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of modern mathematical research on this point appears to he, I shall 
endeavour to explain later on. 
With these preliminary remarks in explanation, I now proceed 
briefly to sketch a system of geometry which, as to its foundations, 
differs from that of Euclid only in the alteration of one (or at most 
two) axioms. Its conclusions will be found to differ very materially 
from his, although this difference is merely in the way of wider 
generality, Euclid’s geometry being contained as a particular case 
in what I shall, for distinction’s sake, call Pangeometry. 
The space which I shall consider is to be tridimensional. I appeal 
to the ordinary conceptions of 
Point, Line or curve, Surface, Solid ; 
and, for the sake of the words, state that a point has no extension, a 
line is once extended, a surface twice, a solid thrice. 
As a test of these distinctions, the idea of motion may be intro- 
duced. I cannot stop now to justify this, but merely remark that 
nothing is to be predicated concerning time. 
Farther, space is to be uniform, in the double sense that it has no 
properties depending either on position or direction. 
The great test of this last statement is congruency,* which I mention 
thus early, because it is the touchstone of geometry. Thus the 
statement that space has no properties depending on position, simply 
means that congruent figures exist, e.g ., that a solid of a certain size 
and shape can be carried from one part of space to another without 
alteration in either respect ; and that two congruent figures can be 
conceived as separately existing in different parts of space. It is 
evident that all space measurement rests on congruency. 
It is essential to be careful with our definition of a straight line , 
for it will be found that virtually the properties of the straight line 
determine the nature of space. 
Our definition shall be that two points in general determine a 
straight line, or that in general a straight line cannot be made to 
pass through three given points. 
It is important to notice the force of the phrase in general. This 
* Two figures are said to be congruent when one can be placed on the other, 
so that every point of one shall coincide with a point of the other, and vice 
versa. The phrase equal in every respect is used in the same sense in most 
English editions of Euclid. 
