Conventional wisdom holds that geometry, as a true discipline, was the creation of the Greek mind. A fair amount of scholarship has been devoted to showing in what ways the surviving written record upholds this belief. Oddly enough, this is not what the Greeks themselves had to say about things. Greek writers such as Isocrates, Plato and Diodorus all credit Egypt as the source of Greek geometrical studies.¹ Sir Thomas Heath, in his work
A History of Greek Mathematics, adds that "the Egyptian claim to the discoveries (of geometry) was never disputed by the Greeks".²

The existence of the pyramids themselves, with all of their complexities and precision, also argue in favor of there having been some form of sophisticated geometrical capability on the part of their architects. I will therefore be accepting as a given that by the time of the building of the Great Pyramid, the art of diagrammatic geometry had become fairly highly developed in ancient Egypt.

The circle is a perfect, though mysterious, shape. It appears daily in the heavens as the sun; and monthly, as the moon. It can appear as an artifact of nature on the surface of the Earth as can be seen, for instance, in a perimeter made in grass by a tethered grazing animal. The ancient Egyptians had gained an awareness of the fact that there is always a constant relationship between the circumference (perimeter) of a circle and the diameter of that same circle. This is the relationship we now call Pi. It is a value equal to 3.14159...., the dots signifying that the fractional part of this number has no known finite end.

There is a surviving document which shows that the Egyptians were in the habit of using a value for Pi equal to the square of (8/9 x 2), or roughly 3.16 in decimal notation. (See footnote 3 for further explanation).³ The scribe writing this document states that he is making a copy of an older work, a work that is judged to have been written in about 1850 B.C. In his treatise on this papyrus, Professor T.E. Peet states, "Surely the complicated fabric of Egyptian mathematics can hardly have been built up in a century or even two, and it is tempting to suppose that the main discoveries of mathematics should be dated to the Old Kingdom.....the Golden Age of Egyptian knowledge".⁴The Great Pyramid was constructed sometime near the date of 2550 B.C., at the height of the Egyptian Golden Age that Peet refers to.

What are the implications to the ancient Egyptians having developed a very accurate means of expressing Pi which was not only easy to remember, but which could also be easily mathematically manipulated by the everyday scribe? I believe it fairly certain that a pragmatic understanding of the Pi relationship had been mastered by this period. As mentioned above in regard to a tethered grazing animal, there is no difficulty in observing that a rope which is allowed to pivot on a post can be employed as a means of marking out a circle. The length of such a rope can then be used to measure the circumference of its associated circle. With the Egyptians having had a penchant for dividing measurements in half, and then in half again, etc., a rope thusly marked out would have enabled them to have made the very accurate estimate of 201 ÷ 64 (= 3.1406) for the circumference to diameter ratio of a circle (Pi). This specific relationship, however, although easily attainable fails to provide any of the mnemonic and manipulative advantages that were clearly of cultural necessity in these ancient times. Also note that the Square Root of 2 derivation presented in the Appendix contains a discussion of yet an additional possible rationale for the Egyptian choice of the square of (8/9 x 2) as the means by which to represent Pi.

In any event, the accompanying insight that the Pi relationship remains constant, regardless of how big or small the circle, is the more crucial point. It is not a result that one might initially have expected prior to its proof by a chance occurrence, or by the exploration of a curious mind. This is a constant numerical relationship which occurs freely in the natural world. Its discovery could only have aroused further curiosity. Pondering over the reality of the Pi phenomenon, it would only have been a matter of time before some insightful priest or scribe began to wonder whether there might be any other such findable constants. From this moment forward, the game was afoot.

A next logical shape to consider would certainly have been the square. In fact, the evolution of the square from a circle is almost inevitable. The circle is halved, and then halved again, and the points at which the diameters intersect the circle are joined to form an interior square.

To perform this task with accuracy, a tool functioning as a drafting compass, or dividers, would be needed. A straightedge, that need not be calibrated, is also necessary.5 Using such equipment, one can empirically assure that all sides of the square are of equal length. One can also empirically show that each of the four interior triangles are identical, and that each central angle is 1/4th of a full rotation. (Such an angle is referred to as a "right angle". Since we use a system in which a full circle is considered to contain 360, a right angle is then an angle having 90).

The issue, then, is to find whether there is a numerically constant relationship between a side of a square and that square's diagonal.

