182 Mr. J. Walsh on the Iwelftk Book of Euclid. 



proved that a straight line is the shortest distance between 

 two points, notwithstanding that by such a way of reasoning 

 the same proposition would always remain to be proved, that 

 a straight line is the shortest distance between two points. I 

 shall observe that, exactly en a similar ground of reasoning, 

 if the curve line be made the base of a triangle, having its 

 vertex in the straight line, then the base or curve line is less 

 than the sum of the two sides ; and if the sides be made the 

 bases of other triangles, having their vertices also in the straight 

 line, and so on as before; then it would appear that any 

 curve line is the shortest distance between two points. To 

 attempt to prove that a straight line is the shortest distance 

 between two points by any intermediate propositions, is to 

 attempt to prove that the whole is greater than its part : it is 

 to attempt to prove, in fact, that a straight line is a straight 

 line. It may be objected here, perhaps, that the calculus of 

 variations proves the proposition under consideration. I shall 

 observe, in reply, that there seems to be a lack of intimate ac- 

 quaintance with some of the results obtained by it among all 

 the writers on that calculus. This will appear by an extract 

 from the Calculus of Functions of Lagrange, page 475 : — 



" L'equation generale donne tout de suite — — — ^r = a une 



constante; d'oii l'on tive y'=b, et de la y = bx + c, b et c etant 

 deux constantes arbitraires ; ce qui est l'equation generale de 

 la ligne droite." The result here obtained is, that for all the 

 points of the axis of x, which axis is the straight line termi- 

 nated by the given points, for all the values of x, the co-ordi- 

 nate of the shortest line has a determinate magnitude; there- 

 fore, the shortest line between two points cannot pass but 

 through one of them. Such is the absurd conclusion to which 

 the reasoning leads. Whether the defect is in the reasoning, 

 or in the calculus, I shall not now inquire. Quite the re- 

 verse of this is the binomial calculus, which is very explicit 



on this subject. The binomial of any curve line is — dx; 



and in the case of maximum or minimum, — is nothing ; 



then n is nothing and y is nothing. The calculation does not 

 go further than to show that this happens when x is nothing, 

 and when x becomes the straight line terminated by the points ; 

 proving only, that the shortest line must pass through the 

 given points ; thus ending at the point at which we set out, 

 and leaving the mind to shift for itself in determining, among 

 all lines terminated by two given points, which is the shortest. 

 I interrogate the binomial calculus on the shortest distance 



between 



