1883
Meyerson, E., "O matematyce starożtnych Egipeyan” [Ancient Egyptian mathematics], Ateneum Pisma Naukowe I Literackie', Warsaw, vol. 32, 1883, pp. 137-151, 268-279, 403-404.
Petrie, W. M. F., 'The Pyramids and Temples of Gizeh, London, 1883, 16 + 250 pp. + plates.
In the chapter commencing on page 162 Petrie discusses “architectural ideas of the pyramid builders” which some seem to think of importance in properly interpreting the sections of the Rhind papyrus dealing with pyramids.
Petrie writes: "The design of the various slopes [of pyramids] that are met with, appears to be always a simple relation of the vertical and horizontal distances. It is important to settle this as it bears strongly on the whole planning of each building; and though the vertical and horizontal distances would seem to be the natural elements for setting out a slope, yet proof of this is necessary; for Ahmes, in his mathematical papyrus, defines pyramids by their sloping heights up the arris edge, and their diagonal of the base beneath that line. Such a method of measurement would naturally be adopted, when the knowledge of the design was lost,as the arris height could be easiest measured; but it is very unlikely as a specification of design."
Petrie then gives the "observed" and "theoretical angle” (that is, angle which the faces of the pyramid make with the square base) of each of "various pyramids, and so-called pyramids," and other material providing the "proof." The angles observed by Petrie are given by him with the range of error ±5' for example: Mastaba 44, Gizeh, a tomb of the Ancient Empire, 76° 0’ with range of error ± 5'. In other words, observations at different parts of the tomb had different values of which the mean was 76° 0'. Then the "theoretical angle" is given as 75° 57' 50", that is, the nearest angle whose cotangent is expressible as the ratio of simple integers,—in this case 1 : 4. For this angle the difference is 2' ± 5’.
Another observation of Petrie which he here records was in connection with the second pyramid of Gizeh 53° 10' (error, ± 4'), theoretical angle 53° 7' 48", 3 :4 (difference, 2' ± 4').
Petrie gives also the following two observations made under Vyse's direction: Mastaba Pyramid, Medum 74° 10' (error, 1° ?),theoretical angle 75° 57' 50", 1 :4 (error, 1°47’ ± 1° ?); South stone pyramid base, Dahshur 54° 14’ 46"
rational fraction is expressed as a finite sorites, we can expand the last term as a limiting sorites, and thus express the fraction as an infinite sorites with this equation holding true for all the terms after a given term. Conversely, if this equation holds true for all the terms of an infinite sorites after a given term, the sorites will represent a rational fraction. When there is no term of an infinite sorites after which this equation holds true for all the terms, the sorites represents an irrational number.
For a discussion of a similar problem involving also the same equation see 0. D. Kellogg, "On a Diophantine problem," American Mathematical Monthly, vol. 28, 1921, pp. 300-303; and D. R. Curtiss, “On Kellogg's Diophantine problem," ibid., vol. 29, 1922, pp. 380-387.
