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This article describes the main mathematical researches performed, in Englandand in the Continent between 1742-1827, on the subject of hyperbolarectification, thereby adding some of our contributions. We start with theMaclaurin inventions on Calculus and their remarkable role in the early mid1700s; next we focus a bit on his evaluation, 1742, of the hyperbolic excess,explaining the true motivation behind his research. To hisgeometrical-analytical treatment we attach ours, a purely analyticalalternative. Our hyperbola inquiry is then switched to John Landen, an amateurmathematician, who probably was writing more to fix his priorities than toexplain his remarkable findings. We follow him in the obscure proofs of histheorem on hyperbola rectification, explaining the links to Maclaurin and soon. With a chain of geometrical constructions, we attach our interpretation toLanden's treatment. Our modern analytical proof to his hyperbolic limit excess,by means of elliptic integrals of the first and second kind is also provided,and we demonstrate why the so called Landen transformation for the ellipticintegrals cannot be ascribed to him. Next, the subject leaves England for theContinent: the character of Lagrange is introduced, even if our interestconcerns only his 1785 memoir on irrational integrals, where the ArithmeticGeometric Mean, AGM, is established by him. Nevertheless, our objective is notthe AGM, but to detect the real source of the so-called Landen transformationfor elliptic integrals. In fact, Lagrange's paper encloses a differentialidentity stemming from the AGM: integrating it, we show how it could bepossible to arrive at the well-known Legendre recursive computation of a firstkind elliptic integral, which appeared in his Trait\'e, 1827, much after theLagrange's paper.

The hyperbola rectification from Maclaurin to Landen and the Lagrange-Legendre transformation for the elliptic integrals