circle beyond three times its diameter as might very well be taken, under an imperfect system of measurement, for the true excess, and certainly sufficiently true for all the purposes of art, or practical geometry and design. It is in fact true within th part of the chord, or about ath part of the circumference to which it belongs. So that if the dome of St. Paul's be 60 feet in diameter it would give the true circumference within th of an inch. These pointing triangles indeed seem to bear an ideal relationship to the circle and its diameter in their own proportions, for the measure of their two sides is as 3 to 1 to their proper base, and, coupled with the office they perform, it would not be wonderful if they were regarded as accessory powers in the developement of the mysteries of proportion, and handmaids of the imperfect science of the day. A people who embalmed their domestic auxiliaries which cleared their granaries of mice, and their rivers of crocodiles, may very easily be conceived to have enshrined such a geometrical auxiliary in the structure of their pyramids. But, before I proceed to the relation of the square figure, permit me to advert in a few words to the division of the pyramidal proportions into the 113 feet of English measure, for we shall not forget that this is the common divisor of Belzoni's measurements. Connected with the previous hypothesis, the use of that number is at least a most singular coincidence with the Metian resolution of the circle, which gives the same number of parts to a diameter whose circumference is 355; making the diameter 113, to an excess of 16 (or 113 x3 + 16). For admitting the proportions which are shewn to exist in the structure of the pyramid to have been contemplated in the original design, it is absolutely necessary that the measure used in laying it out should have been the English foot itself, or some other measure which would commensurate with 113 feet English. But the old Egyptian cubit will not commensurate with 113 feet English. According to the French Encyclopædia of Arts and Sciences all the authorities concur in fixing that cubit at 1ft. 9in. of English measure. Of such cubits it is true that 62 very nearly equals 113 English feet; but the difference amounts to between 6 and 7 inches in the length of the base of the pyramid; a variation which could not possibly have happened by any mistake in the measurement of a base line of 678 feet by a measurer so experienced and accurate as Belzoni was. Can it be, then, that the English measure existed among the Egyptians, and was used by the builders of the pyramids? Do we know whence the English foot was derived? It is detected in very ancient monuments, but more particularly it has been used in the measures of land from all antiquity. But if we may suppose the infant art of geometry to have been brought into England by the Druidical priesthood, of which one order or section was exclusively engaged in the pursuits of science or works of art, and whose connection with the temples of Upper Egypt and the Ammonite Oasis might, I think, be satisfactorily shewn, is it asserting too much if we say that the descent of the English foot from the Egyptian geometers of an extremely ancient date may be classed in the category of probabilities? I think not; and if the measures given by Belzoni be accurate, which need not be doubted, it appears pretty certain that the English measure was the measure used, and no other. By the actual measure of the pyramid, then, the sides of the square Br, Da, which is one-fifth of the radius, or one-tenth of the diameter of that circle of which the quadrant is part, will be expressed by the number 113, and the diameter of the circle in connection with it will be expressed by ten times that number, or 1130, and the excess required in the circle beyond three times that diameter, according to the rule of Metius, will be 160. Thus we find the same diagram giving expression to this excess, and also to the diagonal of the square in relation to its side in these relative numbers, for the side 113 to a diagonal of 160 is correct within th part of the diagonal line, or nearly in the same degree as the excess shewn in that same chord is to the true excess in the circumference of the circle, for that we have said is within th part of that same chord. The chord BD does, therefore, in connection with the number 113, give both the actual measure and the expressed measure of the circle's excess, and the square's diagonal. These combined approximations to the only two unexpressible relations of geometry thus shewn in one nodus was certainly a secret worth knowing; but, in order to satisfy the minds of your readers that I am not speculating upon their credulity, I will state the matter arithmetically, and so plainly that any of them who can fathom the mystery of a rule-of-three sum, and understand the simplest application of decimals to the expression of a fraction, will see that the points are strictly true as stated. As a standard of proportion between the side of a square and its diagonal I will give them the numbers 5,288,400 to 7,478,927, which expresses that proportion so nearly that the variation from truth is only as about one inch in 36,000 miles, or th of an inch in a diagonal a mile long. These proportions are expressible in decimal numbers, as 1 to 1·41421356, &c. which therefore do accurately express the relative quantities of the little square Br, Da, and its diagonal B D. That little square being th of the circle's diameter in which it stands, its diagonal is therefore in proportion to that diameter as 141421356, &c. to 1, while the received proportion of the circle's excess beyond its three diameters is as 141592653 to 1 of that diameter. The difference, therefore, between that excess, and the diagonal of the little square shewn in the chord B D will appear in the difference of those decimals, and will be found to amount in fractional expression, as I have stated, toth part of that diagonal or chord, as any one may prove by reducing the difference of those decimals to the proportion which that difference bears to