 Building on his previous work with tangents, Fermat used the curve

whose tangent at "x" = "a" had a slope of

so the tangent line would have the equation

Next, he increased "a" by a small amount to "a" + "ε", making segment "AC" a relatively good approximation for the length of the curve from "A" to "D".To find the length of the segment "AC", he used the Pythagorean theorem