Paradigm, No 4 (December, 1990)
The Open University,
Milton Keynes MK7 6AA
This is an example of a book that may strike todays innocent reader as strange or bizarre, yet when seen in its historic context makes good and revealing sense. It was in 1648 that John Wilkins (c.1614-1672) published his Mathematical magic: or the wonders that may be performed by mechanical geometry . . . being one of the most easy, pleasant, useful (and yet most neglected part) of the mathematics. It was indeed the first work on mechanics written in English, but is not, to any great extent, a mechanics textbook in the sense that a mathematical practitioner might expect.
Mathematical magic consists of two books: Archimedes; or, mechanical powers, covering things like balances, levers, wheels, pulleys, wedges, screws, ballistae, and catapults; and Daedalus; or, mechanical motions, which discusses automata, sailing chariots, clocks, submarines, and perpetual motion (which he says "does not seem very probable"). The treatment is discursive and literary, with numerous classical references. Despite the nod towards the name of Archimedes, the book has little in common with the detail of Archimedes extant work, and nothing at all with its mathematical style. A typical couple of pages are those where Wilkins explains how to use gearing "to lift up the greatest oak by the roots with a straw, to pull it up with a hair, . . ." Although by Wilkins day there was a well-established tradition of using and interpreting Archimedes mathematics within mathematical-scientific texts, this is not the tradition Wilkins was seeking to promote in this publication.
What was he trying to do then? Wilkins address to the reader claims that there is "much real benefit to be learned" from the book, in particular for two classes of reader: for "such gentlemen as employ their estates in those chargeable adventures of draining mines, coal-pits, &c, who may from hence learn the chief grounds and nature of engines, and thereby more easily avoid the delusions of any cheating imposter: and also for such common artificers, as are well skilled in the practice of these arts, who may be much advantaged by the right understanding of their grounds and theory."
The stream of mathematical wonders and recreations into which Mathematical magic falls is an underrated tradition, yet important for mediating between lay readers and topics which have a great tendency towards to the esoteric. The great applied mathematical sciences of antiquity -- astronomy, statics & mechanics, optics -- had from the earliest days been practices inaccessible to all but the trained initiate, yet it is clear from Wilkins list of intended readers that there were several categories of people who would benefit from an understanding of mechanics that fell short of the full training of a practitioner. This had for centuries from the works of Heron of Alexandria (1st century AD) onwards been provided by texts which promote a fair degree of mathematical and scientific understanding in the guise of mathematical recreations or mechanical curiosities, in a fairly haphazard way but which nonetheless equips readers with cumulative knowledge and understanding.
Looking at the tradition of discourse helps make sense of what may otherwise seem rather over the top -- the pulling up of an oak by a thread, for example, or some of the mechanical devices discussed in the Daedalus section of Mathematical magic come over better as vivid thought-experiments meant to impart an insight into the underlying principles, rather than practical proposals. We find a long tradition of explaining principles via lively examples. Not uncommonly the same illustration is handed down from text to text over the centuries -- not only are most of Wilkins examples and illustrations from earlier continental writers notably Guidobaldo del Monte and Marin Mersenne), but we find the same illustrations in later texts such as Hoopers Rational recreations . And debates on the status of examples are found in the literature of the period. Thus Robert Boyle criticised Pascals discussion of hydrostatics on the grounds that he did not explain how, in one illustration, a man was to sit at the bottom of a 20-foot tub of water with a cupping glass held to his leg. Wilkins himself distinguishes between rational and chirurgical mechanics, the distinction being. between understanding mechanical principles and constructing machines, in a context of concern about the worthiness of different activities, whether certain studies are more honourable than others. The old Platonic discussions of whether working on actual machines is inferior to pure knowledge, such a strong feature of reports of Archimedes work, are evidently still current in the 17th century, and Wilkins is trying to promote the value of mechanical understanding.