NOVEMBER 3, 1899.] 
Legislature which has prevented publication of 
the volume on coal. That interest, owing to 
the sudden expansion of iron manufacture, is 
now paramount, and the state is losing enor- 
mously by this failure to publish the material 
accumulated by Dr. White in extended recon- 
naissances during the last ten years. To those 
engaged in investigating the serious problems 
presented by the Carboniferous, the inacces- 
sibility of this material is a misfortune. 
The map shows the oil fields and productive 
areas of the several coal series. The limits of 
the Pittsburg, as determined by borings, differ 
from the Rogers lines as much for West Vir- 
ginia as for Ohio. The geographical conditions 
during the formation of that bed were evidently 
very unlike those suggested by the older geol- 
ogists. 
The abundance of typographical errors is 
evidence that the author had no opportunity to 
correct the proofs, and reminds the writer of his 
own experience with the West Virginia State 
Printer almost thirty years ago, when Mr. F. 
B. Meek and he were made chargeable with 
statements which afforded some annoyance to 
them and much amusement to their acquaint- 
ances. JOHN J. STEVENSON. 
Introduction & la géométrie différentielle suivant 
la méthode de H. GRASsMAN. Par C. Burali- 
Forti, professeur 4 l’Académie militaire de 
Turin. Paris, Gauthier- Villars. 1897. 8vo. 
Pp. xi + 165. 
This volume contains a brief exposition of the 
geometrical calculus and some of its applications 
to elementary differential geometry. 
The analytical geometry of Descartes (1637), 
operates on numbers which have an indirect re- 
lation with the geometrical elements which 
they represent. Leibnitz* in 1679 recognized 
the advantages of a geometrical calculus opera- 
ting directly on geometrical elements, but the 
operation suggested by Leibnitz does not pos- 
sess the ordinary properties of algebraic opera- 
tions. The idea, however, was fruitful, and in 
1797 Caspar Wessel} gave an analytical repre- 
*Leibnitz, Math. Schriften, II., V., Berlin, 1849. 
ft Caspar Wessel, Om Directionens analytiske 
Betegning, March 10, 1797; published by the Den- 
mark Academy of Sciences, Copenhagen, 1897. 
SCIENCE. 
653 
sentation of direction which contains Argand’s 
(1806) geometrical interpretation of complex 
numbers and several of the operations intro- 
duced by Hamilton (1843-1854) in his method 
of quaternions. Later the barycentric calculus 
(1827-1842) of Mobius and the method of 
equipollences (1832-1854) of Bellavitis brought 
forward two independent methods of geometric 
calculus which their authors applied to various 
questions of geometry and mechanics. In 1843 
Hamilton published his first essay on quater- 
nions ; the complete development of this theory 
in 1854 gives a complete geometrical calculus 
which finds at present its most extensive appli- 
cations in mathematical physics. The works 
of Hamilton were preceded by the Ausdehnungs- 
lehre (1844), of H. Grassmann which, in the 
power and simplicity of its operations, sur- 
passes all other known forms of geometrical 
calculus. The method of exposition adopted 
by Grassmann is exceedingly abstract and this 
fact has stood stubbornly in the way of the 
general adoption of the Ausdehnungslehre to 
such an extent that we use to-day the bar 
tric calculus, the theory of equipollences, 
quaternions, or the Cartesian’geometry, for the 
resolution of geometric questions which are 
capable of much more simple resolution by the 
methods of Grassmann. These classic objec- 
tions to Grassmann’s exposition have been 
met recently by Peano* who has given con- 
crete geometric interpretations to the forms 
and operations of the Ausdehnungslehre. There 
is a splendid account of the importance of this 
discipline in geometry, mechanics and physics 
to be found in the historical memoir of Schlegel. + 
M. Burali Forti gives the elements of Grass- 
mann’s calculus as reconstructed by Peano. 
The latter took the idea of a tetrahedron as his 
starting point and defined the product of two 
and three points ; he then defined the products of 
these elements by numbers and finally gave defi- 
nitions of the sums of these products. The 
theory of forms of the first order gives the bary- 
centric calculus and that of vectors; the geo- 
metric forms of the second order represent 
straight lines, orientations, and systems of forces 
* Peano, Calcolo geometrico, Turin, 1888. 
+ Schlegel, Die Grassmann’sche Ausdehnungslehre, 
Zeitschrift fiir Math. und Physik, 1896. 
