Andronicus of Rhodes having bought them of his heirs, was the firft reftorer of the works of this great philofopher; for he repaired what had been decayed, and got them copied. brafs in it, which, by its falling into a bafon of the fame metal, awaked him. He had feveral conferences with a learned Jew at Athens, who inftructed him in the fciences and religion of the Egyptians, and thereby faved him the trouble of travelling into Egypt. When he had ftudied about 15 years under Plato, he began to form different tenets from those of his mafter, who became highly piqued at his behaviour. Upon the death of Plato, he quitted Athens, and retired to Atarnya, where he married Pythias, the fifter of Hermias, prince of Myfia; whom he is faid to have loved fo paffionately, that he offered facrifice to her. Some time after, Hermias having been taken prifoner by Meranon the king of Perfia's general, Aristotle went to Mitylene, the ca. pital of Lefbos, where he remained till Philip king of Macedon having heard of his great reputation, fent for him to be tutor to his fon Alexander, then about 14 years of age: Ariftotle accepted the offer; and in 8 years taught him rhe toric, natural philofophy, ethics, politics, and a certain fort of philofophy, which, fays Plutarch, he taught nobody elfe. Philip erected ftatues in honour of Ariftotle; and for his fake rebuilt Stagyra, which had been almoft ruined by the wars., The laft 14 years of his life he spent moftly at Athens, furrounded with every affiftance which men and books could afford him for profecuting his philofophical enquiries. The glory of Alexander's name, which then filled the world, infured tranquillity and refpect to the man whom he diftinguished as his friend: but after his premature death, the invidious jealoufy of pricfts and fophifts inflamed the fuperftitious fury of the Athenian populace; and the fame odious paffions which proved fatal to the offenfive virtue of Socrates, fierce, ly affailed the fame and merit of Ariftotle. To avoid their perfecution he withdrew to Chalcis, a measure fufficiently juftified by a regard to perfonal fafety; but leaft his conduct thould appear unmanly, when contrafted with the firinnefs of Socrates, he apologized for his flight, by faying, that he was unwilling to afford the Athenians a fecond opportunity to fin againft philofophy." He feems to have furvived his retreat from Athens only a few months; vexation and regret probably ended his days. (2.) ARISTOTLE, HISTORY OF THE WORKS OF. Befides his treatifes on philofophy, he wrote alfo on poetry, rhetoric, law, &c. to the number of 400 treatifes, according to Diogenes Laertius. An account of fuch as are extant, and of those faid to be loft, may be feen in Fabricius's Bibliotheca Gre ca. He left his writings with Theophraftus, his beloved difciple and fucceffor in the Lyceum; and forbade that they fhould ever be published. Theophraftus, at his death, trufted them to Neleus, his friend and difciple; whofe heirs buried them in the ground at Sceplis, a town of Troas, to fecure them from the king of Pergamos, who made great fearch every where for books to adorn his library. Here they lay concealed 160 years, until, being almoft fpoiled, they were fold to one Apellicon, a rich citizen of Athens. Syila found them at this man's houfe, and ordered them to be carried to Rome. They were fome time after purchafed by Tyrannion a grammarian; and (3.) ARISTOTLE'S PHILOSOPHY, HISTORY OF. Many followed the doctrine of Aristotle in the reigns of the 12 Cæfars, and their numbers increa fed much under Adrian and Antoninus. Alexander Aphrodinus was the firft profeffor of the Peripa tetic philofophy at Rome, being appointed by the emperors Marcus Aurelius and Lucius Verus; and in fucceeding ages the doctrine of Ariftotle prevailed among almost all men of letters, and many commentaries were written upon his works. The firft doctors of the church difapproved of the doctrine of Aristotle, as allowing too much to rea fon and fenfe; but Anatolius, bishop of Loadicea, Didymus of Alexandria, St Jerome, St Auguftin, and several others, at laft wrote and spoke in fa vour of it. In the 6th century Boethius made it known in the weft, and tranflated fome of his pieces into Latin. But from the time