Hertford and Bow; for at Hertford was the nifh fort, and from thence they made frequ excurfions on the inhabitants of London. thors are not agreed as to the method the pursued in laying dry the Danifh fhips: Dead fuppofes that he did it by straitening the chan but Henry of Huntingdon alledges, that he cu veral canals, which exhausted its water. Bet as it may, the bufinefs was done; and fuch the Danish hips as could be got off, the Lond ers carried into their own road; the reft t burnt and destroyed. Alfred enjoyed a prof peace during the three laft years of his re which he chicfly employed in eftablishing and gulating his government, for the fecurity of h felf and his fucceffors, as well as the edle and nefit of his fubjects in general. After a trou fome reign of 18 years, he died on the 48th October, A. D. 900; and was buried at chefter, in Hyde abbey, under a monumen porphyry.

now a comfortable establishment, if they would fubmit and become his fubjects. This propofition was better received than he expected; for multitudes growing weary of a vagabond kind of life, joyfully accepted the offer. After fome years refpite, Alfred was again called into the field: for a body of Danes, being worfted in the weft of France, came with a fleet of 250 fail on the coaft of Kent; and having landed, fixed themfelves at Appletree: fhortly after, another fleet of 80 veffels coming up the Thames, the men landed, and built a fort at Middleton. Before Alfred marched against the enemy, he obliged the Danes, fettled in Northumberland and Effex, to give him hostages for their good behaviour. He then moved towards the invaders, and pitched his camp between their armies, to prevent their junction. A great body, however, moved off to Effex; and croffing the river, came to Farnham in Surrey, where they were defeated by the king's forces. Mean while the Danes fettled in Northumberland, in breach of treaty, and notwithstanding the hostages given, equipped two fleets; and, after plundering the northern and southern coasts, failed to Exeter, and besieged it. The king, as foon as he received intelligence, marched against them; but before he reached Exeter, they had got poffeffion of it.He kept them, however, blocked up on all fides; and reduced them at laft to fuch extremities, that they were obliged to eat their horfes, and were even ready to devour each other. Being at length rendered defperate, they made a general fally on the beliegers; but were defeated, tho' with great lofs on the king's fide. The remainder of this body of Danes fled into Effex, to the fort they had built there, and to their fhips. Before Alfred had time to recruit himself, another Danith leader, whofe name was Laf, came with a great army out of Northumberland, and defroyed all before him, marching on to the city of Werheal in the well, which is fuppofed to be Chefter, where they remained the reft of that year. The year following they invaded North Wales; and after having plundered and deftroyed every thing, they divided, one body returning to Northumberland, another into the territories of the Eaft Angles; from whence they proceed ed to Effex, and took poffeffion of a final ifland called Merefig. Here they did not long remain: for having parted, fome failed up the river Thames, and others up the Lea Road; where, drawing up their fhips, they built a fort, not far from London, which proved a great check upon the citizens, who went in a body and attacked it, but were repulfed with great lofs: at harvest time, the king himself was obliged to encamp with a body of troops in the neighbourhood of the city, in order to cover the reapers from the excurfions of the Danes. As he was one day riding by the fide of the river Lea, after fome obfervation, he began to think that the Danifh fhips might be laid quite dry: this he attempted, and fucceeded; fo that the Danes deferted their fort and hips, and marched away to the banks of the Severn, where they built a fort, and wintered at a place called Quatbrig. This king's contrivance is thought to have produced the meadow between

ALFRED, CHARACTER OF, &c. All our rians agree in diftinguishing him as one of moft valiant, wifeft, and beft of kings, that reigned in England; and it is alfo general lowed, that he not only digefted feveral part lar laws ftill in being, but that he laid the foundation of our prefent happy conftituti There is great reafon to believe, that we ar debted to this prince for trials by juries; an Doomsday Book, which is preferved in the chequer, is thought to be no more than an edition of Alfred's Book of Winchefter, w contained a furvey of the kingdom. It is f fo, that he was the first who divided the dom into fhires: what is afcribed to him is a bare divifion of the country, but the new form of judicature; for after having du his dominions into fhires, he subdivided fhire into three parts, called tythings. 1 are fome remains of this ancient- divation ridings of Yorkshire, the laths of Kent, an three parts of Lincolnshire. Each tything divided into hundreds or wapentakes; and th gaininto tythings or dwellings of ten houfe each of these householders food engaged to king, as a pledge for the good behaviour family, and all the ten were mutually pledz each other; fo that if any one of the ty was fufpected of an offence, if the head bor or chiefs of the tything would not be fe for him, he was imprifoned; and if he mad efcape, the tything and hundred were fin the king. Each fhire was under the govert of an earl, under whom was the reive, his ty; fince, from his office, called shire-re herrif. And to effectual were thefe regula that it is faid he caufed bracelets of gold hung up in the highways, as a challenge to bers, and they remained untouched. In p life, Alfred was the most amiable man in hi minions; of fo equal a temper, that he fuffered either faduefs or unbecoming ga enter his mind; but appeared always of a yet chearful difpofition, familiar to his fr juft even to his enemies, kind and tender He was a remarkable economist of his time Afferius has given us an account of the m

418

ALGEBRA.

faid to have invented the medicine mentioned in laft article.

