1. Each question required the "controlling spirits" to make a
mathematical assumption before answering. For the first, the question of
the non-parallel lines, the assumption concerned whether or not they were
on the same plane. If they were, they would necessarily meet. If they were
not, the correct answer would be no. For the second, the question of determining
the circumference of a circle, the assumption concerned the exactitude of
the answer required. The formula is: circumference = diameter X pi (22/7).
Pi is a transcendental number whose exact value has never been determined.
In fact, there are ongoing projects, involving super computers, still seeking
the value of pi. So, while every "schoolboy" would know how to
find the circumference of a circle, the answer is exact only on those occasions
when the diameter is a multiple of 7. Apparently, the spirits did realize
that pi was a transcendental number. The offer to produce a formula for
"squaring the circle" amounted to an offer to develop the exact
value of pi in thirty days.