368 12. MODERN GEOMETRIESbuilding more and more intricate figures by allowing simpler ones to combine ge-
ometrically was novel and had its uses, but according to Klein, encompassed only
one part of geometry.
2.9. August Ferdinand Mobius. Projective geometry was enhanced through
the barycentric calculus invented by August Ferdinand Mobius (1790-1868) and
expounded in a long treatise in 1827. This work contained a number of very useful
innovations. Mobius' use of barycentric coordinates to specify the location of a
point anticipated vector methods by some 20 years, and proved its value in many
parts of geometry. He used his barycentric coordinates to classify plane figures in
new ways. As he explained in Chapter 3 of the second section of his barycentric
calculus (Baltzer, 1885, pp. 177-194), if the vertices of a triangle were specified as
A, B, C, and one considered all the points that could be written as aA + bB + cC,
with the lengths of the sides and the proportions of the coefficients a : b : c given,
all such figures would be congruent (he used the phrase "equal and similar"). If
one specified only the proportions of the sides instead of their lengths, all such
figures would be similar. If one specified only the proportions of the coefficients,
the figures would be in an affine relationship, a word still used to denote a linear
transformation followed by a translation in a vector space. Finally, he introduced
the relation of equality (in area).
Cauchy, then at the height of his powers, reviewed Mobius' work^16 on the
barycentric calculus. In his review, as reported by Baltzer (1885, pp. xi-xii), he was
cautious at first, saying that the work was "a different method of analytic geometry
whose foundation is certainly not so simple; only a deeper study can enable us to
determine whether the advantages of this method will repay the difficulties." After
reporting on the new classification of figures in Part 2, he commented:
One must be very confident of taking a large step forward in science
to burden it with so much new terminology and to demand that
your readers follow you in investigations presented to them in such a
strange manner.Finally, after reporting some of the results from Part 3, he concluded that, "It
seems that the author of the barycentric calculus is not familiar with the general
theory of duality between the properties of systems of points and lines established
by M. Gergonne." This comment is difficult to explain on the assumption that
Cauchy had actually read Chapters 4 and 5 of Part 3, since this duality (gegenseit-
iges Entsprechen) was part of the title of both chapters; but perhaps Cauchy was
alluding to ideas in Gergonne's papers not found in the work of Mobius. Chap-
ters 4 and 5 contain some of the most interesting results in the work. Chapter 4,
for example, discusses conic sections and uses the barycentric calculus to prove that
two distinct parabolas can be drawn through four coplanar points, provided none
of them lies inside the triangle formed by the other three.
(^16) It might appear that Cauchy was able to read German, not a common accomplishment for
French mathematicians in the 1820s, when the vast majority of mathematical papers of significance
were written in French. But perhaps he read a French or Latin version of the work.