ALGAROTTI, Count, a celebrated Italian, a native of Padua. Led by curiofity, as well as a defire of improvement, he travelled early into foreign countries; and was very young when he arrived in France in 1736. Here he compofed his "Newtonian Philofophy for the Ladies;" as Fontenelle had done his Cartefian Aftronomy, in his work intituled, "The Plurality of Worlds." He was noticed by the king of Pruffia, who gave him feveral marks of his efteem. He died at Pifa in 1764; and ordered his own maufoleum, with this infeription to be fixed upon it: "Hic jacet Algarottus, fed non omnis." i.e. “Here lies Algarotti; but not the whole of him." He was a great connoiffeur in painting, fculpture and architecture; and contributed much to the reformation of the Italian opera. His works, which are numerous, and upon a variety of fubjects, abound with vivacity, elegance, and wit: a collection of them has been made, and printed at Leghorn.

ALGARRIA, a fertile district, in the most northern part of New Caftile, which includes Madrid, the capital of Spain.

ALGARVA, a province in the kingdom of

Portugal, 67 miles in length and 20 in breadth; bounded on the W. and S. by the fea, on the E. by the Guadiana, and on the N. by Alentejo. It is very fertile in figs, almonds, dates, olives, and excellent wines; the fishery alfo brings in large fums. The capital town is Pharo. It contains 4 cities, 12 towns, 67 parishes, and about 61,cos inhabitants.

ALGATE, adv. if fo be; notwithstanding; altogether. Bailey.

*ALGATES. adv. from all and gate. Skinner, Gate is the fame as via; and ftill ufed for up in the Scottish dialect.] On any terms; every way: now obfolete.

Nor had the boafter ever rifen more,
But that Renaldo's horfe ev'n then down fell,
And with the fall his leg opprefs'd fo fore,
That, for a fpace, there muft he algates dwell.
Fairfax.

ALGATRANE, a fort of pitch, found in the bay formed by the point of the cape of St Helena, on the S. of the ifle of Piata.

ALGAVAREIA, the language anciently fpokea by the Moors, or Morifcoes in Spain. It was a dialect of Arabic, and flood contra-diftinguished from the ALJAMEIA, which fee.

ETYMOLOGY, DEFINITION, and HISTORY of
ALGEBRA.

WE
E cannot introduce this fcience better, than
by the derivations and definition of the
word, and the account of the origin of the art,
given by Dr Johnson.

* ALGEBRA. n. f. [an Arabic word of uncer tain etymology; derived, by fome, from Geber the philofopher; by fome, from gefr, parchment; by others, from algelifta, a boncfetter; by Menoge, from algiabarat, the reflitution of things broken.] This is a peculiar kind of arithmetick, which takes the quantity fought, whether it be a number or a line, or any other quantity, as if it were granted, and, by means of one or more quantities given, proceeds by confequence, till the quantity at first only fuppofed to be known, or at leaft fome power thereof, is found to be equal to Lome quantity or quantities which are known, and confequently itself is known. This art was in ufe among the Arabs, long before it came in to this part of the world; and they are fuppofed to have borrowed it from the Perfians, and the Perfians from the Indians. The firft Greck author of algebra, was Diophantus, who, about the year 800, wrote thirteen books. In 1494, Lucas Pacciolus, or Lucas de Burgos, a cordelier, printed a treatife of algebra, in Italian, at Venice. Ile fays, that algebra came originally from the Arabs. After feveral improvements, by Vieta, Oughtred, Harriot, Defcartes, Sir Ifaac Newton brought this art to the height at which it ftill continues. Trevoue. Chambers.-It would furely require no very profound fkill in algebra, to reduce the difference of ninepence in thirty fhillings. Swift.

To be fomewhat more particular, algebra is de

fined, by authors who have wrote expressly ups
the fubject, to be a general method of computs
tion, by means of figns and fymbols, common
the letters of the alphabet, by which numbers cr
any other quantities are reprefented, and hence
is called univerfal arithmetic, literal arithmetic, and
the arithmetic of figns.

Befides the derivations above-quoted from Dr Johnfon, feveral other fanciful etymologies a given of the word Algebra. By the Arabians is coupled with the word macabelah, fignify. oppofition and comparifon. Thus, Algebra macabelah is ufed by then, to exprefs what r properly call algebra; and which is explained be the act of reftitution and comparifon, or c pofition and comparison, or refolution and ec tion. Some, with great probability, derive from geber, by prefixing the article al, wh properly fignifies the reduction of fractions to whole number.

