Upon visiting the Great Pyramid of Egypt I was struck at how clean the huge blocks, some weighing 50 tons, were cut. The interior passageways seemed to have been cut as though by a jeweler with an accuracy of cutting a diamond. So I cannot escape the thought that if the pyramid has been built some five thousands of years ago, there's something far more here than meets the eye. So I take it upon myself to inquire further. Since my profession is in mathematics and engineering I wonder- could these ancients have known about calculus? Or, to put it further, is calculus somehow built into the pyramid in some way? I devise a test of my own although there are other tests. Also the builders may have or may have not known calculus otherwise. This test is simply, as they say, a "shot in the dark". Only with the help of Calculus can this particular test be done.

The Test- Find the angle of inclination of a pyramid in which the surface area is a minimum. That is to say neither a fat nor a thin pyramid should have a minimum surface area for any given volume. But which intermediate angle answers the question? Then compare it with the given angle of inclination of the Great Pyramid, which is given as 51.87 degrees, from 481.4 ft original height by 755.75 ft each base length.*

Given: Volume = 1/3 (Area x Height)

Lateral Surface Area = ½ (Perimeter of the base x Altitude)

V, Volume    Sa, Lateral Surface Area
A, Area        P, Perimeter of the Base
H, Height

The object is to find H / X when the area (of the 4 sides) is a minimum. We are going to need to differentiate the lateral Surface Area, and so we need to express as much as possible in terms of one unknown, X.
We know that S² = X² + H² and S = sqrt ( X² + H² )

and also V = 1/3 (A x H) since A = 4 X² V = 1/3 (4X² H)

And so H = 3V / 4X²

Therefore Sa = ½ (8X) sqrt (X² + 9V / 16X^4 ) and Sa = Sqrt (16X^4 + 9V² X^-2)

Now we are ready to Differentiate because we have only one unknown, X, on the right hand side. The Volume V is a constant. We Differentiate, a process only done by Calculus.

DS / DX = ½ (16X^4 + 9V² X^-2)^-½ o (64X³ - 18 V² X^-3)

When we let DS / DX = 0 we have;

18 V² X^-3 = 64 X³ or, 9V² = 32 X^6 now V = 2 X² H from before

That leaves H = X sqrt 2 or H / X = sqrt 2

And so the angle of inclination from H / X turns out to be 54.74 degrees

Since the actual angle of inclination of the Great Pyramid (as given) to be 51.87 degrees this is 2.87 degrees off of perfect for a Pyramid having the least amount of surface area for its given volume.

Conclusion: Unless some given measurement is wrong, and considering the accuracy with which these ancients built this Pyramid e.g. "The passages are straight to within .013 in per 100 feet", I find that even if it is only 2.87 degrees off of perfect, I can't prove from this particular test that Calculus was built in. Thus we can't tell if these ancients were also telling us about Calculus too besides the amazing amount of other mathematics built in, such as "The perimeter of the base divided by twice the height equals pi to 5 decimal places". From the above given measurements, 481.4 ft in height by 755.75 ft base length, I get not pi exactly, which is 3.14159, but 3.1398, which is very very close. It is correct to ½ of 1%. This cannot be accidental. No wonder this structure is the most measured, analyzed and marveled at structure in the world. And it goes without saying that the ancients, i.e. over 5,000 years ago, are a mystery, as to who they were and how they came by this knowledge and a lot more than we'll ever know. If only the Library at Alexandria hadn't been burned to the ground. What wonderful and amazing knowledge was stored there.

* Encyclopedia Britannica