Dr. Adamson on Geometrical Demonstrations. 409 



cess for the comparison of certain results, and a sound and suffi- 

 cient hypothesis expressed in a definition. The process alluded 

 to will be found in this, that if there be two equal straight lines, 

 and others at their extremities forming angles with them, which 

 are equal each to each, then we show by application and coin- 

 cidence, that if the straight lines attached to one of the given 

 equals meet each other, those attached to the other will also 

 meet. This is a consequence of equality in angles, which rela- 

 tion however we should take care to define. Our argument 

 affi)rds the conclusion, that from a point in a straight line, or out 

 of it, there can be only one perpendicular to the straight line. 



If, in proceeding onwards, we are to reason clearly, we must 

 avoid embarrassing ourselves with the limit as our hypothesis, 

 and also with those properties which require the limit to be in- 

 terposed as a hypothesis in our subsequent deductions. That 

 property of parallelism which satisfies the required conditions 

 may be thus stated. Parallel straight lines are such that every 

 straight line passing through a definite {given or single) point in 

 one of them, must, if prolonged, meet the other. If this defi- 

 nite point be in the one A, it is easy to show that every straight 

 line passing through any other point of A will meet the other B ; 

 and from this truth, combined with the position that there can be 

 only one perpendicular to a straight line at or from any point, 

 the conclusion is obvious, that a straight line perpendicular to B 

 is also perpendicular to A ; and then by the lemma above-men- 

 tioned, the converse position is deduced, that every straight line 

 which meets B must also meet A. Thus the defining test of 

 parallelism is extended to the form as a theorem — " that every 

 straight line which meets one must meet the other.'* 



From this the conclusions readily emerge as theorems, *' that 

 such straight lines never meet,** and that " they are equally in- 

 clined to any straight line meeting them;** and as a corollary, 

 " that through any point a straight line may pass so as to be 

 parallel to another,** according to the principle assumed in the 

 definition. This is a provision for treating the converses, among 

 which, in this mode of proceeding, the instance must occur in 

 which the limit of concurrence is the hypothesis. It is easily 

 deduced from what was stated above as to perpendiculars to both 

 of the parallels. Hence we demonstrate that straight lines 

 equally inclined to a given straight line are parallel, according 

 to the assumed definition. Hence the last converse statement 

 needed to complete the series is easily deduced, viz. if there be 

 two straight lines which, though produced ever so far, do not 

 meet, these lines are parallel ; or are such, that every straight 

 line meeting the one must meet the other. This will follow 

 from the preceding argument, or more shortly, by assuming 



Phil. Mag. S. 4. Vol. 5. No. 34. June 1853. 2 E 



