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      Contacts in London made a huge difference. After the Guide Hutton devised a new, more ambitious publication project: a book on mensuration. This could have been a subject for another slim textbook on the model of The School-master’s Guide. But Hutton had something much grander in mind. Not a little book of practical rules but a veritable encyclopedia covering every aspect of geometry and its practical use. Hutton took to riding over to the village of Prudhoe at weekends to consult with the schoolmaster there, a Mr Young, who coached him in advanced geometry and mensuration and, it was said, worked over drafts of his new book with him.

      Announced in the Newcastle papers in December 1767, Hutton’s Treatise on Mensuration appeared in twenty-eight instalments between March 1768 and November 1770. His publisher diligently promoted it in a range of national and local newspapers. Hutton undertook his own publicity campaign, writing personally to a long list of philomaths culled from The Ladies’ Diary and elsewhere. He obtained permission to dedicate the book to the Duke of Northumberland.

      The results were spectacular. When the Mensuration appeared as a single collected volume at the end of 1770, the list of subscribers contained more than six hundred names. Probably amounting to more than half the active mathematicians and lovers of mathematics in the United Kingdom, from Penzance to Dundee, they included two dukes, one earl, and astronomers from Oxford University and the Royal Observatory. Both the English universities and most of the Scottish ones were represented, as were surveyors and instrument makers, schoolmasters and country curates, surgeons, excise officers and Fellows of the Royal Society.

      This was self-publicity on a scale rare in Hutton’s century, or indeed in any. A good deal of money was involved – 600 subscriptions at fifteen shillings a book were not to be sneezed at – but the visibility had a value that could hardly be measured. It was rapidly becoming impossible to do mathematics, to like mathematics, to be aware of mathematics in Great Britain without knowing Charles Hutton’s name. Hutton was well advanced on the road from provincial schoolmaster with a taste for mathematical puzzles to national celebrity. He was aware of the change himself, of course. Throughout the 1760s he called himself ‘schoolmaster’ or ‘writing master’, but by the 1770s his title pages proclaimed him ‘author’; in 1772 he would switch to ‘mathematician’. And while Tonthu had been ‘of Newcastle’, Charles Hutton could call the town coldly ‘that part of the country in which I reside’, implying choice, impermanence, a lack of decisive ties to his provincial life.

      What the subscribers to the Mensuration got for their money was a fat book which, despite a title that associated it with practical matters, was in fact a comprehensive treatment of theoretical as well as practical geometry – a substitute for Euclid’s Elements, indeed, as far as geometry was concerned. It began with definitions (‘A Line is a length conceived without breadth’), and it ran all the way up to the volumes of polyhedra and the areas contained under algebraically defined curves. Most of the book consisted of increasingly complex geometrical problems with rules for solving them: to find the area of a semicircle; to find the volume of a segment of a sphere; to find the surface area of a hyperboloid. Compared with the Guide, the emphasis was much less on carefully graded examples and much more on comprehensiveness, on a solid, gap-less treatment of a large body of material. It explained how to find the areas and the volumes of certain shapes and solids; how to construct certain curves and surfaces such as those arising from slicing a cone, or from rotating the conic sections that resulted (‘A conoid is a solid conceived to be generated from the revolution of the parabola or hyperbola about the transverse ax[is]’).

      There was elegance and beauty here; there was also a tremendous display of learning. Hutton regularly succumbed to the temptation to exhibit his own cleverness at the expense of relevance or logical structure. Some of the more advanced, more difficult or more important solutions were backed up by proofs given in footnotes – and sometimes the footnotes were long, elaborate, even showy. Attached to a problem about right-angled triangles he indulged in not merely a demonstration but a ‘General Scholium’ (the term irresistibly recalls Isaac Newton’s use of it in his magnum opus, the Principia Mathematica) with ‘some new theorems concerning the relations of the sides and angles of triangles’. These derived from infinite series – never-ending algebraic formulae – for the sine of an angle. A problem aimed at finding the circumference of a circle from its diameter and vice versa occasioned a note which set out the history of infinite series for the circular ratio from the seventeenth century onwards. Finding the length of part of an ellipse became the occasion for another, typical burst of complexity, with the full apparatus of geometry, algebra and calculus deployed to provide five different ways of solving the problem. Passages like this give an impression of extraordinary authorial dexterity: partly because they seem slightly out of place, obtruding upon the reader’s notice.

      So this book, like the Guide, carefully

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