Nero asked me one day how many slaves it would take to complete a pyramid in a time that was equal on each level of the height. He wanted to use the extra slaves as target practice for the Legions. As the leader of the Legions said "Practice makes perfect." and you don't want your soldiers to go into battle and then just cut some persons head half off, it is just sloppy and reflects poorly on the emperor. This was before they had simulation software systems and so this was the best simulation they could devise.
I said that he could conquer some village and bring back one slave for ever block in the base. There after he could use one less for the change in width and height which was applied to the size which would be 2x. Well he wasn't satisfied with that ( even if it was the derivative of a square) and wanted a pyramid that had different widths and heights. On top of that he wanted it to have different step widths on one side for children and wider steps for adults. He said "Think of the children". That is simple, I said , the change in each side is applied to the other sides and so f'(uv) is uv'+vu' with a fudge factor of 1. This is just like the cornered steps that lead to the baths except that it has 4 corners that step. Any of your clerks can use that rule to determine how many slaves to be used for target practice, I will call it the product rule as the it produces slaves for target practice. I was thinking about calling it the chain rule , because death frees the slaves from their chains, but product rule is what sticks. He was happy and sent me a fruit basket stolen from a country that was conquered. I was happy, as it had very little blood on it.
I personally liked working this out for the Pharaoh as he, at least, had some really classy threads and was slightly more entertaining with the whole Isis for the children thing. He rarely used slaves for target practice as there were too many projects to be completed and those temples don't build themselves you know!
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