# 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, for we find him a very respectable contributor to the Ladies Diary in 1744; and he was soon among the foremost of those who then contributed to the support of that small but valuable publication, in which almost every English mathematician who has arrived at any degree of eminence for the best part of this century, has contended for fame at one time or other of his life. Mr. Landen continued his contributions to it at times, under various signatures, till within a few years of his death.

It has been frequently obferved, that the histories of literary men consist chiefly of the history of their writings; and the observation was never more fully verified, than in the present article concerning Mr. Landen.

In the 48th volume of the Philosophical Transactions, for the year 1754, Mr. Landen gave “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 ingenious paper was delivered to the Society by that eminent mathematician Thomas Simpson of Woolwich, a circumstance which will convey to those who are not themselves judges of it, some idea of its merit.

In the year 1755, he published a volume of about
160 pages, intitled *Mathematical Lucubrations.* The
title to this publication was made choice of, as a means
of informing the world, that the study of the mathematics
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. These Lucubrations
contain 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.

About the latter end of the year 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: he also compared these solutions
with others derived from the fluxionary method;
and shewed that the solutions by his new method
were commonly more natural and elegant than the
fluxionary ones.

In the 51st volume of the Philosophical Transactions,
for the year 1760, he gave *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 the late Mr. Thomas Simpson.

In 1764, he published the first book of *The Residual
Analysis.* In this treatise, besides explaining the
principles which his new analysis was founded on, 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 were to have
formed this book lay long by him; but he never found
leisure to put them in order for the press.

In the year 1766, Mr. Landen was elected a Fellow
of the Royal Society. And in the 58th volume of the
Philosophical Transactions, for the year 1768, he gave
*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 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, 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 are and its tangent, and that difference
being not directly attainable when the arc and
its tangent both become insinite, 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 deducible from 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 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 permaneut 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 the year 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, in 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 the year 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 finding 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 1735.

The extreme difficulty of the 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'Alembert, induced many to
think that their solution was right, and Mr. Landen's
wrong; and there did not want attempts to prove it;
particularly a long 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 the year 1790, in which he agrees with the
solutions of Euler 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 Fris<*>'s *Cosmographi<*>
Physicæ et Mathematicæ;* 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, intitled, *Theoria Motûs Corporum
Solidorum seu Rigidorum.* 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 were
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 it. 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 71st year of his age.