the rotation axis. But as yet only one of the three exterior areas (side) is so correlated.

Between the internal and external horizontal planes of the coffer there is this connection, that circuit of interior gives circumference of a circle whose area area of outer base. If the two planes were squares, this would mean that their sides are as π : 4, after the pattern of the alternate squares shown in the Pyramid diagrams of Mr. Latimer and Lieutenant Totten. As they are not squares, nor similar figures, nor directly related in both their homologous dimensions, the object of this connection is a little obscure. It points, however, to this: The circuit within the coffer at about half depth is 209.4395 + (surrounding there an area of 51.51648 circle, as before referred to). But 209.4395 + is circumference to a diameter = 66.6 or 3 100 inches, and to an area of 3490.6585 or 10,000 ÷ 9, which is therefore the area of the outer base.

According to the measures, this equation to base area would come out best in British inches; but the measures do not shew the full original size of the base.† The variations of sectional areas arising from irregularity of figure admit of either inch giving true results, at different levels. Those who prefer British inches might find it worth while to try to correlate all the coffer's measures from this basis. For example:

Assume (length and breadth) ÷ π = height.

and so on with alternative hypotheses.

But adhering to Pyramid inches, we may note with what ease several of the problems may be worked out from known л.

+ Mr. Petrie's lowest measures of outside were four or five inches above base; those of inside, one inch above inner floor.

[Notice in passing that 3490.658504 30 = 116.35528, nearly the length of ante-chamber in British inches.]

The interior horizontal area being that of a circle of 51.51648 diameter, is, of course, of the base area. (The contraction of the coffer within, towards floor, may likewise provide this equation to a lesser base than 9131.052, but at mid-depth the full amount is found.) As the inner end or cross section is

of the horizontal area, the former is = a square with side 91.3105; and the outer side = 4 inner end = a square with

3

side 2% 91.3105 = 60.873 += length of great step.

The annexed diagram is a combined representation of the leading geometrical problems of the coffer; and its connections with the Pyramid are shown by the small pyramid ABD, erected on the same level as the vertical cross and long sections, EFGH, JKLM, the pyramid being of the height of the Great Pyramid, as per Professor Smyth. (I may note that were the small pyramid's base about .84 of an inch lower, or on the level of the adit to chamber, its sloping sides would pass through the coffer's angles K, L.)

AC, the axis, is bisected in D' and trisected in E', F',

The horizontal section through D' at half height is the floor area ZA'B'C'; the section at E' is equal to the inner end RSTU; the section at F' = the outer side JKLM.

The circles equal to these sections respectively are the 6th, 3rd and 8th, counting from the centre.

The outermost circle has circumference 365.242 + Pyramid inches; and this circle I would view as the fundamental origin of the coffer's dimensions. It is of equal area to the four exterior vertical sides. A circle of half this area is figured as passing through the inner angles ZA'B'C'. Its area is the sum of the two vertical sections of the coffer, meridian and prime vertical, and is Too of the sum of the two corresponding sections of the Great Pyramid itself. The radius of this circle is semidiagonal of interior floor, and minimum height of coffer (nearly), and its circumference circuit of base.

=

The diameters of the other circles shown will be found in the margin; and the theorems of which they are exponents will, I

8.

9.=

10.77.27472 and

77.50682

II. = 82.2084 and 83 ±

think, be readily traced, without being detailed here, by those who care to study the diagram. Some of the intersections of the circles with the coffer are also interesting, as in the points O'P', EH, W'X', S"T", L"M". The sarcophagus ledge is defined in plan approximately by the circle passing through ZA'B'C', and in elevation by that passing through A, and (approximately) by the sides AB, AD passing near R and U, Y'. Coffer's depth bisected at E"F"?

The five innermost circles are definitive of the cross sections EFGH, RSTU. The smallest, having diameter = inner breadth (minimum), has circumference equal to horizontal diagonal A'C'. The next, having diameter = of base side, or 30.43685, has area = square of full breadth (26.973972). Then 30.436852 area of inner section RSTU,= circle No. 3 with diameter = depth 34.34432; whose circumference is = perimeter of square of breadth 26.97397. The square of 34. 34432 is again equal to circle No. 4, having diameter 38.75341 += outer breadth. Circle No. 5 is not directly connected with the foregoing. Its diameter, say 45.2, is about half the maximum length of coffer; its area, 1605 = area of outer cross section EFGH.

Circles 6 and 7 give areas of inner floor and side; and Nos. 8 and 9, areas of outer base and side; diameters: 51.51648, 58.13012, 66.66666 and 68.68864.

Will some of your able mathematical contributors take up the theory of the coffer and systematize it as it demands?

Edinburgh, October 27, 1885.

JAS. SIMPSON.

For convenience, my computations are derived from t-year =365.2423396-. A deduction of one, two or three millionths respectively from statements of lineal, areal or cubic measures will reduce to present year values with tolerable accuracy.

At p. 357 of the STANDARD, Vol. II, the cubit of the Turin museum is stated to be .523524 of the metre. That is 6, or the same proportion that the breadth of coffer's interior has to the modulus 51.576 +. Hence a "coffer" measuring within, Xmetre × 1 Turin cubit, would contain I spherical