141 
LAN 
LAN'DEGODE, a fmall ifland in the North Sea, near 
the coaft of Norway. Lat. 67. 25. N. 
LAN'DEHEN, a town of France, in the department of 
the North Coafts : two miles fouth of Lamballe, and five 
north-north-eaft of Moncontour. 
LANDEL'LE, a town of France, in the department of 
the Calvados: five miles north-weft of Vire, and twenty- 
eight fouth-weft of Caen. 
LAN'DEN (John), an eminent Englifh mathematician, 
was born at Peakirk, near Peterborough, in Northamp- 
tonftiire, in the year 1719. That he became an early pro¬ 
ficient in the mathematics, appears from his being a re- 
fpeftable contributor to the Lady’s Diary in 1744.; and 
he was foon ^mong the foremoft of thofe who then fup- 
ported that fmall but valuable publication, in which al- 
moft every Englilh mathematician who arrived at any de¬ 
gree of eminence during a confiderable part of the laft 
century, contended for fame. Mr. Landen communicated 
his contributions to it under different fignatures, till 
within a few years of his death. What further informa¬ 
tion we have been able to colleft relating to this mathe¬ 
matician is chiefly confined to a hiftory of his writings. 
In the 48th volume of the Philofophical Tranfaftions, he 
gave “An Inveftigation of fome Theorems which fug¬ 
ged feveral very remarkable Properties of the Circle, and 
are, at the fame Time, of confiderable Ufe in refolving 
Fractions, the Denominators of which are certain Multi¬ 
nomials, into more Ample ones, and by that Means faci¬ 
litating the Computation of Fluents.” This ingenious 
paper was communicated to the Royal Society by that 
eminent mathematician, the late Thomas Simpfon, of 
Woolwich j which circumftance will convey, to thofe 
who are not themfelves judges of it, fome idea of its me¬ 
rit. In the year 1755, Mr. Landen publiflied a volume, 
called Mathematical Lucubrations; the title of which 
was chofen as a means of informing the world, that the 
ftudy of the mathematics was at that time rather the pur- 
fuit of his leifure hours than his principal employment. 
This, indeed, continued to be the cafe during the greateft 
part of his life ; for, about the year 1762, he was appoint¬ 
ed agent to earl Fitzwilliam, and retained that employ¬ 
ment till within two years of his death. Thefe Lucu¬ 
brations contain a variety of trafls relative to the rectifi¬ 
cation of curve lines, the fummation of feries, the finding 
of fluents, and many other points in the higher parts of 
mathematics. About the latter end of the year 1757, or 
the beginning of 1758, he publiflied propolals for print¬ 
ing by fubfcription, The Refidual Analyfis, a new branch 
of the algebraic art; and in the year laft-mentioned, he 
publiihed a Difcourfe on the Refidual Analyfis, 4I0. in 
which he refolved a variety of problems, to which the 
method of fluxions had ufually been applied, by a mode 
of reafoning entirely new. He alfo compared thefe folu- 
tions with others derived from the fluxionary method ; 
and (bowed, that the folutions by his new method were, 
in general, more natural and elegant than the fluxionary 
ones. In the 51ft volume of the Philofophical Tranfac- 
tions, he gave A new Method of computing the Sums of 
a great number of Infinite Series. The firlt book of The 
Refidual Analyfis made its appearance in 1764. In this 
treatife, befides explaining the principles on which his 
new analyfis was founded, he applied it, in a variety of 
problems, to the drawing of tangents, and finding the 
properties of curve lines ; to delcribing their involutes 
and evolutes, finding the radius of curvature, their greateft 
and leaft ordinates, and points of contrary flexure; to the 
determination of their cufps, and the drawing of afymp- 
totesj and he propofed, in a fecond book, to extend the 
application of this new analyfis to a great variety of me¬ 
chanical and phyfical fubjects. The papers, which were 
to have formed this book, lay long by him; but he never 
found leifure to put them in order for the prefs. 
In the year 1766, Mr. Landen had the honour of being 
elected a fellow of the Royal Society. Two years after¬ 
wards he contributed to the 58th volume of their Trarjf- 
y ; oj.. XII. No. 817 . 
