- MA290_3Topics in the history of mathematics John Napier
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978-1-4730-0788-8 (.epub)IntroductionThis course provides an overview of John Napier and his work on logarithms. It discusses his approach to this lasting invention and looks at the key players who worked with him, including Briggs, Wright and Kepler.This OpenLearn course provides a sample of Level 2 study in MathematicsAfter studying this course, you should be able to:understand the significance of John Napier's contributions to mathematicsgive examples of the factors that influenced Napier's mathematical work.1 Excellent Brief Rules1.1 Background to Napier and his workFor many years, John Napier (1550–1617) spent his leisure time devising means for making arithmetical calculations easier. Just why a Scots laird at the turn of the seventeenth century should have thus devoted the energies left over from the management of his estates remains a puzzle. Up to the publication of his description of logarithms in 1614, three years before his death, Napier was best known to the world for his Protestant religious treatise *A plaine discovery of the whole Revelation of Saint John* (1594), a well-received work which was translated into several foreign languages. Napier's interest in mathematics, and in computational methods in particular, seems to have started in his early twenties (in the 1570s), and continued mostly unknown to the wider world until the flurry of publishing activity forty years later which revealed first his table of logarithms (*Mirifici logarithmorum canonis descriptio*, 1614), then three further computation aids (*Rabdologiae*, 1617), and after his death an account of how logarithms themselves were calculated (*Mirifici logarithmorum canonis constructio*, 1619). The titles of his works form a revealing contrast.**Question 1**Compare the titles of Napier's theological and mathematical writings. Do you notice any difference that may give a clue as to Napier's intended readership?The mathematical titles are in Latin, that of the theological tract is in English. Assuming the contents are likewise in these two languages (which they are, in fact), we might begin to conjecture from this that Napier had different readers in mind. Given the substantial vernacular textbook tradition by this time (seventy or so years after the first appearance of Record's works), we may infer that Napier was not aiming at the homegrown practitioner but at a more learned, possibly international, audience.1.2 Napier's bonesBefore pursuing who logarithms were for (and what they are), we first look briefly at another of Napier's computational aids. For in the years following his death, it was in fact his numerating rods, the so-called *Napier's bones*, that were more widely known and used. These consisted of the columns of a multiplication table inscribed on rods, which could make the multiplying of two numbers easier by setting down the partial products more swiftly. This simple contrivance was derived from an ancient multiplication method (called ‘lattice’ multiplication – see Box 1), which involved a special layout on paper of the two numbers to be multiplied so as to display the partial products of the multiplication, to facilitate adding them in the right way.**Box 1**: Multiplication on Napier's bonesConsider first various ways of laying out the multiplication of 934 and 314, as given in the *Treviso arithmetic* of 1478 (the first printed arithmetic book).Note that the result, 293 276, is reached in the middle configuration by adding the figures in the matrix along the diagonal bands, starting from the bottom right (and carrying a digit over to the next band where appropriate).The same multiplication would be done on Napier's bones by aligning the rods for 9, 3 and 4, and looking horizontally along the *times 3, times 1* and *times 4* rows to form the partial products to be added, as shown here.1.3 Napier's approach to logarithmsNapier's major and more lasting invention, that of logarithms, forms a very interesting case study in mathematical development. Within a century or so what started life as merely an aid to calculation, a set of ‘excellent briefe rules’, as Napier called them, came to occupy a central role within the body of theoretical mathematics.The basic idea of what logarithms were to achieve is straightforward: to replace the wearisome task of *multiplying* two numbers by the simpler task of *adding* together two other numbers. To each number there was to be associated another, which Napier called at first an ‘artificial number’ and later a ‘logarithm’ (a term which he coined from Greek words meaning something like ‘ratio-number’), with the property that from the sum of two such logarithms the result of multiplying the two original numbers could be recovered.In a sense this idea had been around for a long time. Since at least Greek times it had been known that multiplication of terms in a geometric progression could correspond to addition of terms in an arithmetic progression. For instance, considerand notice that the *product* of 4 and 8 in the top line, viz 32, lies above the *sum* of 2 and 3 in the bottom line (5). (Here the top line is a *geometric* progression, because each term is twice its predecessor; there is a constant *ratio* between successive terms. The lower line is an *arithmetic* progression, because each term is one more than its predecessor; there is a constant *difference* between successive terms.) Precisely these two lines appear as parallel columns of numbers on an Old Babylonian tablet, though we do not know the scribe's intention in writing them down.A continuation of these progressions is the subject of a passage in Chuquet's *Triparty* (1484). Read the passage, linked below, now.Click the link below to open the passage from Chuquet's Triparty.Nicolas Chuquet on exponentsChuquet made the same