I had to review more thoroughly the propositions in Book2 and Book 1 since Norman based his WildTrig series on them, in the form of Pythagoras, Archimedes and Euclids Theorems or Propoositions. The Grassmanns, starting with Justus and followed by Hermann and Robert did the same.
The constructive nature of Euclids introduction to philosophy has appealed to many different artisans, who perhaps did not know how deeply Plato thought about the description of our conscious experience. Plato, the original " wide boy", was struck by the Pythagorean philosophy , encountered on his travels, and intrigued by Socratic mind games, and general jollification of life's more complex issues. He recognised the Koanic structure of the Pythagorean praxis, a practice designed to bring the acolyte in direct contact with the Musai. But he also saw how Socrates method also worked!
The idea of his Academy was to bring the best practice to bear on educating the leaders of tomorrow. He wanted politics to transcend the bloody violence of his time, and to become a Utopian hegemony. Governed by the best wisdoms available.
So Euclid was eventually hired by the platonic school to design and teach an introductory course into what is generally called Platonic philosophy, but which is an amalgam tidily and isos-ly stitched together, that is venally constructed. It starts backwards! It builds up to where it comes from, which is the philosophy of the sphere as a representation of Astrological wisdom. We have called it mistakenly spherical Geometry, but even Newton named it Astrological wisdom, or astrology. I call it astrological Spaciometry.
Thus everything found in books 1 and 2 have a prior analytical origin: the analysis of astrological measurements and descriptions and the behaviours of the dynamic planets nd stars.
The seemeia on, the circular disk plane of dual points, the dual point line, the triple point intersections, the parallel line, the right gonu or rotation, the construction lines all derive from this skilful and age long analysis. The synthetic method used by Euclid was the best in class, the clearest introduction, the best preparation for the heavily encrusted mystical meanings attached to these long thought about forms.
In book2 Euclid introduces the lineal Rhetoric and the Gnomon of proportion. This lineal rhetoric from the outset is a sophisticated algebra which connects the observer and the space in which they live and breath to the representation of it in their minds! To be a course on philosophy it must engage the student in mental manipultions of real sensible objects and action. Thus these forms were drawn, and expected to be moved, rotated flipped, conjugated by attentional focus to distinguish the complement, and equated with the rock solidity of kinaesthetic confirmation. These things were equal because they fit exactly!
In book 2 not only were the objects constructed, but once constructed they were mentally manipulated. Today we cn animate these mental actions. This is precisely what Euclid was training his philosophers to do.
Th rhetoric, the " I dy" the constructive instructions, the restatement of the rhetoric in a representational terminology. The mental distinguishing of structures nd the animation of them by translation nd rotation is something Felix Klein is lauded for. He was only mimicking what Euclid was demanding, but pedagogues had forgotten or did not understand. In addition, he was following a line of pedagoguery instituted bt Justus Grassmann, popularised in certain academic circles like Clebsch's, by the presentations of Hermann and Robert Grassmann. Of the 2 hermanns work has been the most read, and the most influential, but all 3 Grassmanns are responsible.
So today, Norman Wildberger has revisited the Euclidean introductory course again, in the mistaken assumption that it is a Maths text, and completely redacted from it a new exposition of what is in there!. Consequently he has presented" mathematically" a set of rhetorical exercises and trainings which have historically been called Algebra, which in todays English means something akin to " Mind Fuck!". Some of us love it though.
This "messing with your head" rhetoric is the kind of training a philosopher needs, and it exemplifies Socratic techniques as well as it does Koanistic Techniques.
The Grassmanns therefore introduced this animated constructive geometry by a happy accident. The Humboldt reforms enabled Justus to work on clearing up what he saw as a pedagogical nightmare for young children. Of course in the holy roman/ Prussian empire, you could not jut teach what you wanted, you had to be licensed. What you taught had to be approved. Documentation and evidence had to be presented to the school commissioner. So Justus and all teachers were constantly pushed to keep up to date, to justify and to write thir own text books once approved!
In this kind of busy scenario , Hermann was born and adopted by his uncle. He learned in the schools set up by his father , and enjoyed good familial communication with his brother and relatives. Hermann was autistic I think, but he worked hard to impress his father, Justus. So it was with some astonishment that the family found out he was gifted mathematically! However the rigid formalism his father had established for mathematics and geometry chimed with Hermann who was able to see things his father had missed, due to his busy schedule, no doubt. The things he saw are all in Euclids books1 and 2 including the lineal Algrbra and the philosophical mind training.
The gnomic algebra was misunderstood because of the rhetoric. When the type of terminology changed to Cartesian and De Fermt's these algebraic forms were redacted into the modern terminology. In so doing, Euclids Gnomic lineal Algebra was overlooked in favour of the Cartesian one. Hermann fid not overlook it, but he struggled many years to distinguish it from the Cartesian overlay. This has been the problem since Regomontus published his work on the triangle. Leibniz felt there was an Algrbra that did not require coordinates, but no one was interested in his ideas regarding that.
However round about the same time the need to develop a mathematics of rotation was felt, Grassmann and Hamilton started to tackle the group structure of arithmetic and then impose it on algebra and geometry. Hamilton went the other way though. He freed himself mentally as much as he could from geometry and set up a description of moments in a progressive time flow. In this way he hoped to avoid the trenchant reliance on Cartesian coordinates.
