a2+2a+1 = (a+1)2
Almost two thousand six hundred years have passed since a young and ambitious mathematician return from his worldly travels. His voyage had taken him to the northern shores of Africa where the Nile Delta bestows her spoils into the Mediterranean Sea. Once on shore he continued his sojourn further south, battling against the rushing flow of the ever-entwining Nile River. Finally! After several months of weary travel, he was standing beside the entrance gates of his destination, the gates leading into the famous Egyptian city…Al-Qahira (Cairo). The goal was to satisfy his curiosity and quench his thirst for further advancement in the studies of higher mathematics. The temple priests understood his wanting and obliged by teaching the mysteries, intricacies, and magic of numbers to this inquisitive northern traveler. They taught him the geometries and spherical measures of nature…he discovered and learned…then, upon completion, they escorted him to the greatest monuments of all time, showing him the perfect examples displayed in stone…the Giza pyramids. And it is here where he spent many a-year studying their grand design before retracing his path back toward home.
But that period of life had passed…he had now returned to his country and its people…ready and willing to teach all that he had learned…and so he did. Traveling throughout the countryside, he was soon to become known as the “Father of Mathematics”, his name…Pythagoras.
His works included several revolutionary mathematical concepts and formulae…one being the “right-angle triangle theory” also known as “The Pythagorean Theorem”.
The most basic example used to illustrate this mathematical formula is the 3-4-5 triangle and we have known for centuries that Khafre’s pyramid (P2) at Giza has a sloped side displaying the identical ratio. Unfortunately, this single ratio was insufficient evidence to indicate the ancient pyramid builders understanding the right-angle theory. What must be considered is the possibility of the builders being completely unaware of the formula; perhaps they had selected this pyramid slope for another reason (Ill. 1).
Illustration 1. The 3-4-5 triangle is designed into Khafre’s pyramid at Giza. This triangle is the most basic example used to demonstrate the right-angle theory. The sum of the squares of the two sides equals to the square of the hypotenuse.
The greatest error was to assume this pyramid side angle being no more than coincidental, that the ancients did not understand the right-angle triangle theory. Closer examination indicates the designers actually had complete understanding…greater understanding than us…the modern intellects we claim to be.
To demonstrate their knowledge, the builders incorporated other basic integer ratios within the design. However, these examples were chosen specifically to verify they knowing of an additional formula for the right-angle triangle (Ill. 2).
Illustration 2. Hidden within the Giza complex are several other examples of the right-angle triangle.
Four examples of the right-angle triangle are illustrated above. The greatest surprise is the well-preserved causeway on site; it actually forms a 9-40-41 triangle, a ratio that has never been realized in the past. The four triangles illustrate ratios of integer values and it is from these examples that a second formula is developed.
Studying the four examples it is noticed:
1) The square of the shorter side is equal to the sum of the other two sides.
2) The hypotenuse has a numerical value one unit larger than the longer side.
Continuing with the series…the next example would be 11-60-61. Eleven squared equals 121 and this value is equal to the total of the other two sides. The hypotenuse is one unit larger than the other side; therefore it is 61 units, leaving the second side measuring 60 units.
To arrange this information into formula context we must begin with the original theory (Ill. 3).
Illustration 3. The formula for the right-angle theory introduced by Pythagoras.
Using the Pythagorean theorem for the right-angle triangle and substituting the values as indicated at the Giza site, only two of the algebraic functions are altered (Ill. 4).
Illustration 4. Substituting the ratios as demonstrated at Giza for the letters of the right-angle theory and squaring their sides, line 1 and 2 total and equal line 3.
At first, the triangle ratios appear complicated, but upon closer examination we realize the ease in calculation. When the values for a, b, and c are substituted with the new values then Pythagoras’s formula expands to (a+1)2 = a2+2a+1…one of the many formulas used in basic algebra.
The system is true…”For any right-angle triangle having the base measure one unit of length shorter than the hypotenuse…then the second side of that triangle is the square root of their sum”.
It is evident that the formula is ideal for preparing right-angled foundations; also it can be used for confirming the accuracy of a formed 90 degrees corner. The amazing advantage is the use of only one value in measure…the three variables: a, b, and c are now replaced with one measure…“a”. The second advantage of this formula is the unnecessary calculation of square roots. Below is a simple example demonstrating the preparation of a right-angle triangle (Ill. 5).
Illustration 5. The calculation and forming of a right-angle triangle.
To begin: The side measure of length is selected (example shows 14.0 units). The measure is squared (14x14 =196). A unit of one (1) is subtracted from the total (196-1 = 195). Dividing this total by two gives the length of the base (195/2 = 97.5). Add the unit value of one (1) to this measure and it will represent the length of the hypotenuse (97.5+1 = 98.5). Scribing arcs from both ends of the side measure will produce the third point of the triangle…the point where the base and the hypotenuse meet.
The system is not restricted to integer values; it also applies to decimals or fractions, but only if the difference of one unit is maintained between the base and the hypotenuse (Ill. 6).
Illustration 6. The formula also applies to decimal and fractional values, but the one unit difference in measure between the base and the hypotenuse must be maintained.
For many years mathematicians have wondered how the ancient Egyptians were capable of construct their monuments to such accuracy. Amazingly, the bases of these structures are very close to “true” square. The accuracy in measure certainly astonished Petrie when recording the dimensions for P1 (Khufu’s pyramid).
The normal procedure for confirming the 90 degrees corners of a square is to measure the length of its diagonals; the square is perfect when both diagonals are equal in length. But, to perform this task, both diagonal paths must be leveled to a depth equal in height to that of the pyramid’s base corners. Channeling through rock to level the diagonal strips across the base of the structure, is time consuming and most impractical. The designers at Giza used the second right-angle formula to confirm the accuracy of the pyramid corner measures. They incorporated “squaring” triangles along the pyramid sides and demonstrated this alignment method in a unique fashion (Ill.7).
Illustration 7. Squaring triangles used to confirm 90 degrees corners at the base of P2 (Khufu). The measures are external to the base boundaries.
Using external triangles eliminated the problem of leveling the rock along the internal diagonal lengths. To confirm that the pyramid designers used the second formula they pre-calculated the width of P2 (Khafre) plus the extended side measure of the squaring triangle to equal the width of P1 (Khufu). From above we see that P2 is 411.25 Rc and the side dimension of the squaring triangle is an additional 28.67 Rc…totaling 439.95 Rc. Khufu’s pyramid side measure is identical…440.0 Rc!
A second formula for calculating the side measures of a right angle triangle has been presented. However, it must be realized that its application is limited to those measures where the hypotenuses is 1 unit of measure greater than one of the sides. This formula has the advantage of simple arithmetic calculation compared to the complexity of square root manipulation. Once the full potential of the formula is understood, then its application would definitely be favored for determining and preparing square foundations in construction. The overall dimensions of the Giza pyramid complex indicated the use of this secondary formula.
Were the pyramid designers aware of the right-angle triangle theory?
From illustration 2 we learned of the four triangles using integer values in measure. The next in the series was demonstrated; it would measure as an 11-60-61 triangle; its calculations were given…now we can witness its location.
Yes…they did know!
The Pyramids and Temples of Gizeh…..1882……………Sir W. M. Flinders Petrie
“1o6” The Dawn of Man………………………….1999 sb……….C. Ross
Clive Ross Amitron2001@yahoo.com