observation as above, that the product of 4 and 8 (‘whoever multiplies 4 which is a second number by 8 which is a third number’) gives 32, which is above the sum of 2 and 3 (‘makes 32 which is a fifth number’). Chuquet seems virtually to have said that a neat way of multiplying 4 and 8 is to add their associated numbers (‘denominations’) in the arithmetic series and see what the result corresponds to in the geometric series. (Of course, had he wanted to multiply 5 by 9, say, Chuquet would have been stuck.) And in *The sand-reckoner*, long before, Archimedes proved a similar result for any geometric progression.So the idea that addition in an arithmetic series parallels multiplication in a geometric one was not completely unfamiliar. Nor, indeed, was the notion of reaching the result of a multiplication by means of an addition. For this was quite explicit in trigonometric formulae discovered early in the sixteenth century, such as:Thus if you wanted to multiply two sines, or two cosines, together – a very nasty calculation on endlessly fiddly numbers – you could reach the answer through the vastly simpler operation of subtracting or adding two other numbers. This method was much used by astronomers towards the end of the sixteenth century, particularly by the great Danish astronomer Tycho Brahe, who was visited by a young friend of Napier, John Craig, in 1590. So Napier was probably aware of these techniques at about the time he started serious work on his own idea, although conceptually it was entirely different.Napier's definition of logarithm is rather interesting. We shall not pursue all its details, but just enough to see its approach and character. Imagine two points, *P* and *L*, each moving along its own line.The line *P*_{0} Q is of fixed, finite length, but *L*'s line is endless. *L* travels along its line at constant speed, but *P* is slowing down. *P* and *L* start (from *P*_{0} and *L*_{0}) with the same speed, but thereafter *P*'s speed drops proportionally to the distance it has still to go: at the half-way point between *P*_{0} and *Q*, *P* is travelling at half the speed they both started with; at the three-quarter point, it is travelling with a quarter of the speed; and so on. So *P* is never actually going to get to *Q*, any more than *L* will arrive at the end of *its* line, and at any instant the positions of *P* and *L* uniquely correspond. Then at any instant the distance *L*_{0}L is, in Napier's definition, the *logarithm* of the distance *PQ*. (That is, the numbers measuring those distances.) Thus the distance *L* has travelled at any instant is the logarithm of the distance *P* has yet to go.How does this cohere with the ideas we spoke of earlier? The point *L* moves in an arithmetic progression: there is a *constant difference* between the distance it moves in equal time intervals – that is what ‘constant speed’ means. The point *P*, however, is slowing down in a geometric progression: its motion was defined so that it was the *ratio* of successive distances that remained constant in equal time intervals.**Question 2**Compare what you have gleaned of Napier's concept of logarithm with earlier ideas about progressions and trigonometrical formulae. What seems to you the most striking difference in overall approach? What problem do you see Napier's work solving?The major difference is surely in his use of the concept of motion, of points moving along lines with speeds defined in various ways. Both exponents of Chuquet's kind and trigonometrical formulae are quite ‘static’ objects by comparison – there is evidently a deep difference of mathematical style here.What seems so clever about Napier's approach is that he can cope with any number, in effect, not just the ones that happen to form part of some particular discrete geometrical progression. This is effected by his intuition springing from the *continuous* nature of the straight line and of motion.1.4 Napier and motionSo where did the idea of *motion* which is found in Napier's work come from? It was again a concept used by Archimedes, in his study of spirals, so there was a classical precedent for propositions about points moving along lines (see, for example, Proposition 1 of *On spirals*, linked below). Further, although much of the Western mathematical tradition had been rather nervous of the concept of motion hitherto, there had been exceptions to this three centuries or so earlier: both the Merton School in fourteenth-century Oxford and Nicole Oresme at the University of Paris had made prolonged study of issues involving this concept. The details of Napier's education are obscure – we know he spent a year at the University of St Andrews in his early teens, but not what he did or learned thereafter – but it is not implausible that he became aware of mediaeval studies of motion at some stage.Click the link below to open Proposition 1 of On spirals.Proposition 11.5 Napier and BriggsThe fact remains that few mathematical inventions have burst on the world so unexpectedly as Napier's logarithms. Although various disparate strands – the idea of doing multiplication via addition, the idea of comparing arithmetic and geometric progressions, the use of the concept of motion – had all been floated at some stage, the enthusiasm with which Napier's work was received makes it clear both that this was perceived as a novel invention and that it fulfilled a pressing need. Foremost among those who welcomed the invention was Henry Briggs, Professor of Geometry at Gresham College, who wrote to the biblical scholar James Ussher in 1615:Naper, lord of Markinston, hath set my Head and Hands a Work with his new and admirable Logarithms I hope to see him this Summer if it please God, for I never saw Book which pleased me better or made me more wonder.(Quoted in D. M. Hallowes (1961–2) ‘Henry Briggs, Mathematician’, Transactions of the Halifax Antiquarian Society, pp. 80–81.)

