of Edinburgh , Session 1879 - 80 . 643 
will be best understood from an illustration. We all know from 
the case of a three legged stool, if not from any more scientific source, 
that three points determine a plane. Yet not any three points ; for, 
if the third foot were put in line with the other two, the one stool 
would be as unsafe a seat as the proverbial two. Yet again, and very 
near indeed to our case, two points on a sphere in general determine 
a great circle on it. But there are exceptions; a point and the 
diametrically opposite point do not determine a great circle, and yet 
it would be a good definition of a great circle to call it that line on 
a sphere which is in general determined when two of its points are 
given, no other condition being assigned.* 
We recognise therefore that, although in general, any two points 
being taken, a line will thereby be determined, yet it may happen 
that, one point being taken, another point may exist which along 
with the first does not determine a straight line. The necessity for 
this admission appears when we consider space in which two straight 
lines have more than one point of intersection. 
Here let it be mentioned, to avoid misconception, that it follows 
from our definition of a straight line, and from the uniformity of 
space (the test being congruency), that space is symmetrical round 
every straight line. This is at once an answer to those who say 
that pangeometry is merely an analogy drawn from the theory of 
surfaces of constant curvature. 
A plane may be defined as Euclid defines it, and the conclusions 
drawn, that two intersecting lines, a point and a line, or a line passing 
through a given point and moving perpendicular to a given line, all 
in general determine a plane. The last form of definition of course 
presupposes the definition of a right angle. 
Farther, we adopt all Euclid’s definitions up to the definition of an 
'* It is interesting to notice that any curve already conditioned a number of 
times less by two than the whole number of conditions that completely deter- 
mine it, fulfils in many respects the definition of a straight line, for any two 
points completely determine the curve. A very interesting particular case is 
that of a series of circles which always pass through a given fixed point. Such 
a series of circles may take the place of straight lines in many of Euclid’s pro- 
positions. Most of the propositions as to congruency hold for them. The sum 
of the three angles of a triangle formed by three such circles is two right angles; 
the perpendiculars from the vertices of such a triangle on the opposite 
sides are concurrent; and so on, as is otherwise evident by the theory of 
inversion. 