of Boethius to the 8th age, Joannes Damafcenus was the only man who made an abridgement of his philofophy, or wrote any thing concerning him. The Grec ans, who took great pains to rettore learning in the 11th and following centuries, ftudied much the works of this philofopher, and many learned men wrote commentaries on his writings: amongit thefe were Alfaragius, Algazel, Avicenna, and Averroes. They taught his doctrine in Africa, and at Cordova in Spain. The Spaniards intro duced it into France, with the commentaries of Averroes and Avicenna; and it was taught in the univerfity of Paris, until Amouri, having supported fome particular tenets on the principles of the philofopher, was condemned of herefy, in a cou cil held there in 1210, when all the works of Ar ftotle that could be found were burnt, and realing of them forbidden under pain of excommuni cation. This prohibition was confirmed, as to the phyfics and metaphyfics, in 1215, by the pope's legate; though at the fame time he gave leave for his logic to be read, inttead of St Auguftin's, ufed at that time in the univerfity. In 1265, Simon, cardinal of St Cecil, and legate from the holy fee, prohibited the reading of the phyfics and meth phyfics of Ariftotle. All these prohibitions how. ever, were taken off in 1366; for the cardina's of St Mark and St Martin, who were deputed Pope Urban V. to reform the univerfity of Pars permitted the reading of thofe books which bad been prohibited: and in 1448, Pope Stephen proved of all his works, and procured a new tra lation of them into Latin. ARIS FOTUS, in ichthyology, the thad, called by Albertus and others. See CULPEA THRISSA. ARISTOXENUS, the moft ancient mosca writer, of whofe works any tracts are come dow to us, was born at Tarentum. He was the fa of a musician, whom fome call Afnejjas, others Spintharus, He had his first education at Marnaa, under his father, and Lamprus of Erythre he next ftudied under Xenophilus, the Pythago rean; and lastly under Ariftotle, in company wit Theophraftus. Suidas adds, that Amterenu enraged at Ariftotle having bequeathed his fchoo 1. ARI ARI ( 487 ) But to Theophraftus, traduced him ever after. Ariftocles the Peripatetic, in Eufebius, exculpates Ariftoxenus, and affures us that he always fpoke with great respect of Ariftotle. Ariftoxenus lived under Alexander the Great and his first fucceffors. His Harmonics in 3 books, (all that have reached us,) together with Ptolemy's Harmonics, were first published by Gogavinus, at Venice, in 1562, 4to, with a Latin verfion. John Meurfius next tranflated thefe 3 books into Latin from the MS. of Jof. Scaliger. With these he printed at Leyden, in 1616, 4to, Nicomachus and Alypius, two other Greek writers on mufic. Meibomius collected thefe mufical writers together; to which he added Euclid, Bacchus fenior, and Ariftides Quintilianus; and published the whole, with a Latin verfion and notes, from the elegant prefs of * ARITHMANCY. n. f. [from agiluos, number, and μa, devination.] A fortelling future events by numbers. Dia. ARITHMETIC. INTRODUCTION. SECT. I. ETYMOLOGY, and DEFINITION of cerning the dimenfions of the ark, leave us no room to doubt that he had a knowldge of both numbers, and measures. When Rebekah was fent away to Ifaac, Abraham's fon, her relations •) ARITHMETIC, [from acids, number, wifhed the might be the mother of thousands and (1.) SECT. II. HISTORY of ARITHMETIC. (2.) At what period Arithmetic was first introduced into the world, we can by no means determine. That fome part of it, however, was coeval with the human race is abfolutely certain. We cannot conceive how any man endowed with reafon can be without fome knowledge of numbers. We are indeed told of nations in America who have no word in their language to exprefs a greater number than three, which they call patarraro rincouroac: but that fuch nations fhould have no idea of a greater number than this, is abfolutely incredible. Perhaps they may compute by threes, as we compute by tens; and this may have occafioned the notion that they have no greater number thar three. (3.) But though we cannot fuppofe any nation or indeed any fingle perfon, ever to have been without fome knowledge of the difference between greater and smaller numbers, it is poffible