Lancaster is 58 miles west of Philadelphia, Lat. 3', north ; Long. 76° 10' west. The population, as returned by the census of 1800, was 4292 And by that of 1810, 5405 The inhabitants are principally of German descent.
It was formerly the seat of the state government, but, in 1812, the assembly passed an act for its removal to Har ristm rgh.
In the year 1764, he published the first book of the Re sidual Analysis, in which he applies it to the drawing of tangents ; to the finditig 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 fiuxure; and to the determination of their cusps, and the drawing of asymptotes. He proposed in a second book, to show its application to a great variety of mechani cal and physical problems; but he never found leisure to complete this part of his plan.
On the 16th of January 1766, Mr. Landen was elected a member of the Royal Society ; and in 1768, he published his Specimen of a new method of comparing Curvilineal Areas, by which many such areas may be compared, as have not yet appeared to be comparable by any other method. In the same work tor 1770, he gave Some new Theorems for completing the Areas of certain Curve Lines.
In the Phil. Trans. tor 1770, he published A disquisition concerning certain Fluents, which are assignable by the Arcs of the Conic Sections ; w/i're are investigated, some new and useful Theorems for computing such Fluents. This sub ject had been previously treated by Maclaurin iu his Fluxions, and by D'Alembert in the Menzoirs of the Berlin Academy ; hut Landen had the merit of removing a very great defect in their methods. In the same year he pub lished his Animadversions on Dr. Matthew Stewart's com putation of the Sun's distance from the Earth, a work writ ten in a style of acrimony disgraceful to a man of genius.
In the same work for 1775. he published an Investiga tion of a general Theorem. for finding the length of any Arc of any Conic Hyperbola by means of two Elliptic Arcs ; with some other new and useful Theorems deduced from it. In this paper he has shewn, that both the elastic curve, and the curve of equable recess from a given point, with many others, may be constructed by the rectification of the ellipsis only, without failure in any point ; whereas the elegant method by Maclaurin, of constructing them by the rectification of the hyperbola and the ellipse, fails, when some principal point of the curve is to be determined, as the hyperbolic arc and its tangent then become infinite, though their difference be at the same time finite.
In 1777, he published in the Philosophical Transactions, 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. In this paper, he considers only the motion of a sphere, spheroid, and cylinder ; but in consequence of his having after wards found, that D'Alembert had treated of the same subject, he purchased the Opuscules of that eminent author ; w here he bound it stated, that some mathematician doubted, whe ther there is any solid whatever besides the sphere, in which any line passing through the centre of gravity will be a per manent axis of rotation. He was thus led to resume the sub ject, and he succeeded in pointing out several bodies, which, under certain dimensions, have that remarkable property. This paper was published in a volume of memoirs, which appeared in 1780, and which contains also a large appen dix, with a complete collection of Theorems for Me calcu lation of Fluents, principally investigated by himself.
Mr. Linden published three small tracts on the summa tion of converging sines, in which he explained and ex tended the theorems of De Moivre, Stirling, and Thomas Simson.
Ahout the commencement of the year 1782, Mr. Lan di n had made such additions to his theory of rotatory mo tion, that he thought himself capable of resolving the ge neral problem, namely, to determine 46 the rotatory notion of a body of any form whatever, revolving without restraint about any axis passing through its centre of gravity." Hav ing found, however, that the result was materially different from that given by d'Alembert, he declined publishing his solution. The solution of the same problem given by litt ler, in the .Memoirs of the Berlin Academy for 1757, hap pened to fall into his hands, and he saw that the result was the same as that of d'Alembert ; but the great perspicuity of Eulei's investigations enabled him to discover the point in which Bee solution differed from his own. After repeated examinations of his own solution, he was firMly convinced of its correctness, and at last published it in the Philoso phical Transactions for 1783. The Rev. Charles Wildbore, a respectable mathematician, attacked Landen's solution in the Phil Trans. for 179u, in a paper on Spherical Motion, and obtained the same solution as that of Euler and d'A leriabert. Mr Landen was thus led to revise and extend his solution, and having found a result similar to that of Euler, in nisi's Cosmographia ; and having also learned that Euler had revised his own solution, and obtained the same result, in his Theoria Mot us corporum Solidorum sou rigidorum, which appeared in 1765, he set about a full ex planation of his own views.
He was now severely afflicted with the stone ; but dur ing the painful intervals of that agonizing disorder, he con tinued to write the second volume of his Memoirs, which appeared after his death. This volume contains, among other important papers, a solution of the general problem concerning rotatory motion, the resolution of the problem relative to the motion of a top, and an investigation of the precession of the equinoctial points, in which he had the honour of detecting, for the first time, Sir Isaac Newton's mistake in his celebrated solution of the same problem. Mr. Landen had the satisfaction of receiving a copy of this work on the day before his death, which took place at Milton, near Peterborough, on the 15th of January, 1790, in the 71st year of his age. A flat stone in the north aisle of the church of Castor, in Northamptonshire, is the only monument to the memory of this distinguished ma thematician. .
As a mathematician, Mr. Landen is entitled to a very high reputation ; but he possessed, in no small degree, a coarseness of mind, and a disposition to contemn the pur suits of others, which, we 'egret to say, is too often exhi bited by those who exclusively cultivate the mathematics. His treatment of Euler is by no means handsome ; and the controversy with Dr. Henry Clarke, respecting that gen tleman's translation of Lorgna's treatise on series, was carried on with an asperity of language unworthy of a man of genius. From the singular contrast between the man ners of Mr. Landen and those of his noble friend the Earl of Fitzwilliarn, the villagers often exclaimed, when they were seen to pass together, There }foes Lou! Landen and Mr. Fitzwilliam." Mr. Landen left one daughter, who, we believe, is still alive. His manuscripts Were sold to the shop-keepers of Peterborough for waste paper.