Page 10 - J. C. Turner - History and Science of Knots

Preface ix

It seems that it was not until the late 17th century and just beyond,
when mathematics flourished and scientific reasoning began to lead the minds
of great thinkers, that knots became worthy elements of study, in searches for
rational nautical knowledge. In the mid-18th century the grand encyclope-
dia of Diderot and D'Alembert appeared, and knots and their uses became
recorded as part of man's heritage. It was about this time that shipbuilding
was becoming more than an art: methodical thinking was being given to hull
shapes, strengths of spars, and ropes and riggings. Studies of principles of sea-
manship began to be made too. By the end of the 18th century, books such
as Hutchinson's A Treatise on Practical Seamanship, with Hints and Remarks
Relating Hereto: designed to contribute something towards fixing rules upon
philosophical and rational principle (1777), and Steel's Elements and Practice
of Rigging and Seamanship (1794), had appeared. Textbooks bearing instruc-
tion on seamen's knots found ready sale, as the need for trained ship's officers
grew rapidly with the great expansion of the British Navy during the period of
the Napoleonic Wars. As is so often observed in history, advances of a science
are greatest in times of war.
In view of the venerable and extensive use of knots over the long mil-
lennia, it is somewhat surprising that they did not become a topic deemed
worthy of mathematical study until the late eighteenth century. No doubt
there had been many attempts in the past to discover the most efficient kinds
of knot, and the best kinds of rope to use, in specific applications-for exam-
ple in seamanship. But it seems that before about A.D. 1771 no-one saw any
possibility* of modelling knots mathematically; one certainly couldn't apply
Euclidean geometry to do so-and that generally was firmly believed to be
the mathematics of God's plan for the Universe and all that it contained.
It took the genius of a man like Gauss, one of the greatest mathematicians
and astronomers the world has known, to break this taboo. In 1794 he prepared
sketches of thirteen knots (Fig. 2), with English names, perhaps copied from an
English book of seafarers' knots; this note, together with other papers bearing
sketches of knots, was found amongst his papers after his death in 1855. It is
cleat from his notes that he considered the study of knots to be an important
task for mathematicians to attempt; and he encouraged some of his students
to engage in that study. He himself published only one paper in the field.
The story of these beginnings of topological knot theory, and of its gradual
development into what it has now become, a major branch of Topology, is told
in Part IV, Chapter 11.

*In 1771, A. T. Vandermonde (1735-1796) published a paper in Memoires de 1'Academie
Royale des Sciences, Paris, in which he suggests that `the craftsman 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 title
of his paper was `Remarques sur les problemes de situation.'