654 
applied to rigid bodies; the forms of the third 
order represent planes and the plane at infinity. 
Among the operations, the progressive and re- 
gressive products give the geometric operations 
of projecting and cutting; the inner product 
gives the orthogonal projections and the ele- 
ments which we designate in mechanics by the 
terms moment, work, ef cetera. 
In ordinary differential geometry simple prop- 
erties most frequently yield themselves only 
after very complicated calculations. This com- 
plication is due in general to the use of coordi- 
nates ; with these codrdinates algebraic trans- 
formations are made on numbers in order to 
obtain certain formule, namely, invariants, 
which are susceptible of geometric interpreta- 
tions. On the other hand the geometrical 
calculus makes no use whatever of codrdi- 
nates; it operates directly on the geomet- 
ric elements ; each formula which it produces 
is an invariant, capable of a simple geomet- 
ric interpretation and leading directly to the 
graphic representation of the elements con- 
sidered. Burali-Forti’s work, though by no 
means a pioneer in the application of Grass- 
mann’s theories to differential geometry (note 
for example the memoirs of the younger Grass- 
mann in the theory of curves and surfaces), 
shows the elegant power and simplicity of the 
geometrical calculus in elementary differential 
geometry and points the student to a vast field 
of transformations and researches in higher 
geometry. 
The work is designed after the following plan 
which exhibits the skeleton of its contents: 
I. The geometric forms.—1° Definitions and 
rules of caleulus:—tetrahedron, geometric forms, 
equality of forms, points, segments, triangles, 
sum and product by a number, progressive 
product; 2° Vectors and their products :— 
vectors, bivectors, trivectors, rotation, opera- 
tion index; 3° Reduction of forms:—forms of 
the first order, forms of the second order, forms 
of the third order, projective elements, identity 
between forms of the first order ; 4° Regressive 
products :—forms of the second and third orders, 
forms of the third order, general properties of 
products, duality, regressive products in a pro- 
jective plane ; 5° Coordinates. 
II. Variable forms.—1° Derivatives :—defi- 
SCIENCE. 
[N. S. Vou. X. No. 253. 
nitions, limit of a form, limit of a projective 
element, derivatives, mean forms, Taylor’s 
formula, continuous forms ; 2° Lines and en- 
velopes :—lines and envelopes of straight lines 
on a projective plane, space curves and en- 
velopes of planes; 8° Ruled surfaces :-—ruled 
surfaces in general, skew ruled surfaces, de- 
velopable surfaces ; 4° Frenet’s formule :—arcs, 
curvature and radius of curvature, torsion and 
radius of torsion, formule of Frenet, spherical 
indicatrix and angle of contingence. 
III. Application.—1° Helix; 2° Surfaces 
ruled relative to a curve—polar surface, rectify- 
ing surface, surface of principal normals, surface 
of binormals, skew ruled surfaces whose line of 
striction is given, developable ruled surface de- 
scribed by a straight line whose position is fixed 
with regard to the tetrahedron PTNB; 3° 
Orthogonal trajectories :—orthogonal trajec- 
tories of the generatrices of a ruled surface, 
evolutes, involutes, orthogonal trajectories of 
planes of an envelope; 4° Curves of Bertrand. 
Nores.—1° Forms which are functions of two 
or more variables; 2° Tangent plane; 3° Dif- 
ferential parameter of first order; 4° Curvilinear 
coordinates. E. O. Lovett. 
PRINCETON, NEW JERSEY. 
Chemical Experiments. By JOHN F. WOoODHULL, 
Professor of Physical Science, Teachers Col- 
lege, Columbia University, and M. B. VAN 
ARSDALE, Instructor in Physical Science in 
Horace Mann School and Assistant in 
Teachers College. New York, Henry Holt 
& Co. 1899. Pp. 136. Price, 50 cents. 
This book gives a series of very elementary 
experiments dealing chiefly with the elements 
oxygen, hydrogen, chlorine, sulphur, nitrogen 
and carbon. The apparatus recommended for 
the experiments is simple, and in several cases, 
quite ingenious, For pupils of a certain grade 
the book will doubtless prove useful, but the 
introduction of a few more quantitative experi- 
ments designed to illustrate fundamental princi- 
ples seems desirable. 
A Laboratory Outline of General Chemistry. By 
ALEXANDER SMITH. Chicago, Kent Chem- 
ical Laboratory of the University of Chicago. 
1899. Pp. xii+90. 
The work before us represents a very dis- 