that mankind may have fubfifted for a confiderable time without bringing this fcience to any perfection, or computing by any regular fcale, as 10, 60, &c. That this, however, was very early introduced into the world, even before the flood, we may gather from the following expreffion in Enoch's prophecy, as mentioned by the apoftle Jude: "Behold, the Lord cometh with ten thou fands of his faints." This shows, that even at that time men had ideas of numbers as high as we have at this day, and computed them alfo in the fame manner, namely by tens. (4.) The directions alfo given to NOAH, con millions; and if they were totally unacquainted with the rule of multiplication, it is difficult to fee how fuch a wifh could have been formed. It is probable, therefore, that the four fundamental rules of arithmetic have always been known to fome nation or other. (5.) Some nations, however, like the Europeans formerly, and the Africans and Americans now, have doubtlefs been immersed in the most abject and deplorable state of ignorance; and might therefore remain for fome time unacquainted with numbers, except fuch as they had immediate occafion for. And, when they came afterwards to improve, either from their own industry, or hints given by others, they might fancy that they themfelves, or thofe from whom they got the hints, had invented what was known long before. (6.) Dr Chambers thinks it highly probable, that arithmetic, as a science, 'must have taken its rife from the introduction of commerce; and confequently that it fhould be of Tyrian invention. (7.) From Afia it paffed into Egypt, says Josephus, by means of ABRAHAM. Here it was greatly cultivated and improved; infomuch that a large part of the Egyptian philofophy and theology feems to have turned altogether upon numbers. Hence thofe wonders related by them about unity, trinity, the numbers feven, ten, four, &c. In fact, Kircher, (in his Oedip. Ægypt. tom. 2. p. ii.} fhews, that the Egyptians explained every thing by numbers. Pythagoras affirms, that the nature of numbers goes through the whole universe; and that the knowledge of numbers is the knowledge of the Deity. (8.) From Egypt arithmetic was tranfmitted to the Greeks, who were doubtlefs the firft European nation among whom arithmetic arrived at any degree of perfection. M. Goguet is of opinion, they ufed pebbles in their calculations: 3 proof proof of which he imagines, is, that the word np in the common way: then 60 was called a fexage which comes from 4, a little ftone, or flint, among other things, fignifies to calculate. The fame he thinks, is probable of the Romans; and derives the word calculation from the ufe of little stones (calcali) in their firit arithmetical operations. (8.) This method, however, must have been but for a fhort time, fince we find the Greeks very early made ufe of the letters of the alphabet, taken according to their order, at firft denoting the numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 100, 200, 300, 400, 500, 600, 700, and 800; to which they added the three following,,,), to reprefent 6, 90, and 900. The difficulty of performing arithmetical operations by fuch marks as these may easily be imagined, and is very confpicuous from Archimedes's treatise Concerning the dimenfions of a circle. (10.) A a fimilar method was followed by the Romans: and befides characters for each rank of claffes, they introduced others for five, fifty, and five hundred. Their method is still used for diftinguishing the chapters of books, and some other purposes. Their numeral letters and values are the following: Five hundered, one thousand. (11.) Any number, however great, may be reprefented by repeating and combining thefe according to the following rules: ift, When the fame letter is repeated twice, or oftener, its value is reprefented as often. Thus II fignifies two: XXX thirty, CC, two hundred. 2d, When a numeral letter of leffer value is placed after one of greater, their values are added thus XI fignifies eleven, LXV fixty-five, DCVIII fignifies fix hundred and eight, MDCXXVIII one thoufand fix hundred and twenty eight. 