The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection. In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured that
Although Gregory proved the conjecture in 1668 in his Exercitationes Geometricae, the proof was presented in a form that renders it nearly impossible for modern readers to comprehend; Isaac Barrow, in his Geometrical Lectures of 1670, gave the first "intelligible" proof, though even that was "couched in the geometric idiom of the day." Barrow's proof of the result was the earliest use of partial fractions in integration. Adapted to modern notation, Barrow's proof began as follows:
Substituting for reduces the integral to
The second of these follows by first multiplying top and bottom of the interior fraction by . This gives in the denominator and the result follows by moving the factor of 1/2 into the logarithm as a square root.
The third form follows by replacing by and expanding using the identities for . It may also be obtained directly by means of the following substitutions:
The conventional solution for the Mercator projection ordinate may be written without the modulus signs since the latitude (φ) lies between −π/2 and π/2:
The integral can also be derived by using the tangent half-angle substitution, also known as the Euler or Weierstrass substitution. A somewhat non-standard version of the tangent half-angle substitution, which is simpler in the case of this particular integral, published in 2013, is as follows:
^Edward Wright, Certaine Errors in Navigation, Arising either of the ordinaire erroneous making or vsing of the sea Chart, Compasse, Crosse staffe, and Tables of declination of the Sunne, and fixed Starres detected and corrected, Valentine Simms, London, 1599.
^H. W. Turnbull, editor, The Correspondence of Isaac Newton, Cambridge University Press, 1959–1960, volume 1, pages 13–16 and volume 2, pages 99–100.
^D. T. Whiteside, editor, The Mathematical Papers of Isaac Newton, Cambridge University Press, 1967, volume 1, pages 466–467 and 473–475.