The Residual Analysis; A New Branch of the Algebraic Art, Of very extensive Use, both in Pure Mathematics, and Natural Philosophy

Author: John Landen (1719–1790)

Year: 1764

Publisher: Printed for the author, and sold by L Hawes, W Clerke and Collins

Place: London

Description:

viii+218 [i.e., 128] pages with five engraved folding plates and list of subscribers. Quarto (9 1/4" x 7") bound recent tan calf over marbled boards, Gilt-ruled spine with crimson morocco label. First edition.

The Residual Analysis; A New Branch of the Algebraic Art is a landmark mathematical treatise published in 1764 by the English mathematician John Landen. The work represents a historically significant attempt to provide a purely algebraic foundation for calculus, completely avoiding the controversial use of infinitesimals, fluxions, or limits that characterized the methods of Isaac Newton and Gottfried Wilhelm Leibniz. Instead of using traditional dynamic limits, Landen focused on rigorous algebraic identities to find the rates of change. His "residual" method relied on expanding terms through polynomial division and factoring differences of powers.

Landen originally planned a multi-book series, but only Book I was published in 1764. The book features various applications to geometry, trigonometry, and mechanics ("Natural Philosophy"). It contains an early form of what is known today as Landen's Transformation, an important tool in the study of elliptic integrals. While the system proved too cumbersome for complex, higher-order calculus compared to the elegant limit notation developed later, it heavily influenced other major mathematicians. Notably, Joseph-Louis Lagrange pursued a similar philosophy of purely algebraic foundations for calculus, and Karl Marx later analyzed Landen's work extensively in his own Mathematical Manuscripts.

John Landen was an influential English amateur mathematician and land surveyor who made foundational contributions to mathematical analysis, particularly in the study of elliptic integrals. Despite never working as a professional academic, he became one of the most prominent British mathematicians of the late 18th century, developing concepts that later stimulated pioneering work by Euler, Lagrange, and Legendre.

Condition:

Text a bit toned, occasional foxing, crease (apparent printer's error) on outer margin of title-page else very good.