For further information we may refer to the work from which the preceding abstract is taken. The anther 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 this 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 of 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 .nothing of the question which /nay 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 M3L 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 c linear unite, and the opposite angle c angular units, the other angles containing A and n 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, e, A, and n being the side and adjacent angles of a really existing triangle, c is given when c, A, and B are given. There must then be some algebraical mode of expressing o in terms of c, A, and 13, such as • c= 4 (e, A, /3).
From such an equation, if it exist, c can be found in terms of A, B, and c, 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 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 o=p (A, B1, so that the third angle of a triangle is given when the other two are given, whatever the side may be, provided the triangle be known to exist. Let there be a right triangle A c B, and let C D be perpendicular to A n ; then the triangles A C c D, have a common angle at A, and a right angle in each : consequently their third angles are equal, or ACD=ADC. Similarly D CB=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 equal 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 iu fact assume Euclid's axiom : but if, as we apprehend, be 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 be considered himself as having proved that axiom.