This remarkable identity was known to Cotes, Newton and De Moivre, before Euler derived an exponential form of it.
Of the many identites in trigonomery this has to be the most fundamentally significant. When it was derived it was thought to link imaginary quantities to trigonometric ratios. That was fine for those who believed in the doctrine ofthe imaginaries, the mathesis of the imaginaries. And for many reasons this identity was a good missionary tool ro convert unbelievers!
There were and always have been some reluctant disciples of the mathesis of the imaginaries, Gibbs was one of them. Hamilton on the other hand was the high priest of the imaginaries. Gauss was a reluctant and extremeely cautious convert. Grassmann was the prophet of weirness in the n dimensions! He prepared for the coming of a more perfect manifestation of the divine mathematics. They say aprophet is not honoured in his own country, nor among his own people, and that was certainly true for Grassmann.
Gibbs fought invain to exterminate the quaternions and the mathesis of the imaginaries, and he was not alone amongst those of a pragmatic turn of mind. Yet he could not see the very tool the imaginarians were using was the one he needed to establish his vector ideas.
The Cotes Euler identity links an imaginary exponent to an imaginary trigonometric magnitude in Gauss form, But in fact it also links a vector in a direction or orientation specifed by i the unit vector i, to a trigonometric vector in gauss form made up of two unit vectors e and i, where e and i are perpendicular unit vectors. In fact by Demoivre's theorem it goes further and can link generalosed vector forms to vectors in gauss form. Thus the nth roots of unity become generalised vectors in Gauss form.
How can the distance along a perpendicular vector be related to a general vector orientation? The answer is remarkable and is due to Cotes. The rotation of a circle has one point that moves in synchrony with the circumference, the centre. Thus a distance along the arc of a circle, a radian is equivalent to the circle centre moving i radian in a given direction. Rotation is thus associated with a given direction, i say. Thus the radian measure of a vector orientation is associated with a linear translation of the centre along a vector path i
Among people with some mathematical background, Euler may well be best known for Euler’s formula: eix = cos x + i sin x (also often referred to as the Euler-Cotes formula). In examining the relation between exponentials and trigonometry, Roger Cotes (1682-1716) came to the formula ix = log(cos x + i sin x). This appeared in his Logometria of 1714 (printed in the Philosophical Transactions of the Royal Society, then a widely read publication) and reprinted in his posthumous 1722 work Harmonia Mensurarum. In this work Cotes studied logarithms and their relation to hyperbolas. Defining the “modulus and modular ratio” as the ratio of the number 1 to the factorials, he found the same terms as Euler did in his series expansion above. In particular, Cotes stated the ratio of 2.718281828459 to 1. Thus even our attribution of the decimal expansion of e to Euler is erroneous. But as we saw above, Euler did originally use c. If he had continued with that, urban legend might now say that he named it after Cotes, which would be correct in that Cotes was the first to explicitly write out the numerical approximation for the series expansion.
Euler-Cotes identity enable us to relate a general gauss vector to an exponential vector function.
It goes further. The exponential vector function represents rotating and translating vectors. Thus the exponential vector function represents general trochoidal paths of points in space . Additionally the vector argument for the exponent can be any dynamic vector, and so we have a function that is a vector function of general vectors, enabling us to derive plots of points in space in general; for example epitrochoids or hypotrochoids.
Of course it goes further. This remarkable formula can be extended to quaternions. Thus 3 dimensional vectors can control the motion of a sphere as it rotates.
This image gives a general but approximate idea
When you look at Lazaus Plaths Circa app you see this remarkable identity put to extraordinary use.
Few realise the onnection etween logarithms and the trigonometric ratio sine. Even fewre realise the Arab initiate project in producing accurate sine tables, an immense calculation effort that was engaged in over a number of centuries. Uf course it was the greek Ptolemy whose efforts in the Almagest inspired this revision heavily influenced by indian innovations.
So in the enlightened east many identities were known that the darkened west had little knowledge of. Napier as a keen traveler and dabbler in the knowledge of the east was gifted with enough insight to pursue an identity that others had known by prosthapharesis to its tortuous conclusion. This would not have been possible without the ready availability to astronomers of some version of the sine tables.
What Napier imagined was a dynamic rotating vector !07 units long. This meant he used whole numbers for the sine of an angle, and consequently could multiply whole numbers by prosthapharesis. What he noted was that proportion was related to unit change. Thus if A is an angle and it is increased by a unit then the product of the sines decreases proportionately. The way he envisaged it precedes Descartes Cartesian coordinates,by about a decade. He could see as the vector rotated the sine proportion followed a logarithmic curve. However he could not explain it in these terms. Instead he used a comparison between a ratio(logos) and a uniformly increasing form(arithmos) to convey the dynamic locus relationship.
But behind the sine tables is a simple geometrical relationship: a gnomon in a Semicircle. What Cotes picked up on, that Napier had not enunciated was that the uniformly changing form was in fact the arc length along the circumference of the circle. By defining the arc measure by the unit circle, Cotes was into the most astonishing harmony of all measurements one could imagine.
Although he published his relationship in the logometria it was only the tip of the iceberg. His collaboration with Newton and De Moivre was going to prove extraordinarily fruitful, and then he died. The unit circle relationships with the trigonometric functions was in fact a standard Newtonian and Wallis idea. Wallis in fact participated in the new versions of the logarithms that were being produces, and so was familiar with their construction. Cotes therefore was advancing well established ideas in this close circle of collaborators. De Moivre provided Cotes with the trigonometric factors of the √-1 for example.
The identity is in fact a comparison of sequences and when the terms are compared as Napier advises the relationship just drops out without any complication. The proof of the relationship involves solving multinomials which themselves simplify. De Moivre was particularly adept at this, and so was Cotes. The significance of these identities were perhaps not appreciated by De Moivre and Cotes, as Cotes was focusing on Newton's Principaea, and De Moivre was working hard on developing Probability theory as an application of the trigonometric tables.