MA290_3Topics in the history of mathematics John NapierAbout this free courseThis free course provides a sample of Level 2 study in Mathematics: www.open.ac.uk/courses/find/mathematicsThis version of the content may include video, images and interactive content that may not be optimised for your device.You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/john-napier/content-section-0.There you’ll also be able to track your progress via your activity record, which you can use to demonstrate your learning.
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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, consider and 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 P0 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 P0 and L0) 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 P0 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 L0L 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 2Compare 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.
1.5 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 that Click the link below to open Clifford Truesdell's full comment.Clifford Truesdell on EulerBox 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 PoQ whose logarithm is 0, and further that the shorter the length (in the geometric series) the larger its logarithm. Napier chose P0Q 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 was It 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.