GEORGE BRUCE HALSTED NEW ELEMENTS OF GEOMETRY. 
13 
This property, however, does not appertain to the straight, but 
to the curve which in my paper On . the Foundations of Geometry I 
have designated as circle-limit. 
Finally if the difficult problem of parallelism is to be resolved 
by experiment, Legendre’s proposal, to put a radius six times in a 
circle, must be declared much too insufficient. 
In my Foundations of Geometry I have shown, using- astronomical 
observations, that in a triang-le whose sides are approximately 
equal to the distance of the earth from the sun, the angle-sum can 
not differ from two rig-ht angles by more than 0.0003 of a second. 
This difference increases in geometric ratio with the sides of 
the triangle, and consequently up to the present, as I remarked 
before, is the ordinary g-eometry more than sufficient for the 
measurements in practice. 
We can arrive at this result by the aid of propositions sufficiently 
simple and conformable to the first notions of the science, though 
of course the complete theory requires a wholly chang-ed order of 
exposition and besides the addition of trig-onometry. 
To the imperfections of the theory of parallels must also be 
reckoned the definition of parallelism itself. However this imper- 
fection depends nowise, as Legendre supposes, upon a faulty defi- 
nition of the straig-ht line, nor even on certain faults in primary 
notions, faults which I propose to indicate here, attempting- to cor- 
rect them so far as I can. 
Ordinarily one begins g-eometry with attributing- to bodies three 
dimensions, to surfaces two, to lines one, while to the point is al- 
lowed none. 
In calling- the three dimensions leng-th, breadth, heig-ht, and un- 
der these designations actually understanding the three coordi- 
nates, we hasten in this way to impart immature ideas bywords to 
which ordinary speech gives already a certain, but indeed for rig- 
orous science still indefinite sense. 
In fact, how is it possible to represent clearly to one’s self the 
measurement of length, if we as yet do not know what a straight 
line is? 
How can one speak of breadth and height without previously 
having said something of perpendiculars, of the plane, and besides 
how are related perpendiculars in one and in different planes? 
Finally, if the point has no extent, what remains over to it then, 
that it may be the object of a conclusion? 
Suppose even that every one represents clearly to himself the 
straight line, though indeed unable to give an account of his idea; 
but the question is, how now with help of the straight line to de- 
