Origins of Early Calculus

        Not only India, the ancient Egypt was also one of the origins of early calculus. The earliest known mathematical text discovered in the history is the Moscow papyrus, “an Egyptian Middle Kingdom papyrus dated c. 2000-1800 BC.” It was written by an unknown mathematician in ancient Egypt and purchased by an Egyptologist V.S. Golenishchev, who sold to the Moscow Museum of Find Art in 1947. According to McGill University (2008), the papyrus was about 15 feet long and 3 inches wide and it consists of “what are today called ‘word problems’ or ‘story problems’, which were apparently intended as entertainment.” However, problem #14 in the papyrus was considered to be of exceptional importance, which gives a method for calculating the volume of a pyramidal frustum. According to the Moscow Papyrus (1700BC), the problem 14 says: “Volume of a frustum. The scribe directs one to square the numbers two and four and to add to the sum of these squares the product of two and four. Multiply this by one third of six.” This can be easily interpret into the equation:

[pic 1]

This looks like a simple geometry question but actually in order to find the volume of the pyramidal frustum, we need to introduce integral calculus.

The formula for the volume of a frustum was well known by the ancient Egyptians to be:

 [pic 2]

in which a is the length of the tope and b is the length of the bottom.

Then, the problem is, how we get this formula. According to Analyzemath.com, “A frustum may be obtained by revolving y=x between x=a and x=b around x-axis as shown below. The height h=b-a.” In this problem, b is 6 and a is 0. Starting with the same way as what we do to find the area under the curve, we need to know the area of the cross section of this frustum, and then take the sum of an infinite number of pieces of the cross section of this frustum, which gives us the volume of it. According to the AP Calculus Textbook, “we partition [a,b] into subintervals of length  and slice the solid, as we would a loaf of bread, by planes perpendicular to the x-axis at the partition points.”[pic 3]

 (The fundamental theorem of calculus)[pic 4]

[pic 5]

[pic 6]

 (Difference of cube)[pic 7]

 ( b-a=h)[pic 8]

[pic 9]

        From the problem above, which was in the Moscow Mathematical Papyrus, it is safe to say the ancient Egyptians technically had the awareness of calculus. However, even though they found the formula for the volume of the frustum by using integral calculus, they did not know what they did was called calculus.

        In summary, the ancient Egyptians were able to find integrals and summation, and the problem #14 in the Moscow Mathematical Papyrus was a fairly crusted form of integral calculus. However, they did not discover anything that was significantly close to the pure calculus.