The following diagram shows how this relationship may have first been discovered.⁶

The diagram begins with the drawing of the smallest square to the left. Each side has a length of one unit, and so the area contained within that square is said to be 'one square unit'. (Area within a square - or any rectangle - is determined by multiplying the length of a side by the length of an adjacent side. In a square, the lengths of these two sides are equal, and so the formula becomes: Area = S x S = S². Since S equals 1 in this first square, the area equals 1 x 1 = 1 square unit.) The next step is to draw in the diagonal to this first square, and when done, it is seen that the diagonal creates two equivalent right angled triangles each of which contain an area of one half of one square unit.

What now happens if a square is drawn using this diagonal as a side? As the diagram shows, this next square (in green) will contain exactly four of these same triangles. As a result, it is visually apparent that the area of this second square must be two square units. From the understanding that Area equals the length of the square's side multiplied by itself, for this second square we get : Area = S² = 2 sq. units. As a result, for this second square "S" must then equal that number which when multiplied by itself yields the number 2. This number is then the "square root" of 2, and is written as 2. It can be closely approximated in decimal notation by the value of 1.4142.

The diagram above continues to show that the length of the diagonal for any square can be found by simply multiplying the length of that square's side by the 2. Notice that each side of the green square has a length of 2, and its diagonal equals 2. Each side of the orange square has a length of 2, and the diagonal of this square is 2 x 2, and so on. We have here, then, another freely occurring numerical constant in nature. The first, Pi, was found in the context of a circle, and now we find 2 in the context of a square.

Two very important questions now arise and must be addressed: 1) Did the ancient Egyptians have an awareness of the2 relationship between a square's side and its diagonal? And 2) Did they have the mathematical capability to determine square roots? (For instance, did they have a way to find that the 2 has an equivalent value to 1.4142?) The answer to both of these questions is "yes".

That the Egyptians knew of the 2 relationship is evidenced by the fact that they had a unit of measurement, the "double-remen", which they defined as being the length of the diagonal of a square whose sides were each one royal cubit in length. (In other words, the double-remen equalled the length of a royal cubit multiplied by the 2).⁷ There is also ample evidence that, not withstanding their reliance upon a computational system much removed from our own, they were indeed quite capable of handling the sometimes complex chore of finding square roots.⁸(I include in the Appendix a diagrammatic square root method which fits in remarkably well with the ancient Egyptian unit fraction system, and which could quite conceivably have been devised at the time in question. In addition, utilizing this square root diagrammatic method, I present an interesting coincidence of number between the ratio of the perimeter of a square to its diagonal and the Egyptian choice for the manner in which they represented Pi. See the aforementioned Square Root of 2 derivation .)

Having found these two constant relationships of Pi and the 2, would not the thought have occurred that perhaps there were still other similar constants to be found? Would not those involved with line drawing and building design have next wondered whether there might be some way to discern a relationship between the sides and diagonal in non-square rectangles? I believe it entirely possible that not only would this question have been pursued, but that it would have been pursued in a manner similar to the method used in the previous diagram.

Instead of starting with a square, Diagram 3 begins with a rectangle (AB). The diagonal (here denoted "C" ) is drawn within this rectangle, and a square is drawn using this diagonal as the length of its side.

If one could find the area contained by the square on C in terms of the values A and B, the length of the diagonal C could be determined by taking the square root of this amount.

Diagram 3 shows how the right triangle ABC may distributed to best advantage inside the square on C. It is clear that the area of this square can be represented as being equal to the area of four times the area of triangle ABC plus the area of the small square on (B-A). Since each triangle ABC contains half the area of the rectangle AB, and since the area of the rectangle AB equals A multiplied by B (i.e., AxB), the area of four times the area of triangle ABC is the same as the area of twice (AxB), or 2AB. It remains now to find the area of the square on (B-A).⁹

To do this, it is likely that the empirical advantages inherent in the drawing of a gridded square would have been utilized.¹⁰

In Diagram 4, line B and line A are shown at the top of a square whose side is the length B + A. Lightly drawn in are the squares on each of these line segments.

In the following diagram, Diagram 5, the square on the line segment B - A has been added.

This diagram may perhaps appear at first glance to be a little more forbidding than it actually is. An AB rectangle has been placed horizontally inside the green-sided square on B, and the magenta-sided square on (B-A) has also been delineated inside the square on B. Left unaccounted for inside the square on B, then, is the small rectangle containing six small-square units to the right of the square on (B-A). Investigation shows that this amount, when added to the area of the square on A (added above these six units as a dotted orange-sided square), proves to be exactly enough to constitute a second AB rectangle.

