Binomial Theorem And Compound Interest And John Napier.

It came to mind this morning, having written a short comment on Normans video on Lagrangian Differential calculus the night before, That compound interest played a crucial rolr in the development of Napierian Logarithms, and Briggs Logarithms.

At first i thought that i had written about this before and forgotten about it. But a quick search on google showed me that my principle concern had been to alert others to the fundamental impact on Calculus that the Binomial theorem and compound interest had had.

So an eqqually important point can be made bout the effect of hese 2 on the Napierian notion of Logarithms.

The Binomial theorem has a long influential history, but it took on its ancient significnce in 2 ways. The first is the construction of Vedic Hymn patterns , which in fact develops into the familiar Triangle of Pascal after a long and noble history in the east and far east.

The second is in the userers trade of extracting interest on loans over periods of days weeks months and years. That the binomial system and theorem should show up in this context seems remarkable. but in fact, Eudoxus in jis theory on Proportions, or more generally Logos Analogos, Reveals a Pythagorean conception of compounding relationships. Since it is Pythagorean, we can expect that this represents an extant Tradition prior to his times.

It is often remarked, in reference to the Golden Mean, how ubiquitous it was in Ancient Architecture and Aesthetics. In greece we know this was due to a style and design standard of measurements called the Summetria. In India we know that the temple builders passed down these kinds of standards in the form of ganitas or sulbasutras. The cultural interactions between India and China , particularly through Buddhism makes for an easy passage of these standards into that civilisation.

Trade and commerce and Merchants and Userers have similarly passed information of this nature to one another, and so have different schools of Astrology.

The binomial theorem starts out in a rhetorical guise, describing certain relationships between magnitudes. In fact in Euclid's Stoikeioon 2 we can see how starting with a rectangle , Gnomonic forms are not only constructed but theorems demonstrated showing dualities between various multiple forms. The binomial form soon arises and is demonstrated for a square. Later it was demonstrated for a cube, but that is later on inor after book 7, when stereos arithmoi are defined.

The growth demonstrated by this remarkable geometric pattern is visually remarkable, and it is of course only 2 shakes of a rats tail before Commercial interest and usury took this form to its own.

The principal driving force for developing this form beyond the stereos Arithmoi is financial gain. However, as i pointed out, in India, the aesthetics of religious hymnary , that sense of Brahma or divine order was also a motivating drive.

It is not clear, but it seems from the Pythagorean point of view, it was an important Astrological tool also for calculating patterns in the circular arcs for measurement of the movement of planets etc. The striking resemblance of divisions of arcs and the behaviour of their chords as they cut the diamaeter into segments visually reflects he Binomial Expansion or the Table of Pascal.

A Napierian logarithm before Napier

At its meeting of 15 July 1912, the Council of the Royal Society of Edinburgh resolved to commemorate the tercentenary of the publication in 1614 of Napier's Mirifici Logarithmorum Canonis Descriptio. From Saturday 25 July 1914 to Monday 27 July 1914 the Royal Society of Edinburgh held a Congress in Edinburgh to honour the Tercentenary. A fine volume was published in the following year C G Knott (ed.), Napier Memorial Volume (Royal Society of Edinburgh, London, 1915). Knott writes in the Preface:-

As regards the Congress itself it is pleasant to recall the goodwill and friendliness which characterised its meetings, attended though these were by men and women whose nationalities were fated to be in the grip of war before a week had passed.

Our library in St Andrews contains at least two copies of this Napier Memorial Volume , one of which still retains many uncut pages.

A number of interesting articles in this volume are difficult to obtain elsewhere. We produce below one by Giovanni Vacca. When he wrote the article Vacca was Professore Incarito of Chinese in the Royal University of Rome. The article makes an interesting comment on the appearance of a logarithm in Pacioli's Summa de Arithmetica.

In the ordinary histories of mathematics there are very few suggestions about the way in which John Napier conceived the idea of his great discovery, truly one of the most beautiful made by man, not only As supplying a new method for saving time and trouble in tedious calculations, but also as forming one of the most important steps towards the discovery of the infinitesimal calculus.

Generally the only reference made is to … Archimedes.

I have lately observed that in the Summa de Arithmetica of Fra Luca Pacioli, printed in Venice in 1494, there is the following problem:

(Fol. 181, n. 44.) 'A voler sapere ogni quantità a tanto per 100 I'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 I'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale.'

Here is a rough translation of the Italian: You want to know for every percentage interest per year, how many years will be required to return double the original capital, you hold the rule 72 in mind, which always you will divide by the interest, and the result will determine in how many years it will be doubled. Example: When the interest is 6 per 100 per year, I say that you divide 72 by 6; that is 12, so in 12 years the capital will be doubled.

Luca Pacioli says that the number of years necessary to double a capital placed at compound interest, is the number resulting from the division of the fixed number 72 by the rate of interest per 100.

If we try to explain the mystery of this number 72 (and the reason of this mystery was impenetrable to the succeeding arithmeticians, for instance, Tartaglia), we easily see in modern notation that

(1 + r/100)x = 2

or, taking Napierian logarithms

x log(1 + r/100) = log 2

and to a first approximation, if r is small:

x = 100 log 2/r

therefore 72 is only a rough calculation of the number 100 log 2.

