Page 221 - J. C. Turner - History and Science of Knots

A History of Topological Knot Theory 211

his study Analysis Situs or Geometria Situs, and it comes closest to what we
now would call Combinatorial Topology [57], the discipline in which geometri-
cal figures are considered as aggregates of smaller building blocks. Leibniz did
not go so far as to study knots; but his endeavours at finding a geometry of
this kind, different from the only one known at the time, predated other work
in this direction by more than half a century.
Although thinkers like Leibnitz recognized the need for different geome-
tries, it was not until 1771 that the birth of knot theory occurred. In that year
Alexandre Theophile Vandermonde (1735-1796) wrote a paper [90] (see also
[30]), in which he specifically places knots into the arena of the geometry of
position. In the opening paragraphs, Vandermonde includes the lines:

Whatever the twists and turns of a system of threads in space, one
can always obtain an expression for the calculation of its dimen-
sions, but this expression will be of little use in practice. The crafts-
man who fashions a braid, a net, or some knots will be concerned,
not with questions of measurement, but with those of position: what
he sees there is the manner in which the threads are interlaced.

The possibility for a mathematical study of knots was probably first rec-
ognized by the truly great mathematician and physicist , Carl Friedrich Gauss
(1777-1855), of Gottingen, Germany. One of the oldest notes found amongst
his papers after his death was on a sheet of paper dated 1794, which bore
the caption A Collection of Knots. It contains thirteen sketches of knots with
English names written beside them . It is probably an excerpt he copied from
an English book. With it are two additional pieces of paper with a few more
sketches of knots. One is dated 1819, the other some eight years later [31].
Notes of Gauss referring to the knotting together of closed curves appear in
his collected works [38]. During the period of 1823 -1827 he was working on
Geometria Situs about which he later wrote, on 22 January 1833:

Eine Hauptaufgabe aus dem Grenzgebiet der Geometria Situs and
der Geometria Magnitudinis wird die sein, die Umschlingungen
zweier geschlossener oder unendlicher Linien zu zahlen.*

His work on electromagnetism had led him to compute inductance in a system
of two linked circular wires; and he introduced the concept of winding numbers
(or linking numbers), which are now a basic tool in knot theory and other

*[One of the main tasks in the borderland between Geometric Situs and Geometria Magni-
tudinus will be to count the `windings around' of two closed or infinite lines.]