May 10, 1912] 
are corrected the better. Let not the name of 
a world hero be bungled in the world language, 
English. 
The theory of parallels as it exists in hyperbolic 
space has no application in elliptic geometry. But 
another property of Huclidean parallel lines holds 
in elliptie geometry, and by the use of it parallel 
lines are defined. Thus throughout every point of 
space two lines can be drawn which are lines of 
equal distance from a given line l. 
This property was discovered by W. K. Clifford. 
The two lines are called Clifford’s right and left 
parallels to 7 through the point. 
In both elliptic and hyperbolic geometry the 
spherical geometry is the same as the ‘‘spherical 
trigonometry’’ in Huclidean geometry.’ 
The historical sketch is blemished by the 
unwarranted prominence it gives to Gauss. It 
says: 
We find him in 1804 still hoping to prove the 
postulate of parallels. In 1830 he announces his 
conviction that geometry is not an a priori science; 
in the following year he explains that non- 
Euclidean geometry is free from contradictions, 
and that, in this system, the angles of a triangle 
diminish without limit when all the sides are 
increased. He also gives for the circumference of 
a circle of radius r the formula mk(e"/k — er/k), 
[In this formula the Encyclopedia has a 
misprint. ] 
But all that and immensely more had been 
given by John Bolyai in 1823 and by Loba- 
cheyski in 1826, and published in 1829, while 
as our authors themselves say, “ Gauss pub- 
lished nothing on the theory of parallels.” 
Then comes the most offensive clause: 
It is not known with certainty whether he in- 
fluenced Lobachevski and Bolyai, but the evidence 
we possess is against such a view. 
But it zs known that he did not, and the evi- 
dence we possess against any such influencing 
is absolute and final. The very next sentence 
is the opening one of my Translator’s Preface, 
1891: ‘ 
Lobachevski was the first man ever to publish 
a non-Euclidean geometry. 
Of Bolyai’s work is said: 
1See chapter XVI., Pure Spherics, in my ‘‘Ra- 
tional. Geometry.’ 
SCIENCE 
137 
Its conception dates from 1823. It reveals a 
profounder appreciation of the importance of the: 
new ideas, but otherwise differs little from Loba- 
chevski’s. Both men point out that Euclidean 
geometry is a limiting case of their own more: 
general system. 
[The Encyclopedia, by a misprint, has as: 
for is.] 
The works of Lobachevski and Bolyai, though 
known and valued by Gauss, remained obscure and 
ineffective until, in 1866, they were translated into 
French by J. Hoiiel. 
Bolyai was not translated until 1868. Not 
only were these known to Gauss, but I called’ 
attention to the very significant fact that the 
striking work of Saccheri, truly a non-eucli- 
dean geometry, was in the Gottingen library 
and freely accessible to Gauss during the years 
1790-1800. See Gino Loria,’ who says of 
Gauss: 
Ignoto fino a qual punto egli siasi spinto nella 
nuova via, come 6 ignoto se egli abbia ricevuto 
qualche ispirazione dall’ opera del Saccheri che 
esisteva a Gottinga negli anni 1790-1800 (essendo- 
segnata con un asterisco nella Bibliotheca mathe- 
matica del Murhard).$ 
If figures are to be freely movable, it is neces- 
sary and sufficient that the measure of curvature 
should be the same for all points and all directions 
at each point. Where this is the case, it a be the 
measure of curvature.... 
This z¢ should be 7f. 
If a be positive, space is finite, though still 
unbounded, and every straight line is closed—a 
possibility first recognized by Riemann. 
This, as it stands, is a mistake. On page 24 
of von Staudt’s “ Geometrie der Lage” (1847) 
we read: 
Eine Gerade erscheint hiernach .. . 
geschlossene Linie. 
als eine: 
The possibility first recognized by Riemann: 
is that straight lines may be finite. 
On page 729 occurs the long dead phrase: 
“anharmonic ratio,’ now happily superseded 
everywhere by Clifford’s “ cross ratio.” 
?Tl passato ed il presente delle principali teorie: 
geometriche. Terza edizione, 1907, pp. 286-287. 
3 Osservazione fatta dall’ Halsted nell’ articolo. 
‘‘The Non-Huclidean Geometry Inevitable’? in-- 
serto in The Monist, July, 1894. 
