Dr. Adamson on Geometrical Demonstratiom. 407 



equality of inclination of two straight lines crossed by another, 

 establishes the fact of their non-concurrence, so that their con- 

 dition is the limiting one of all concurrent straight lines passing 

 through the intersections of the crossing line. Instead there- 

 fore of the introduction of the doctrine of limits being a matter 

 of doubt as to its propriety, it is the only logically legitimate 

 mode in which the end can be attained, when we make the non- 

 concurrence of the lines to be the hypothesis. The limiting case 

 is in our hands as the only foundation of our reasoning. The 

 argument has to predicate, "Because the straight lines never 

 meet therefore,'^ — the hiatus never has been, and never can be 

 supplied in any other than the mode now stated. 



The late Sir John Lesslie introduced the doctrine of limits in 

 this form, — that if there be a base and a perpendicular, and 

 straight lines crossing the summit of the perpendicular and 

 meeting the base, then, regarding the angles formed by these 

 crossing lines on one side of the perpendicular, one group of 

 them will form angles less than a right angle, and the other 

 will form angles greater than a right angle; and since they 

 diverge from the same point and approach one another, making, 

 as they approach, angles which are more and more nearly equal, 

 therefore the limit between the two groups, which does not 

 meet the base on either side, must make angles which are equal, 

 or are right angles. This seems the clearest mode of stating 

 the argument, as rendering the notion of revolution unnecessary. 

 But the argument fails in establishing the singleness of the limit 

 of concurrence, for though the angles approach to right angles 

 as the points of concurrence retire from the perpendicular, there 

 is nothing in the reasoning to show that concurrence may not 

 have ceased before the angles become right angles. It therefore 

 leaves the point undetermined, whether there can, or cannot be 

 more than one straight line through a point non-concurrent with 

 another, and therefore fails to demonstrate that the straight line 

 is necessarily perpendicular to both the non-concurrent straight 

 lines, simply in virtue of their never meeting. 



The defect however may be supplied as follows : — Let there 

 be through the summit of the perpendicular a straight line which 

 never meets the given base, then, if this be not also at right an- 

 gles to the perpendicular, through the same point, i. e. the sum- 

 mit of the perpendicular, a straight line may make right angles 

 with the perpendicular, and therefore never meet the given 

 base. Hence there are two through the same point which never 

 meet the given base, and since they form an angle, there must 

 be an indefinite number having the same character. But through 

 the same point there may be drawn innumerable straight lines 

 which meet the given base, therefore there must be a limiting 



