Whenever I read summary biographies of theoretical physicists or mathematical engineers of the late 19th early 20th century, those who are now regarded as pioneers are always said to have created their own, idiosyncratic vector algebra. This is so of Heaviside and in this particular instance P.A.M. Dirac whose bra-ket notation is regularly inflicted on young impressionable minds.

I suppose it is easier to make such statements than it is to delve into the murky and nefarious misdeeds in academia!

Vectors in principle as I now understand are inherent in the ancient Greek notion of a segmented line, especially as described in the Stoikeia and elaborated on by Apollonius and Archimedes. Many Pre Socratic philosophers from Thales, Herakleitos , Parmenides Luycippius and Democrites, considered the plenum to be eternally dynamic, and their drawn lines just to represent instaneous form or motion,: skesis or sketches and schematics of ephemeral manifestations in the flow of Rhea or Hekate. Phusis was that grand system of opposing entities which alone maintained a sensible equilibrium or a system of equilibria with an attendant Harmonia. These two were represented as sisters and members of the 9 muses, the attendants of Apollo the great Sun God whose beauty was admired by all!

Nice myths, and memorable stories, which in the absence of television game shows occupied the minds of young impressionable people of the time. Now these myths reappear in some detail in the console games loved by do many of today's youth.

To the wise they encoded deep philosophical truths, and Newton as an example devoted considerable effort in his youth to deciphering the physical and alchemical information encoded therein. But the line segment as a symbol belongs to the Pythagorean Eudoxus, who encoded the understanding to be found in the mosaic arrangements that decorated the Pythagorean temple walls and floors. These temples to the Pythagorean Muses were called Mousaion, whence we derived Museums and also the epithet mosaic. But ib Euclid these patterns are called epipedoi, and in the literature speripedoi. They were also called Epiphaneia, because light fell upon them and cast shadows. These shadows were marked ti designate certain times of the day or year, and so crazy "abstrac" patterns were noted in early examples of mosaics.

Eudoxus took these principles to the Pythagorean conclusion that all forms can be reduced to these dynamic shadows cast onto these mosaic floor patterns. The shadows could be quickly sketched round thus giving grammai or drawn lines in the earth. The embedded pebbles, later baked tiles enabled these lines to be counted or measured, and this was the meaning of geometry.

Geometry used these drawn lines on mosaics to measure space out to the sun, and the stars, and this was known as Astrology. Thus geometry was not about surveying the earth, rather the earth was measured to survey the stars and planets in their motions and positions .

Aristotle who through circumstance never completed his Pythagorean studies under Plato, investigated this dynamic relationship between all things that moved in his great work on metaphysics. His ideas were greatly influential on Islamic scholars and notably on Newton and many others. However they were criticised by Gilbert in his work on Magnetism and on the empirical basis of human knowledge. In this Gilbert predates Bacon, who somewhat witheringly disparaged Gilbert's great insight.

Suffice it to say Aristotle was misleading on many many points and was taken with a pinch of salt by many later philosophers, who found safety and clarity in Euclids Stoikeia. Later some questioned even Euclid!

But it must be acknowledged that in Newton all that was best in Aristotles reasonings were combined into Newtonian astrological principals, ie the Principia Mathematica. Thus the Greek notions of dynamic line segments pass on to Newton and others who following Archimrdes, Apollonius and Parcletus, to name a few , gave us a system of analytical Mrchanics. However it is Newton who combines this with the analysis of Aristotle on motion to give to us the new Mevhanical philosophy that avoids many of the ancient extremes and mistakes..

From this source most of the scientific clubs and societies in the world drew their concepts of how this world order might best be described, and what mathematical principles might best be employed in doing do. Buried deep within this apprehension was the ignoble line segment and it indissoluble link to mosaic ratios of line segments and to Eudoxian proportions. Amongst these line segments and mechanical motions of drawings, the ubiquitous circle was also preserved, and its utility and fundamental constructive power was seen as the ultimate goal of all analysis. It's perfection could hardly be described without all it's children of which the most important was the segmented line.

Notions of straight, parallel and rotation were fundamentally and mechanically defined in terms of the circle and the rigidity of space. Even though it was impossible to find such a design in nature yet it easily dropped out of rotating every rigid object no matter how convoluted! Eudoxian proportions fundamentally rely on it, as do geometric extremes and means.

Thus the great secret and harmonium Mensurarum of Sir Roger Cotes, that which Newton intuitively perceived, and De Moivre also skilfully analysed both in trigonometric form and probability measure form is the unit sphere and it's projected shadow the unit circle.

It is the unit circle which defines every aspect of the notion of line segments and thus every spect if the notion of so called vectors. And while Newton addressed these properties in his geometrical reasonings, yet he did not make fully clear the underlying Algebra. And While Hamilton by brute force extended the underlying algebra into 3 dimensions from his noteable concepts in his essay on the science of Pure Time, that is conjugate functions or couples., it is remarkable that he did not fully apprehend what he was pursuing so ardently.

On the other hand, both Lagrange and Euler casually cast these things about as if they were playthings, but it is not certain that they grasped the fundamental nature of what they had adduced to Newtons great works, nor those of his acolytes De Moivre and Cotes. In any case they did not seem concerned to fundamentally give insight into how these things may be best understood.

