perpendicular to BD at any point, BE and DE, if producéd will meet. To demonstrate this, it is supposed, that B9, BC are two parts taken in BE, of which BC is greater than BO and that the perpendiculars ON, CL are drawn to BD; then shall BL be greater than BN. For, if not, that is, if the perpendicular CL falls either at N, or between B and N, as at F; in the first of these cases

B

F N

E

the angle CNB is equal to the angle ONB, because they are both right angles, which is impossible; and, in the second, the two angles CFN, CNF of the triangle CNF, exceed two right angles. Therefore, adds our author, since, as BC increases, BL also increases, and since BC may be increased without limit, so BL may become greater than any given line, and therefore may be greater than BD; wherefore, since the perpendiculars to BD from points beyond D meet BC, the perpendicular from D necessarily meets it. Q. E. D,

Now it will be found, on examination, that this reasoning is no more conclusive than the preceding. For, unless it be proved, that whatever multiple BC is of BO, the same is BL of BN, the indefinite increase of BC does not necessarily imply the indefinite increase of BL, or that BL may be made to exceed BD. On the contrary, BL may always increase, and yet may do so in such a manner as never to exceed BD: In order that the demonstration should be conclusive, it would be necessary to show, that when BC increases by a part equal to BO, BL increases always by a part equal to BN; but to do this will be found to require the knowledge of those very properties of parallel lines that we are seeking to demonstrate.

LEGENDRE, in his Elements of Geometry, a work entitled to the highest praise, for elegance and accuracy, has delivered the doctrine of parallel lines without any new Axiom. He has done this in two different ways, one in the text, and the other in the notes. In the former he has endeavoured to prove, independently of the doctrine of parallel lines, that all the angles of a triangle are equal to two right angles; from which proposition, when it is once established, it is not difficult to deduce every thing with respect to parallels. But, though his demonstration of the property of triangles just mentioned is quite logical and conclusive, yet it has the fault of being long and indirect, proving first, that the three angles of a triangle cannot be greater than two right angles, next, that they cannot be less, and doing both by reasonings abundantly subtle, and not of a kind readily apprehended by those who are only beginning to study the Mathematics.

The demonstration which he has given in the notes is extremely in

Pp

genious, and proceeds on this very simple and undeniable Axiom, that we cannot compare an angle and a line, as to magnitude, or cannot have an equation of any sort between them. This truth is involved in the distinction between homogeneous and heterogeneous quantities, (Euc. v. def. 4.) which has long been received in Geometry, but led only to negative consequences, till it fell into the hands of Legendre. The proposition which he deduces from it is, that if wo angles of one triangle be equal to two angles of another, the third angles of these triangles are also equal. For, it is evident, that, when two angles of a triangle are given, and also the side between them, the third angle is thereby determined; so that if A and B be any two angles of a triangle, P the side interjacent, and C the third angle, C is determined, as to its magnitude, by A, B and P; and, besides these, there is no other quantity whatever which can affect the magnitude of C. This is plane, because if A, B and P are given, the triangle can be constructed, all the triangles in which A, B and P are the same, being equal to one another.

But if the quan ities by which C is determined, P cannot he one; for if it were, then C must be a function of the quantities A, B, p; that is to say, the value of C can be expressed by some combination of the quantities A, B and P. An equation, therefore may exist between the quantities A,B, C, and P; and consequently the value of P is equal to some combination, that is, to some function of the quantities A, Band C; but this is impossible, P being a line, and A, B, C being angles so that no function of the first of these quantities can be equal to any function of the other three. The angle C must therefore be determined by the angles A and B alone, without any regard to the magnitude of P the side interjacent. Hence in all triangles that have two angles in one equal to two in another each to each, the third angles are also equal.

Now this being demonstrated, it is easy to prove that the three angles of any triangle are equal to two right angles.

Let ABC be a triangle right angled at A, draw AD perpendicular to BC. The triangles ABD, ABC have the

angles BAC, BDA right angles, and the angle B common to both; therefore, by what has just been proved, their third angles BAD, BCA are also equal. In the same way it is shewn, that CAD is equal to CBA; therefore the two angles BAD, CAD are equal to the two BCA, CBA; but BAD +CAD is equal to a right angle, therefore the angles BCA, CBA are together equal to a right angle, and consequently the three angles of the right angled triangle ABC are equal to two right angles.

B

D

C

And since it is proved that the oblique angles of every right angled triangle are equal to two right angles, and since every triangle may be divided into two right angled triangles, the four oblique angles of which are equal to the three angles of the triangle, therefore the three angles of every triangle are equal to two right angles. Q. E. D.

