:Indeed, to get an equilateral hyperbola one has to take all line segments whose X and Y intercepts have constant product, not sum.
2.
When the foci of the Cassini ovals are the points ( a, 0 ) and (-a, 0 ), the equilateral hyperbolas centered at ( 0, 0 ) and passing through the foci are curves orthogonal to the Cassini ovals.
3.
The one demanded by his reasoning is the reciprocal of that of the natural system . " Further along in the same book we read, " In the study of quadratures Gregory St . Vincent found the grand property of the equilateral hyperbola which connected the hyperbolic space between the asymptotes with the natural logarithms, and ied to these logarithms being called'hyperbolic'" }}
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