- PROJECTIVE AND DESCRIPTIVE GEOMETRY 367
and declared a military secret. He wrote a book on descriptive geometry and one
on the applications of analysis to geometry, whose influence appeared in the work
of his students. Klein says of the second book that it "reads like a novel." In this
book, Monge analyzed quadric surfaces with extreme thoroughness.
Monge is regarded as the founder of descriptive geometry, which is based on
the same principles of perspective as projective geometry but more concerned with
the mechanics of representing three-dimensional objects properly in two dimensions
and the principles of interpreting such representations. Monge himself described the
subject as the science of giving a complete description in two dimensions of those
three-dimensional objects that can be defined geometrically. As such, it continues
to be taught today under other names, such as mechanical drawing; it is the most
useful form of geometry for engineers.
Monge's greatest student (according to Klein) was Jean-Victor Poncelet (1788-
1867). He participated as a military engineer in Napoleon's invasion of Russia in
1812, was wounded, and spent a year in a Russian prison, where he busied himself
with what he had learned from Monge. Returning to France, he published his
Treatise on the Projective Properties of Figures in 1822, the founding document of
modern projective geometry. Its connection with its historical roots in the work of
Desargues shows in the first chapter, where Poncelet says he will be using the word
projective in the same sense as the word perspective. In Chapter 3 he introduces the
idea that all points at infinity in a plane can be regarded as belonging to a single
line at infinity.^14 These concepts brought out fully the duality between points and
lines in a plane and between points and planes in three-dimensional space, so that
interchanging these words in a theorem of projective geometry results in another
theorem. The theorems of Pascal and Brianchon, for example, are dual to each
other.
2.8. Jacob Steiner. The increasing algebraization of geometry was opposed by
the Swiss mathematician Jacob Steiner (1796-1863), described by Klein (1926, pp.
126-127) as "the only example known to me... of the development of mathematical
abilities after maturity." Steiner had been a farmer up to the age of 17, when he
entered the school of the Swiss educational reformer Johann Heinrich Pestalozzi
(1746-1827), whose influence was widespread, extending through the philosopher-
psychologist Johann Friedrich Herbart (1776 1841) down to Riemann, as will be
explained in the next section.^15 Steiner was a peculiar character in the history
of mathematics, who when his own originality was in decline, adopted the ideas of
others as his own without acknowledgement (see Klein, 1926, p. 128). But in his
best years, around 1830, he had the brilliant idea of building space using higher-
dimensional objects such as lines and planes instead of points, recognizing that
these objects were projectively invariant. He sought to restore the ancient Greek
"synthetic" approach to geometry, which was independent of numbers and the con-
cept of length. To this end, in his 1832 work on geometric figures he considered
a family of mappings of one plane on another that resembles somewhat Newton's
projection. Klein (1926, p. 129) found nothing materially new in this work, but
admired the systematization that it contained. The Steiner principle of successively
(^14) Field and Gray (1987, p. 185) point out that Johannes Kepler (1571 1630) had introduced
points at infinity in a 1604 work on conic sections, so that a parabola would have two foci.
(^15) Klein (1926, pp. 127-128), has nothing good to say about the more extreme recommendations
of these men, calling these recommendations "pedagogical monstrosities."