LANDEN, a town of the Netherlands, in the province of Liège and arrondissement of Huy, on the river Beake. It is remarkable for the battle fought near it, usually called the battle of Neerwinden, in 1693, in which the allies were defeated by Marshal Luxembourg. Long. 4. 55. Ľ. Lat. 50. 45. N.

Landen, John, an eminent mathematician, was born at Peakirk, near Peterborough, in Northamptonshire, in January 1719. He early became a proficient in the mathematics, for we find him a respectable contributor to the Ladies Dairy in 1744; and he was soon amongst the foremost of those who then contributed to the support of that small but valuable publication, in which a number of English mathematicians at one period or other contended for fame. Mr Landen continued his contributions to it under various signatures till within a few years of his death.

In the forty-eighth volume of the Philosophical Transactions for the year 1754, Mr Landen published an Investigation of some theorems which suggest several very remarkable properties of the circle, and are at the same time of considerable use in resolving fractions, the denominators of which are certain multinomials, into more simple ones, and by that means facilitate the computation of fluents. This paper was handed to the Society by Mr Thomas Simpson of Woolwich; a circumstance which will convey to those who are not themselves judges some idea of its merits. In the year 1755 he published a volume entitled Mathematical Lucubrations. It contains a variety of tracts relative to the rectification of curve lines, the summation of series, the finding of fluents, and many other points in the higher parts of the mathematics.
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About the latter end of 1757, or the beginning of 1758, he published proposals for printing by subscription The Residual Analysis, a new branch of the algebraic art; and in 1758 he published a small tract in quarto, entitled A Discourse on the Residual Analysis, in which he resolved a variety of problems, to which the method of fluxions had been usually applied, by a mode of reasoning entirely new; compared these solutions with solutions of the same problems investigated by the fluxionary method; and showed that the solutions by his new method were, in general, more natural and elegant than the fluxionary ones. In the fifty-first volume of the Philosophical Transactions for the year 1760, he published a New Method of computing the sums of a great number of Infinite Series. This paper was also presented to the society by his ingenious friend Mr Thomas Simpson. In 1774< he published the first book of The Residual Analysis, in 4to, with several copperplates. In this treatise, besides explaining the principles on which his new analysis was founded, he applied it to drawing tangents and finding the properties of curve lines; to describing their involutes and evolutes, finding the radius of curvature, their greatest and least ordinates, and points of contrary fluxure; to the determination of their cusps, and the drawing of asymptotes ;—and he proposed in a second book to extend the application of this new analysis to a great variety of mechanical and physical subjects. The papers which were to have formed this book lay long by him; but he never found leisure to put them in order for the press. In January 1766 Mr Landen was elected a fellow of the Royal Society, and admitted in the April following. In the fifty-eighth volume of the Philosophical Transactions for the year 1768, he published a specimen of a new method of comparing curvilineal areas, by means of which many areas are compared, that did not appear to be comparable by any other method; a circumstance of no small importance in that part of natural philosophy which relates to the doctrine of motion. In the sixtieth volume of the same work, for the year 1770, he gave some new theorems for computing the whole areas of curve lines, where the ordinates are expressed by fractions of a certain form, in a more concise and elegant manner than had been done by Cotes, Demoivre, and others, who had considered the subject before him. In the sixty-first volume, for 1771, he investigated several new and useful theorems for computing certain fluents, which are assignable by arcs of the conic sections. This subject had been considered before, both by Mr Maclaurin and Μ. d’Alembert; but some of the theorms which were given by these celebrated mathematicians, being in part expressed by the difference between an arc of a hyperbola and its tangent, and that difference being not directly attainable when the arc and its tangent both become infinite, as they will do when the whole fluent is wanted, although such fluent be finite; the theorems therefore fail in those cases, and the computation becomes impracticable without further help. This defect Mr Landen has removed by assigning the limit of the difference between the hyperbolic arc and its tangent, whilst the point of contact is supposed to be removed to an infinite distance from the vertex of the curve. And he concludes the paper with a curious and remarkable property relating to pendulous bodies, which is deducible from these theorems. In the same year he published an animadversion on Dr Stewart’s computation of the Sun"s distance from the Earth. In the sixty-fifth volume of the Philosophical Transactions for 1775, appeared the investigation of a general theorem, which he had promised in 1771, for finding the length of any arc of a conic hyperbola by means of two elliptic arcs; and by the theorems there investigated, both the elastic curve and the curve of equable recession from a given point may be constructed in those cases where Mr Maclaurin’s elegant method fails. In the sixty-seventh volume, for 1777, he gave a new theory of the motion of bodies revolving about an axis in free space, when that motion is disturbed by some extraneous force, either percussive or accelerative. At this time he did not know that the subject had been handled by any other person; and he considered only the motion of a sphere’s spheroid and cylinder. The publication of this paper, however, was the cause of his being told that the doctrine of rotatory motion had been considered by Μ. d’Alembert; and having purchased that author’s Opuscules Mathématiques, he there learned that Μ. d’Alembert was not the only one who had considered the matter before him; for Μ. d’Alembert speaks of some mathematician, though he does not mention his name, who, after reading what had been written on the subject, doubted whether there existed any solid whatever, besides the sphere, in which a line, passing through its centre of gravity, would be a permanent axis of rotation. In consequence of this, Mr Landen took up the subject again; and though he did not then give a solution of the general problem, namely, to determine the motions of a body of any form whatever revolving without restraint about any axis passing through its centre of gravity, he fully removed every doubt of the kind which had been started by the person alluded to by Μ. d’Alembert, and pointed out several bodies, which, under certain dimensions, possess that remarkable property. This paper is published, amongst many others equally curious, in a volume of Memoirs which he gave to the world in 1780. But what renders that volume yet more valuable, is a very extensive Appendix, containing theorems for the calculation of fluents. The tables which contain these theorems arc more complete and extensive than any which are to be found in other authors, and are chiefly of his own investigating; being such as had occurred to him in the course of a long and curious application to almost every branch of the mathematical sciences. In the years 1781, 1782, and 1783, he published three little tracts on the summation of converging series, in which he explained the extent of some theorems which had been given for that purpose by Demoivre, Mr Sterling, and Thomas Simpson, in answer to some things which he thought had been written to the disparagement of these mathematicians.

