PARALLELS. 



PARALLELS. 



252 



directions make the same angle with any other direction. 18. Mr. 

 Exley (1818) proposes to assume that if four equal straight lines, each 

 at right angles to the preceding, do not meet and enclose a space, a 

 fifth such line would do so. 19. Dr. Creswell proposes to assume that 

 through any point within an angle less than two right angles, a straight 

 line may be supposed to be drawn cutting the two straight lines which 

 contain the angle. 20. Professor Thompson makes it an axiom that 

 " if a triangle be moved along a plane, BO that its base may always be 

 on the same straight line, its vertex describes a straight line equal to 

 that described by either extremity of the base. 21. M. Legendre (in 

 the earlier editions of his Elements ) makes a direct appeal to the 

 senses. 22. In the seventh edition he assumes (as in instance 15) that 

 a magnitude increases without limit, where perpetual increase is all 

 that is demonstrable. 23. In the twelfth edition he fairly brings the 

 disputed proposition to rest upon the axiom, that if two angles of a 

 triangle diminish without limit, the third (whatever the base may be) 

 approaches without limit to two right angles, a proposition not admis- 

 sible when, as in M. Legendre's final construction, the base at the same 

 time increases without limit. 24. In a note to the same edition, he 

 demands as an axiom that no straight line can be entirely included 

 between two straight lines which make an angle less than two right 

 angles, which may easily be shown to be nothing more or less than 

 Euclid's axiom. 25. He attemps a proof of the last, which fails. 26. 

 M. Legendre's analytical proof, which we shal) presently examine. 27. 

 M. Lacroix would confine the assumption of Euclid to the case in which 

 one of the internal angles is a right angle and the other less. 28. H. 

 Bertrand extends the use of the term equality ; we shall afterwards 

 examine his proof. 29. Mr. Ivory assumes a right to construct a 

 series of triangles in a manner which cannot be certainly done unless 

 an assumption as difficult as in (20) be made : and 30, Professor Young 

 makes a modification of the preceding, which does not remove the 

 difficulty. 



For further information we may refer to the work from which 

 the preceding abstract is taken. The author of it proposes his own 

 system, the latent assumption of which is, that if equal straight lines 

 make an angle, and other straight lines equal to them be attached to 

 their extremities at the same angle, the remaining extremities of the 

 second pair of straight lines will not * always meet. 



The same author, whose erudition on thia subject would alone 

 entitle any attempt of his to attention, has published a new view of 

 the subject, in which he proposes to deduce the properties of the 

 equiangular spiral, and to make them the foundation of a proof of 

 Euclid's axiom. It assumes the doctrine of limits, ^and the theorem 

 that velocities (in one case at least) are to one another in the limiting 

 ratio of spaces described in the same times. Whatever may be thought 

 f this method as evidence for producing conviction, we cannot take 

 such an assumption as removing the geometrical difficulty, since, by 

 the introduction of a totally new line, it leaps the conventional boun- 

 daries of geometry ; to say Clothing of the question which may fairly 

 arise, as to whether the axioms of the theory of limits are not as diffi- 

 cult as that of Euclid. 



Two proofs have been referred to as requiring further explanation : 

 those of MM. Legendre and Bertrand. We take them successively. 



The first assumes all that knowledge which is derived from algebra 

 and the theory of algebraical operations. We premise that the theory 

 of parallels may be strictly deduced, though not without some trouble, 

 if it can previously be shown that the three angles of a triangle are 

 equal to two right angles. Let there be a triangle of which the base 

 contains r linear units, and the opposite angle c angular units, the 

 other angles containing A and B units. Then it can be easily shown 

 that any other triangle which has the same base c and the same ad- 

 jacent angles A and B must be in all respects equal to the first : that 

 is, r, A, and B being the side and adjacent angles of a really existing 

 triangle, c is given when c, A, and B arc given. There must then be 

 some algebraical mode of expressing c in terms of c, A, and B, such as 



c=f (<;, A, B). 



From such an equation, if it exist, c can be found in terms of A, B, 

 an.l ( , that is, the length of a straight line can be expressed by means 

 of angles only. Now it is known that no equation can determine a 

 magnitude by means of magnitudes no one of which is of the same 

 C kind with it : and the only way of avoiding this 



supposition is by supposing that c does not enter 

 the equation at all, or that c=<f> (A, Bl, so that the 

 third angle of a triangle is given when the other 

 two are given, whatever the side may be, provided 

 B I/it triangle be tnovm to exitt. Let there be a right 

 angled tiiangle ACB, and let CD be perpendicular to AB; then the 



* In the edition at 1833, thil assumption was eiplicitly made. A writer in 

 the ' Journal of Education (rol. rii. p. I 05) having; pointed this out, the words 

 " provided the straight lines do not meet " were added (in the edition of 1 834) 

 to the enunciations of the leading propositions of the proof. This certainly 

 removes the difficulty from the propositions themselves, but throws it upon 

 their consequences ; and the final result only remains proved, provided tho.e 

 lines do n< t always meet : that is, the proportion that they do not always meet, 

 still remains an axiom. The work is well worth the perusal of the student, 

 though we hare no doubt whatever that in respect to the theory of parallel* it ii 

 ao amendment of Euclid. 



triangles A c , A c D, have a common angle at A, and a light angle in 

 each : consequently their third angles are equal, or A D = A B c. 

