OCTOBER 20, 1899. ] 
figures as he has rigorously constructed. 
They understand that problems of con- 
struction play an essential part in a scien- 
tific system of geometry. Far from being 
solely, as our popular text-books suppose, 
practical operations, available for the train- 
ing of learners, they have in reality, as 
Helmholtz declares, the force of existential 
propositions. Therefore is evident the high 
import of Gérard’s work to establish the 
fundamental propositions of non-Huclidean 
geometry without hypothetical construc- 
tions other than the two assumed by 
Euclid: 1. Through any two points a 
straight line can be drawn; 2. A circle 
may be described from any given point as a 
center with any given sect as radius. 
Gérard adds explicitly the two assumptions : 
3. A straight line which intersects the 
perimeter of a polygon ina point other than 
one of its vertices intersects it again; 4. 
Two straights, or two circles, or a straight 
and a circle, intersect if there are points of 
one on both sides of the other. 
Upon these four hypotheses, perfecting a 
brilliant idea of Battaglini (1867), Gérard 
establishes the relations between the ele- 
ments of a triangle. 
Lobachévski never explicitly treats the 
old problems changed by transference into 
the new geometric world, such as ‘ Through 
a given point to draw a parallel to a given 
straight”; nor yet the seemingly impossi- 
ble problems now in it capable of geometric 
solution, such as ‘‘ To draw to one side of 
an acute angle the perpendicular parallel 
to the other side ’’; ‘‘ To square the circle.”’ 
These would be sought in vain in the 
two quarto volumes of Lobachévski’s col- 
lected works. Bolyai Janos, in his all too 
brief two dozen pages, gives solutions of 
them startling in their elegance. 
But in establishing his theory, he uses, 
for the sake of conciseness, the principle 
of continuity even more freely than does 
Lobachévski. 
SCIENCE. 
549 
Gérard, in the second part of his memoir, 
gives the elements of non-Euclidean analy- 
tic geometry, and in the third part, a strict 
treatment of equivalence. 
Even Euclid, in proving his I., 35, ‘ Par- 
allelograms on the same base, and between 
the same parallels, are equal to one an- 
other,’’? does not show that the parallelo- 
grams can be divided into pairs of pieces 
admitting of superposition and coincidence. 
He uses rather the assumption explicitly set 
forth by Lobachévski, ‘‘ Two surfaces are 
equal when they are sums or differences of 
congruent pieces.”’ But Creswell in his 
Treatise of Geometry, showed how to cut 
the parallelograms into parts congruent in 
pairs. The same can be done for Euclid 
I., 43, ‘‘ The complements of the parallelo- 
grams, which are about the diagonal of any 
parallelogram are equal.”’ Hence, we may 
use the definition : Magnitudes are equiva- 
lent, which can be cut into parts congruent 
in pairs. This method I applied to the 
ordinary Euclidean geometry in my Ele- 
mentary Synthetic Geometry before the ap- 
pearance of Gérard’s work, where it is ex- 
tended to the non- Euclidean. 
Regarding the first assured construction 
of Euclid and Gérard: ‘A straight line 
can be drawn through any two points,’’ W. 
Burnside has given us a charming little 
paper in the Proceedings of the London Math- 
ematical Society, Vol. XXIX., pp. 125- 
182 (Dee. 9, 1897), enitled ‘The Construc- 
tion of the Straight Line Joining Two 
Given Points.’ Euclid’s postulate implies 
the use of aruler or straight-edge of any 
required finite length. The postulate is 
clearly not intended to apply to the case in 
which the distance between the two points 
is infinite. In fact, Huclid I., 31, gives a 
compass and ruler construction for the line 
when one of the points can be reached while 
the other cannot. The other exceptional 
case when neither point can be reached, 
i. €., When two given points are the points 
