Landen, John

, an eminent mathematician, was born at Peakirk, near Peterborough in Northamptonshire, in January 1719. He became very early a proficient in the mathematics, as we find him a contributor to the “Ladies Diary” in 1744, to which useful publication he continued to send articles until a few years before his death. In the “Philosophical Transactions” for 1754, he wrote “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, &c.” In 1755, he published a small volume, entitled “Mathematical Lucubrations,” and containing 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. The title “Lucubrations,” was supposed to intimate that mathematical science was at that time rather the pursuit of his leisure hours, than his principal employment and indeed it continued to be so during the greatest part of his life for about the year 1762 he was appointed agent to earl Fitzwilliam an employment which he resigned only two years before his death.

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 entitled “A Discourse on the Residual Analysis,” in which he resolved a variety of problems, to which the method of fluxions had usually been applied, by a mode of reasoning entirely new; and | in the “Philosophical Transactions” for 1760 he gave a “New method of computing the sums of a great number of infinite series.” In 1764 he published the first book of <c The Residual Analysis/' in which, besides explaining the principles on which his new analysis was founded, he applied it, in a variety of problems, 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 flexure; 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 formed this book lay long by hi in; but he never found leisure to put them in order for the press.

In 1766, Mr. Landen was elected a fellow of the royal society, and in the “Transactions” for 1768 he wrote “A specimen of a new method of comparing Curvilinear 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 60th volume of the same work, for 17 70, 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, De Moivre, and others who had considered the subject before him.

In the 61st volume, for 1771, he has 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 Maclaurin and d’Alembert; but some of the theorems that were given by these celebrated mathematicians, being in part expressed by the difference between an hyperbolic arc 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; these theorems therefore fail in these cases, and the computation becomes impracticable without farther help. This defect Mr. Landen has removed, by assigning the limit of the difference between the hyperbolic arc and its tangent, while 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 deduciblefrom those theorems. In the same year he published “Animadversions on Dr. Stewart’s Computation of the Sun’s Distance from the Earth.”

In the 65th volume of the Philosophical Transactions, for 1775, he gave the investigation of a general theorem, which he had promised in 1771, for finding the length of any curve of a conic hyperbola by means of two elliptic arcs: and he observes, that by the theorems there investigated, both the elastic curve and the curve of equable recess from a given point, may be constructed in those cases where Maclaurin’s elegant method fails.

In the 67th 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 that time he did not know that the subject had been treated by any person before him, and he considered only the motion of a sphere, spheroid, and cylinder. After the publication of of this paper, however, he was informed, that the doctrine of rotatory motion had been considered by d’Alembert; and upon procuring that author’s “Opuscules Mathematiques,” he there learned that d‘Alembert was not the only one who had considered the matter before him; for d’Alembert there speaks of some mathematician, though he does not mention his name, who, after reading what had been written on the subject, doubted whether there be any solid whatever, beside the sphere, in which any line, passing through the centre of gravity, will 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 to the general problem, viz. “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 d’Alembert, and pointed out several bodies which, under certain dimensions, have that remarkable property. This paper is given, among many others equally curious, in a volume of “Memoirs,” which he published in 1780. That volume is also enriched with a very extensive appendix, containing “Theorems for the calculation of Fluents;” which are more complete and extensive than those that are found in any author before him. | In 1781, 1782, and 1783, he published three small tracts on the “Summation of Converging Series;” in which he explained and shewed the extent of some theorems which had been given for that purpose by De Moivre, Stirling, and his old friend Thomas Simpson, iii answer to some things which he thought had been written to the disparagement of those excellent mathematicians. It was the opinion of some, that Mr. Landen did not shew less mathematical skill in explaining and illustrating these theorems, than he has done in his writings on original subjects; and that the authors of them were as little aware of the extent of their own theorems, as the rest of the world were before Mr. Landen’s ingenuity made it obvious to all.

About the beginning of 1782 Mr. Landen had made such improvements in his theory of rotatory motion, as enabled him, he thought, to give a solution of the general problem mentioned above; but rinding the result of it to differ very materially from the result of the solution which had been given of it by d‘Alembert, and not being able to see clearly where that gentleman in his opinion had erred, he did not venture to make his own solution public. In the course of that year, having procured the Memoirs of the Berlin academy for 1757, which contain M. Euler’s solution of the problem, he found that this gentleman’s solution gave the same result as had been deduced by d’Alembert; but the perspicuity of Euler’s manner of writing enabled him to discover where he had differed from his own, which the obscurity of the other did not do. The agreement, however, of two writers of such established reputation as Euler and 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; and being every time more convinced that his own. solution was right, and theirs wrong, he at length gave it to the public, in the 75th volume of the Philosophical Transactions for 1785.

The extreme difficulty of th% subject, joined to the concise manner in which Mr. Landen had been obliged to give his solution, to confine it within proper limits for the Transactions, rendered it too difficult, or at least too laborious a task, for most mathematicians to read it; and this circumstance, joined to the established reputation of Euler and d’Al-embert, induced many to think that their solution was right, and Mr, Landen’s wrong; and there did not | want attempts to prove it; particularly along and ingenious paper by the learned Mr, Wildbore, a gentleman of very distinguished talents and experience in such calculations; this paper is given in the 80th volume of the Philosophical Transactions for 1790, in which he agrees with the solutions of Kuler and d’Alembert, and against that of Mr. Landen. This determined the latter to revise and extend his solution, and give it at greater length, to render it more generally understood. About this time also he met by chance with the late Frisi’s “Cosmographia Physica et Mathematica;” in the second part of which there is a solution of this problem, agreeing in the result with those of Euler and d’Alembert. Here Mr. Landen learned that Euler had revised the solution which he had given formerly in the Berlin Memoirs, and given it another form, and at greater length, in a volume published at Rostoch and Gryphiswald, in 1765, entitled “Theoria Motus Corporum Solidorum seu Rigidorwn.” Having therefore procured this book, Mr. Landen found the same principles employed in it, and of course the same conclusion resulting from them, as in M. Euler’s former solution of the problem. But notwithstanding that there was thus a coincidence of at least four most respectable mathematicians against him, Mr. Landen was still persuaded of the truth of his own solution, and prepared to defend ifc. And as he was convinced of the necessity of explaining his ideas on the subject more fully, so he now found it necessary to lose no time in setting about it. He had for several years been severely afflicted with the stone 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 extinguish his ardour for mathematical studies; for the second volume of his “Memoirs,” lately published, was 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 relating to the motion of a top; with an investigation of the motion of the equinoxes, in which Mr. Landen has first of any one 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 just 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. Though Mr. Landen was one of the greatest mathematicians of the age, his merit, in this respect, was not more conspicuous than his moral virtues. The strict integrity of his conduct, his great humanity, and readiness to serve every one to the utmost of his power, procured him the respect and the esteem of all who knew him. ¹