I mentioned last week how the period 1550-1650 is the crucial century in understanding the causes of the acceleration of innovation. From 1550, when England was an indisputable backwater, by the 1590s it could claim craftsmen, artists, mathematicians, and navigators of international renown. It had not yet begun to inch ahead of its rivals, but it had at least begun to catch up.

What caused this? I have a number of hypotheses I’m currently researching, and I’ll share each of them with you as I explore them. Something that is certainly part of the answer, however, lies in the use of mathematics. Almost all of the late sixteenth-century English innovations seem to involve the application of geometry, following the maxims set down almost two thousand years earlier by Euclid of Alexandria. (Though at the time, most people mistakenly attributed his work to the even more ancient philosopher Euclid of Megara.) Manuscripts of Euclid’s work had circulated in western Europe throughout the Middle Ages, but they were only accessible to the handful of people who could read Latin, or the even fewer who could read Arabic or ancient Greek. In 1482, however, the Latin version was printed in Venice, and throughout the sixteenth century it was translated into more and more modern languages.

The full edition would eventually appear in English in 1570, but Euclid’s geometry was revealed to English-speakers much earlier, through the intervention of the mathematician Robert Recorde. Recorde embarked on a project to comprehensively unveil the mysterious mathematical arts in print. He started in 1543 with *The Ground of Arts*, a basic introduction to arithmetic, which in 1551 he followed up with *The Pathway to Knowledge*, an introduction to geometry. Once the ground had been stood upon, and the pathway to knowledge had been followed, there was then a *Gate of Knowledge *(on practical geometry, sadly now lost), which revealed a *Treasure of Knowledge* (also lost, but presumably also on applied geometry), which was then kept in a *Castle* - you guessed it - *of Knowledge* (on applying the learning from the other books to some complex astronomical instruments). At this point he apparently ran out of places to go with his metaphor, returning in 1557 with a book on advanced arithmetic called the *Whetstone of Wit*. He unfortunately died the following year.

But his *Pathway of Knowledge* was the first book to introduce English-speakers to Euclid’s geometry. In fact, it was the first book in English on geometry ever. And he wrote it in a way that he thought would be more accessible than reading Euclid raw. The effect was revolutionary. He created the market for books on mathematics, opening the way to books on its applications by common gunners and navigators and makers of navigational instruments, as well as by scholars. And geometry began to creep into invention after invention. Recorde’s *Pathway* extolled the seemingly extraordinary achievements of the ancients - Archimedes, Daedalus, and others - which he argued had all been made possible by their understanding of geometry. Naturally, it inspired others to emulate or even exceed them.

In agriculture, you began to see the use of more accurate techniques for surveying lands - something that was not always appreciated, as when a lord found that their land was smaller than they had supposed and should thus have a lower rent, or the opposite, when a tenant was told that they should have been paying more. In navigation, and the closely-related field of map-making, you also began to see the development of better instruments and methods. Without them, the English expeditions of the late sixteenth century would not have been possible - in fact, Recorde himself advised some of the explorers, and he invented at least one kind of navigational instrument (unfortunately, he described it in one of the books that are now lost). His *Whetstone of Wit* was dedicated to the Muscovy Company, and he apparently had a book on practical navigation in the works when he died.

But the applications did not end there. Mathematics had a host of applications to warfare, which made it especially appealing to the war-like nobility. Geometry was used to improve the accuracy of artillery, as well as to defend against it in the design of fortifications. Surveying techniques were even used to calculate how many soldiers a given patch of land might hold when drawing up battle lines, or when deciding where to hold camp. By the early 1570s, geometry was considered by some to be an essential part of the education of the warrior elite. And in 1588, when the threat of a massive Spanish invasion was still fresh, how was it that London tried to turn its merchants into a militia? By instituting a series of mathematical lectures.

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