Measuring America. Andro Linklater

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      And so have all Lordes that landes do possesse: But Tennauntes I feare will like me the lesse. Yet do I not wrong, but measure all truely, And yelde the full right to everye man justely. Proportion Geometricall hath no man opprest, Yf anye bee wronged, I wishe it redrest.

      It was against this background – an urgent and growing need for the accurate measurement of land – that Edmund Gunter devised his chain. Born in 1581 to a Welsh family, Gunter had been sent to Oxford University to be educated as a Church of England priest, but by the time he was ordained he had discovered that numbers were more inspiring to him than religion. In twelve years as a divinity student he preached just one sermon, and its reputation endured long after his death because, according to Oxford gossip, ‘it was such a lamentable one’. What really interested him was the relationship of mathematics to the real world, and consequently he spent most of his time making instruments to illustrate the way in which numbers worked.

      Ratios and proportions were his passion. He invented an early slide-rule, known as Gunter’s scale, to demonstrate proportional connections between numbers, and worked out to seven places of decimals the logarithms for sine and cosine. The point at which this enthusiasm for numerical ratios touched upon concrete reality was trigonometry, which allowed mathematicians to calculate the length of two sides of a triangle, when only the third side and two angles were known; it also, as we have seen, enabled surveyors to work out the distance between two objects without having to walk between them.

      Since only the most basic instruments existed at that time, mathematicians and astronomers were expected to design their own. To demonstrate the solutions to problems in geometry, Gunter was constantly adapting and improving nautical instruments like the quadrant and cross-staff, which measured vertical angles between the sun and the horizon, or horizontal angles between towers, trees and churches. Indeed his enthusiasm for new gadgets cost him the best scientific job in the land. In 1620 the wealthy but earnest Sir Henry Savile put up money to fund Oxford University’s first two science faculties, the chairs of Astronomy and Geometry. Gunter applied to become Professor of Geometry, but Savile was famous for distrusting clever people – ‘Give me the plodding student,’ he insisted drearily – and the candidate’s behaviour annoyed him intensely. As was his habit, Gunter arrived with his sector and quadrant, and began demonstrating how they could be used to calculate the position of stars or the distance of churches, until Savile could stand it no longer. ‘Doe you call this reading of Geometrie?’ he burst out. ‘This is mere showing of tricks, man!’, and according to a contemporary account, ‘dismisst him with scorne’.

      Fortunately Gunter was supported by the Earl of Bridgewater, who did like brilliance, having grown up in a house where poets like Edmund Spenser and Ben Jonson were guests and where Othello was first performed. Since his father had inherited huge estates on the Welsh border and acquired valuable land north of London, the Earl was also even richer than Savile, and it seems probable that the surveyor’s chain that Gunter designed in about 1607 was first used to measure the immense Bridgewater property.

      Aided by aristocratic influence, Gunter was then appointed rector of the wealthy parish of St George’s, Southwark, in London, and, in 1619, Professor of Astronomy at Gresham’s College, London. However, both his congregation and his students were utterly neglected in favour of his scientific instruments. Like electronic devices today, these were sold with a book of incomprehensible instructions. Gunter at least had the excuse that few could understand his instructions because they were written in Latin. In 1623 the chorus of complaints persuaded him to produce a translation. ‘I am at the last contented that it should come forth in English,’ he wrote. ‘Not that I think it worthy either of my labour or the publique view, but to satisfy their importunity who not understanding the Latin yet were at the charge to buy the instrument.’

      The complete collection of Gunter’s instructional books issued in 1623 was called The description and use of the sector, the cross-staffe and other instruments for such as are studious of mathematical practise. By then Gunter must have known that the last bit of the title was nonsense. The reason the book had to be in English was because his instruments were being used not by maths students but by surveyors for measuring and by sailors for navigating – and, unlike mathematicians, neither group could read Latin. Nevertheless, it contained so much new information on logarithms, trigonometry and geometry that one of his contemporaries paid him this tribute: ‘He did open men’s understandings and made young men in love with that studie [mathematics]. Before, the mathematical sciences were lock’t up in the Greeke and Latin tongues and so lay untoucht. After Mr Gunter, these sciences sprang up amain, more and more.’

      It was in this book that Gunter first described the chain that was to bear his name: ‘for plotting of ground, I hold it fit to use a chaine of foure perches in length, divided into an hundred links’. Four perches measured twenty-two yards, and the fact that this strange distance eventually became integral not only to the game of cricket in his own country (it is the length of the pitch), but to the town planning of almost every major city in the United States (the lengths of most city blocks are multiples of it), was a tribute to the chain’s versatility. Its practical advantage was simply that, unlike a rod, its links made it flexible enough to be looped over a person’s shoulder, and that being made of metal it neither stretched nor shrank as cords always did. Yet there was more to it than mere practicality. As a passionate believer in the usefulness of maths, Gunter built into his chain the most advanced intellectual learning of the time, until it could almost be compared to a primitive calculating machine.

      His cleverness lay in dividing the chain into one hundred links, marked off into groups of ten by brass rings. On the face of it, the chain’s dimensions make no sense – each link is a fraction under eight inches long, ten links make slightly less than six feet eight inches, and the full length is sixty-six feet. In fact this is a brilliant synthesis of two otherwise incompatible systems: the traditional English land measurements, which were based on the number four, and the then newly introduced system of decimals, based on the number ten.

      It was the Dutch engineer Simon Stevin who first published an account of decimals in 1585, and Gunter was quick to grasp the concept, using them in his logarithmic tables. Where the chain was concerned, he realised that units of ten made for simple calculation, hence the hundred links with the brass rings grouped in tens; but the overall length was no less important. His twenty-two-yard chain measured four rods long, which integrated it into traditional English measurements.

      The rod’s inconvenient length of 16½ feet was derived from the area of land that could be worked by one person in a day. This was reckoned to be two rods by two rods (thirty-three feet by thirty-three). Thus, there were four square rods in a daywork. Conveniently there were forty dayworks in an acre, the area that could be worked by a team of oxen in a day, and 640 acres in a square mile. All these once variable units became fixed in the sixteenth century, and it was significant that all of them were multiples of four, a number that simplified the calculation of areas.

      Gunter’s chain produced the happy result that ten square chains measured precisely one acre. Thus, if need be, the entire process of land measurement could be computed in decimals, then converted to acres by dividing the result by ten. With an understandable hint of satisfaction Gunter concluded his description of its use: ‘then will the work be more easie in Arithmetick’. It was that ease in calculating acreages, as much as its accuracy and straightforward practicality, that earned Gunter’s chain its popularity among surveyors using the old four-based system of measurements. Even the least competent could come close to the standards of exactness that were now expected of them.

      It would be difficult to exaggerate the need that the growing army of surveyors had for this kind of assistance. Even an oblong field, where the length was longer than the breadth, made the maths go shaky, and repeatedly the manuals are forced to remind their readers that the area of any square or rectangular field can be calculated by multiplying the length by the breadth. Yet the same mistakes kept recurring: as late as 1688 John Love, who spent many СКАЧАТЬ