1. Prove that the shoe may be represented by an equation of the fifth
degree. Find the equation to a man blacking a shoe: (1) in rectangular
co-ordinates; (2) in polar co-ordinates.
2. A had 500 shoes to black every day, but being unwell for two days he
had to hire a substitute, and paid him a third of the wages per shoe
which he himself received. Had A been ill two days longer there would
have been the devil to pay; as it was he actually paid the sum of the
geometrical series found by taking the first _n_ letters of the
substitute’s name. How much did A pay the substitute? (Answer, 13_s._
6_d._)
3. Prove that the scraping-knife should never be a secant, and the brush
always a tangent to a shoe.
4. Can you distinguish between _meum_ and _tuum_? Prove that their
values vary inversely as the propinquity of the owners.
5. How often should a shoe-black ask his master for beer notes?
Interpret a negative result.