Egyptian algebra was rhetorical, that is, the problems and their solutions
were expressed in words. There were symbols for addition (a pair of legs
walking from left to right) and for subtraction (a pair of legs walking
from right to left). Their algebra was limited almost entirely to linear
equations in one variable, which were solved by the method of
false position.
This is most easily explained in modern terms. Given a formula
of the form ax + x/b = c, where a, b
and c are known, they would guess an answer for x, almost
invariably the wrong one. Substituting this value into the left hand side
of the equation they would get some value d (generally different
to c) on the right hand side. Because the equation is linear, we
can use proportions to deduce that the correct value of x is c/d
times the false value.
Their knowledge of basic algebra was fairly advanced. They knew how
to find the area of a rectangle, an isoscles triangle, and an isosceles
trapezium. The area of a circle was taken as equal to that of a square
on 8/9 of the diameter. The volumes of rectangular boxes and circular cylinders,
all conceived of concretely as capacities of barns, were known. But
perhaps the most remarkable result of Egyptian mensuration is to be found
in Problem 14 of the Moscow Papyrus, where we find the equivalent, in modern
notation, of the formula
V = (h/3)(a² + ab + b²)
for the volume V of the frustum of a square pyramid, where a and b are
the sides of the square bases, and h is the height. Problem 56 of the Rhind
Papyrus hints at the rudiments of trigonometry, and even a theory of similar
triangles.