# Books - Mathematics

# An Introduction - Anon

Description: This book is a facsimile of the second edition of the earliest printed work in English entirely devoted to Arithmetic. The author is unknown, although the book comprises a compilation of material from two other works, one Dutch and one French, translated into English and with the addition of some new material. It was imprinted in Aldersgate Street, London by Nicolas Bourman in 1539. The first edition was produced two years earlier by John Herford, whose press was located in the abbey of St Albans and who printed under the patronage of the abbot, Richard Boreman. It was long thought that no copies of the first edition were extant, with the exception of a small fragment in the British Library. However, in 2005 a complete text turned up at a Sotheby's auction in New Bond Street, London. This rare survivor sold under the hammer to the British Library for the staggering sum of £97,500.

Among the problems posed in An Introduction is the "rule and question of a catte". This concerns a cat which climbs a 300 foot high tree, ascending 17 feet each day but descending again 12 feet each night. The problem to be solved is - how long does the cat take to reach the top? The answer given is 60 days, which of course is quite wrong. Another problem concerns "The rule and question of zaracins, for to cast them within the see". Given that on a sinking ship there are thirty merchants, 15 of whom are Christian and the other 15 Saracens, half of whom must be thrown overboard to save the ship, how should they be ordered so that counting off by nines will always result in a Saracen being sacrificed and never a Christian? Not a problem that fits easily with current ideas about political correctness. Then there is "a dronkart who drynketh a barell of bere in 14 days", but "when his wife drinketh with him" they empty it in 10 days How quickly, the reader is asked, could his wife drink it alone? These are just a few of the beguiling puzzles set within the pages of this fascinating book.

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# The Grounde of Artes - Robert Recorde

Description: The first edition of Robert Recorde's The Grounde of Artes was printed in London, at the sign of the Brazen Serpent, by Reynold Wolfe in 1543. The book teaches the rules and operations of arithmetic and provides many simple examples. It was probably intended as a textbook for the rapidly increasing number of mercantile clerks, but also for mariners engaged in the newly important science of celestial navigation. Recorde first shows how to carry out numerical operations using pen and paper, which in his time was a comparatively new and potentially confusing way of performing calculations. He goes on to demonstrate arithmetic done with counters, the centuries-old method of manipulating tokens on a ruled board. Finally, he shows how to indicate numbers with the hands, a system practised by merchants in market halls and on quaysides since antiquity.

In a preliminary discussion Recorde defines the art of arithmetic and claims it to be the basis of all learning, not only of geometry and astronomy but also of music, physic, law, grammar, philosophy and even theology - hence the title, The Grounde of Artes. The book is written in the form of a dialogue between a master and a somewhat precocious scholar. Recorde makes an effort to reproduce the speaking voice, within the limits of his didactic purpose, in the question and answer sessions. To the modern reader his prose is delightfully colloquial, if always straight to the point and never unnecessarily chatty. In places he injects statements of principle, for example this warning of the dangers of rote learning: Scholar. Sir, I thank you: but I think I might the better doe it, if you did shew me the working of it. Master. Yea, but you must prove yourself to doe some things without my aid, or else you shall not be able to doe any more than you are taught: And that were rather to learn by wrote (as they call it) than by reason.

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# The Pathway to Knowledge - Robert Recorde

Description: The first edition of Robert Recorde's The Pathway to Knowledge was printed in London, at the sign of the Brazen Serpent, by Reynold Wolfe in 1551. This book is the earliest work on geometry in the English language and was used as a standard textbook well into the middle of the seventeenth century. Recorde's prose is delightfully rhythmical and his poetical phrasing perhaps made learning less of a chore than otherwise for his studious readers. That he well knew this book, although modelled after Euclid, was breaking new ground is evidenced by his statement in the preface to the theorems: 'For nother is there anie matter more straunge in the english tongue, than this whereof never booke was written before now, in that tongue, and therefore oughte to delite all them, that desire to understand straunge matters, as most men commonlie doo'.

