Demanet devised an interesting method
of solution of trinomial equations which depends on
the use of communicating vessels of convenient forms.
To solve an equation of the third degree of the form:
x3 + x = c
where c is a constant, an inverted
cone and a cylinder, joined together by means of a tube,
As shown below.
radius r of the cone and its
height h are in the ratio:
r/h = √3/√π
while the base of the cylinder is taken as 1 cm2
If c cubic
centimeters of water are poured into one of the two vessels,
the water will rise to the same height h in
both. The volume of water contained in the cone will be h3,
that in the cylinder h, so that
h3 + h = c
Therefore, by measuring the height h of
the water we obtain a solution of the equation.
the case of the equation
x3 - x = c
the cone alone is used, and a solid cylindrical piece whose
base is 1 cm2 is introduced. The volume c of
water poured in will thus be the difference between h3 and
h, and therefore h, the height
of the liquid, is again a solution.
a substitution z = x√p we
can reduce all reducible equations of the third degree
x3 + pz = q
where p and q are
given positive numbers, to the form x3 + x = c