ALGEBRA feems not to have been wholly known to the ancient mathematicians. We the traces and the effects of it, in many places though it looks as if they had defignedly conc ed it. Something of it there feems to be in 1 clid, or at leaft in Theon upon Euclid, who ferves that Plato had begun to teach it. A there are other inftances of it in Pappus, 272 more in Archimedes and Apollonius. But th naly fis ufed by thofe authors is rather geome than algebraical: as appears by the examples find in their works: fo that we make no for to fay, that Diophantus is the first, and only thor among the Grecks, who has treated of: gebra profeffedly. It was known, however, mong the Arabs, much earlier than amar Greeks; and the Arabs carried it into Sper

whence, fome are of opinion, that it paffed into England, before Diophantus was known among us. Only fix of Diophantus's books are extant, which were tranflated into Latin, by Xylander, in 1575; and published in 1621, in Greek and Latin, by M. Bachet and Fermat. This algebra of Diophantus only extends to the folution of arithmetical indeterminate problems. The firit European writer on algebra is Lucas Pacciolus, or Lucas de Burgos, a Minorite friar, who having previously given fmall treatifes, publifhed at Venice, in the year 1494, his principal work in Italan on algebra. He makes mention of Leonardus Pifanus, and fome others, of whom he had learned the art, and adds, that algebra came originally from the Arabs; but as he never mentions Diophantus, it is probable, that that author was not then known in Europe. His algebra goes no farther than fimple and quadratic equations, and politive roots only are ufed. After him came Stifelius, Scipio Ferreus, Cardan, Tartaglia, and fome others, who reached as far as the folution of fome cubic equations. Bombelli followed the fe, and went a little farther. Nunnius, Ramus, Schoner, Salignac, Clavius, &c. took different courfes, but none of them went farther than quadratics. in 1390, Vieta introduced his Specious Arithmetic, denoting the quantities, both known and unknown, by fymbols or letters. He alfo gave an ingenious method of extracting the roots of equations, by approximations; fince greatly improved and facilitated by Raphfon, Halley, Maclaurin, Simpfon, and others. Oughtred, an Englishman, in his Clavis Mathematica, printed in 1631, improved Vieta's method, and invented several compendious characters, to fhow the fums, differences, rectangles, fquares, cubes, &c. Harriot, another Englishman, cotemporary with Oughtred, left at his death, an Analyfis, on Algebra, printed in 1631, in which Vieta's method is brought into a fill more commodious form, which is yet in eftimation. In 1657, Des Cartes publifhed his geometry, in which he applied Harriot's method to the Eigher geometry, explaining the nature of Curves by equations, aud adding the confucions of cubic, biquadratic, and other higher equations. From the time of Des Cartes, continual improvements have been made in the fcience, by Baker, Fermat, Schooten, Slucius, Wallis, Newton, Mercator, Demoivre, Maclaurin, Landen, Euler, Waring, Lorgna, Bernouilli, Preftet, Ozanam, Kerfey, Roberval, Guifnee, Ghetaldus, Pell, Ward, Hammond, Saunderson, &c.

PART I.

INTRODUCTION.

ALGEBRA is of two kinds, nuneral and literal. ALGEBRA, numeral, or vulgar, is that of the ancients, which only had place in the refolution of arithmetical queftions. In this, the quantity fought is reprefented by fome letter or character; but all the given quantities are exprefled by numbers. This is thought by fome to have proved an introduction to the art of keeping merchants accounts by double entry.

ALGEBRA, fpecious or literal, or the new algebra, is that wherein the given or known quanti

419

ties, as well as the unknown, are all expreffed or reprefented by their species, or letters of the alphabet. This eafes the memory and imagination of that vaft fires or effort, required to keep feveral matters, neceffary for the difcovery of the truth in hand, prefent to the mind: for which reafon, this art may be properly denominated metaphyfical geometry.

A quantity which can be measured, and is the object of mathematics, is of two kinds, number and extenfion. The former is treated of in arith metic; the latter in geometry. Numbers are ranged in a fcale, by the continued repetition of fome one number, which is called the Root; and, in confequence of this order, they are conveniently expreffed in words, and denoted by characters. Investigations by common arithmetic, are greatly limited, from the want of characters to exprefs the quantities that are unknown, and their different relations to one another, and to fuch as are known. Hence letters and other convenient fymbols have been introduced to fupply this defect; and thus gradually has arifen the feience of Algebra, properly called Univerfal drithmetic. In the common arithmetic too, the given numbers difappear in the courfe of the operation, fo that general rules can feldom be derived from it; but, in algebra, the known quantities, as well as the unknown, may be exprefied by letters, which, through the whole operation, retain their original form; and hence may be deduced, not only general rules for like cafes, but the dependence of the feveral quantities concerned, and like wife the determination of a problem, without exhibiting which, it is not completely refolved. This general method of expreffing quantities alfo, and the general reafonings concerning their connections, which may be founded on it, have rendered this fcience not lefs ufeful in the demonftration of theorems, than in the refolution of problems. If geometrical quantities be fuppofed to be divided into equal parts, their relations, in refpect of magnitude, or their proportions, may be expreffed by numbers; one of these equal parts being denoted by the unit. Arithmetic, however, is ufed in expreffing only the conclu fions of geometrical propofitions; and it is by algebra that the bounds and application of geometry have been of late fo far extended. The proper objects of mathematical fcience are number and extenfion; but mathematical enquiries may be inftituted alfo concerning any phyfical quantities, that are capable of being meatured or expreffed by numbers and extended magnitudes: And, as the application of algebra may be equally univerfal, it has been called The Science of Quantity in general.