Hamilton is a good read.
He is prolix and Dickensian, but clear nonetheless. Patience you seekers after knowledge! You will find such freedom and flowing in his conceptions that you will be carried along in a current so powerful as to upend the foundations of all mathematics and science!
From his pen flows inexorably words of such purport, that the tenor of your very conceptions will be changed! So troubling are his gentle Irish Murmurings that Maxwell of Scotland, and Lewis Carrol of England, And god knows who in Wales took up arms against him, sounding the alarm to all stalwarts, men and true:" Defend the line! Hold the line, by God!Defend your very lives!"
Of course Hamilton was swamped eventually by the tide of change sweeping across the continent to America. And those who joyfully welcomed him there also drew his blood in venomous deceit, and shameful acts of ignominy.
The flowing of Pure Time was for a moment stopped, but its ground shaking revolution could not be held back and, transformed, its lava streams run hot and free throughout the domains of dynamical sciences.
I am grateful to Hamilton, and to his archives, for he above all Astronomers, scientists, and those who call themselves mathematicians preserved the essential link to those who were children of the Arabic invention Al Jibr, most notably the quiet work of the engineer Rafael Bombelli. And though we may see glimpses of Brahmagupta's brilliance ,Hamilton took governance from Newton and his school, and Lagrange and his thought, to set up Algebra as a science independent of analysis and geometr- from which all induction and deduction arose.
In my estimation he did not achieve what he set out to do, because what he set out was a dismembering of the indissoluble link which flows from Dynamic spaciometry through Algebraic analysis to utilitarian arithmetics which require further interpretive analysis.
This flow, this movement and progression he captured well, but analogised it to time. Newton had analogised it to flow as in a fluidic motion, Lagrange as to a continuous indivisible motion Napier as a progressive and varying motion. Thus i have found that we agree if not in name at least in the action of dynamic motion.
I have often heard it parroted that Greek geometry in Euclid was fixed. But Greek geometry was never fixed. It was and always had been a science of dynamic magnitudes equal to our own.
Notation is not everything. The conception of differentials of fluxions preceded Newton and Leibniz. Notation however belong s to them both. But the essence of the calculus , of the Cartesian coordinates lies within he Indian/ Greek trigonometric ratios. The flow of the motion of the stars and planets in the very heavens is mediated through these gnomonic entities and the unities of our own conception.
Let it not be said that Hamilton was not any the wiser, but rather his intuition was to take the dynamism of the heavens and base a logical Algebraic science on it, and in so doing to introduce the ground for ordered sets of moments, sets which became ordered real numbers, which became ordered entities of all sorts from, brackets to vectors to n-tuples to tensors and by multiplexing:matrices, and tensor matrices…. Such an outflowing of invention because he dared to make algebra a founded science of "Pure Time".
So i am indebted to him for had he not taken Euclid's Axioms as a model, and like Newton written them so sweetly into sensibility i would not have had the ideas or the terminology to lay out my inductive and deductive explorations of the very foundations of mathematics, that is i would not have had the luxury of defending an alternative mathesis, were it not for Hamilton, bringing Brahmagupta and Bombelli to Life.
We switch, through Hamiilton from the ordered number line to the ordered set, enabling us to freely measure wherever we wish, unimpeded by axes determinig our very motions. Now through Hamilton we may determine the very axes we wish to apply to our ordered set! Thus our set remains invariant, but our geometrical relation through axes determines the geometry of flow in space.
I now am truly fee to use order to determine any flow. Alternatively i am free to use any flow to determine order, and by order i mean sequence in each and every case.