mercredi 20 avril 2011

Selon H.C. Agnew (XIXe s.), les Égyptiens ont appliqué la “quadrature du cercle” pour la configuration du site de Guizeh

Une fois encore, il m’est impossible d’avoir accès à des informations précises sur l’auteur présenté ici : H.C. Agnew. Quelques références sont disponibles sur internet, mais faute de pouvoir vérifier leur exactitude, je préfère les omettre.
Cet auteur voit son nom accompagné de l'abréviation “Esq” (“Esquire” - “cavalier”), signifiant une origine nobiliaire. On sait également qu’il était membre de la Egyptian Society.
L’ouvrage, relatif aux pyramides de Guizeh, que l’on connaît de lui est un opuscule d’une cinquantaine de pages : A Letter from Alexandria, on the evidence of the practical application of the quadrature of the circle, in the configuration of the Great Pyramids of Gizeh, 1838. Il comporte de savants calculs basés sur les mesures, la forme, l’inclinaison des faces des pyramides et leur emplacement respectif sur un même site ayant fait l’objet d’un projet global (un “immense système”), fixé par une “loi” (law), et non par “pure commodité” (mere convenience), source d’une extraordinaire harmonie d’ensemble, “preuve du goût et du jugement des architectes”. Bref, les “matheux” pourront se délecter à la lecture de cette Lettre.
Cumulant à la fois une certaine aversion et une réelle incompétence dès lors qu’il s’agit de chiffres et de proportions, je suis dans l’incapacité d’assurer une présentation au moins honnête de la théorie d’Agnew. Il se trouve par contre qu’un certain Howard Vyse a déjà fait le travail à ma place, au terme de son ouvrage Operations carried on at the pyramids of Gizeh in 1837 (1840), présenté dans ce blog. Il m’était impossible de ne pas profiter de l’aubaine, pour m’abriter derrière l’autorité du savant colonel...
Je souligne enfin le conseil donné par Vyse lui-même : pour mieux apprécier la démonstration d’Agnew, il est indispensable de se référer à son propre texte, ce que j’ai illustré ici par quelques exemples. On trouvera au bas de cette note un lien vers ce texte dans son intégralité (en anglais).

Extrait de Google Earth

Résumé de la théorie de H.C. Agnew par Howard Vyse
“Mr. Agnew published, in 1838, a treatise on the application of the quadrature of the circle to the configuration of the Great Pyramids of Gizeh, which, together with the causeway, he considers to be "component parts of one immense system" (1) ; and says that “the whole of the immense scheme of the three Pyramids proceeded from a chief circle of origin, the properties of which were more especially to be represented". He also observes in another place, that "the Pyramids of Egypt appear in general to have been emblems of the sacred sphere, and of its great circle, exhibited in the most convenient architectural form" ; and adds that "the chief object of these buildings being to serve for sepulchral monuments, the Egyptians sought, in the appropriate figure of the Pyramid, to perpetuate, at the same time, a portion of their geometrical science".
He imagines that the three in question were built in succession during the course of sixty years, and that each was begun before the preceding was finished ; that the third monarch, under whom the whole design was completed, dismissed the people from their labours, and again opened the temples for the national worship. He then goes into a variety of calculations demonstrated by geometrical figures to support these opinions, and to prove that the quadrature of the circle was known, "with all practicable approach to exactness", by the ancient Egyptians (2) ; and that in the bases, proportions, and relative positions of these three Pyramids, and also in the size and course of the northern causeway, and in those of the adjoining pits, certain geometrical rules were attended to. For the causeway belonging to the Third Pyramid he attributes other reasons.
He assigns to the interior of the Pyramids also mysterious properties ; but observes that in their construction, "the proportion of five to four is very dominant" ; and he endeavours to prove that it might have referred to astronomical calculations, which may shew the aera when these buildings were erected. He likewise supposes that the labyrinth near lake Moeris was constructed on "some curious combination of geometrical figures, relating to a sphere and circle, the radius of which was the perpendicular of the forty-fathom Pyramid, which stood at one end" ; and that all the great buildings of Egypt, besides their primary and special use, elucidated in their construction geometrical science. To understand clearly, however, the author's ideas, a reference to the book is absolutely necessary.
The following remarks on the Three Pyramids are, therefore, only added.

The Great Pyramid
The author having quoted from Pliny, " est autem saxo naturali elaborata et lubricata," in support of his opinion that the exterior of the Pyramid had been originally saturated with some fluid, such as oil or varnish, (*) observes that the liquid, whatever it may have been, appears to have been transparent, and nearly colourless ; and that the brownish tinge upon the stones, which formed the casing, has been acquired by time. He does not conceive that any part of the Pyramids had been painted red, or it would have been mentioned by Herodotus, and by other ancient authors, although the pavements and steps around them may have been stained with some dark colour. He is of opinion, that the Great Pyramid had been revetted in the same manner as the Second, because he has seen fragments of stones precisely similar to those, which covered that building, the external faces of which had been "lubricated".
With respect to the interior of this Pyramid, he found the angle of entrance to be 26° 34' ; and, referring to a diagram, proceeds with many conjectures and calculations, which it is not necessary to detail. He is of opinion that many other apartments, besides those already known, exist ; and that more than one person was interred in the edifice ; that several pits and deep channels, with places on their sides for the reception of mummies, have been excavated in concentric squares, the area of each diminishing by one-half successively ; and that, by the extent of these works, the length of time taken up in their construction might be ascertained. He imagines that there must be a second entrance at the base, either at the northern, or eastern side; and mentions the grotto in the well, and several other points, where apartments and passages might probably be found.

