lines are every where equidistant; that is, if from any two points of the one, perpendiculars be drawn to the other, those perpendiculars shall be equal to one another. Let AB, C D be two parallel straight lines; and from the points A, B, of AB, let AC, BD be drawn perpendicular to CD (12.): AC shall be equal to B D.

B

D

Join B C. Then because AC and B D are perpendicular to the same straight line CD, they are parallel (14.). Therefore, the alternate angles ACB, DBC, are equal to one another (15. Cor. 2.). Again, because A B is parallel to CD, the alternate angles ABC, DCB are equal to one another (15. Cor. 2.). Therefore the triangles A B C, D C B, having two angles of the one equal to two angles of the other, each to each, and the same line, BC, lying between the equal angles, are equal in every respect (5.). Therefore, AC is equal to B D. Therefore, &c.

Cor. It appears from the demonstration, that if à C be only parallel to BD, AC and B D will be equal to one another. Therefore, the parts of parallel straight lines, which are intercepted by parallel straight lines, are equal to one another.

PROP. 17. (EUc. i. 30.)

Straight lines, which are parallel to the same straight line, are parallel to one another.

PROP. 20. (EUC. i. 32. Corr. 1. and 2.) All the exterior angles of any rectilineal figure are together equal to four right angles: and all the interior angles, together with four right angles, are equal to twice as many right angles as the figure has sides.

For, if from any point in the same plane, straight lines be drawn, one after the other, parallel to the sides of the figure, the angles contained by these straight lines about that point, will be equal to the exterior angles of the figure

(18.), each to each, because their sides are parallel to the sides of the figure. Thus, the angles a, b, c, d, e, are respectively equal to the exterior angles A, B, C, D, E. But the former angles are together (3. Cor.) equal to four right angles; therefore, all the exterior angles of the figure are together equal to four right angles (ax. 1.).

Again, since every interior angle, together with its adjacent exterior angle, is equal to two right angles (2.); all the interior angles, together with all the exterior angles, are equal to twice as many right angles as the figure has angles. But all the exterior angles are, by the former part of the proposition, equal to four right angles; and the figure has as many angles as sides: therefore all the interior angles together with four right angles are equal to twice as many right angles as the figure has sides.

Therefore, &c.

Cor. The four angles of a quadrilateral are together equal to four right angles.

PROP. 22. (EUc. i. 34, first part of.)

The opposite sides and angles of a parallelogram are equal, and its diagonals bisect one another: and, conversely, if, in any quadrilateral figure, the opposite sides be equal; or if the opposite angles be equal; or if the diagonals bisect one another; that quadrilateral shall be a parallelogram.

Let ABCD be a parallelogram (see the last figure), and let its diagonals AC, BD cut one another in the point E: the sides AD, BC, as also, AB, CD, shall be equal to one another the angles A and C, as also B and D shall be equal; and the diagonals A C, BD shall be bisected in E.

For, in the first place, that the opposite sides, as AD and B C, are equal, is evident, because they are parts of parallels intercepted by parallels (16. Cor.). Also, the opposite angles are equal, as at D and B; for the angle at D is equal to the vertical angle formed by CD, AD produced (3.), and the latter to the angle B (18.).

Lastly, with regard to the bisection of the diagonals: because AD is parallel to B C, the two triangles E AD, ECB have the two angles EA D, E DA of the one equal to the two angles EC B, EBC of the other, each to each (15.); and it has been shown, that the interjacent sides AD, B C are equal to one another; therefore, (5.) E A is equal to EC, and ED to EB, that is, AC, BD are bisected in E.

Next, let the opposite sides of the quadrilateral ABCD be equal to one another: it shall be a parallelogram. For, in the triangles A B D, C D B, the three sides of the one are equal to the three sides of the other, each to each; therefore, the angle ABD is equal to CDB (7.), and (15.) A B is parallel to CD. And, for the like reason, A D is parallel to B C.

Or, let the opposite angles be equal: then, because the angles at A and B together are equal to the angles at C and D together, and that the four angles of the quadrilateral (20. Cor.) are equal to four right angles, the angles at A and B are together equal to two right angles, and

(15. Cor. 3.) AD is parallel to B C. And, for the like reason, A B is parallel to C D.

Or, let the diagonals A C, BD, bisect one another in E: then, because the triangles E A D, E C B, have two sides of the one equal to two sides of the other, each to each, and the included angles A ED, CEB, (3.) equal to one another, the angle E A D is (4.) equal to EC B, and, therefore, (15.) AD is parallel to B C. And, for the like reason, A B is parallel to CD.

Therefore, in each of the three cases, the figure is a parallelogram.

Therefore, &c.

Cor. 1. (Euc. i. 34. second part of.) A parallelogram is bisected by each of its diagonals; for the triangles into which it is divided are equal to one another.

Cor. 2. The diagonals of a rhombus bisect one another at right angles. For the triangles into which it is divided by either of its diagonals are isosceles triangles, of which that diagonal is the base (6. Cor. 4.)

Cor. 3. (Euc. i. 46. Cor.) If one angle of a parallelogram be a right angle, all its angles will be right angles.

Cor. 4. By help of this Proposition more complete notions may be acquired of the rhombus, rectangle, and square: for, hence it appears, that a rhombus has all its sides equal to one another; that a rectangle has all its angles right angles; and that a square has all its sides equal, and all its angles right angles.

PROP. 23. (EUc. i. 43.)

The complements of the parallelograms, which are about the diagonals of any parallelogram, are equal to one another.

PROP. 24. (Euc. i. 35.) Parallelograms upon the same base, and between the sume parallels, are equal to one another.

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Let the parallelograms A B C D, EBCF, be upon the same base B C and between the same parallels AF, BC; the parallelogram A B C D shall be equal to the parallelogram EBCF.

Because AD and EF are each of them (22.) equal to BC, they are (ax. 1.) equal to one another. Therefore, the whole or the remainder AE is equal to the whole or the remainder DF (ax. 2. or 3.). Therefore, the two triangles EA B, FDC, having two sides of the one equal to two sides of the other, each to each, and the included angles E AB, FDC equal, are equal to one another (4. Cor.). Therefore, taking each of these equals from the whole figure A B C F, there remains (ax. 3.) the parallelogram EBCF equal to the parallelogram A B C D.

If the points D, E coincide, A E is the same with AD, and DF the same with EF; therefore, A E and D F, being each of them equal to BC, are equal to one another; and, hence, the triangle E AB is equal to the triangle FD C, and the parallelogram EBCF to the parallelogram A B CD, as before.

Therefore, &c.