He succeeded in publishing the first vector algebra since Newton, which embedded the arithmetics of all the types of numbers and which he successfully ( with a bit of trickery!) used to embed the complex numbers as conjugate functions. This he called "The Science of Pure Time", published in 1834 . He then set himself the task of extending this Algrbra to deal with rotations in 3d.
In the meantime Grassmann is very slowly uncovering/ intuitively discovering the Gnomonic algebra in Euclid, and testing it out. Origins is solving the rotation problem using spherical trigonometry and Half angles, but hardly telling any body and Hamilton is stuck until 1843 when he suddenly realises that there was a solution based on 8 axes rather than 6! He called these solutions he found Quaternions for mystical Pythagorean reasons, and presented them to the royal society in Ireland and devoted the rest of his life to studying them and promoting them. The following year Hermann Grassmann published Ausdehnungslejre 1844 to an almost deathly silence! Only a few young philosophers picked up on it , one being Peano!
For nearly 17 years thereafter he tried to promote it, without any success. However in about 1853 Hamilton received a copy and at once recognised its genius. He acknowledged Hermann as his master, and strove to generalise his work on Quternions to catch him up!
The redaction and republishing effort inspired by Robert Grassmann is probably what saved the book. The Grassmanns also seemed to have sparked off an international reform movement !
Since then, only a handful have gone back to Euclid to see if the Grassmanns got it right! In the min, they agreed , but still brought out more from Euclids treasure chest!
Amongst those in current times is Norman Wildberger, who has morphed the gnomic algebra again, to create a hybrid Cartesian form. He has engaged in a mammoth redaction and rewrite and has been able to extend the work of the Grassmanns in a positive and authentic direction..while it is true that Hermann uses the continuos numbers or real numbers, it was not as a Dedekind cut. The notion of continuity and contiguity in Grassmann is therefore Eucldean and Newtonian.
What is the difference? Well today we derive functions on real interval, they served functions on rational intervals with interpolations! They used tabulated sines as functions, not a computer generated approximation. Thus a tabulated sine, sometimes given to over 20 places, is in effect a very large integer or natural number. The pragmatics forced a realism in calculus. The results looked for we're always approximations not exactitudes. Norman's rational trigonometry is therefore historically appropriate and pragmatic, and does divert away from sloppy thinking.
The gnomonic Algrbra in book 2 develops the notion of a gnomon from scratch, but it develops it in a standard rectangular form, while defining it in a general parallelogramic form. The reason is that the standard form is the one the algebra is reduced to by constructions, whatever the form of the gnomon. In fact the real standard gnomon is the circular or lunar one, but few have ever heard of it!
The gnomon in a parallelogram form , by use of parallel line constructions can be reduced to a rectilinear one. Also by " gonu" rotations a form can be adjusted to a standard form.
The last few constructions show how the student is expected to imagine the gnomon in his or her mind in order to make the required reduction or association to a prior result. The training in Books 1 and 2 take a student from simple on the page or Epiphania constructions to full on mental constructions based on those exercises. Thus Norman points out how some sophists have tied to how that relying on diagrams leads to logical mistakes! In fact it is quite the reverse! Relying on mental constructions leads to mistakes! That is why practice and training is necessary and why we always go back to irst principles of construction!
Now we see why these 2 books inspire such creativity. They are not a definitive algebraic geometry! They are a set of training exercises to develop the philosophical thinking of the student. In the course of doing the training, many have noticed new possible distinctions which either simplify or complicate! So it was they Hermann came upon the interior product as a projected construction by vertical projection( book1) . Within book 2 a line representing the cosine appears. This is usually converted into a cosine of an angle, but in the construction it is just another Strecke. Thus book 2 demonstrates that it is possible to construct an algebra that is based purel on Strecken and the forms thy produce. It also demonstrates from the outset the summation or combinatorial sum of these forms!
Now book 2 is directly related to books 6 and 7 which I will review again. But the point is that ratios and proportions have to be defined on an embedded mosaic. The geometric algebra shows much can be done to simplify any formal construction Before any arithmetical calculus ( Arithmos or mosaic based calculus) is employed. But ultimately we have to bed down in some mosaic to actually construct pragmatically.
It was always Pythagoras belief that these embedded mosaics preceded Astrological reasoning, but Euclid shows that for the gnomic Algebra , even the circular gnomic Algebra there exists a constructive method that actual underlies the proportion schemes in the Arithmoi. I used to think and teach that geometry leads to algebra which leads to arithmetic. This is the Axiomatic Dogma in a sense. But Euclid shows that it is possible for philosophical and pragmatic concerns to generate or construct forms and relationships in space and mentally which motivate the design and use of Arithmoi, and lead to the definition of ratio and proportion.
Thus an philosophical, metaphysical mixing of inner and outer experience which may perhaps be defined as intuitive, but also must be recognised as profoundly constructivist is what underpins ratio, proportion, through a fractal construction of the Arithmoi. It is the Algebra, the rhetoric and terminology that animates this process of intuition and construction , and it truly is a convoluted, self referencing tautological ball of WAX!
We tend to call it creativity. The best response to adversity is creativity. But this is a general solution! The best response to living is creativity!