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The integral of the secant function was first correctly conjectured by Henry Bond in the 1640s, and Isaac Newton was aware of his conjecture in 1665, although no proof was published until 1668. Of course, once you've guessed the right integral you can prove it works by differentiation, using the Fundamental Theorem of Calculus; was it unknown to Newton at that time?

But my main question concerns James Gregory's 1668 proof in Execitationes Geometricae, which Wikipedia says "was presented in a form that renders it nearly impossible for modern readers to comprehend". By contrast, Isaac Barrow's 167 proof is a fairly simple one, the $u=\sin\theta $ substitution I discuss here. But what was Gregory's proof? I can only find his book in Latin.

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According to Rickey and Turchinsky (1980, p. 165; my emphasis):

The first to prove the conjecture was, to quote Edmund Halley, “the excellent Mr. James Gregory in his Exercitationes Geometricae, published Anno 1668, which he did, not without a long train of Consequences and Complication of Proportions, whereby the evidence of the Demonstration is in a great measure lost, and the Reader wearied before he attain it” ([7], p. 203). Judging by Turnbull’s modern elucidation [19] of Gregory’s proof, one would have to agree with Halley. At any rate, Gregory’s proof could not be presented to today’s calculus students, and so we omit it here.

So I’d seek a copy of Turnbull’s rare book. (I have before for other reasons, but never successfully.)

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