Hamilton begins his development by sytematically laying out all the common relationships between moments of time. Hamilton is able to draw on a consensus of these "time" relationships and therefore is able to "perceive" these things as being in time. However this is a "mis-perception" as these relations are a refined subset of spatial relations used to construct the notion of time.
It was an interesting find of mine when reading some presentation on the arguments of Leibniz against Descartes philosophical notions of motion. The lecturer noted in passing that it was not until Galileo that our forefathers had any modern notion of time. Acceleration for example was as easily related t distance traveled as "time" 'traveled', and some examples were given of ow ancients described time in terms of travel distance or distance in terms of days traveled.
So it is not surprising that such a "close" linkage exists between time and space, as one is in fact a refined suset of the other!
It is not enough to say that they are distinguishable, which by instrumentation they are. One must recognise the tautological construction of time from spatial concepts and notions. In fact, the concept of ratio is a powerful enough matrix to comprehend the notion of "time" in all its forms. We may easily conclude that "time" i the motion of space in space relative to a standard periodic spatial motion. that we tend to use rotation is not strange either, as all motions are examples of rotational motion, or/and rotational expansion/contraction motion /and/or reflected motion in the "centre" of rotation between one region and its π radian opposite. This last is a special motion that i will write again: reflection occurs only in a "point" that is a centre of rotation between a region and its 'mirror" opposite. Thus any region within any subsuming region will have a reflection pi radians from it around a centre of rotation. AS we investigate subregion by subregion we will note that the subsuming region is reflected in an AXIS made up of these subregion centres of rotation. The behaviour of a mirror axes reflection is entirely different to the behaviour of a reflection through a centre of rotation for an infinitesimal subregion.
In fact if we reflect all the subregions through a common point we get a different behaviour to reflecting a subregion through its unique point connecting it with its rotation/reflection by π radians.
I have only recently elucidated this relationship, and so am still exploring it. Nevertheless, these 3 motions are necessary and sufficient to describe all motions, and therefore are to be found in any utilitarian construction of the "time" ratio.
Concisely, then, Hamilton is expounding on dynamic Spaciometry by developing his "pure Time" conception, beginning with the systematic survey of geometrical relationships.
Now some may confuse his systematic approach with Logic, and they would be justified because Logic is precisely a systematic survey of the relationships between all forms of expression and there attendant referents. In addition it provides on the basis f these studied relationships rules of conduct to "win" an argument, including"ad hominem".
While many do not regard its full field of study, Logic has become a weapon in the hand of able and articulate rhetoricians, while not in the least validating anything at all as "true" except in the specific sense of "consistent".
While consistency may e a prize worth having, it is salutary to realise that truth at the last evades us as it is an abstract notion of little substance, but major consequence.I myself prefer the empirical notion of "true": congruent,exact, in line with, coherent etc.