SEPTEMBER 15, 1899. ] 
more than a specialist, one who has not 
only studied a portion minutely, but has 
also taken a comprehensive glance over the 
whole. From the preface to the Ausdeh- 
nungslehre of 1844 we get an insight into the 
origin and development of his course of in- 
vestigation, and we find that it was in a 
manner the reverse of Hamilton’s. The for- 
mer started from a variety of geometrical 
facts and developed a method which is in- 
dependent of space, and has perhaps suf- 
fered from its philosophische Allgemeinheit ; the 
latter started from general philosophical 
ideas and developed an algebra which is 
uniquely adapted to space of three dimen- 
sions. But, as their subjects were largely 
the same, their results, so far as they in- 
volve truth, must also be capable of unifica- 
tion to a large extent. : 
In the preface quoted, Grassmann informs 
us that he started from the treatment of 
negatives in geometry; he observed that 
the straight lines AB and BA were oppo- 
site, and that AB + BC= AC, whether the 
point C is beyond B or between A and B. 
This led him to the principle of geometrical 
addition—namely, that AB+ BC=AC, 
whether A, B, C are in one straight line or 
not. It may be remarked here that this 
principle is all right so long as the compo- 
nents have no real order, such as forces ap- 
plied at a point or the coordinates of a 
point; but that it does not apply where the 
components have a real order, as, for ex- 
ample, the sides of a polygon. In succes- 
sive addition the straight line from the ori- 
gin to the end of the polygon is the scalar 
result, but the area enclosed is another re- 
sult, which depends on the form of the path. 
Then turning to the product in geometry, 
he adopted the view that the parallelogram 
is the product of its two sides, whether 
these are at right angles or not. He next 
found that the geometrical ideas of a sum 
and a product which he had adopted satis- 
fied the principle of distribution, but not 
‘SCIENCE. 307 
the principle of commutation so far as the 
factors of a product were concerned. In- 
the case of the products commutation could 
be made, provided the sign of the product 
were changed also—that is, they were sub- 
ject to negative commutation. Another set 
of basal facts was taken from the doctrine 
of the center of gravity. He observed that 
the center of gravity may be considered as 
the sum of several points, the line joining 
two points as the product of the points, the 
triangle as the product of its three points, 
and the pyramid as the product of its four 
points; and from these facts he developed a 
method similar to the ‘ Barycentric Calcu- 
lus,’ of Mobius. 
He also considered the geometrical mean- 
ing of the exponential function. He ob- 
served that if a denote a finite straight line 
and 2 an angle in a plane through the line, 
then ae“ denotes the line a turned through — 
the angle «. The treatment of angles in 
one plane is easy, but on attempting to 
treat of angles in space he encountered 
difficulties which he was unable to sur- 
mount. This fact has been cited as in- 
dicating the superiority of Hamilton’s 
method ; while that is true, it must not be 
forgotten that Hamilton failed to generalize 
the exponential theorem. 
What is the view which Grassmann takes 
of the fundamental principles of algebra? 
An answer to this question is found in the 
introduction to the Ausdehnungslehre of 1844. 
He divides the sciences into the real and 
the formal ; the former treat of reality, and 
their truth consists in the agreement of 
thought with reality; the latter treat of 
thought only, and their truth consists in 
the agreement of the processes of thought 
with one another. Pure mathematics is 
the doctrine of forms. As a consequence 
he is obliged to place geometry under ap- 
plied mathematics, for it has a real subject, 
and should anyone think otherwise he must 
deduce from pure thought the tridimen- 