3d, When a numeral letter of leffer value is placed before one of greater, the value of the leffer is taken from that of the greater: thus IV fignifies four, XL forty, XC ninety, CD four hun dred. (12.) Sometimes I is ufed inftead of D for 500, and the value is increased ten times by annexing to the right hand. Thus I fignifies 500. Alfo CIO is used for 5000 1000 for 10000 50000 CC for 100000 (13.) Sometimes thousands are reprefented by drawing a line over the top of the numeral, V being used for five thoufand, L for fifty thousand, CC two hundred thousand. (14.) A new kind of arithmetic, called SEXAGESIMAL, was invented, as is fuppofed, by CLAUDIUS PTOLOMEUS, about A. D. 200. The defign of it was to remedy the difficulties of the common method, efpecially with regard to fractions. In this kind of arithmetic, every unit was fuppofed to be divided into 60 parts, and each of thefe into 60 others, and fo on: hence any number of fuch parts were called fexagefimal fractions; and to make the computation in whole numbers more eafy, he made the progreflion in thefe alfo fexagefimal. Thus from one to 59 were marked fima prima, or firft fexagefimal integer, and had one fingle dafh over it; fo 68 was expreffed thus I'; and fo on to 60 times 60, or 3540, which was thus expreffed LIX'. He now proceeded to 60 times 60, which he called a fexagefima fecunda, and was thus expressed I". In like manner, twice 60 times 60, or 7200, was expreffed by II"; and fo on till he came to 60 times 3600, which was a third fexagefimal, and expreffed thus, I". If any number lefs than 60 was joined with these sexagefimals, it was added in its proper characters with out any dash; thus l'XV reprefented 60 and 15, or 75 I'VXXV is four times 60 and 25, or 265; X"II'XV, is ten times 3600, twice 60 and 15, or 36,135, &c. Sexagefimal fractions were marked by putting the dash at the foot or on the left hand of the letter; thus I,, or 'I, denoted ; In or "I, too, &c. (15.) The numeral characters which we now ufe, and which are doubtless the moft perfect me. thod of notation, came into Europe from the ARABIANS by the way of Spain. The Arabs however, do not pretend to be the inventors of them, but acknowledge that they received them from the Indians. (16.) Some, indeed, contend that neither the Arabs nor the Indians were the inventors, but that they were found out by the Greeks. But this is by no means probable; as Maximus Planudes, who lived towards the close of the 13th century, is the first Greek who makes ufe of them and he is plainly not the inventor; for Dr Wallis mentions an infcription on a chimney in the parfonage houfe of Helendon in Northamp tonfhire, where the date is expreffed by M° 133, inftead of 1133. Mr Luff kin furnishes a ftill ear. lier inftance of their ufe, in the window of a house, part of which is a Roman wall near the marketplace in Colchester; where between two carved lions ftands an efcutcheon with the figures 1090 Dr Wallis is of opinion that these characters mutt have been used in England at leaft as long ago as the year 1050, if not in ordinary affairs, at least in mathematical ones, and in the aftronomical tables. ed by the Indians we are entirely ignorant. How these characters came to be originally invent (17.) The introduction of the Arabian characters in notation did not immediately put an end to the faxgefimal arithmetic. As this had been ufed in all the astronomical tables, it was for their fakes retained a confiderable time. The fexa gefimal integers went first out, but the fractions continued till the invention of decimals. (18.) The most ancient treatifes extant upon the theory of arithmetic are the 7th, 8th, and 9th books of EUCLID's elements, where he treats of proportion and of prime and compofite numbers; both of which have received improvements fince his time, efpecially the former. (19.) The next of whom we know any thing is NICOMACHUS the Pythagorean, who wrote a treatise of the theory of arithmetic, confifting chiefly of the diftinctions and divifions of numbers in claffes, as plain, folid, triangular, quadran gular, and the rest of the figurate numbers as they are called, numbers odd and even, &c. with fore of the more general properties of the feveral kinds. This SECT. II. III. ARITHMETIC. This author is by fome, faid to have lived before the time of Euclid; by others, not long after. His arithmetic was published at Paris in 