The Second Pyramid
He considers, that the exterior is steeper by one degree than that of the Great one (3) ; and that its summit is higher above the horizon, although the Great Pyramid is the more lofty, owing to a base upon which the Second is built, as it is situated upon a high perpendicular platform beyond the pavement, and is surrounded with three squares or steps, which bear certain proportions to the sphere or circle ; and this, he imagines, to have been likewise the case with the steps around the other two Pyramids. The step of the Second Pyramid is five feet high, and seven feet from the perpendicular of the edge of the base, and the platform of rock is about ten or eleven feet above the inner surface. (...) He conceives that there were two ranges of granite, and that the inner step was also of that material. After other observations, he says, that the dark colour of the casing has led many people to believe that it was painted.

The Third Pyramid
He is of opinion that the granite casing, mentioned by ancient authors as extending half-way, was ninety-two feet high ; and that it had some mysterious signification : that the size of the building was regulated by that of the other two (4) ; and that, from an examination of the granite blocks, which formerly composed its revetment, he found that the angle of its elevation indicated a perfection of form superior to that of each of the other Pyramids, and that it was nearly a medium between them : that its perpendicular height was the radius of a circle, the circumference of which was equal to the square of the base ; and that this Pyramid was an emanation, or spirit, and essence from the first great principle of the system, namely, the circle of origin ; and also that its relative position was determined by some fixed law, and not by mere convenience.
The exterior of the upper part did not appear to have been covered with painted stucco, but to have been saturated with some fluid like those of the other Pyramids ; and the contrast of the two colours must, in his opinion, have had a good effect.”

(*) The word "lubricata" simply means a surface polished, or made slippery, and does not infer that oil or liquid of any kind had been applied to produce the effect.


Textes de H.C. Agnew, dans A Letter from Alexandria, on the evidence of the practical application of the quadrature of the circle, in the configuration of the Great Pyramids of Gizeh, 1838

(1) “Following the common fable, that each pyramid had its own peculiar builder, and that each was a separate monument, unconnected with its neighbours except by the casual contiguity of position, my first idea was that the constructors of the first and second (call them Cheops and Cephren) had each built his pyramid on the geometrical plan most accordant with his conceptions of propriety, and that their successor Mycerinus, finding his father's monument too flat and his uncle's too steep, had discovered, or believed he had discovered, the rule of perfection, and formed his pyramid accordingly. This notion was correct only in so far as that the third pyramid was the most perfect geometrical figure ; but if the deductions in the following pages be admitted, we must arrive at the remarkable conclusion, that the three great pyramids of Gizeh were component parts of one immense system, members of a vast united triad, each in itself admirable, but all three so connected with the first principle of the system as to form but one perfect whole. If then, in the contemplation of one of these sublime structures, we are lost in astonishment at the greatness of the undertaking, how must our wonder be increased when we find that all were planned at once ! that before a stone of the great causeway was laid, the precise proportions of the second and third pyramids, as well as of the first, were unalterably determined by the necessary effect of the rule which fixed the length and breadth of the causeway itself !”

(2) “Here then we find the quadrature of the circle exemplified in a curious manner, with all practicable approach to exactness, by the Egyptians a thousand years perhaps before the birth of Archimedes or of Hippocrates of Chios ; I say approach to exactness, for I am far from supposing that they had really solved the question geometrically. Its arithmetical solution is known now to be impossible; the geometrical solution, in all probability, is so likewise ; but whether the Egyptian priests were of this opinion I cannot venture to say. In the instance before us of the practical exemplification of the problem by the united properties of the two great pyramids, the method may appear a little complicated ; but the principle is one, whether a single square be made equal to a single circle, or two squares to two circles, when all are defined. We shall, however, presently see that a single and unmixed elucidation was exhibited in the third pyramid, the perpendicular of which was the radius of a circle equal to the perimeter of its base.”

Photo de Simon Cog (avec son aimable autorisation)
(3) “Most people appear to take it for granted that the two great pyramids of Gizeh were built at the same angle of elevation, and differed only in size, and not in shape ; but the French, as well as Belzoni, had long ago given measurements, the deductions from which made the second pyramid to be steeper than the first. They do not however agree in their statements of the difference, nor attempt to give a reason for the variation of the angles. The second pyramid does in fact appear steeper to a good eye ; but as it still retains the upper portion of its casing and its pointed top, while the great pyramid has lost all its outer stones and many feet of its top, the observer is apt to suppose himself deceived by these circumstances, and concludes the pyramids were of the same shape: but the truth is that the faces of the second were really steeper than those of the first by more than one degree. If you stand somewhere near the third pyramid and bring the lines of the south-east edges of the two great pyramids in contact as high up as possible, the line of the first will be observed to diverge towards the base from that of the second, showing a lower angle.”

(4) “I now felt persuaded that the third pyramid formed part of the grand system, and that the circumstance so expressly mentioned by Herodotus, Diodorus, and Strabo of the granite casing reaching half way up only, had a special meaning. It was reasonable to conclude that this third pyramid had its size necessarily determined by some properties of the first great pyramid, or of the two great pyramids jointly. The angle of the inclination of the faces which I had measured on the granite stones, gave me something under 52°, and it was evident that the true angle, were it possible to ascertain it with sufficient accuracy, would be found to have been 51° 51' 14", and that therefore this pyramid presented in itself a perfection which neither of the two great pyramids separately possessed, namely, that its perpendicular was the radius of a circle, the circumference of which was equal to the square of its base.
Texte intégral de la Lettre d’Agnew