You may not know that Norman Wildberger is one of the most forward thinking Mathematicians on the Internet and youTube today. So I put the wrd rant in quotes more to connect it to Viharts tite than anything else.
Check out his video below.
Pi has a very important role to play in mathematics as the premier ratio! Just as the sphere and thus the circle are the premier forms of ancient Spaciometry. The process to these platonic forms , at least in the Stoikeia of Euclid, go through Eudoxus. In books 5 and 6 in particular Eudoxus sets out the Logos Analogos calculus. Later it is claimed that HippRchus another prominent Pythagorean set down the Trigonometric calculus. These are both a landmark philosophical forms and to be distinguished completely from the circular functions of the Indian mathematicians, the Newtonian triad of Newton De Moivre and. Cotes, and the preeminent Euler who clearly laid down the radian or “Halbmesser Bogen” measure intimated by Cotes who died before publication of his Harmonium Mensurarum.
Having made ths distinction I can now turn to Napiers Logarithms with some clarity!
The calculation of the sines was a centuries long enterprise, perhaps beginning with Brahmagupta, but certainly the Indian Mathematicians, and then continuing through the work of the Islamic ( Arabic and Persian) scholars. Thus by Napier’s time, voluminous details, formulae, methods and tabulated results existed in multiple forms. While the angle was associated to this enterprise, it was of astronomical significance only. For most, the pure ratio was the significant part. In fact stretching back to Ptolemy, the spread of 2 rays, lines or line segments were recorded as the ratio of chord to the diameter of the circle. The sine is precisely half that ratio.
The arc subtended on the chord( there are2, so the smaller one) was not related to the sine ratio until Euler, who characteristically set it out very clearly. It was only later that scholars found that Cotes had been moving in precisely this direction, and had discussed this on a European scale with some other mathematicians besides Newton and De Moivre. In relation to the calculus of Fluxions, especially of elliptical functions.
So, at the time of Napier, the prostate resins that was a method of using the sine tables to do calculations, was not as is presented , based on angle Arcs or degrees, but on angle ratios. The difference is profound for a modern mind to understand. For the most part Napier worked directly with a set of tables of sines. These were interpretable as arc measures, but not especially so. They were and are the ratio of 2 orthogonal lengths. Thus when Napier declares that his method depends on the proportion of diminishing length by a unit as one length increases by a unit he is comparing these orthogonal lengths at a very small scale. Typically 10^-7.
In his actual introductory comments he uses the isosceles triangles in a small sector of a circle!
Where he differed from Tycho Braes method was in basing his selection of sines on a geometric progression. This meant that instead of using prostate resins as described in the ancient texts he modified it by Approximation. By his time the sine tabulation was sufficiently rich for him to choose an entry sufficiently close to his values calculated by tha geometric progression. Tabulated values were often written to 20 + digits!
Once he started with a geometric progression his work was relatively straight forward addition of the sine entries. A couple of calculation properties relating to factors were important for logarithms to be useful: thus 4×3 must have the same logarithm sum as 6×2, and this is a straightforward property to demonstrate in genera
Hence by a lot of work and careful annotation he was able to write the first table of calculating sines turning multiplication into addition and back again.
Briggs was much taken with this discovery and collaborated with Napier on many other logarithmic tables using different” bases” that is different ratios written in tabulated form.nby avoiding prostatharesus Napier had made a breakthrough in calculation speed accuracy and amen ability. Many around the world were very much relieved to have access to his tables for calculations .
Later, through Eulers work the circular arc measure could be related to logarithms, but are to be distinguished from Napierian or Natural logarithms based on the ” natural” sine ratio not the arc sine ratio!
Using. Napiers method Euler was able to derive the exponential constant e used by actuarial professions in calculating compound interest, work previously done and advised upon by Newton. Based on this work De Moivre had moved into the formulation of his work on probability. But it was Cotes, drawing on De moivre’s wrk on Multinomials and their relationship to the sine tables, something only he and of course Newton his mentor fully appreciated; it was Cotes who went on to tie together the trigonometric series to the logarithmic series using i and to suggest it use in the gravitational description of orbits!
In this regard I want to look at how pi becomes i !
I need the notion of Logarithm described by Napier.
A series of articles contextualising Napiers ideas and Burghi’s. However Burghi’s tables were more difficult to use for non mathematicians because of not properly clarifying the characteristic of the logaarithm. The mantissa behaved in the same way as it must.