the whole. The number 113, therefore, found in the measures of this pyramid, carries also the other Metian number of 16 as part of the mystery intended to be recorded. Whether that is at all connected with the sixteen cupids, which are found sporting on the River God, the ancient Nilus, in the museum of the Capitol, I do not know: but it has appeared to me that the 16 cubits in the rise of the Nile, which those cupidons are generally thought to represent, do not answer the idea which those emblems convey, for the 16 cubits were not the fruitful cubits, but those which preceded the opening of the waters into the country. But take the 16 degrees in which the ancient river was known between the Delta and the Ethiopian Meroë to be intended, those figures would very accurately express that number of regional divisions through which the river flowed; and if we take these as the proximate number of degrees which in the circumference of the globe fills up the geometrical measure of it beyond its 3 diameters, there seems an additional reason for thinking it possible that this number may have had that meaning in it. It is very singular that we find this same number again on the summit of the great pyramid, which, according to Thevenot, terminates in a square table, which, he says, measures 16 feet and rds. Now that pyramid is greater than the one of Belzoni's measurement, and, if the charmed proportion was intended to be preserved in the measure of this table, the difference in the magnitude of the pyramid may account for the excess of the table beyond the proximate 16 of Belzoni's pyramid. The rds of a foot would be a twenty-fourth of 16 feet; but say the measure is a twenty-sixth, which would make the table to be a little less than Thevenot's measure, that would require the lesser pyramid to be increased th in its base to equal the larger one, making the base of the greater one in effect the same as the lesser one, plus the square root of the base line of that lesser one; for 26 is the square root of 678, which is the base of the lesser pyramid. The larger pyramid according to this rule ought therefore to measure 678 +26, or 704 feet. What its true measure is will probably never be known, until another Belzoni arises; it has been stated at 693 feet, but all the old measures have proved upon examination to be too little, and that is probably the case with this. The object would be worth a visit to Cairo, and I offer it as a boon to some of your travelling readers, Mr. Fellows or Mr. Auldjo for instance, who would find it as profitable as ascending Mont Blanc, or digging stones in the Troad. It would prove demonstratively the knowledge of the square root among these people. To return to our diagram, the question arises, whether, in an imperfect state of science, the measures of the chord or diagonal pointed out by the pyramidal pointers may not have been regarded as essentially that which formed the circle's excess, and which, as Metius did, might be expressed by the number 16 in relation to the diameter of 113. The difference from truth in the Metian numbers of 113 and 16 is so small that it amounts only to about 12 feet in the circumference of the whole globe; and from the actual measure of the chord the real excess differs by about th part of the circumference, as I have stated, which is a practical infinitesimal; but, taking it to have been so regarded, it would have given to that chord and its pointers the credit of a geometrical proposition, which may be thus stated that the excess of the circumference of a circle over its three diameters is proportional toth of its diameter, as the diagonal of a square is to its side, and consequently the circumference would be found by 3 times the diameter, plus the diagonal of a square, formed on one-tenth of that diameter. In conclusion, I may observe, that it is no answer to this hypothesis, that the relations thus shewn are not mathematically true; for, though not mathematically true, they are proportional, and, therefore, geometrically true; and they are as true as the arts of geometry enabled the people of those periods to approach the truth; and until the discovery of the great problem of mathematics, by which the proportions of right-angled figures are alone adjustable, and a new method of reasoning upon the nature of the circle, grew up in the improved state of science, any conclusions upon the proportions of these figures could only have been attained by actual measurement or accidental observation; and I apprehend that the approximations to the truth presented in the above figure, connected with the natural GENT. MAG. VOL. XXVI. index which the twin pointers afford, were quite sufficient to attach all the importance to them which is here ascribed. One word more, and I have done. The nodus of these relations lies in the pyramidal apex B D X, comprehending one-fourth of the height of the pyramid from its summit; and it is precisely that portion which, in one of the pyramids, is covered or sealed up from all access by having a smooth surface, instead of the stepping stones of the lower portion. Surely there must have been a reason for this: the lower regions of the building were appropriated to a use: may not the upper regions have had theirs? I believe no attempts have ever been made to investigate the interior of these pyramids, except at the base; but in the most ancient idolatry of these countries, the Jupiter Belus, which prevailed both in the Egyptian Thebes and Babylon, there was a double consecration of the temples in their lower and upper stories. The former held the inanimate image of the God, the latter his bed and table, and was the strong hold of his living appetites; and is it not equally probable that if the Egyptian Pharaohs buried their dead divinity in the foundation of the pyramids, they may have made these sealed summits the depository of their true worship? In fact, that these were their treasure cities, removed by their height above the reach of the waters, and guarded by the superstitious veneration attached to the sepulchred divinity below from the approach and violence of