272 PROFESSOR KELLAND ON SUPERPOSITION. 



paratively useless as elements of training. The loose way in which many of 

 these treatises are compiled, causes us to be astonished at the celebrity they have 

 attained. In the treatise of Legendre, for instance, the author attempts, in his 

 first edition, to place the doctrine of parallels on a more simple basis, — at that 

 time, it may be presumed he was tolerably ignorant of the real state of the ques- 

 tion, — but when his book became pretty generally adopted, he gave his close at- 

 tention to the subject, and became convinced that his emendation required 

 amendment. New editions were called for, and new amendments succeeded the 

 old ones, until, in the tenth edition, we are presented with the following signifi- 

 cant words as an advertisement: " By the advice of several distinguished pro- 

 fessors, I have determined to re-establish, in this tenth edition, the theory of pa- 

 rallels nearly on the same basis as Euclid's." 



The author has thus sailed round nearly all the points of the compass, and 

 agreeably to his own confession arrives nearly at the point at which he started. 

 Nearly, he says; and, truth to tell, not so very nearly at all. There is certainly 

 an abandonment of all novelty in the exposition of the doctrine; but any one may 

 see how very far the author is behind Euclid, even in his tenth edition. If we 

 could examine the last edition, we should probably find Euclid's method com- 

 pletely restored. At any rate, Legendre tells us, in his note, that he is not 

 satisfied with the theory as it stands, and he attributes its imperfection to the 

 definition of a straight line ; but whether he means his own definition, which is 

 imperfect enough, no doubt, or Euclid's, does not appear. 



But it is time we should leave Legendre, and offer one positive argument in 

 favour of the method of demonstration by superposition. It is admitted, I think, 

 that a chain of reasoning upon an abstract definition is the most healthful exer- 

 cise of the mental powers, at least in the days of youth. Viewed as such, Eu- 

 clid's Elements stand above all other writings; but a class of objectors of a totally 

 different order from Legendre has arisen, who, with considerable show of reason, 

 urge against geometry, as based on superposition, that it excludes all exercise 

 of ingenuity, inasmuch as it only demands a uniform and unvaried march, to 

 deviate from which is to wander into error. There is some truth in this, — but it 

 is not altogether true ; and I have here exhibited, in reply to it, some of the dif- 

 ferent solutions of a single problem where we are bound down by a specific re- 

 quirement ; and it will be seen that the solutions present themselves in tolerable 

 variety, and that their discovery must have brought out some ingenuity. 



The problem was proposed to me by the late lamented Secretary of this So- 

 ciety, Sir John Robison. I gave him the first solution. The others have arisen 

 out of it, partly from my own suggestions, partly from my students' exercises. 

 It is certainly a very remarkable problem. I have never met with one which 

 presents such a variety of altogether independent solutions. There can be no 

 doubt that the problem has appeared in the Ladies' Diary, or elsewhere, but I 



