GEORGE BRUCE HALSTED NEW ELEMENTS OF GEOMETRY. 
3 
Application of the new theory to analysis is to be found in me- 
moires which I printed in the Kazan Courier of the years 1829 and 
1830 under the title On the Foundations of Geometry. 
The principal result at which I have arrived under the assump- 
tion that lines are dependent upon angles, is the possibility of the 
existence of Geometry in a wider sense than that in which Euclid 
first expounded it to us. 
In this broadened form I g*ave to the science the name Imaginary 
Geometry , in which as a special case the Customary Geometry is 
contained under limitations in the general hypotheses recom- 
mended by practical measuring-. 
That the new foundations are sufficient, I have undertaken to 
show in a paper recently published in the Scientific Memoirs of 
the University of Kazan. * 
In the endeavor to attain this aim, not so much directly as rather 
by the shortest inverse way, I chose then, g-oing- over from certain 
assumed foundations, to arrive at equations for all relations and at 
expressions for all magnitudes of g-eometry. 
Even should my discovery have produced no other advantag-e 
than the filling- of the g-ap in the elements, yet at least the interest 
which this subject has always aroused oblig-es me now to treat it 
with detail. I will begin with an examination of the earlier 
theories. 
It is easy to show that two straig-hts making- equal angles with 
a third never meet, since they are then perpendicular to a certain 
straig-ht. 
Euclid assumed inversely, that two straig-hts unequally inclined 
to a third always meet. 
To demonstrate this latter assumption, recourse has been had 
to many different procedures; such as trying- first to find the sum 
of the angles of a triangle, or comparing- infinite areas comprised 
between the sides of an angle and between perpendiculars to a 
straig-ht, or supposing- that the angles depend only on the ratio of 
the sides, or finally attributing- to the straig-ht new properties, in 
order to complete its definition. 
All these demonstrations, some ing-enious, are without excep- 
tion false, defective in their foundations and without the necessary 
rig-or of deduction. 
There is not even one of them sufficiently simple and sufficiently 
specious to present usefully to beginners. 
In 1800 Leg-endre, in the third edition of his g-eometry, set up 
*In Book I of the Scientific Memoirs, for the year 1835, under the title Im- 
aginary Geometry. 
