^560 Remarhs on the Lor/ic of Elementary Geometry. 



Definition. The position of one straight line in regard to 



Unother is determined hy the angle of inclination, or by that 



angle which is formed when the lines arc produced till they meet. 



Definition. When the adjacent angles so formed are equal 



the position is perpendicular. 



Hence, all perpendiculars to a line have the same position in 

 respect of it. 



Hence, all portions of any straight line have the same position 

 in respect of any other straight line, or have the same inclina- 

 tion to it. 



Definition. The complement of inclination is the angle formed 

 by any straight line, having a given position in respect to another, 

 with the perpendicular to that other. 



Hence, since the position of every part of a straight line in 

 respect to another is the same, the complement of the inclination 

 IS every where the same. 



Hence, all straight lines perpendicular to one straight line 

 are equally inclined to every other which intersects them. 



However, in regard to parallels, we are not logically restricted 

 to one definition, hut are bound to choose that which is n>ost 

 convenient as the foundation for argument. Of the two which I 

 have mentioned as being commonly and illogically employed, it 

 is far more convenient to choose the latter, or rather the converse 

 of it, and to announce it in this form as our original hypothesis, 

 viz. Parallel straight lines are such that all straight lines passing 

 through a point in the one will meet the other. From this the 

 whole system of properties may be logically evolved. It can by 

 a short demonstration be extended to this form, viz. Lines are 

 parallel when every straight line which meets the one also meets 

 the other : and the test of parallelism in this form has two 

 advantages ; it is a property constantly occurring in geometrical 

 reasoning, while the other occurs comparatively seldom ; and it 

 enables us to establish the properties of parallels in which angles 

 alone are concerned, withoat interfering with figures. 



Theorems may be of two sorts, according as the hypothesis is 

 single or compound. In all of those which are derived directly 

 from the definitions, the hypothesis must be single, and in all 

 cases when the general relations of quantity are applied to 

 geometrical magnitudes the hypothesis is single. These ought 

 perhaps to be considered as corollaries or axioms rather than 

 theorems ; and a theorem ought to be a conclusion derived from 

 at least a two- fold hypothesis (i. e.) it will present some property 

 of a compound object of which the elements are two objects 

 having separate definitions : thus — if parallel straight lines 

 intersect a circumference, the definition of parallelism is com- 

 bined with that of a circle, to establish a property belonging 

 to a compound object. A strictly geometrical theorem ought 