This fell into the hands of a great an innovative teacher named Justus Grassmann, who was charged with the responsibility of bringing the academic standing go the average Prussian child to the level required for Prussia to compete with the French empire in the modern industrial world. Indeed it was the duty of all educators under the imperial seal to implement the Humboldt reforms throughout the length and breadth of the Prusdian holy roman empire.

As such, the classically trained Grassmann, educated out of the best of the French Ecole tradition, which was principally disseminated by Lagrange from Berlin, who saw to it that Legendre and the work of many others became the text books in Prusdian higher education; as such Grassmann deconstructed the work of Legendre who himself had reconstructed the works of Euclid from what sources he had. This lead Grassmann into a deep logical analysis and synthesis, that resulted in him training students in a constructive and dynamic geometry, which he could show logically hung together and underpinned Aruthmetic. Rather he was keen to show that arithmetic was the fundamental of all geometry! But he came unstuck at multiplication. Under his analysis there was no logical precursor or analogue ti multiplication. It stood in its own right as a purely geometrical construction!

In fact, in my opinion, this was a misreading of the concept of logic, perpetrated by Aristotle. Aristotle based his definition of logic on the grammar of language. While tis is a very powerful consonance it is also a fundamental dissonance. Language in the end is a verbalisation of non verbal experiences. Much of what is communicated in language is non verbal, and that means it has a referent in one or many of the other sensory modalities. Thus to understand language we have to start in the nonverbal experiential modalities. This means we have to start in our experience of interacting with space.

To cover this concept I defined the notion of Spaciometry. This covers an individuals logos, analogos, sunthemata and summetria response. These non verbal interactions with space underpin the comparison, distinction and declarations of those distinctions we call language. Thus multiplication is without doubt a non verbal primitive. In fact I claim we have mis apprehended the primitive by calling it multiplication!

In any case this intense scrutiny and synthesis lead to Hermann developing a unique constructivist approach. This Förderung was so innocuous that it literally requires the individual to go back to the wide eyed suspension of critical faculties common among children. This concept is expressed in 2 Laws of Grassmann

AB=-BA

And AB+ BC=AC

This is the fundamental product sum laws( nb! Product sum not product and sum) of what Grassmann called his Aussere product.. The inner product is based on a different kind of line segment, what I have called the trig line segments. This produces a " Normal" or usual product but the product sim is still Grassmanns product sum law.

This product sum law makes no sense as a measurement, but as a process description of spatial behaviours it is absolutely fundamental to any algebra of space.

None of this was notated as precisely as this. In fact it was not even considered as needing notation because it was sufficiently clear in the rhetoric of any geometrical " proof" or exposition. The fact that Grassmann set out an algebraic notation for it, which is really a symbolic arithmetic for it, is one of those rare moments of wacky thinking that just proved to be so right! Recall Bombelli saying that his idea of adjugate numbers was so crazy he had to shut his eyes and muddle through! Again this was described as Akgebra, by Bombelli. As you will read in the links in the thread on Bombellis operator, if they are still active, Bombelli went on to solve quintic and degree 6 equations using his notational algebra.

This of course begs the question what is algebra? Newton hated the idea of notating all his muddled thinking by means of these symbols, but Wallis believed this could help others to emulate the genius of the very best mathematical minds.

It should be noted that despite protestations to the contrary algebra represents muddled, muddied and contortional thinking in feeling ones way to a solution. Often it requires one to assume the impossible or the unexplainable, to hold ones breath or shut ones eyes and hope to god it all works out and cancels to some easy result.

It is a tortuous exercise which only a few actually enjoy. In particular, the great savants who could calculate pi in their heads to several hundred places would find the formulaic patterns intriguing and even " nice"! But to the vast majority of us these algebraic symbols and patterns are literally " all Greek" to us.

So I say that the Algebra that Grassmann and Bombelli perceived is a symbolic procedural algebra best expressed as rhetoric. Although this is much longer to read or explain it is more user friendly. The best communicators explain the symbolic arithmetic with inviting analogies, as I remember Lancelot Hogben doing in his book Mathematics for the millions, which I read as a child and which confirmed me in my decision to become a " great" mathematician!

As you see I failed in that goal round about entering into university, when my relatively sheltered mathematical training was introduced to the harsh realities of pontification! No one would explain how all these mathematical symbols and juggling actually came into existence and had any relevant meaning in everyday life. I was astute enough to realise that my mathematics should be able to describe a door! And yet I could not describe a door mathematically.

The Grassmann concept of line segments as the basic primitives of an algebra is so far reaching that you need to see how it reaches into the mind of Dirac to realise this. You need to ubpnderstand that Hamilton recognised Grassmann as a Master, that Gibbs recognised the work ofbGrassmann as fundamental to replacing the arcane Alice in wonderland world of Quatenions, that Heaviide realised that Grassmanns principles properly understood gave him physical insight, and that Bill Clifford determined to spend his life promoting Grassmann work and analysis.

Today Norman Wildberger, David Hestenes and others are busy continuing the programme and work of Grassmann. Of course Grassmann is dead, so why name your work after him! In all fairness Grassmann is referred to as the source, but few really explain how much what they discover by using his methods of Analysis and synthesis has already been written down long ago by Grassmann.

In this thread Grasdmann deals with his issue in that particular instance of Quatenions. He does not clim to have discovered quatenionic, but rather that what can be discovered by Quatenions can more rapidly and more cogently be discovered by his algebra, and indeed already had bern by him!