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 Lecenpre, 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, | have determined to re-establish, in this tenth edition, the theory of pa- 
rallels nearly on the same basis as Evc.ip’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 Evcuip, even in his tenth edition. If we 
could examine the last edition, we should probably find Evciin’s method com- 
pletely restored. At any rate, Lecenpre 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 Evciin’s, does not appear. 
But it is time we should leave LeGeNnpDRE, 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, Ev- 
cLip’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 Joun Ropison. 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 
