406 Dr. Adamson on Geometrical Demonstrations. 



it must happen that boundlessness of extent in two non-concur- 

 rent lines, combined with straightness in the individual lines, 

 must occur as a hypothesis, from which, unaided by anything 

 else, we must deduce positive conclusions as to equality of angles 

 or of lines. And it is as evident that whether we take as the 

 defining criterion of parallelism, that two perpendiculars are 

 equal, or that two alternate angles are equal, the difficulty equally 

 awaits us, with regard to other perpendiculars and other alter- 

 nate angles. 



Assuming the hypothesis that two straight lines are equally 

 inclined to a straight line crossing them, it is easy by a short 

 argument to prove that if they met each other on either side of 

 the crossing line, they would meet also on the other side of it, 

 and that therefore they cannot meet, though produced on either 

 side. It would then be the natural mode of proceeding, and it 

 has ever been the aim of geometricians, to take in hand for de- 

 monstration the converse statement, in which the hypothesis 

 consists of the two facts of straightness and non-concurrence. 

 The failure in this has mainly arisen from inattention to the 

 nature of one of the facts of the hypothesis, and of its presence 

 among the data, as determining the only mode of argument 

 which the case admits of. It has been hesitatingly suggested 

 whether the doctrine of limits, as it is termed, can be admitted 

 as an element in the proof. Now the doctrine of limits means 

 either that the existence of a limit to a set of varying objects 

 exists, and may be demonstrated, or that there are opposite 

 variations, real or imaginary, between which, following the law 

 of continuity, a limit must exist. In the first instance, that 

 there is a limit is to be established, on a given hypothesis, as a 

 fact. The second affords undoubtedly the only mode by which, 

 conversely, any conclusion can be deduced from the nature of a 

 limit possessing the character of infinitude. An unreal limit 

 determined by unreal variations may be compared with a real 

 one, so as to lead to a contradiction, when the limit so deter- 

 mined results from a false supposition. We thence determine 

 the solitariness of the real limit, or else that it alone can have 

 the property which it is the object of our demonstration to attach 

 to it. It is thus that a curve is shown to be the solitary limit 

 between interior straight lines enveloped by it, and exterior 

 straight lines enveloping it, or that it is (greater than the sum of 

 the one set, and less than the sum of the other set; and its 

 solitariness as the limit of any exterior set is demonstrated by 

 the contradiction evolved from the assumption, that any set of 

 exterior straight lines is the limit. If however this solitariness 

 be not established in such a case, the demonstration is a failure. 



Now in regard to parallelism, it is to be observed that the 



