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certain third line, or, more simply, lines which are perpendicular to a given line. It must then be proved, 1. That straight lines which are equally inclined to a certain line or perpendicular to a certain line, must be equally inclined to all the other lines that fall upon them; and also, 2. That two straight lines which do not meet when produced, must make equal angles with any third line that meets them.
The demonstration of the first of these propositions is not at all facilitated by the new definition, unless it be previously shewn, that all the angles of a triangle are equal to two right angles.
The second proposition would hardly be necessary if the new definition were employed; for when it is required to draw a line that shall not meet a given line, this is done by drawing a line that shall have the same inclination to a third line that the first, or given line bas. It is known that lines so drawn cannot meet. It would no doubt be an advantage to have a definition that is not founded on a condition purely negative.
2. As to the Mathematicians who have rejected Euclid's Axiom, and introduced another in its place, it is not necessary that much should be said. Clavius is one of the first in this class ; the Axiom he assumes is, “ That a line of which the points are all equidistant from a cers tain straight line in the same plane with it, is itself a straight line.” This proposition he does not, however, assume altogether, as he gives a kind of metaphysical proof of it, by which he endeavours to connect it with Euclid's definition of a straight line, with which proof at the same time he seems not very well satisfied. His reasoning, after this proposition is granted (though it ought not to be granted as an Axiom), is logical and conclusive, but is prolix and operose, so as to leave a strong suspicion that the road pursued is by no means the shortest possible,
The method pursued by Simson, in his Notes on the First Book of Euclid, is not very different from that of Clavius. He assumes this Axiom, “ That a straight line cannot first come nearer to another “ straight line, and then go farther from it without meeting it.” (Notes, &c. English Edition.) By coming nearer is understood, conformably to a previous definition, the diminution of the perpendiculars draws from the one line to the other. This Axiom is more readily assented to than that of Clavius, from which, however, it is not very different; but it is not very happily expressed, as the idea not merely of motion, but of time, seems to be involved in the notion of first coming nearer,
and then going farther off. Even if this inaccuracy is passed over, the reasoning of Simson, like that of Clavius, is prolix, and evidently a circuitous method of coming at the truth.
Thomas Simpson, in the second edition of his Elements, has presented this Axiom in a simpler form. “ If two points in a straight line are * posited at unequal distances from another straight line in the same
plane, those two lines being indefinitely produced on the side of the is least distance will meet one another.”
By help of this Axiom it is easy to prove, that if two straight lin
AB, CD are parallel, the perpendiculars to the one, terminated by the other, are all equal, and are also perpendicular to both the parallels. That they are equal is, evident, otherwise the lines would meet by the Axiom. That they are perpendicular to both, is demonstrated thus :
'If AC and BD, which are perpendicular to AB, and equal to one another, be not also perpendicular to c
D CD, from C let CE be drawn at right an
E gles to BD. Then, because AB and CE are both perpendicular to BD, they are parallel,
and therefore the perpendiculars AC and BE are equal. But AC is equal to A!
B В BD, (by hypothesis,) therefore BE and BD are equal, which is impossible; BD is therefore at right angles to CD.
Hence the proposition, that “ if a straight line fall on two parallel lines, it makes the alternate angles equal," is easily derived. Let
FH and GE be perpendicular to CD, then they will be parallel to one another, and also at right angles to AB, and therefore FG and HE are equal to one another, by the last proposition. Wherefore in the triangles EFG, EFH, the sides HE and EF are equal to the sides GF and FE, each to each, and also the third side HF to the third side EG, therefore the angle HEF is equal to the angle EFG, and they are alternate angles. Q. E. D.
This method of treating the doctrine of parallel lines is extremely plain and concise, and is perhaps as good as any that can be followed, when a new Axiom is assumed. In the text above, I have, however, followed a different method, employing as an Axiom; " That two
straight lines, which cut one another, cannot be both parallel to the
same straight line.” This Axiom has been assumed by others, particularly by Ludlam, in his very useful little tract, entitled Rudiments of Mathematics.
It is a proposition readily enough admitted as self-evident, and leads to the demonstration of Euclid's 29th Proposition, even with more brevity than Simson's.
3. All the methods above enumerated leave the mind somewhat dissatisfied, as we naturally expect to discover the properties of parallel lines, as we do those of other geometric quantities, by comparing the definition of those lines, with the properties of straight lines already
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 ļine 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, bę 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 bis Opuscoli Mathematici, at Bassano in 1787.
The ditficulty is there reduced to a proposition nearly the same with this, That-if BE make an acute angle with BD, and if DE be
perpendicular to BD at any point, BE and DE, if produced, will meet. To demonstrate this, it is supposed, that BO,
L 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 nce 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 shew, 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 bis 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.
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 tri. angle 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 P is 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 te BC. The triangles ABD, ABC have the
А angles BAC, BD, 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
D 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.
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.