Gregory / integration of hyperbola

In his treatise entitled "Opus geometri-cum quadraterae circuli et sectionum coni", Gregory (1584 - 1667) had successfully computed the case of , which Fermat was unable to calculate.

The computation was that he plotted points along the axis in such a way that the length from increased in equal ratio and he drew vertical lines to (hyperbola) from the respective points; in this case, the superficial contents of the segments separated by vertical lines were in arithmetical progression.

However, I don't know this proof.

The property of the superficial content is expressed below in a typical mathematical expression of our own time.

Let , and make it the first term of the geometric progression.

is taken as the common ratio so as to lead to .

Therefore, the interval is in geometric progression but the superficial content is in arithmetical progression.

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