The author’s starting point consists of the two oddities attached to the name of Euclid: on one hand the trouble around appreciation of his Postulate, on the other doubts about his identity and nationality. Pointing out precise errors and lies, in the Western culture, and fighting injustice, C. Velpry calls for improvements both in science and in ethics.

 
 
 
 

ABSTRACT of the BOOK:

 
 

The Prologue opens with a brief survey of the matter (we translate):

 
 

 

 
 

« As an individual Euclid was concealed by history. His working conditions are left in the dark; on antecedents of the work deceitful accounts are given; in the work itself, the geometrical theory, which its supreme achievement consists of, stays misunderstood, and wrongly read. More the task of editing the text must be considered as unfinished too. »

 
 
 
 

Next it is right away announced that Euclid, to be sure, was an Egyptian priest; the Alexandrine society, wherein he lived and worked, never intended, as the author accuses, to keep his memory: Greeks denied acknowledgement to his person and moreover to the whole Egyptian civilization. C. Velpry points out that his own work is situated « within the line reopened by Cheikh Anta Diop », and he is himself « a leucoderm, Catholic-and-French born ». He recalls that the ancient Egyptian people was melanoderm, and he will from now on refer to it through the name « Kemet », derived from one they used, with respect to their skin colour. He gives notice that, having started just with the desire to practise geometry in a way « rationally satisfactory », he was led to the discoveries here transmitted; and he remarks that the limits which he collided with were those revealed through the Westerns’ bad reception of

N. Lobačevskiĭ’s and J. Bolyai’s works.

 
 

As for Euclid’s time, he invites the reader to revisit the first Ptolemy’s policy (1). The creation of the Library and Museum is to be seen as the Egyptian science seizure organized by the sovereign: the scholars, which were priests, were obliged to dwell in the king’s neighbourhood (i.e. under his surveillance), while science documents were pulled out of the temples and became his property. No improvement resulted in science research or teaching. Tendentious presenting of the facts, from Antiquity and above all in the history written during Modern Times by the Westerns, was intended, the author says, to conceal that Europeans are indebted for acknowledgement to Euclid and broadly to the Kemet‑s. Stress is laid on omission being at work in the writing of history. Lastly the author takes the liberty of restoring the sense of one political understatement, that in Euclid’s famous reply to Ptolemy about the lack of any « royal direct way » in geometry.

Chapter 1 is devoted to study Euclid’s own contribution to geometry theory, in a still fundamental work of his, the Elements. The importance of the famous 5th Postulate, in First Book, is emphasized (let us recall the wording: « If a straight [line] falling on two straight [lines] makes the interior angles on the same side less than two right [angles], the two straight [lines] produced to infinity meet, on that side on which are the angles less than two right [angles]. ») The geometer put it in the limelight throughout Books 1 & 3, since in each one a whole first part (prop. 1 to 28 in the former, 1 to 19 in the latter) is devoted to results proved without the help of the postulate, while the end consists of those stated with it; let us note among these the constructing of a square and both theorems, direct and reciprocal, about the hypotenuse square. Euclid is precisely the first geometer having realized that square is no self-evident datum, but its construction cannot be proved without any special postulate. More generally, Euclid showed capable for knowing whether such and such theorem is depending on 5th Postulate or not, for which performance his successors have not shown capable before a long time. Yet it is to be noticed that in Book 11, where stereometry is introduced, the author gave the proposition sequence without submitting to a same requirement; even he supplied some properties not depending on the postulate with proofs relying on others, which depend on it (a detailed table is given). Having noticed the fact, one is led to conceive that Book 11 was written before Books 1 & 3 received the shape transmitted to us; and that, having resumed his text and rewritten both books in order to give them their ultimate form, Euclid brought to light 5th Postulate (and also the refined definition of straight line parallelism); having thus provided it (in these books) with a spectacular rôle, he brought out the concept of a « theory relying on postulates », of which his work showed the first example.

 
 

First, the postulate was a new invention, through that its precise wording enabled us to effect: especially, to use systematically triangle congruence cases, replacing the so-called « Pythagora’s » theorem with them, and to obtain first results in the geometry theory improperly named « non-Euclidean ». (One may note, besides, that in the postulate terms Euclid inscribed the reducing of a global problem [solvable through an infinitary scheme of trials] to a local [and finite] one.) Second, postulate was a new invention: having chosen, to erect theory, the concept « ait

çma » — that means « request, asking », Latin « postulatum » — in preference to « axiōma » — « unfalsifiable principle » — (which term Aristotle was fond of), Euclid ran the decisive tiny step aside making a new path open to exact sciences. There had been to wait until second half of XIXth century A. D. for mathematicians having completely taken in account Euclid’s invention. What they did, unfortunately, denying Euclid’s rôle since they did not yet, despite Bolyai’s and Lobačevskiĭ’s acquiring, “cleaned out Euclid of every suspicion”, and made it their practise, contrary both to etymology and history, to call the postulates “axioms” and

« axiomatical » such theories as the one constructed by him. (By the way, the author makes the counter-proposal intending to better name them « etematical ».) The facts here reported are hushed up in most of works about history of mathematical logic, and so the analysis of improvements effected by Euclid in the domain of proof logic. Far ahead, indeed, of the Logicists in his time, he advanced, especially through the use of equality relation, up to equip his text, written in the « Propositional Calculus », with a very precise tendency towards the « Predicate Calculus »: from some aspects, the « variable » concept is already underlying in it… Then is given exposition of Euclid’s connexion to Plato and of his dissimilarities from Aristotle. To come to an end, the author notes that Euclid used to leave numerical matter out, preferring to deal with magnitudes, and, although he was much interested in using infinite in logic and arithmetic, he looks timorous before the use of infinitesimal calculus methods applied to varied lengths, areas and volumes; yet he was eagerly fond of what we call algebraic geometry, peculiarly the rigorous constructions effected by means of straight lines and circles.  
 

