Perhaps after researching these sources, the reader will appreciate the extraordinary creative influence Newton and his Students shared with each other, tucked away from the rages of the European turmoil. The extraordinary freedom they had to investigate and experiment, in major part is due, not to a lack of rigour, but a focus on magnitude, not "number". The creation of "number" by Dedekind has had many unfortunate and unforeseen effects, not the least of which is an inability to interact with the space around you.
While later generations were bemused by these imaginary "numbers", even up to Euler's time they were not confused with numbers. They were and always had been some kind of constant magnitude. The only question was "which one?"De Moivre, following Newton's advice viewed the magnitude as that within the unit circle. Cotes realised before he died that it was the arc length of the quarter circle and his Cotes Euler formula revealed the precise relation. He was so convinced of its harmonising effect that he called it Harmonium Mensuraram, and invented "the radian" to take full advantage of it. The radian is a ratio that is in fact quite familiar in astronomy and may be linked back to Ptolemy and the Indian Mathematicians. And Cotes merely defined it and used it. He did not even name it! Nevertheless it was crucial to his view of what fundamentally unified all magnitudes when quantified by any system of measurement.
This was such an important discovery and consequence of Newton's work that Newton, now feeble of mind, at once perked up and became reanimated. However, without Cotes to continue the insightful investigation due to his premature death, and because De Moivre was not fully au fait with Cotes thinking, the topic was rounded off but not developed further.
Fortunately about 50 or so years later Euler took these magnitudes to heart, after being advised by Bernoulli to perhaps investigate the unit circle. Bernoulli had just completed some major research on the circle, developing on Wallis' s Conic section formulae. He was particularly keen to know if Newton had found all the solutions for the gravitational orbits, and how his subtle mind had come to that conclusion. In some correspondence with Newton, through Cotes, Bernoulli was convinced that Newton was indeed correct in hi conclusions, so he left off the deep study of the circle to continue an argument with Leibniz about how to represent the logarithms of negative magnitudes. In so doing he handed off the way to the resolution of that heted debate to Euller. Neither were aware of Cotes work in this area which already solved he issue for
ln(-x)= 2*ln(ix) using Eulers notation
and ix = ln (cosx + i*sinx)
Euler's solution for -1 and his general exponential solution completed the tautological identity, revealing plainly that the arc of the circle was this fundamental magnitude.
However, nobody really understood what Cotes and Euler in particular were positing about magnitudinal measres. All magnitudinal measures are founded on the trigonometric ratios and relations, even those of straigt lines. In particular, the measures in 3d space fundamentally rely on the arcs in the spheres of the heavens.
While perhaps some may have had a chance of fully understanding this when spherical trigonometry was in its hey day, the decline in spherical trigonometry meant the decline in understanding these magnitudinal relationships, and the obfuscation by the concepts and concerns of "number" sealed these magnitudes into a walled garden called the imaginary numbers.
Rather than say "number", say label; and consequently think in terms of labeling. Do not think "Number Theory" instead think The Theory of and Terminology for Labelling Magnitudes. Give yourself some creative room to breathe!
Fortunately for us Hamilton was intrigued to explore in that garden, and to blaze a lighted path into the most wonderful exotic we know! Like little Jack Horner he pulled out a plumb called he quaternions which later proved to be a peach!
Grassmann's more general approach would perhaps not have had such potency without the sauce of the Quaternions. For while it is desperately true that Grassmann's work is superior to Hamilton's. like all general theories i requires one killer application, and that as it happened turns out to be Hamilton's quaternions, not Gibbs bowdlerized and concocted version of his work now called Vector Algebra.