The Demands of Euclid “itemised”!
1530s, "to demand, to claim," from L. expostulatus, pp. of expostulare "to demand urgently, remonstrate," from ex- "from" (see ex-) + postulare "to demand" (see postulate). Friendlier sense is first recorded in English 1570s. Related: Expostulated; expostulating.
postulate (v.)
early 15c. (implied in postulation), "nominate to a church office," from M.L. postulatus, pp. of postulare "to ask, demand," probably formed from pp. of L. poscere "ask urgently, demand," from *posk-to-, Italic inchoative of PIE root *prek- "to ask questions" (cf. Skt. prcchati, Avestan peresaiti "interrogates," O.H.G. forskon, Ger. forschen "to search, inquire"). Use in logic dates from 1640s, borrowed from M.L. The noun is first recorded 1580s.
1759, from Fr. postulant, from L. postulantem (nom. postulans), prp. of postulare (see postulate).

The word "axiom" comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

The root meaning of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that some things can be done, e.g. any two points can be joined by a straight line, etc.[2]

Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid's books Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property".[3] Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.

This postulate effectively specifies that we are dealing with the geometry of flat, rather than curved, space.
Common Notions

Common understandings about comparisons in which Meizonos, Elassoon, Isoon and Olon are used comparatively when comparing objects of magnitude, static or dynamic.

It is important to realise that there is no "equal" as such, just tne notion of duality or copy as something divided like a vein off of some other thing, both of which together form the original whole. The comparison then is always between parts of a whole, the whole being greater than the parts. If parts are dual then we distinguish many other attributes in those parts. Duality is a much more pervasive and subtle notion than equality which is derived from it.

ask, beg

(Show lexicon entry in LSJ Middle Liddell Slater Autenrieth) (search) αἰτέω verb 1st sg pres subj act epic doric ionic aeolic
αἰτέω verb 1st sg pres ind act epic doric ionic aeolic parad_form

Word frequency statistics

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αἰτ-έω (Aeol. αἴτημι Pi.Fr.155, Theoc.28.5), Ion. impf.
A. “αἴτεον” Hdt.: fut. αἰτήσω: aor. ᾔτησα: pf. “ᾔτηκα” 1 Ep.Jo.5.15: plpf. “ᾐτήκει” Arr.An.6.15.5: pf. Pass. ᾔτημαι, etc.:—ask, beg, abs., Od.18.49, A.Supp.341.
2. mostly c. acc. rei, ask for, demand, Il.5.358, Od.17.365, etc.; ὁδὸν αἰ. ask leave to depart, Od.10.17; αἰ. τινί τι to ask something for one, 20.74, Hdt.5.17: c. acc. pers. et rei, ask a person for a thing, Il.22.295, Od.2.387, Hdt.3.1, etc.; δίκας αἰ. τινὰ φόνου to demand satisfaction from one for . . , Hdt.8.114; “αἰ. τι πρός τινος” Thgn.556; “παρά τινος” X.An.1.3.16; “τὰ αἰτήματα ἃ ἠ̔τήκαμεν παρ᾽ αὐτοῦ” 1 Ep.Jo.5.15.
3. c. acc. pers. et inf., ask one to do, Od.3.173, S.OC1334, Ant.65, etc.; “αἰ. παρά τινος δοῦναι” Pl.Erx. 398e.
4. c. acc. only, beg of, D.L.6.49.
5. in Logic, postulate, assume, Arist.APr.41b9 (Pass.), Top.163a6, etc.
II. Med., ask for one's own use, claim, “Λύσανδρον ἄρχοντα” Lys.12.59; freq. almost = the Act., and with the same construct., first in Hdt.1.90 (παρ-), 9.34, A.Pr.822, etc.; αἰτεῖσθαί τινα ὅπως . . Antiphol.12 codd.; “πάλαισμα μἠποτε λῦσαι θεὸν αἰτοῦμαι” S.OT880; freq. abs. in part., “αἰτουμένψ μοι δός” A.Ch.480, cf. 2, Th.260, S.Ph.63; “αἰτουμένη που τεύξεται” Id.Ant.778; “αἰτησάμενος ἐχρήσατο” Lys.19.27; “οὐ πῦρ γὰρ αἰτῶν, οὐδὲ λοπάδ᾽ αἰτούμενος” Men.476; αἰτεῖσθαι ὑπέρ τινος to beg for one, Lys. 14.22.
III. Pass., of persons, have a thing begged of one, “αἰτηθέντες χρήματα” Hdt.8.111, cf. Th.2.97, etc.; “αἰτεύμενος” Theoc.14.63: c. inf., to be asked to do a thing, Pi.I.8(7).5.
2. of things, to be asked, “τὸ αἰτεόμενον” Hdt.8.112; ἵπποι ᾐτημένοι borrowed horses, Lys. 24.12.