The correct result is 100 log 2 = 69.31471806. However, the result of the approximation made above actually often makes 72 a better value than 70. For example for Pacioli's own example of r = 6, the correct answer is 11.89566105. We see here that 72/6 = 12 is a better approximation than 70/6 = 11.66666667

This problem is to be found, without explanation, in modern treatises, for instance in the introduction to the Tables d'intérêt composé of Pereyre.

Sometimes the number 70 is given instead of 72.

Note that 70 is a much better approximation to 100 log 2 = 69.31471806 than 72 but as we noted above that often 72 gives more accurate answers to the problem

If this problem were known to Napier, might it not have been a suggestion leading to his further discovery? Perhaps a research in his manuscripts can explain this point.

In any case it is curious to note that the Napierian logarithm of 2 was printed before the year 1500, with an approximation of 3 per 100.

JOC/EFR March 2006

The quote hows that the documentary evidience is out there if we look for it.

Napier, by reason of his travels, interests in Eastern wisdoms and conversations with Merchants was influenced by this background activity and research. By his time a centuries long calculation of the Sines, initiated by the Indian and Greek Mathematikos, that is Astrologers, and redacted and revised and updated and improved in Earnest by the Expanding Arab Islamic empire, had resulted in some fundamentally important tables being produced. Alongside hese tables was always placed the method of interpolation .

These methods of interpolation were the basis of finite differences which of course underpin the notions of calculus, but they were noteable for their reliance on the binomial theorem. In addition, as Alluded to in the quote, Userers tables were also a fundamental reference tool. And again, the binomial expansion features heavily in that table construction.

It is Noteable that Richard Witt published his groundbreaking work a year before Napier published his Groundbreaking work on Logarithms.,d.d2k

It is clear, that at the time a great interest in calculation was being shown, principally because much "reckoning" was going on. The solution arrived at by Witt and Napier, was to publish and explain Tables of previously calculated figures, making the actual calculations required by Merchants and Astrologers much easier, faster, and more reliable.

The binomial Theorem is absolutely fundamental to both these sets of tables, as well as to the intepolation schemes in the extant sine tables.

Where Napier shows a remarkable insight is in using an Old Idea, reputably Pythagorean, of dividing The Quarter Arc or the Quarter turn into Equal segments, even into seconds of arc or more, resulting in a fabulously large numeral 10^7.
For all intents and purposes this straightened the curve of the circumference of the quarter turn, so that he could consider [t as if it were made op of equal straight lines.

This notion finds its way into some descriptions of the method of calculus, but it was Newton who corrected this prevailing notion in a Lemma or 2 in the Principia. As much as the curve may appear straight, it is not, otherwise it loses its curvature. Instead he traps the curve between the secant and the tangent and uses this ratio in substitution for the curve.

However, Napier considers these segments of arc or sectors of arc as approximately straight and so he is able to compare the ratio of the sines and the sectors. This gives him his famous idea of the diminishing of a length by a certain amount In Proportion(logos Analogos) to that diminishing amount..

When i first came upon this explanation by Napier, i was confounded. But now i know of Eudoxus and Book 5 of the Stoikeioon, i recognise immediately the language of Eudoxus. By Eudoxus we also expect a chain of proportions which came to be called a sequence of results of a calculation. This was distinguished by the word series,, which also came to be applied to any summed sequence of multiple or calculable forms.

The next step in the story of logarithms was to note that the sequence of seggments of the arc had the same numbering as the power or exponent of the proportion. Thus the exponent became the index to the resultant.

Finally adding or subtracting the indices or exponents was the same as multiplying by those resultants.

This was quite complex to explain, but rapid and accurate to perform. In addition one soon understood that to multiply or divide the sines, something Astronomers were always in labour to do, they now simply had to look up the sine, find its index , sum them or subtract them and read off the result.. The work and labour of others, thus tabulated, made the work of their heirs that much easier.

It was Briggs of Gresham College, who reading the proofs of Napiers works, travelled to Scotland to assist him in its publication and further development. Not only was this a remarkable calculating aid, but itwas also a publishing goldmine!

Briggs was easily able to extend the principles through the formulary to other bases. This was because it was soundly based on the binomial theorem. The work of Tycho Brahe's assistantürgi , though based on exponents, was not keyed directly into the work on the sines, thus the work of calculating the resultants from scratch for each exponent was laborious and difficult. in addition the use of the binomial for interpolation loses its significance through pressure of calculation, especially when the calculation is a private enterprise.

Although Prosthapharesis is known and documented in the story, Logarithms are something else. Burgi was developing on both the prosthapharesis and the exponent front, but in a confused overlapping method, which he was refining. Napiers approach was quite different, drawing on the binomial expansion inaddition to the work of Eudoxus. Thus he was "outside of" the nitty gritty of the sine tables and could take a different view that was more accessible.

Richard Witt's popularisation of compound interest tables created a table ready audience hungry for easier ways of calculating. The remarkable similarity in table design and structure made the Logarithms a worldwide success with both Astrologers and Merchants Who could at once see how he two ideas were related by the binomial expansion and the compound interest expansions.

Two remarkable instruments of calculation were thus made public within a year of each other, and that literally changed the face of civilisation.


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