1538. (20.) The next remarkable writer on this fubject is BOETHIUS, who lived at Rome in the time of Theodoric the Goth. He is fuppofed to have copied moft of his work from Nicomachus. A compendium of the ancient arithmetic, written in Greek, by PSELLUS, in the ninth century from our Saviour, was given us in Latin by XYLANDER, in 1556. (21.) From this time no remarkable writer on arithmetic appeared till about the year 12co, when JORDANUS of Namur wrote a treatife on this fubject, which was published and demonftrated by Joannes Faber Stapulenfis in the 15th century, foon after the invention of printing. The fame author alfo wrote upon the new art of computa tion by the Arabic figures, and called this book Algorifmus Demonftratus. Dr Wallis fays this manufcript is in the Savillian library at Oxford, but it hath never yet been printed. (22.) As learning advanced in Europe, fo did the knowledge of numbers; and the writers on arithmetic foon became innumerable. About the year 1464, REGIOMONTANUS, in his triangular tables, divided the Radius into 10,000 parts inftead of 60.000, and thus tacitly expelled the fexagefimal arithmetic. Part of it, however, ftill remains in the divifion of time, as of an hour into 60 minutes, a minute into 60 feconds, &c. (23.) RAMUS in his arithmetic, written about A. D. 1550, and published by Lazarus Schonerus in 1586, ufes decimal periods in carrying on the fquare and cube roots to fractions. The fame had been done before by our countrymen BUCKLEY and RECORD; but the first who published an exprefs treatife on decimals was SIMON STEVINIUS, about the year 1582. (24.) BARLAAMUS MONACHUS alfo gave a theory for demonftrating the common operations, both in integers and broken numbers, in his Lo giftica, published in Latin by J. Chambers, an Englishman, in 1600.-And LUCAS DE BORGO, in an Italian treatise, published in 1523, gave the feveral divifions of numbers from Nicomachus, and their properties from Euclid; with the algorithm, both in integers, fractions, extractions of roots, &c. (25.) The first entire body of practical arithme- (26.) We omit other more practical authors, 489 He is (27.) As to the circulating decimals, Dr WALLIS, (28) The greateft improvement, however, which SECT. III. DIVISIONS of ARITHMETIC. (29.) This Science has been variously divided by different authors, according to the limits they prefcribed to their treatifes, or the peculiar mode of arrangement they adopted; fuch as Vulgar and Decimal; Theoretical and Practical; Numerous and Specious; bivary, duodecimal, dynamical, infrumental, logarithmical, fexagefimal, tetracical, &c. upon which it is unneceflary here to enlarge, as none of them seem properly adapted to a work of this kind, and the terms are explained in their order. (30.) The mode of divifion and arrangement we have adopted in this treatife, we are perfuaded will be found, not only in fome degree new, but full as proper, as any yet attempted by our predeceffors, or cotemporaries. PART I. SIMPLE ARITHMETIC. SECT. I. NOTATION and NUMERATION. (31.) NOTATION is the expreffing of any propofed number, either by words or characters. All numbers are expreflible by thefe ten characters or figures; 8 I 2 3 4 6 7 One, two, three, four, five, fix, feven, 9 eight, nine, cypher. The nine firft are called fig- 2 a cypher is written on the right of it thus, 20, it reprefents 2 tens, or twenty; and if another cypher be affixed thus, 200, it will reprefent 2 hundreds, &c. (32.) It has been faid, that "there does not feem to be any number naturally adapted for conftituting a clafs of the loweft, or any higher rank to the exclufion of others; that however, as ten has been univerfally ufed for this purpose by moft nations who have cultivated this fcience, it is probably the most convenient for general ufe. Other fcales (it is alledged,) may be affined: thus, if eight were the fcale, 6 times 3 would be two claffes and two units, and the number 18 would then be reprefented by 22. If 12 were the feale, 5, times 9 would be three claffes and nine units, and 45 would be reprefented by 39, &c. (33.) But this theory feems far from being fupported by fact. The univerfality of the practice of reckoning by tens is allowed even by thote who plead for it. The antiquity of