known. The most ancient writer who appears to have attempted to do this is Ptolemy the astronomer, who wrote a treatise expressly on the subject of Parallel Lines. Proclus has preserved some account of this work in the Fourth Book of his commentaries: and it is curious to observe in it an argument founded on the principle which is known to the moderns by the name of the sufficient reason.

To prove, that if two parallel straight lines, AB and CD be cut by a third line EF, in G and H, the two interior angles AGH, CHG will

be equal to two right angles, Ptolemy reasons thus: If the angles AGH, CHG be not equal to two right angles, let them, if possible, be greater than two right angles; then, because the lines AG and CH are not more parallel than the lines BG and DH, the angles BGH, DHG are also greater than two right angles. Therefore, the four angles AGH, CHG, BGH, DHG are greater than four right angles; and they are also equal to four right angles, which is absurd. In the same manner it is shewn, that the angles AGH, CHG cannot be less than two right angles, Therefore they are equal to two right angles. But this reasoning is certainly inconclusive. For why are we to suppose that the interior angles which the parallels make with the line cutting them, are either in every case greater than two right angles, or in every case less than two right angles? For any thing that we are yet supposed to know, they may be sometimes greater than two right angles, and sometimes less, and therefore we are not entitled to conclude, because the angles AGH, CHG are greater than two right angles, that therefore the angles BGH, DHG are also necessarily greater than two right angles. It may safely be asserted, therefore, that Ptolemy has not succeeded in his attempt to demonstrate the properties of parallel lines without the assistance of a new Axiom. Another attempt to demonstrate the same proposition without the assistance of a new Axiom has been made by a modern geometer, Franceschini, Professor of Mathematics in the University of Bologna, in an essay, which he entitles, La Teoria delle parallele rigorosamente dimonstrata, printed in his Opuscoli Mathematici, at Bassano in 1787.

The difficulty is there reduced to a proposition nearly the same with this, That if BE make an acute angle with BD, and if DE be

E

B

FN

perpendicular to BD at any point, BE and DE, if produced, will meet. To demonstrate this, it is supposed, that BO, 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 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.

L

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 shew, that when BC increases by a part equal to BQ, 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

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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 two 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 plain, 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 of the quantities by which C is determined, P cannot be 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 Pis equal to some combination, that is, to some function of the quantities A, B and 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, BDÁ 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 constantly 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. 1.

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, FAB, GBC; these angles are together equal to four right angles.

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 car

ried 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 coincide 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 come

F

E

B

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 CAR, 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.