Length of an arc
Determining the length of an irregular arc segment was historically difficult. Although many methods were used it wasn't until calculus that a universal formula became available.
Modern Methods
Consider a function 
on a 2-dimensional Cartesian plane. The length L
Where and 
are continuous on [a,b].
Historical Methods
Ancient
For much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a rectangular approximation for finding the area beneath a curve with his method of exhaustion, few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation.
Fermat
Fermat was the first to propose a formula for calculating arc lengths. Building on his previous work with tangents, he used the curve
whose tangent at x=a had a slope of
so the tangent line would have the equation
Next, he increased a 
, he used the Pythagorean Theorem:
which, when solved, yields
In order to approximate the length, Fermat would sum up a sequence of short ACs.
