*How to Draw a Straight Line* by A B Kempe

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 85

Have you ever encountered such an intriguing title? I encountered it in 1957 in Mathematical Models by Cundy and Rollett (OUP) and, ever since then I have kept an eye open for it. Its subtitle is A LECTURE ON LINKAGES, it is fifty-one pages long and it was published by Macmillan in 1877. Nice to see Vector is up to date!

I tried to get a copy from the national lending library, but failed and when I mentioned it to my son-in-law he downloaded it from Cornell University! It was worth the wait. I have nothing but praise for the book. Here is how Kempe begins:

The great geometrician Euclid, before demonstrating to us the various propositions contained in his Elements of Geometry, requires that we should be able to effect certain processes. These Postulates, as the processes are termed, may roughly be said to demand that we should be able to describe straight lines and circles. And so great is the veneration that is paid to the master-geometrician, that there are many who would refuse the designation of "geometrical" to a demonstration which requires any other construction than can be effected by straight lines and circles. Hence many problems - such as, for example, the trisection of an angle - which can readily be effected by employing other simple means, are said to have no geometrical solution, since they cannot be achieved by straight lines and circles only.

But how do you draw a straight line? A circle is easy. In principle the method is perfect although in practice your pencil may be blunt. Creating a straight edge is as fraught with trial and error as making an engineer’s surface table. How odd that Euclid didn’t notice how different it is from a circle. Until 1874 no-one in England knew of a method for drawing a straight line that was, in principle, perfect. The first solution was found by a French army officer called Peaucellier and was brought to England by Professor Sylvester (the same J J Sylvester you have heard Donald McIntyre speak about) in a lecture at the Royal Institution in January 1874. Kempe treats us to descriptions of Watt’s Parallel Motion, better linkages by Richard Roberts of Manchester and "Professor Tchebicheff" of St Petersburg, then M. Peaucellier’s perfect solution with seven links and Mr Hart’s (of Woolwich Academy) improved version with five links. Kempe’s note on Watt’s Parallel Motion is worth quoting in full.

I have been more than once asked to try and get rid of the objectionable term "parallel motion." I do not know how it came to be employed, and it certainly does not express what is intended. The apparatus does not give "parallel motion," but approximate "rectilinear motion." The expression, however has become crystallised, and I for one cannot undertake to find a solvent.

Kempe tells of his correspondence with Sylvester and their invention of the "Quadruplane" and describes other linkages, including those for dividing an angle into any desired number of equal parts. Beautifully written and fascinating!

How to find it: Google found it straight away, given the title in double quotes. http://moa.cit.cornell.edu/math_K.html takes you to the index by authors, section K, where it is easy to find Alfred Bray Kempe. The address of the first page of download was:

http://moa.cit.cornell.edu/Hunter/hunter.pl?handle=cornell.library.math.Kemp009&id5