the people? One cannot help thinking, therefore, that if future Belzonis would direct their labours to these upper regions for the treasures which Cambyses may have missed in his predatory visit to the ancient of days, rather than the sepulchral vaults of the dead Apis underneath, it might better answer their purpose; a hint which, perhaps, Ibrahim Pasha might use with advantage. Yours, &c. H. M. G. P. S. Your correspondent of Lichfield did not assist my exposition of the pyramidal problem by his statement of it in another, as it struck me, not a more simple form. I am obliged by his good intentions, but the manner 3 R in which he put his observations was certainly calculated to mislead a casual reader into a supposition that the proportions had been already discovered in the pyramids. I am aware he did not intend this, because it is not pretended that the proportions themselves are a new discovery, which is all that is disclosed by the "toy" he refers to in the Egyptian squares, of the existence of which I confess I was not aware when I wrote my letter. These proportions are perfectly well known, as all the relations of geometrical figures are, and, though the inventor of the Egyptian squares applied them to the ingenious use he has made of them, and given them an Egyptian name, probably because of their relation to the pyramidal form, yet it never was surmised nor suspected that these proportions were observed in the structure of the pyramids themselves, a fact which has been disclosed incidentally by the accurate measurements of Belzoni. ANCIENT ALTAR-SCREEN AT REIGATE. THE annexed wood-cut is a representation of an ancient stone rere-dos, or altar-screen, discovered in the course of last year, and occupying the eastern wall of the chancel of Reigate Church, Surrey. The height of that portion which extends from the window-sill to the pavement is 8 feet. On each side of this rise two lofty niches one above another, with brackets and canopies flanking the window, to a height of more than 16 feet. The total breadth is 20 feet. angle of the altar to the large niches on either side. The whole was surmounted by a range of that ornament known as the "Tudor flower," which was continued along the north wall of the sacrarium, below the window, and in both cases supported by a moulding, enriched at intervals by other carved foliage. The plan of this rere-dos consists of a series of 13 niches, richly worked within, and surmounted by crocketed canopies of the ogee form. Between each canopy a pinnacle, also crocketed, was introduced. These, together with the exterior angles of the brackets terminating the niches below, and all other projecting ornaments, were found to have been struck off, and over the even surface a coat of plaster was laid, on which were the remains of an inscription in black letter. Thus the whole of this rere-dos was probably mutilated in the 16th century. The work itself is assigned to the perpendicular, or third pointed period of Christian art. Twelve of these niches are of equal size, and were doubtless occupied by figures of the Apostles; the central one, of somewhat larger dimensions, by the Virgin and infant Saviour. Below, and occupying more than 10 feet of the entire width, projected the altar slab, supported probably on the three sides by solid stone-work of plain panelling, similar to that which still runs below the tier of niches, from the That which, perhaps, gives the chief interest to this discovery, is the variety and harmony of the colours employed on the work. It was on this account described at the Winchester meeting of the Archæological Association as "polychromatic.” The prevailing colours are red and green, but these of several shades, and some few lines are picked out in white. Over the face of the work ran a delicate tracery in colour of tendrils and flowers. Fragments of the crockets and other carefully wrought ornaments were found mortared into the recesses themselves-these were richly painted and gilt. Gold flowers, stars, fleursde-lis, &c. occurred frequently in various parts, and the sacred name of ibs seemed to be repeated beneath every one of the thirteen niches. It would appear that one of the upper side-niches had been entirely diapered in gold, and thus the general effect when the whole was lighted up must have been magnificent. The side-altar in the north chancel seems to have had a similar decoration on a smaller scale. On removing the plaster on either side of the east window in that chancel, a niche was discovered, filled in with masonry. One of these was enriched with the name of ibs repeatedly inscribed; the other was ap MR. UPCOTT'S LIBRARY OF MSS. &c. WE now proceed to give our promised account of the Sale of the Manuscripts and other Collections of the late Mr. William Upcott. His property was divided into three Catalogues, the first of Books, for five days, beginning on the 15th of June; the second, of Manuscripts and Autograph Letters, for three days, beginning on the 22d June; the third of Prints, Pictures, and Curiosities, on the 25th of June and two following days. The sale took place in the rooms of Messrs. Evans in New Bond Street, but, in consequence of the peculiar circumstances of that firm, was conducted by Messrs. Leigh Sotheby and Wilkinson. The collection of Books was very miscellaneous in character. Many of the volumes had been preserved only on account of the autograph name or notes inscribed in them.* The most * Lot 557 was a copy of the " Parallele de l'Architecture Antique et de la Moderne. Par R. Freart. Paris, 1702," a volume which had been presented by the Earl of Cork to Dean Swift, with the following inscription, to which Pope was a party "I give this Book to Dr. Jonathan Swift, Dean of St. Patrick's, Dublin, in order to constitute him the Director of Architecture in Ireland, especially upon my own Estate in that kingdom. "CORKE BURLINGTON. "July 27, 1726. "Witness, A. POPE. "Which Book I do hereby give to my ingenious and worthy friend Francis Bindon, Esq. hereby delegating him Director of Architecture through all Europe." This book was sold for three guineas. |