In the Chapter 2, the author starts denouncing one calamity in geometry, which consists of the « equidistance » property having been received as an alternative definition for parallelism: in ancient times, the only al-Khayyam smelt out the trap

. Although J. Bolyai et N. Lobačevskiĭ, recovering the real meaning of 5

th Postulate, were geometry’s true liberators, their works were left in a semi-darkness up to present time by the Westerns (except some specialists). Their task opening to hyperbolics was well prepared, as for the « etematic » concern, by Euclid. Less natural, in the opposite, was the opening to spherics; this relates to the fact that the Elements’ first proposition (« on a given segment to construct an equilateral triangle ») is supplied with an uncompleted proof, and cannot be performed in every case on the sphere. Uneasy to be removed, the lapse (sic) of Euclid’s is studied among others, revealing repairable. The author ends with yielding a trigonometry unification of the three metrical geometries, including universal formulas from his own, with the help of which he produced contributions in physics (referenced).
 

An Appendix presents, extracted from the Elements and translated word for word by the author, the series of proposition enunciations in Books 1 & 11 (excluding proofs and figures).

In the Chapter 3, coming to history and remarking the lack of ethical landmarks in the domain, the author, in a very unusual manner, starts requiring the « friend whom he speaks to » to confess whether he agrees himself to discrimination and racism or not, prior to the debate being continued.

Then he recalls that Kemet geometers most probably obtained, in the second half of Vth century B.C., rigorous proofs in the calculation of cone and pyramid volumes and areas, in a time when no geometry theory is proved to have existed in Hellas. Explaining a passage ordinarily bypassed in Plato’s Laws (818-822), C. Velpry proves the Kemet‑s are the inventors of the theory of irrationnals attributed to Eudoxos of Knidos. Archimedes’s witness is invoked in order to show in how high position Euclid and his work were as soon as in the time just after him.

Besides, from the absence of any ethnical epithet attached to the famous Euclid’s name (no city claims for having given him birth), we are definitively stopped from seeing him as a Hellene; moreover, geometry having remained up to his time the Kemet‑s’ speciality, hence follows the conclusion: he must be an Egyptian priest. One English mediaeval poem, literally quoted, gives credit to the idea according to which western peoples were long conscious of the fact. One act of purloining Euclid’s identity is proved: in the first printed edition of the Elements (Venice, 1482), which yields the Latin text (from the Arabic tradition) by Campanus of Novara (XIIIth century), the title was altered by the editor, E. Ratdolt. Originally « Euclidis Elementorum quindici Libri », the title is added by Ratdolt the term « Megarensis », put between the two first words, though what the geometer Euclid is mistaken for Euclid of Megara, a known philosopher, himself a Hellene, and one century more ancient. Although Clavius, coming afterwards, erased the « Megarensis », the mathematician curiously remained a Greek to the Westerns’ eyes, at least up to Christian Velpry’s work. This man evokes other examples, notably that of the Black Virgin’s pictures (which the Celts were already interested in), facing to which the Moderns gave a proof of how difficult to them is to acknowledge what they received from Africans. He recalls such an oddity as, while in Russia or Hungary they could admit Euclid’s Postulate to its function as a postulate, in France, Britain or Germany, they persisted pretending it was a prime principle. Is thus brought to evidence how much the political rejection of the « Black » could have weighed on scientists’ thinking and damaged science development, facts going on up to present day.

In the Conclusion, which first sums up results obtained by the author, remarks are inscribed against Heiberg’s edition of the text. Then the author, having for the last time questioned the Western policy, from Rome to present days, says again what danger, for those practising it, consists of lying by omission, denounces the traditional way to look at the Africans, and calls everyone to develop towards them feelings joined with the sense of acknowledgement.

A poetical Épilogue evokes around Euclid’s ghost, some ethical patterns.

In the Supplément, the author puts some questions arisen from his African encounters and friendships: 1) family vs. society, 2) illness, death, medicine, solidarity, 3) bank, money, and Tchundjang Pouemi’s teaching, 4) sexual oppressions and exclusions.

In the Notes put at the back, some complements are given; and the intellectual skiddings of some scholars (dead or alive) there receive suitable treatments.

To come to an end, the Bibliography gives a large extent, by the side of mathematics, to historical references.

 
 
 
 

(As for an English title to the work, we should propose:

 
 

« Euclid, an African

About the Purloined Geometry » ,

a significant reference to E. Allan Poe’s famous « the Purloined Letter ».)

 
 

CV, 30-I-2006

The book is edited by Menaibuc (Paris), and distributed by Fnac (France).

ISBN: 2-911372-55-7 ; EAN: 978 2911372551.

 

Notes.
(1). Alexander’s general and, nearly sure, bastard half brother, Ptolemy Lagid moved in to Egypt at the latter’s death (323 B.C.); he there founded a dynasty that will have longed till the celebrated Cleopatra. The astronomer Claudius Ptolemy (2nd century A.D.) did not pertain to the family.

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