 Similarly D c B = c A B ; whence the angles at A and B are together equal 

 to a right angle. And if the two acute angles of a right angled triangle 

 be eqxial to one right angle, it is readily shown that all the three angles 

 of any triangle are equal to two right angles. 



It is not our intention to go fully into the objections which have 

 been made to this proof, nor into Legendre's answers ; all which may 

 be found in the notes to Sir David Brewster's translation of Legendre. 

 It has the disadvantage of being founded upon a science which requires 

 more and harder axioms than geometry itself, and of which the par- 

 ticular process employed, namely, inversion of a function, is in many 

 cases full of unexplained difficulties ; while it has the advantage of not 

 appealing to any new notions of space. As an illustration of the con- 

 nection between algebra and geometry, it must always be valuable : 

 but we suppose no one would think of making it the foundation of 

 geometry. Some objectors imagined that Legendre would infer that 

 a base c, with two adjacent angles together less than two right angles, 

 must be the base of a triangle ; or that because the formula applies 

 wherever there is a triangle, that there must be a triangle wherever 

 the formula applies. If this were the case, undoubtedly they were 

 right in saying that Legendre did in fact assume Euclid's axiom : but 

 if, as we apprehend, he would have applied the proposition thus proved 

 of existing triangles, to the proof of Euclid's axiom, he should certainly 

 have stated his intention more distinctly in his reply. It seems to us 

 that he took it as admitted on all sides how to deduce Euclid's axiom, 

 while his opponents imagined that he considered himself as having 

 proved that axiom. 



It is of much import-ince, in connection with Legendre's view, to 

 remember the point mentioned at the end of ANGLE, namely, that 

 angle, of all magnitudes, is the only one of which number is a function, 

 and the converse. 



The proof of M. Bertrand is as follows : Let it be granted that two 

 spaces, whether finite or infinite, are equal,* when one can be placed 

 upon the other so that any point whatsoever of either lies upon a point 

 of the other : that is, let it be legitimate to say, of the word equal as 

 thus used, that spaces which are equal to the same space are equal to 

 one another. Granting this, it is easily shown 1, that the infinite 

 spaces of equal angles are equal ; 2, that of two angles, the infinite 

 space contained in the greater is greater than the infinite space con- 

 tained in the less. 



Let there be two lines, OF, A o, making with o A internal angles 

 r o A, a A o, equal to two right angles. Then o P and A c are parallel. 



T;ike A B, B c, c D, &c., each equal to o A, and make the angles HBO, 

 K c D, L D E, &c., all equal to F o A or GAB. Let all lines with arrow- 

 heads be produced without limit in the direction to which the arrow- 

 head points. Then if o A be placed on A B, the lines o F and A o will 

 in their new positions coincide with A o and B it, or the infinite space 

 FOAO is equal to the infinite space OABR: and similarly IIBCK, 

 K c D L, Ac., are all equal to one another and to F o A a. But it is 

 obvious that no number of these spaces, however great, will fill up the 

 infinite space of the angle F o . Now let a line o 1 be drawn in such 

 a manner that 1 o A and o A o are together less than two right angles ; 

 whence o 1 falls nearer to o E than does o F. Take F o 2 double of F o 1, 

 F o 3, treble of F o 1 and so on : whence, since of two quantities which 

 bear a ratio, the less may be multiplied so as to exceed the greater, 

 some multiple of F o 1 must be greater than FOE, whence some 

 multiple of the infinite space r o 1 is greater than the infinite space 

 FOE. But no multiple of F o A o will be so great as F o E ; whence the 

 infinite space F o 1 must exceed the infinite space FOAO. Therefore 

 o 1 produced must cut A o ; for if not, the space F o 1 would be 

 entirely contained in FOAO, and the former could not exceed the 

 latter. 



We have not noticed the numerous attempts at the solution of the 

 difficulty which proceed by tacit assumption or illogical process, under 



The author of ' Geometry without^ Axioms ' cites Plato for the axiom that 

 equality is only to be predicated of finite magnitudes, lint ivithou tlooking at 

 the authority of Plato, or any one else, it is for the reader now to ask himself 

 whether he can, by comparing the infinite spaces contained in equal and 

 unequal angles, obtain a distinct conception of equality, of greater, and of less : 

 If so, the proof of M. Bertrand must be admitted by him ; if not, no one has a 

 right to demand his acquiescence. We are inclined to think that it a proof 

 to some, but not to others ; to us it certainly i* a proof. But we do not contend 

 for the admission of considerations respecting infinity into elementary geometry : 

 we should hold such a plan altogether opposed to every principle of sound 

 teaching. 



