Hamilton derived "couples" from Pure time and did so to support his friends theory. What he did was ponder and apprehend the fundamental nature of relations. As i follow Hamilton's development i realise that at the level of foundation is motion -sequence and from that relation, before i get anywhere near to structure and aggregation structure and measurement.
The Logos Response is a response to signals and fundamentally the comparison of signals. The comparison of signals is the essential notion of relation and relationship: relation, relationship; the notion of relations derives from comparison.
It is amazing how fundamental comparison is and how we have a rich vocabulary and set of notions fr comparison which may be systematically organised, sequenced and arranged into a foundation for Manipume, an extraction from spaciometry of the tissue of relationships necessary and sufficient to model spaciometry mentally,
It is a breathtaking experience to see what Hamilton saw and so carefully communicates: to look on a set of relations and to say, is hat all that it is? The realisation of the essential simplicity of it all.
Hamilton made sense of something that others had no way of making sense of, and yet no one ever teaches Hamilton's Mathesis, His doctrine, except that it is everywhere in modern mathematics ans science in one bowdlerized form after another.
FRom Hamilton we have the quintessential notion of vectors, versors, quaternions and octonion and couples. Without him ideas such as these , which indeed have a long provenance would not have had such a utilitarian impact on our understanding of the world.
When Hamilton discovered quaternions, he first checked them out in Spherical trigonometry (r,χ,ψ,ω) and (r,x,y,z). These coordinate systems are measuring tools first and foremost. They dimension space differently one in sperical arcs on the surface of spheres, the other in orthogonal axes in the body of space with a direction radial vector.
Both remind me of umbrellas.
Some including Levi, Ricci and thus Einstein have used the geodesic tool as a measuring tool. Insofar as they confine themselves to 4vectors, they confine themselves to Hamilton,s Umbrella. Doug Sweetster has realised this to good effect.
Hamilton is not perfect, nor could he explore all in his lifetime, neither could his school win out against the more robust protestant old guard. But nevertheless everyone of them has taken from his work without credit in some cases, and vitriolic diatribe in some, and built their own extensions of his work.
This is not to say that Hamilton would have objected, for in the end he has influenced mathematics and science beyond calculation, but rather to say that knowing where the influence has come from enables a proper estimation and revision.
Some, to whit may have engaged with his ideas to perfect them, but i have yet to find the one who truly acknowledges his mathesis, and uses it to correct where he is lacking, and also to retrace as Hamilton did the roots of Algebra back to the Arabs to correct mistakes they made.
No one has at yet reduced the notion of number to its proper place and revised the basis of our accounting as Hamilton allows us to do. I can build a better Manipume because of him.
If i compare Brahmagupta and Hamilton i would say i do not know enough about Brahmagupta to state with certainty, but i suspect Brahmagupta began what Hamilton finished: an exploration of the fundamental relationships in geometry prior to the formulation of aggregation structures in Algebras, and thus arithmetics.