Recorde encountered an unexpected difficulty when setting out to teach Euclidean geometry to English readers. He found that the English language did not (at that time) have a sufficiency of technical terms. But rather than use longstanding Latin or Greek words, he invented his own English equivalents. So for example, obtuse angles are 'blunt corners', an equilateral triangle is a 'threelike' and a square is a 'likeside'. Unfortunately, Recorde's terminology was not taken up and did not survive the passage of time. Hence schoolchildren in geometry lessons today have to wrestle with difficult Latin words like tangent, instead of Recorde's much more homely and easily understood 'touch line'. The mathematical text itself is extremely lucid in both exposition and diagrams, proceeding from a list of definitions through forty-six constructions and seventy-seven theorems. At the start of the definitions is the statement that 'Geometry teacheth the drawyng, measuring and proporcion of figures' and history produced no finer or more eloquent tutor in the subject than Robert Recorde.

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# The Whetstone of Witte - Robert Recorde

Description: The sole edition of Robert Recorde's The Whetstone of Witte was printed at London by John Kingston in 1557. One of Recorde's concerns in this book is to develop not only a means of representing powers of numbers, but also a means of naming them. Prior to the development of a numerical index notation, the names given to the powers was of considerable importance. Hence in these pages we find terminology which is now archaic, for instance the strange word zenzizenzizenzike, the name for the eighth power of a number. It is generally acknowledged that Recorde's treatise on algebra, in the section entitled The arte of cossike numbers, is the first to be printed in the English language. Although this work owes much to the German mathematicians Christoff Rudolff and Michael Stifel, it does have one well known claim to originality; the first use of two parallel lines as the sign for equality (because noe 2 thyngs, can be moare equalle). Recorde's invention of the equals sign =, together with his adoption of the + sign (which betokeneth more) and the minus sign − (which betokeneth less) placed him at the very forefront of European practice.

Like most of Recorde's books, The Whetstone is written in the form of a dialogue between a learned master and a clever, but rather precocious, scholar. After being patiently encouraged through the seconde parte of arithmetic (begun by the scholar in Recorde's first book, The Grounde of Artes) followed by the extraction of rootes, the scholar remarks 'I am moche bounde unto you … Trusting so to applie my studie, and emploie my knowlege, that it shall never repente you of your curtesie in this behalfe'. To which the master, about to start an exposition on the difficult and strange cossike arte (algebra), replies 'Then marke well my words, and you shall perceive, that I will use as moche plainesse, as I maie, in teaching : And therefore will beginne with cossick numbers first'. Here Recorde is again using terminology that is now archaic. In his day algebra was called the cossic art, derived from the Latin cosa, meaning 'thing'. The Whetstone also includes a lengthy treatise on the arte of surde nombers, that is, on irrational numbers.

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# A Description of the Admirable Table of Logarithmes - John Napier

Description: John Napier published his treatise on the discovery of logarithms in 1614. It was written in Latin, the scholarly language of his day, under the title Mirifici Logarithmorum Canonis Descriptio. The importance of the work was quickly perceived and an English language translation by Edward Wright followed two years later, with the title A Description of the Admirable Table of Logarithmes. A further English edition followed in 1618. It is said that this book freed the world from a logjam of calculations. John Napier spent more than twenty years working alone on his system of logarithms, during a time when the multiplication and division of large numbers, as well as the finding of square roots, was considered to be extremely difficult. Because of his discovery of logarithms, these tedious mathematical operations could be replaced by the much easier processes of simple addition, subtraction and division by two. Never again would astronomers, architects, merchants and navigators become bogged down with calculations that were simply too difficult or time consuming to carry out.

Seeking a name for his discovery, Napier turned to Greek, coining the word Logarithm from logos (Greek for ratio or reckoning) and arithmos (Greek for number). Johannes Kepler, the imperial mathematician and astronomer at Prague, was one of the first to realise the enormous importance of Naperian logarithms. Initially indifferent, his attitude was quickly changed to one of great enthusiasm when he saw that tables of logarithms could considerably ease the burden of difficult astronomical calculations. The French mathematician and astronomer Pierre Simon Laplace said that logarithms, ‘…by shortening the labours, doubled the life of the astronomer.’ At a congress held in Edinburgh to celebrate the 300th anniversary of the publication of this book, it was remarked that ‘…no previous work had led up to it; nothing had foreshadowed it or heralded its arrival. It stands isolated, breaking upon human thought abruptly, without borrowing from the works of other intellects or following known lines of mathematical thought.’ Thus has posterity judged the worth of John Napier, Baron of Merchiston, and his logarithms.

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