*Note:* Ussher was the scholar whose studies of biblical chronology revealed that the Creation took place at 9 a.m. on October 23rd, 4004 BC.Briggs did indeed ‘see him this Summer’, in a visit recorded some time later by the astrologer William Lilly. Please read this famous story now, linked below.Click the link below to read the story.William Lilly on the meeting of Napier and BriggsDuring this visit, and a further one the next year (1616), Briggs and Napier discussed some simplifications to the idea and presentation of logarithms. It is fortunate they profited from each other's company in this way, for their world-views were very different. Napier not only considered the Pope to be the Antichrist and expected the Day of Judgement quite shortly (probably between 1688 and 1700), but also was ‘a great lover of astrology’ (according to Lilly), and may even have practised witchcraft, as contemporary rumour had it. Briggs, on the other hand, represented what one might call the Yorkshire common-sense school of thought on occult matters, and like his friend Sir Henry Savile (also a son of Halifax) had no time for astrological practices.It was these two very different people who worked together on simplifying logarithms. They agreed it would generally be more useful if the logarithm of 1 were to be 0, and the logarithm of 10 were to be 1 (see Box 2). Briggs spent several subsequent years recalculating the tables on this basis. Thus the early history of logarithms exemplifies well a remark which the historian Clifford Truesdell has made in another context: ‘the simple ideas are the hardest to achieve; simplicity does not come of itself but must be created’ (his full comment is linked below). Napier and Briggs had to work hard to create even the ‘simple’ major logarithm property thatClick the link below to open Clifford Truesdell's full comment.Clifford Truesdell on Euler**Box 2**: Napier's, and later, logarithmsNapier's original presentation of logarithms differed markedly from that generally adopted later, and it is worth spelling out the differences.It follows from our earlier description that it was the length *P*_{o}*Q* whose logarithm is 0, and further that the shorter the length (in the geometric series) the larger its logarithm. Napier chose *P*_{0}Q to be 10 000 000, a conventional figure for the ‘whole sine’, because he intended the table of logarithms to be used trigonometrically – it was the logarithms of sines and tangents he calculated and tabulated in the 1614 *Descriptio*, not the logarithms of numbers in general. A further difference from later practice was that what we think of as the major logarithm property did not hold in our simple form, but wasIt was as a result of Briggs' discussions with Napier that log 1 was redefined to be 0, thus simplifying this formula, and making the use of logarithms easier. Briggs, too, developed the calculation of logarithms of ordinary numbers, using the correlation log 10 = 1, log 100 = 2, log 1000 = 3, and so on.1.6 Spreading the word about logarithmsAnother person besides Briggs to recognise immediately the importance of Napier's concept was the navigational practitioner Edward Wright, who translated Napier's *Descriptio* into English, as *A description of the admirable table of logarithmes*. The extract linked below comprises the Preface to that work (the translation of Napier's original Preface, with further sentences added by Napier himself).Click the link below to open the extract.John Napier’s Preface to A Description of the Admirable Table of Logarithms**Question 3**What explanation does Napier give of why his work appeared first in Latin, then in English? What light does this throw on your answer to Question 1?Napier says his book was written in Latin ‘Tor the publique use of Mathematicians, to ensure it shall be the more common’, that is, presumably, more widespread. The English translation is for the benefit of ‘our Countrymen in this Island’ and ‘the more publique good’. This confirms our hypothesis, from Question 1, that Napier had in mind as his primary audience the international mathematical community. He sounds pleased, all the same, that non-Latin speakers in the United Kingdom might find logarithms useful too.Knowledge of logarithms spread rapidly in various ways. Wright's English translation was one source of knowledge; it was dedicated (by Wright's son, for Wright died before publication) to the East India Company, another of the great trading and exploration companies of the time. This dedication suggests an audience for whom the knowledge was thought to be useful – Wright was navigational consultant to the Company for the last year or two of his life. Knowledge of logarithms spread by word of mouth too; Briggs lectured on them at Gresham College, as did Edmund Gunter, who was appointed Professor of Astronomy there in 1619. Within a decade, editions or similar tables had been published in France, Germany and the Netherlands. As an example of the impact of logarithms abroad, let us consider the response of the astronomer Johannes Kepler.1.7 Kepler and logarithmsKepler was precisely the kind of practitioner for whom logarithms were of greatest benefit: a professional astronomer (and, of course, competent Latin scholar) driven at times to distraction – he tells us – by the magnitude and complexity of the calculations he needed to do. So when he was able to study a copy of Napier's *Descriptio* in about 1619 he welcomed it warmly, dedicating his next book to Napier (not realising he had been dead for two years); and he went further. Napier's book contained the definition of logarithms, and a set of logarithm tables, but not how to get from one to the other. (It is worth bearing in mind that although logarithms *once calculated* are a considerable saving of time, the calculation of the tables themselves was extraordinarily laborious – especially as one did not have the assistance of logarithm tables to help with the necessary computations.) So Kepler set to work recalculating their construction from first principles, basing his argument upon the classical theory of proportion of Euclid's *Elements*, Book V. This approach was consciously quite different from Napier's geometrical and kinematic considerations; Kepler wrote that logarithms were not associated, for him,inherently with categories of trajectories, or lines of flow, or any other perceptible qualities, but (if one may say so) with categories of relationship and qualities of thought.(Quoted in Yu. A. Belyi (1975) ‘Johannes Kepler and the development of mathematics’, in A. Beer and P. Beer (eds), Kepler: Four Hundred Years, Pergamon Press, p. 656.)