LAN 
aCIions, A Specimen of a new Method of comparing Cur¬ 
vilinear Areas ; by means of which many areas are com¬ 
pared that did not appear to be comparable by any other 
method : a circumftance of no fmall importance in that 
part of natural philofophy which relates to the doCtrine of 
motion. In the 60th volume of the fame work, he gave 
“Some New Theorems for computing the whole Areas of 
Curve Lines, where the Ordinates are expreffed by Frac¬ 
tions of a certain form,” in a more concife and elegant 
manner than had been done by Cotes, De Moivre, and 
others who had confidered the fubjeCl before him. In 
the 61 ft volume of the TranfaCtions, he has inveftlgated 
feveral new and ufeful theorems for computing “ Certain 
Fluents which are affignableby Arcs of the Conic Sections.” 
This fubjeft had been confidered before, both by Maclaurirc. 
and d’Alembert; butfomeof the theorems which were given 
by thefe celebrated mathematicians, being in part expreffed 
by the difference between an hyperbolic arc and its tangent, 
and that difference being not direitly attainable when the 
arc and its tangent both become infinite, as they will do> 
when the whole fluent is wanted, although fuch fluent be 
finite; thefe theorems, therefore, fail in thofe cafes, and 
the computation becomes impracticable without farther 
help. This defeft Mr. Landen has removed, by afligning 
the limit of the difference between the hyperbolic arc and 
its tangent, while the point of contaCt is fuppofed to be. 
removed to an infinite diftance 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 thofe theorems. In the year 1774, he 
publiflied “ Animadverfions on Dr. Stewart’s Computa¬ 
tion of the Sun’s Diftance from the Earth;” in which he 
not only pointed out the doftor’s errors, but (howed that 
a true folution of the problem was not to be expeCVed, 
either from his method of reafoning, or from the data on 
which he had founded it. Mr. Landen’s next contribu¬ 
tion to the Philofophical TranfaCIions is to be found in 
the 65th volume, and confifts of the inveftigation of a ge¬ 
neral theorem, which he had promifed in 1771, “ for find¬ 
ing the length of any arc of a conic hyperbola by means 
of two elliptic arcs, with fome other new and ufeful the¬ 
orems deduced therefrom and it concludes with ob- 
ferving, that thefe theorems, properly applied, will evince, 
that both the elaftic curve, and the curve of equable recefs 
from a given point, with many others, may be conftruft- 
ed, by the rectification of the ellipfis only, in thofe cafes 
in which Maclaurin’s elegant method is defective. In the 
67th volume of the fame work, he gave “A New Theory 
of the Rotatory Motion of Bodies affefled by Forces dif- 
turbing fuch Motion.” At that time he did not know 
that the fubjeft had been handled by any perfon before 
him, and he confidered only the motion of a fphere, fphe- 
roid, and cylinder. After the publication of his paper, 
however, he was informed, that the dodtrine of rotatory 
motion had been confidered by d’Alembert; and, upon 
procuringthat author’s Opufcules Mathematiques, hethere 
learned that d’Alembert was not the only perfon who had 
preceded him in this inveftigation; for he found him 
(peaking of fome mathematician, though he does not 
mention his name, who, after reading what had been 
written on the fubjedt, doubted whether there be any fo- 
lid whatever, befides the fphere, in which any line, pall¬ 
ing through the centre of gravity, will be a permanent 
axis of rotation. In confequence of this, Mr. Landen 
took up the fubjedt again; and, though he did not then' 
give a folution to the general problem, namely, “ to de¬ 
termine the motions of a body of any form whatever, re¬ 
volving without reftraint about any axis palling through 
its centre of gravity; 5 ’ he fully removed any doubt of the 
kind which had been advanced by the peifon to whom 
d’Alembert had alluded, and pointed out feveral bodies 
which, under certain dimenfions, have that remarkable 
property. This paper may be fcen, among many others 
equally curious,ina volumeof Memoirs, which our author 
publiflied in the year j;8o. That volume is alio enriched 
Q * iviti| 