it, which as above obferved, (§ 3,) is antediluvian, might also be urged, as an evidence that it is the moit natural claffification of numbers. But this is not all. There feems to be a regularity of gradation from the loweft to the higheft poffible numbers, and uniformity of proportion in reckoning them, upon the decadary plan, that is unattainable, if not impracticable by adopting any other mode, or any higher or lower number as the limit of a clafs. An additional argument may be drawn from confidering how we firft acquire our ideas of numbers. (34) The first elements of arithmetic are acquired during our infancy: for, when a child gathers as many ftones together as fuits his fancy, and then throws them away, he, acquires the firft elements of the two capital operations in arithmetic, addition and fubtraction. Small numbers are most easily apprehended: a child foon knows what two and what three is; but has not any diftinct notion of taventy-three. Experience removes his difficulty by degrees, and he becomes accuftomed to handle larger collections, and to form many units into a clafs, and feveral of thefe claffes into one of a higher kind, and thus to advance through as many ranks of claffes as occafion requires. If a boy arrange an hundred ftones in one row; he would with difficulty reckon them; but if he place them in ten rows of ten stones each, he will reckon an hundred with cafe; and if he collect ten fuch parcels, he will reckon a thousand. Quintillions. Quadrillions. Trillions. (35.) But fuppofe a teacher should adopt this mechanical method of teaching a boy arithmetic, and thould at the fame time take it into his head to reckon by fevens or nines inftead of tons, we may readily believe, he would find it a very diffi. cult talk to make his pupil entertain any accurate idea of the proportions between the larger and fmaller numbers, whatever denomination fuch a fanciful arithmetician might give them. The ancient Greeks and Romans would have brought the fcience of arithmetic to a much greater degree of perfection, than they ever did, had they hit upon the method of expreffing by TEN DISTINCT CHARECTERS the numbers by which they reckoned. But the idea of a CYPHER, which can only be introduced into the decadary fyftem, and which may be filed the KEY-STONE of ARITHMETIC, feems never to have ftruck them; and thus, though they reckoned properly enough by tens, yet not having characters proportionate to express their numbers, they involved their arithmetic in a labyrinth of confufion, from which neither a EUCLID, nor an ARCHIMEDES, with all their wonderful mechanical powers, were able to extricate it for want of this clue. In a word, it is to the cypher, in uniform alteration with the nine digits, that the moderns owe the honour of having PERFECTED A SCIENCE, in which the ancients, with ail their great attainments, had made but small progrefs. And perhaps, if all our modern weights and meafures, were divided and fubdivided upon the decadary plan, instead of into fourths, eighths, twelfths, fixteenths, &c. that general uniformity of both, fo long wanted, might be foon attained. (36.) NUMERATION implies the numbering or reading of numerical characters; or the reckoning any number of things by them. For the more caly numbering, and expeditious reading of large num bers, when they are expreffed by figures, they are divided from the right hand towards the left, into periods and half periods, and each period confifting of three figures; the common name of the first period being units, or ones; of the fe cond, millions, of the third billions; of the fourth trillions, &c. Alfo the first half of any period is fo many ones of it, but the latter half is fo many thoufands of it. The following example exhibits a summary of this whole doctrine, and may be extended to fextillions, feptillions, octillions, nonillions, &c. ad infinitum. 9 8 7 6 5 4 3 2 1,9 8 7,6 5 4 3 2 1,9 8 7 6 5 4 3 2 1, 9 8 7,6 5 4,3 A number expreffing a quantity of one name or denomination, is called a fimple number, as 20 pounds, or 17 gallons, or 5 days; and that repreunting a quantity of feveral names, is called a compound number, as 13 pounds 5 fhillings and 6 pence, or 17 gallons and 2 pints, or 3 hours and 50 minutes. (37) RULE |