It is interesting to notice that Kepler felt that the appeal to kinematical intuition in Napier's work lacked rigour. He emphatically asserted that what he was supplying was *rigorous proof*, which was, for him, something cast in Euclidean mould. In the event, he had to approximate (just as Napier had done) to express potentially endless numbers to a suitable finite approximation, but Kepler's overall approach was influential nonetheless. You can see something of his style of argument in the extract linked below, which you should look at now.Click the link below to open the extract.Charles Hutton on Johannes Kepler’s construction of logarithmsConclusionThis free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.Keep on learning Study another free courseThere are more than **800 courses on OpenLearn** for you to choose from on a range of subjects. Find out more about all our free courses. Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus.If you are new to university study, you may be interested in our Access Courses or Certificates. What’s new from OpenLearn?Sign up to our newsletter or view a sample. For reference, full URLs to pages listed above:OpenLearn – www.open.edu/openlearn/free-coursesVisiting our online prospectus – www.open.ac.uk/coursesAccess Courses – www.open.ac.uk/courses/do-it/accessCertificates – www.open.ac.uk/courses/certificates-heNewsletter – www.open.edu/openlearn/about-openlearn/subscribe-the-openlearn-newsletterThe material acknowledged below is Proprietary (not subject to Creative Commons licensing) and used under licence. See terms and conditions.Grateful acknowledgement is made to the following:Course image: Kim Traynor in Wikimedia made available under Creative Commons Attribution-ShareAlike 2.0 Licence.Figure 1 Bodleian Library;Figure 2 Keele University, Turner Collection;Figure 7 Deutsches Museum, Munich.The material acknowledged below is contained in *The History of Mathematics – A Reader* (1987) J Fauvel and J Gray (eds), published by Macmillan Education in association with The Open University. This text forms part of the course material for MA290 *Topics in the history of mathematics*. Copyright © in the editorial selection The Open University.*7.B2 N. Chuquet*: tr. and ed. H.G. Flegg, C.M. Hay, B. Moss, *Nicolas Chuquet Renaissance Mathematician*, Reidel, 1985, p. 144, 151-153, published in Chapter 7 ‘Mathematics in Mediaeval Europe’;*Proposition 1*: Chapter ‘Archimedes and Apollonius’;*9.E3 W. Lilly*: *Mr Lilly's History of his Life and Times*, London, 1715, pp. 105-106, published in ‘William Lilly on the meeting of Napier and Briggs’, Chapter 8 ‘Mathematical Sciences in Tudor and Stuart England’;*Clifford Truesdell on Euler*: Chapter ‘Euler and his Contemporaries’;*9.E1 J. Napier*: *A Description of the Admirable Table of Logarithms*, preface R.E. Wright, London, published in ‘John Napier's Preface to A Description of the Admirable Table of Logarithms’, Chapter 9 ‘Mathematical Sciences in Tudor and Stuart England’;*9.E4*: C. Hutton, *Mathematical Tables: … to which is prefixed, a large and original history of the discoveries and writings relating to those subjects; …*, London 1785, 1822 edition, pp. 49-54, published in ‘Charles Hutton on Johannes Kepler's construction logarithms’, Chapter 9 ‘Mathematical Sciences in Tudor and Stuart England’.**Don't miss out:**If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University - www.open.edu/openlearn/free-courses