Mr. J. P. Hennessy on the Theory of Parallels. 285 



^'' Now when we regard a pair of parallel lines, just as we regard 

 a "Square or a circle, as a simple geometrical figure, and when we 

 seek for its definition, we find that there are a large number 

 of properties (propria) from which to choose. A different 

 property has been chosen by Wolfius, D'Alembert, Varignon, 

 R. Simson, and a host of others who have written on the sub- 

 ject. The definition given by Wolfius, but which a distinguished 

 writer* suggests was borrowed by him from Clavius, seems to 

 me, according to analogy, to be the most philosophical. None 

 of these geometers attempted to solve the difficulty by going 

 back to the fundamental principles of definition, or by carefully 

 analysing those other definitions which have been found to suc- 

 ceed. It is, however, a remarkable fact, that if we now adopt 

 this course we in one case get a definition exactly the same 

 as that borrowed from the Geometria Practica. "> *5i*i iif ^j/io 



Euclid defines a circle to be, — iwjrrf*! yb 



" A plain figure bounded by one continued line, called its cir- 

 cumference; and having a certain point within it, from which 

 all right lines drawn to its circumference are equal." ^^O! 



It thus has, like a system of parallel lines, two essential ele- 

 ments. In the one there is a continued line and a point, in the other 

 there are two right lines. The definition of the circle depends 

 on a property possessed by every possible right line drawn from 

 the point to the continued line. That property is, mutual equa- 

 lity. Now we know that the right lines from the point to the 

 continued line are all perpendicular to it. So that Euclid's de- 

 finition depends essentially upon the equality of all perpendi- 

 culars drawn from one element of the circle to the other. This 

 leads us very naturally to Wolfius's definition : — 



" Parallel right lines are such, that perpendiculars from every 

 point in one upon the other are equal." 



From this, as well as from the one I formerly gave, all the 

 properties of parallel lines may be deduced ; the only difference 

 is, that this definition makes the demonstrations rather complex. 



If, in the same way, an analysis is made of Euclid's definition 

 of a square, it will be seen to bear a very close analogy to the 

 definition of parallels in which an equality of angles to a con- 

 stant quantity is made the difibrentia. Having defined a rhombus 

 to be " an equilateral quadrilateral figure," he defines a square 

 as '^ a rhombus each of whose angles is equal to a right angle." 



No serious objection has ever been made to this definition ; 

 and yet when we treat parallel lines in a similar way, we are 

 charged with assuming something that ought to have been 

 proved. I do not think such a charge can be upheld for a single 

 moment. To say, when two parallel lines are intersected by a third 

 * Dr. Larduer. -riuv; -, ^« ... 



