LAN 



524, 



LAN 



Lancaster About a mile to the north-east of the town, is the aque- 

 duct-bridge of the Lancaster canal, which consists of 

 five arches, and cost 4-5,000. Barges of 60 tons can 

 pass over it. This town has communication, by in- 

 ternal navigation, with the rivers Mersey. Kibble, 

 Ouse, Trent, Derwent, Severn, Humber, Thames, 

 Avon, &c This navigation, including its windings, 

 extends about 500 miles into the counties of Lincoln, 

 Nottingham, York, Westmoreland, Cheshire, Stafford, 

 Warwick, Leicester, Oxford, Worcester, &c. There is 

 scarcely any manufactory of consequence in Lancaster. 

 It is chiefly celebrated for its cabinet ware. The spin- 

 ning of twine, printing of cotton, and weaving of sail 

 cloth, is also carried on to some extent. Ship-building 

 has been carried on to a considerable extent, and ves- 

 sels of 450 tons have been launched here. The river 

 being obstructed by shoals, only vessels of 250 tons can 

 reach the quay. Its trade is principally to America and 

 the West Indies. In 1799, 52 vessels cleared out tor 

 the latter, with cargoes estimated at upwards of two 

 millions ; but at present the trade is not so considerable. 

 On the 30th of September 1 800, the registered ship- 

 ping consisted of 140 ships, 19,094 tons, navigated by 

 1926 men. In the immediate neighbourhood of the 

 town, is an excellent salt marsh, of about 501 statute 

 acres, which belongs to 80 of the oldest burgesses, or 

 their widows. In the year 1801, Lancaster contained 

 1601 houses, and 9030 inhabitants. In the year 1811, 

 the returns were as follow : 



Houses inhabited, . . 

 Families occupying them, 

 Houses uninhabited, 



Families employed in agriculture, 

 i)o. do. in trade, . 



AIL other families, . . 



Jlules, 

 Females, 



Total 



lC9t 



1906 



37 



182 

 12fiO 



464. 

 4237 

 5010 



9247 



LA\ 7 CEROTTA. See CANARY ISLES, vol. v. p. 356. 



LANDEN, JOHN, a celebrated mathematician, was 

 born at Peakirk, near Peterborough, in Northampton- 

 shire, in January 1719- So early as 1744, he was a 

 contributor to the Ladies Diary ; but it was not till the 

 year 1754, that he published in the Philosophical Trans- 

 actions, his first paper, entitled, An investigation of some 

 Theorems, which suggest several very remarkable proper- 

 lies of the Circle, and are at the same time of consider- 

 able use in resolving Fractions, the denominators of which 

 arc certain multinomials into more simple ones, and by 

 that means facilitate the computation of Fluents. In 

 1755, he published a small volume, entitled Mathema- 

 tical Lucubrations, containing a variety of tracts relative 

 to the rectification of curve lines, the summation of 

 series, the method of finding fluents, and other branches 

 of the higher mathematics. 



About the end of 1757, he published proposals for 

 printing by subscription, the Residual Analysis ; and 

 in 1758, he published his Discourse on the Residual 

 Analysis, in which he resolved a great variety of pro- 

 blems by an entirely new mode of reasoning, and point- 

 ed out the superior elegance of his method to that which 

 had been derived from the fluxionary calculus. 



In the Transactions of the Royal Society for 1760, 

 he published A new met/tod of computing the sums of 

 certain Series, a subject which he afterwards pursued in 

 his Mal/tematical Memoirs, which appeared in 1780. 



Hitherto Mr. Landen had lived as a farmer at the 

 village of Walton, near Peterborough ; but in 17W, he 

 removed to Milton, the seat of Earl Fitzwilliam, where 



he discharged the duty of land steward to that noble- 

 man till within a few years of his death. 



In the year 1764, he published the first book of the 

 Residual Anuli t six, in which he applies it Jo the drawing 

 of tangents to the finding the properties 6T curve lines ; 

 to describing their involutes and evolutes ; finding the 

 radius of curvature, their greatest and least ordinate?, 

 and points of contrary flexure ; and to the determination 

 of their cusps, and the drawing of asymptotes. He pro- 

 posed in a second book, to shew its application to ;i 

 great variety of mechanical and physical problems ; but 

 he never found leisure to complete this part of his plan. 



On the 16th of January 1766, Mr. Landen was elect- 

 ed a member of the Royal Society ; and in 1768, he 

 published his Specimen of a new method' of comparing 

 Curvilintal Areas, by which many such areas may be 

 compared, as liave not yet appeared to be comparable by 

 any other method. _ In the same work for 1770, he gave 

 Some new Theorems Jbr completing the Areas of certain 

 Curve Lines. 



In the Phil. Trans, for 1770, he published A disqui- 

 sition concerning certain Fluents, tchich are assignable 

 by the Arcs of the Conic Sections ; where are investiga- 

 ted, some new and useful Theorems for computing such 

 Fluents. This subject had been previously treated by 

 Maclaurin in his Fluxions, and by D'Alembert in the 

 Memoirs of the Berlin Academy ; but Landen had the 

 merit of removing a very great defect in tlicir methods. 

 In the same year, he published his Animadversions on 

 Dr. Matthew Stewart's computation of the Sun's distance 

 from the Earth, a work written in a style of acrimony 

 disgraceful to a man of genius. 



In the same work for 1775, he published an Investi- 

 gation of a general Theorem, fur /inding the length of 

 any Arc of any Conic Hyperbola by means of two Ellip- 

 tic Arcs; with some other new and useful Theorem 1 ; de- 

 duced 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 Maclau- 

 rin, of constructing them by the rectification of the hy- 

 perbola and the ellipse, fails when some principal point 

 of the curve is to be determined, as the hyperbolic 

 arc and its tangent. then Jbecome infinite, though their 

 difference be at the same time finite. 



In 1777, he published in the Philosophical Transac- 

 tions, A new Theory of the Motion of Bodies revolving 

 about an Axis in free S/jace, when that Motion is disturb- 

 ed by some extraneous force either percussive or accelera- 

 tive. In this paper, he considers only the motion of a 

 sphere, spheroid, and cylinder ; but in consequence of 

 his having afterwards found, that D'Alembert had 

 treated of the same subject, he purchased the Opus- 

 cules of that eminent author, where he found it sta- 

 ted, that some mathematician doubted, whether there 

 is any solid whatever besides the sphere, in which any 

 line passing through the centre of gravity will be a 

 permanent axis of rotation. He was thus led to resume 

 the subject, and he succeeded in pointing out several 

 bodies, which, under certain dimensions, have that re- 

 markable property. This paper was published in a vo- 

 lume of memoirs, which appeared in 1 780, and which 

 contains also a large appendix, with a complete collec- 

 tion of Theorems for the calculation of Fluents, princi- 

 pally investigated by himself. 



Mr. Landen published three small tracts on the sum- 

 mation of converging sines, in which he explained and 

 extended the theorems of De Moivre, Stirling, and 

 Thomas Simson. 



