The integral is evaluated as follows:  formula_27  as claimed. This was the formula discovered by James Gregory. 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 formula_29 for formula_6 reduces the integral to Therefore,formula_32as expected. The formulas for the Weierstrass substitution are as follows. Let formula_33, where formula_34. Then  Hence,  by the double-angle formulas. As for the integral of the secant function,formula_37as before. The integral can also be derived by using the a somewhat non-standard version of the Weierstrass substitution, which is simpler in the case of this particular integral, published in 2013, is as follows:  The integral of the secant function defines Lambertian function, which is the inverse of the Gudermannian function:  This is encountered in the theory of map projections: the Mercator projection of a point with longitude "θ" and latitude "φ" may be written as: