332 
isomorphisms existing between these groups 
are tabulated. As many as six forms of a 
single group of order 25,920 are given, this 
group having applications in various geo- 
metric problems. 
Professor Asaph Hall, Jr., of the Uni- 
versity of Michigan, communicated to the 
Section certain results of a series of obser- 
vations of the meridianal zenith of Polaris 
made by him between May, 1898, and July, 
1899, with a view to determining the lati- 
tude variation at Ann Arbor and the aber- 
ration constant. The observations were 
made above and below the pole, both direct 
and reflected. The direct observations, at 
upper and lower culmination, respectively, 
give the values for the aberration constant 
20.60 and 20.58, and for the parallax 
0.32 and 0.29. The reflected observa- 
tions show a close agreement with the di- 
rect observations. The observations are 
being continued. 
A paper entitled ‘ Ancient Eclipses and 
Chronology’ was presented by R. W. Mac- 
Farland, of Oxford, O. It is, in the main, 
a critical examination of the sources in an- 
cient history from which the commonly ac- 
cepted dates of various events are deter- 
mined, and especially such sources as 
involve references to eclipses. The author 
of the paper reaches the conclusion that in 
each case examined the historical statement 
connecting a specified important event with 
an eclipse is either inadequate to establish 
accurately the relation of the two in time 
or else that the computations of the astron- 
omers of the present day are not of sufficient 
accuracy: to fix the eclipse in question 
within several years of the truth. 
Professor H. C. Lord, of the Ohio State 
University, gave an interesting account of 
an investigation in which he has lately been 
engaged as to the best relative dimensions 
for different parts of a spectroscope which 
is to be used photographically, not visually. 
His account was fully illustrated by photo- 
SCIENCE. 
[N.S. Vou. X. No. 245. 
graphs and tables of results. He insisted 
especially upon certain advantages to be 
gained by using a camera of considerable 
focal length rather than one of short focus. 
The proof of Grassmann’s fundamental 
theorem, that there can be but two kinds 
of lineal multiplication of two factors, is 
somewhat long and rather difficult to fol- 
low. The object of the paper presented by 
Mr. Jos. V. Collins was to show how this 
proof may be shortened and simplified. 
Professor G. J. Stokes, of Queen’s Col- 
lege, Cork, Ireland, was prevented by sick- 
ness from finishing his paper on ‘The Theory 
of Mathematical Inference.’ The abstract 
of the incomplete paper indicates that the 
theory is advocated that the fundamental 
truths of mathematics are logical conse- 
quences of the mere fact or possibility of 
synthesis generally, and that ordinary 
mathematical inference is compounded of a 
logical or analytical element which has 
been reduced to mathematical form in 
Boole’s Laws of Thought, and a synthetic 
element represented by Algebras of the 
type of Grassmann’s Ausdehnungslehre. 
A paper entitled ‘ Practical Astronomy 
during the First Half of the Present Cen- 
tury,’ by Professor T. H. Safford, of Wil- 
liamstown, Mass., unfortunately arrived too 
late to be read before the Section. Itisashort 
and interesting account of the relations of 
the eminent astronomers Gauss, Bessel, the 
elder Struve and Airy to the astronomical 
progress made during the period stated. 
Dr. G. A. Miller, of Cornell University, 
presented a short and interesting paper ‘ On 
the Commutators of a Group.’ The fol- 
lowing relations were brought out and com- 
mented upon by Dr. Miller: (1) If with a 
given group (G) commutators are formed 
with a fixed operator and all the operators 
of that group these commutators will gen- 
erate a group which is transformed into itself 
by all the operators of the group G. (2) 
When the fixed operator transforms the 
