What was the point of it all? What was the use of the huge labour of copying down two hundred pages of text and diagrams with such care, when there were perfectly good geometry textbooks in print, many of them at very reasonable prices?
You’ll probably know the famous story about the death of Archimedes: Archimedes was,
as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.
That account tells us, in a way, the same story as Thomas Porcher’s copy-book. Geometry was often thought of, in the ancient and the early modern worlds, as an absorbing discipline, involving both the mind and the hand and enabling the student to be indifferent to all kinds of distractions. It was often held up as a useful and an ennobling study for that very reason: by fixing the mind on higher, immaterial things it trained the student in proper methods of reasoning and taught a virtuous detachment from the distractions of everyday life. It could ‘charm the passions, restrain the impetuosity of imagination, and purge the Mind from error and prejudice’, ‘accustoming it to attention’, and ‘giving it a habit of close and demonstrative reasoning’, according to the Scot John Arbuthnot, whose essay on the ‘Usefulness of Mathematics’ was reprinted several times during the first half of the eighteenth century.
Those were reasons for teaching geometry to well-to-do students, whether in schools or universities (it was often lamented during the eighteenth century that geometry had fallen too much out of the university curriculum). They were reasons for studying geometry in the meticulous, deductive way Thomas Porcher did: if you wanted to beautify your mind and train yourself in proper habits of thought and reasoning.
But this was also the age of Newtonian science, and of rapidly increasing confidence about what mathematics could do in practical life. For an increasing range of trades it was vital to learn some mathematics: for some it had always been vital. Consider surveying, or navigation, the construction of sundials (‘dialling’) or the estimation of liquid volumes in barrels (‘gauging’). They all required a use and perhaps an understanding of geometrical ideas. And they were all rather far removed, both intellectually and socially, from the ideas about mental purity and abstraction we’ve just been talking about. It’s sometimes said that mathematical education split into two halves in Georgian Britain: deductive methods for the sons of gentlemen; practical rules for their social inferiors. Although there were exceptions, that pattern was particularly clear in the case of geometry, where there really were two different ways of learning the subject depending on why you were doing so.
A second example will show what I mean.