This is, of course, what is today known as the Pythagorean Theorem. The truism that is represented in this finding is simply that there is a definable relationship between the lengths of the sides in every right angled triangle (i.e., in a triangle which is, in effect, one half of a rectangle, be the rectangle a square or non-square). If the lengths of two of a right triangle's sides are known, then there is one and only one length that the third side can have, and that length is determinable via this equation.

The discovery of this theorem would not have immediately satisfied the goal of finding other naturally occurring constants such as are found in Pi and the 2, but as we will see, it would have provided the means by which this pursuit could have been vigorously continued. First things first, however. If, supposedly, Pythagoras (in 500 B.C., approximately) discovered the formula that bears his name, then how is it possible the ancient Egyptians are not generally credited with having known of it some 2,000 years earlier?

To start with, it must be noted that there simply is not a large body of ancient Egyptian mathematical material that has survived down to our era. Additionally, the few papyri that have survived are largely student training exercises, and they present us with an incomplete picture of the full capabilities of the master scribes. Some scholars do feel, however, that reasonable inferences can be drawn from existing data to support the possibility of Egyptian knowledge of the workings of the Pythagorean Theorem.¹¹

It is also of interest to note that a Babylonian clay tablet, dating from roughly the same time period as the afore-referenced Rhind Papyrus, reveals that an extremely sophisticated understanding of the Pythagorean Theorem and associated number theory existed during the Old Babylonian period.¹² This tablet leaves no doubt that the Pythagorean Theorem was well understood by this contemporaneous civilization at least 1,200 years before the birth of Pythagoras. I will later present evidence of substantial contact between this culture and that of ancient Egypt prior to this date.

Lastly, I want to stress the inherent simplicity and power of an empirical diagrammatic approach. Movement from one understanding to another proceeds organically, with each new result open to easy corroboration either by direct measurement or by a comparison of shapes. Such a diagrammatic technique is well within the context of ancient Egyptian capabilities and understandings. The Egyptians may not have used exactly the same diagrams as derived in this essay, and they may not have written their results in an identical symbolic formula format, but the same basic understandings as we have reached thus far, and will reach in the following sections, would have been well within their grasp.

Following an analysis of the proportions inherent in a square, the search would certainly have continued to find what other shapes can be created within the context of a circle, and to see what (if anything) could be learned from them.

1. See Isocrates (ca. 390 B.C.), Busiris, Vol. III, p. 119. See Plato (ca. 380 B.C.), Phaedrus 274d. See Diodorus (ca. 60 B.C.), On Egypt, I, 98.
2. Sir Thomas Heath, A History of Greek Mathematics, p. 121
3. See Richard Gillings, Mathematics in the Time of the Pharaohs, pp. 140-6. To find the area of a circle, the scribe would "subtract from the diameter its 1/9th part, and square the remainder." Since a circle's diameter is equal in length to 2 radii, then the circle's area, instead of being seen as equal to the square of 8/9 x the diameter, can be said to be equal to the square of 8/9 x 2 radii. Pi therefore is represented as the square of (8/9 x 2).
4. T.E. Peet, The Rhind Mathematical Papyrus, p. 9
5. Right angle squares, drafting triangles and compasses only appear in the archeological record at a point dating much later than Old Kingdom times. I.E.S. Edwards mentions in reference to such implements that "there is no reason to doubt they preserve an ancient tradition". (The Pyramids of Egypt, p. 245). For a similar sentiment, see also Alexander Badawy, Ancient Egyptian Architectural Design, pp. 40 - 45.
6. This diagram closely follows that presented by Robert Lawlor on page 26 of his work, Sacred Geometry.
7. R.J. Gillings, op. cit., p. 208.
8. Ibid., pp. 214 - 217.
9. That the Egyptians well knew how to find the areas of rectangles and triangles, see Gillings, pp. 137-9.
10. Such grids were indeed known to the scribes and artists of the Old Kingdom period. See Gay Robins, Proportion and Style in Ancient Egypt, p. 70
11. See A. Seidenberg, "The Ritual Origin of Geometry", Arch. Hist. of Exact Sciences Vol. I, no. 5, p. 510. Also B. Lumpkin, "The Egyptians and Pythagorean Triples", Historia Mathematica 7, pp. 186-7.
12. See J. Friberg, "Methods and Traditions of Babylonian Mathematics", Hist. Mathematica 8, pp.277-318.

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