Thus we see that Euclids Teaching material has a list of demands, things the student is required to be able to do and things they are required to accept.
If you could not draw a good line or a circle of any radius, then you were off the course! Thus it is taken for granted that you could do the requested things and accept the required assumption about good lines. Thus it is implicit that you know how to use the tools of the tradesman, the artisan, the builder , the architect, the technical drawer and he engineering drawer.The tools are: the isos, the pair of compasses, the right gnomon(ortesgnomon), the plane surface and the straight edge of the gnomon. The isos is a tool like the gnomon but for leveling off like a spirit level. It is used to produce a plane surface.

The requirement or a good line is a reference to a skilled artisan who can draw a good line or even a circle unaided. Michaelangelo was reputedly of this level of skill. Thus this was not a course for beginners, but a university level course for skilled craftsmen.

The aim at the outset appears to be the indoctrination of the Eudoxian system of reasoning and the inculcation of the properties of the sphere, and disk.

The common inferences are actually common judgements made when comparing things, but the structure of the section called common inferences is a reverse of the isis model.

Euclid has been redacted many times over the 2500 years, so there is much that may be shaped by others, or by Euclid's own revisions, but the dividing vein structure works both ways: Thus the whole is constructed from the parts. In the so called "oroi" the structure is stylised, flowing and veinlike: previous definitions are carried forward into the next level of exposition, so the meaning builds up from previous ideas, the direction is controlled by previous definitions, the development flowers like a plant from earlier seeds and notions, the branches and veins grow thicker the more definitions are established.

This connected structure of offshoots is the isis structure that leads to finer and finer distinctions and more and more relative cases.

The Common inferences use this structure in reverse. The idea is the use of Isos, which though translated equal actually better means dual. Dual however does not mean two if isos is used, it means split from the same vein into 2 similar or identical veins which remain connected to the original larger vein. Thus one starts with a whole{olon} that is divided into 2 similar or identical(isos) parts(meros).

The first inference then is the whole is greater (meizonos) than the part (meros).

The next common inference is identicalness: If one part coincides with the other part, then that is defined as Isa (dual, twin, identical). This notion and definition of dual is carried forward to the next common inference: removing a dual part from a dual part leaves the remaining parts dual. This description is fractal in nature: The dual at one level has a dual at another level removed. The remaining parts with their duals removed themselves form a dual.

This description is deliberately exposited in this way because it is a mystery of the Isis, a consequence of the attributes of the sphere that requires due reverence and meditation. However, the structure of every written sum is based on this pattern, and it is a fractal pattern.

The next inference follows from this duality: adding duals to duals makes the new wholes (greater than the old parts) dual The use of the term aphaphethoo meaning "to divide off" and prostetheuo meaning "to place onto" represent the actions of division and combination. These actions create remainders or new wholes, and the way of duals builds or reduces the magnitudes according to the isis way.

The last common inferences extends duals out wards to cover all things. Clearly duals form a powerful system of combinations and divisions.

There is one unmentioned inference: if you do exactly the same procedural steps, using the same constituents and methods of construction and implementation then the identical result will be produced. And this can be tested by identicalness inference.

Both these inferences are needed to demonstrate collinearity and any other coincidence or identicalness. In order to demonstrate that a radius can section a periphery of a circular disc into six equal arcs requires this also, as does the definition of a dual line.

The question is why this arrangement of the stanzas in the first part of the Teaching material? If Euclid did not arrange it , and it seems odd to place the entry requirements and the common notions After the definitions, then the dual nature of things was either not understood or deliberately obscured. For example the dual nature of seemeioon in relation to a line seems strange, but once one knows the secret, the spherical relationships, and in the plane the circular relationships between a "good" line and two fixed points it becomes clear that the crossing arcs are duals.

The nature of of Euclid's teaching material is deeply Pythagorean and deeply Theurgical.It is therefore an advanced study of our relationship with the so called "God's of Nature.


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