LANDEN, John A Disquisition concerning certain Fluents, which are assignable by the Arcs of the Conic Sections; wherein are investigated some new and useful Theorems for computing such Fluents.
London: Lockyer Davis & Peter Elmsly, 1771. First edition.

The paper in which “Landen implicitly gave the first example of the technique known as transformation of elliptic integrals, which was to form and essential part of both the theoretical and computational aspect of elliptic functions.” (Roger L. Cooke). “In reducing integrals to a standard form, Euler, Maclaurin and d’Alembert had frequently expressed such integrals as the difference between an arc of a hyperbola from the vertex to a given point P and the projection of the line segment from the centre of the hyperbola to P onto the tangent at P. This difference, however, assumes the indeterminate form &infinity; - &infinity; when the point P is infinitely distant. To remove this indeterminacy, J. Landen showed in 1771 [the offered paper] that the indeterminate quantity could be expressed as the sum of two such finite differences decreased by the projection of a line from the centre of an ellipse onto the tangent by the projectionof a line from the centre of an ellipse onto the tangent to the ellipse. The techniques developed in solving this problem enabled Landen to express an arc of the hyperbola x²/(m-n) - y²/2(mn) y^1/2 = 1 in terms of an algebraic function of position and the arcs of two ellipses, one with semi-axes m and n, the other with semi-axes 1/2(m+n) and (mn) ^1/2; that is, the semi-axes of the second ellipse are the arithmetic and geometric means of the axes of the original ellipse. In discovering this relationship, Landen implicitly gave the first example of the technique known as transformation of elliptic integrals, which was to form and essential part of both the theoretical and computational aspect of elliptic functions.” (Grattan-Guiness [Ed.]: Companion Encyclopedia of the Mathematical Sciences, p.533).

Small 4to: 220 x 180 mm. Extracted from: Philosophical Transactions of the Royal Society of London, vol. 61, pp. 298-309 with 1 folding engraved plate. Stapled into early 20th century blue wrappers, bookseller’s ticket of Henry Sotheran on upper cover.