derivatio​

The general form of the equation is

y = Ωx^α …………………………………………………………………………………………………………………(1)

Constants α and Ω are determined from the ancient text.

The phrase five of you shall chase a hundred implies that y = 100 when x = 5. Substitution of these values into Equation 1 results in

100 = (Ω)5^α ………………………………………………………………………………………………………..(2)

The solution of Equation 2 for Ω is

Ω = 100/5^α ………………………………………………………………………………………………………….(3)

Substitution of Equation 3 into Equation 1 results in

y = (100/5^α)x^α…………………………………………………………………………………………………….(4)
Combination of the terms with the α exponent results in

y = 100(x/5)^α ……………………………………………………………………………………………………..(5)

The phrase a hundred of you shall put ten thousand to flight implies that y = 10,000 when x = 100. Substitution of these values into Equation 4 result in

10000 = 100(100/5)^α ………………………………………………………………………………………(6)

Upon simplification, Equation 6 becomes

100 = 20^α ……………………………………………………………………………………………………………(7)

Application of the natural logarithm on both sides of Equation 6 produces

ln(100) = α ln(20)……………………………………………………………………………………………..(8)

Finally, the solution of Equation 8 for α is

α = ln(100)/ln(20)……………………………………………………………………………………………..(9)

Alpha is a very unique number. It is an irrational number, expressed by an infinite number of digits in the decimal system:

α = 1.53724357368 •••

The final form of the equation is

y = 100(x/5)^{1.53724357368 •••}