Though this method of treating the subject is strictly demonstrative, yet, as the reasoning in the first of the two preceding demonstrations is not perhaps sufficiently simple to be apprehended by those just entering on mathematical studies, I shall submit to the reader another method, not liable to the same objection, which I know, from experience, to be of use in explaining the Elements. It proceeds, like that of the French Geometer, by demonstrating, in the first place, that the angles of any triangle are together equal to two right angles and deducing from thence, that two lines, which make with a third line the interior angles, less than two right angles, must meet if produced. The reasoning used to demonstrate the first of these propositions may be objected to by some as involving the idea of motion, and the transference of a line from one place to another. This, however, is no more than Euclid has done himself on some occasions: and when it furnishes so short a road to the truth as in the present instance, and does not impair the evidence of the conclusion, it seems to be in no respect inconsistent with the utmost rigour of demonstration. It is of importance in explaining the Elements of Science, to connect truths by the shortest chain possible; and till that is done, we can never consider them as being placed in their natural order. The reasoning in the first of the following propositions is so simple, that it seems hardly susceptible of abbreviation, and it has the advantage of connecting immediately two truths so much alike, that one might conclude, even from the bare enunciations, that they are but different cases of the same general theorem, viz. That all the angles about a point, and all the exterior angles of any rectilineal figure, are con stantly of the same magnitude, and equal to four right angles.

DEFINITION.

IF, while one extremity of a straight line remains fixed at A, the line itself turns about that point from the position AB to the position AC, it is said to describe the angle BAC contained by the lines AB and AC.

COR. If a line turn about a point from the position AB till it come into the position AB again, it describes angles which are together equal to four right angles. This is evident from the second Cor, to

the 15th.

PROP. I.

All the exterior angles of any rectilineal figure are together equal to four right angles.

1. Let the rectilineal figure be the triangle ABC, of which the exterior angles are DCA, FÅB, GBC; these angles are together equal to four right angles.

F

A

Let the line CD, placed in the direction of BC produced, turn about the point C till it coincide with CE, a part of the side CA, and have described the exterior angle DCE or DCA. Let it then be carried along the line CA, till it be in the position AF, that is in the direction of CA produced, and the point A remaining fixed, let it turn about A till it describe the angle FAB, and eoincide with a part of the line AB. Let it next be carried along AB till it come into the position BG, and by turning about B, let it describe the angle GBC, so as to coincide with a part of BC. Lastly, Let it be carried along BC till it coincide with CD, its first position. Then, because the line CD has turned about one of its extremities till it has

B

E

C

come into the position CD again, it has by the corollary to the above definition described angles which are together equal to four right angles; but the angles which it has described are the three exterior angles of the triangle ABC, therefore the exterior angles of the triangle ABC are equal to four right angles.

2. If the rectilineal figure have any number of sides, the proposition is demonstrated just as in the case of a triangle. Therefore all the exterior angles of any rectilineal figure are together equal to four. right angles. Q. E. D.

COR. 1. Hence, all the interior angles of any triangle are equal to two right angles. For all the angles of the triangle, both exterior and interior, are equal to six right angles, and the exterior being equal to four right angles, the interior are equal to two right angles.

COR. 2. An exterior angle of any triangle is equal to the two interior and opposite, or the angle DCA is equal to the angles CAB, ABC. For the angles CAB, ABC, BCA are equal to two right angles; and the angles ACD, ACB are also (13. 1.) equal to two right angles; therefore the three angles CAB, ABC, BCA are equal to the two ACD, ACB; and taking ACB from both, the angle ACD is equal to the two angles CAB, ABC.

COR. 3. The interior angles of any rectilineal figure are equal to twice as many right angles as the figure has sides, wanting four. For all the angles exterior and interior are equal to twice as many right angles as the figure has sides; but the exterior are equal to four right angles; therefore the interior are equal to twice as many right angles as the figure has sides, wanting four.

PROP. II.

Two straight lines, which make with a third line the interior an-* gles on the same side of it less than two right angle, will meet on that side, if produced far enough.

Let the straight lines AB, CD, make with AC the two angles BAC, DCA less than two right angles; AB and CD will meet if produced towards B and D.

In AB take AF-AC; join CF, produce BA to H and through C draw CE, making the angle ACE equal to the angle CAH.

Because AC is equal to AF, the angles AFC, ACF are also equal (5. 1.); but the exterior angle HAC is equal to the two interior and op posite angles ACF, AFC, and therefore it is double of either of them, as of ACF. Now ACE is equal to HAC by construction, therefore ACE is double of ACF, and is bisected by the line CF. In the same manner, if FG be taken equal to FC, and if CG be drawn, it may be shewn that CG bisects the angle ACE, and so on continually. But if from a magnitude, as the angle ACE, there be taken its half, and from the remainder FCE its half FCG, and from the remainder GCE its half, &c. a remainder will at length be found less than the given angle DCE.*

E

D

H

G

K

Let GCE be the angle, whose half ECK is less than DCE, then a straight line CK is found, which falls between CD and CE, but nevertheless meets the line AB in K. Therefore CD, if produced, must meet AB in a point between G and K. Therefore, &c. Q. E. D.

This demonstration is indirect; but this proposition, if the definition of parallels were changed, as suggested at p. 302, would not be necessary; and the proof, that lines equally inclined to any one line. must be so to every line, would follow directly from the angles of a triangle being equal to two right angles. The doctrine of parallel lines would in this manner be freed from all difficulty.

PROP. III. 29. 1. Euclid.

If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another; the exterior equal to the interior

*Prop. 1. 1. Sup. The reference to this proposition involves nothing inconsistent with good reasoning, as the demonstration of it does not depend on any thing that has gone before, so that it may be introduced in any part of the Elements.