About the beginning of the year 1782 Mr Landen had made such improvements in his theory of rotatory motion, as he thought would enable him to give a solution of the general problem specified above; but finding the result to differ very materially from that of the solution which had been given by Μ. d’Alembert, and not being able to see clearly where that gentleman had erred, he did not venture to make his own public. In the course of that year having procured the Memoirs of the Berlin Academy for 1757, containing Μ. Euler’s solution of the problem, he found that it gave the same result as had been deduced by Μ. d’Alembert; but the perspicuity of Μ. Euler’s manner of writing enabled him to discover where he had erred, which the obscurity of the other did not do. The agreement, however, of two writers of such established reputation as Μ. Euler and Μ. d’Alembert made him long dubious of the truth of his own solution, and induced him to revise the process again and again with the utmost circumspection; but being every time more convinced that his own solution was right and theirs wrong, he at length gave it to the public in the seventy-fifth volume of the Philosophical Transactions for 1785.

The extreme difficulty of the subject, joined to the concise manner in which Mr Landen had been obliged to give his solution in order to confine it within proper limits for the Transactions, rendered it too difficult, or at least too laborious a piece of business, for most mathematicians[7:13:57]to read it; and this circumstance, joined to the established reputation of Euler, induced many to think that his solution was right and Mr Landen’s wrong, an opinion which attempts were made to establish by proof. But although these attempts were manifestly abortive, as every one who perused them saw, yet they convinced Mr Landen that there was a necessity for giving his solution at greater length, in order to render it more generally understood. About this time also he met by chance with Frisi’s work on cosmography, physics, and mathematics, in the second part of which there is a solution of this problem, agreeing in the result with those of Euler and D’Alembert; which is not surprising, as Frisi employed the same principle that they did. Here Mr Landen learned that Euler had revised the solution which he had formerly inserted in the Berlin Memoirs, and given it in another form and at greater length in a volume published at Gryphiswald in 1765, entitled Theoria Motus Corporum solidorum seu rigidorum. Having procured this book, Mr Landen found the same principles employed in it, and of course the same conclusion resulting from them, which he had observed in Euler’s former solution of the problems; but as the reasoning was given at greater length, he was enabled to see more distinctly how Euler had been led into the mistake, and to set that mistake in a stronger point of view. As he had been convinced of the necessity of explaining his idea on the subject more fully, he found it necessary to lose no time in setting about it. He had for several years been severely afflicted with calculus in the bladder, and towards the latter part of his life to such a degree as to be confined to his bed for more than a month at a time, yet even this dreadful disorder did not abate his ardour for mathematical studies, the second volume of his Memoirs having been written and revised during the intervals of his disorder. This volume, besides a solution of the general problem concerning rotatory motion, contains the resolution of the problem concerning the motion of a top; an investigation of the motion of the equinoxes, in which Mr Landen first pointed out the cause of Sir Isaac Newton’s mistake in his solution of this celebrated problem; and some other papers of considerable importance. He lived to see this work finished, and received a copy of it the day before his death, which happened on the 15th of January 1790, at Milton, near Peterborough, in the seventy-first year of his age.