A L G 



linprovfmfiiu, particularly with rcfcrcncf to the ineJucihle 

 ciilc m cubic- equations. A fifth edition of this valuable trta- 

 tife, with notes and additions uaspubliflu'd at Pans in 1797, 

 in 2 vols. A\a. He has alfo fcveral papors on analytics, in 

 the Memoirs of the Academy of Sciences. In 1747, M. 

 I'ontaine gave, in the fame memoirs, a paper on the refolu- 

 tion of equations, and otiur papers in fubfcquent memoirs. 

 In 17+8, Maden-.oifelle M. G. Agnefi, pubhihed at Milan 

 in Itahan, " Analytical Intlitulions, in 2 vols. 410. M. Cal- 

 tilion, in 1761. pubhihed in 2 vols. 4to, Newton's Uni- 

 verfal Arithmetic, with an an>ple commentary. In T-'i.?, 

 Mr. Emerfon publiflied hi; " Increments," and in 1764 his 

 " Algebra." Mr. Landen publifticd his " Rcfidual Ana- 

 Kfis," in 17(>4. his " Mathematical Lucubrations," in 

 I'^iSj, and his " Maihematical Memoirs," in :78o. M. 

 liuler publiflied liis " Elements of Algebra," in the Ger- 

 man language in 1770, and in 1774, a French tranflation 

 was publilhed, by J. Bernouilli, with the analyfis of indeter- 

 minate problems, by M. de la Grange. An Englifh tranf- 

 Intion was pubU'.hed in 1797, in 2 vols. The memoirs of 

 lierlin and Peterlburgh abound with various improvements 

 on feries and other branches of analyfis by this celebrated 

 mathematician. Dr. Waring, late of Cambridge, has com- 

 municated feveral valuable papers to the Philofophical Tranf- 

 actions, and many of his improvements, arc contained in his 

 fepai-ate publications, particularly the " Meditationes AU 

 gcbi-aicsE," publilhed in 1770; the " Prbprietates Alge- 

 braicarum Curvarum," in 1772 ; and the " Meditationes 

 Analyticae," in 1776. The firft of thefe publications dc- 

 fcrvci particular notice. The firil chapter treats of the 

 transformation of algebraical equations into others, of which 

 the roots have given algebraical relation to the roots of the 

 given equatitms. The limits and number of impoffible and 

 affirmative and negative roots of algebraical equations are the 

 fubjeds of the fecond chapter. The third chapter compre- 

 hends the inveftigation of the roots of equations or irrational 

 quantities, which have given relations to one another, the 

 rcfolution of equations, &c. ccc. The fourth chapter is 

 principally converfant concerning more algebraical cq\iations 

 and their reduftion to one ; and the fifth chapter trcits of 

 rational and integral values of the unknown quantities of given 

 tquations. Francis Maferes, Efq. claims honourable mention, 

 not only as an original writer, who has contributed to the ex- 

 plication and improvement of fome of the moil abllruieand 

 yet moft mtcrelling branches of algebra and analyfis, but 

 on account of the labour and expence which he has bellowed 

 on the publication of the " Scriptores Logarithmici," in 

 three vols., 410., 1791, 1796? containing many curious and 

 uftful tradts, which are thus prtfcrved from being loft, and 

 many valuable pnpers of his own on the binomial theorem, 

 ftries, &c. After this detail, for which we are in a con- 

 fider.ible degree indebted to the relearch of Montucla and 

 Llr. Hutton, manv authors who have, in feparate treatiles 

 or in occafional elTays, contribute.d to the improvement of 

 algebra in generSl, or fome particular branches of it, or wlio 

 have publilhed treatifes on the Icience, ftill remain unno- 

 ticed ; and we mull content ourfelvcs with merely mentioning 

 Francifcus Cahgarius, Rudolphus, Adam Gigas or Rifeu, 

 butio, R. Wentworth, Ant. Maria Floridus, I.a/arus 

 Schonerus, Bernard Salignac, Leonard, Digges, and Ro- 

 bert Norman, in the i6th century, Chrillopher Clavius, in 

 160S, Georgius Hcnefchius, in 1609, Seballian Kurt/., 

 C'oignet, Laloubere, Degraave, Mefcher, the BernouiUis, 

 Malbranche, Wehs, Dodlon, Manfredi, Regnault, Rown- 

 ing, Hammond, Lorgna, Hellins, de la Grange, de la 

 Place, Bcrtrand, Kuhnius, Hales, Malkelyne, Viuce, Wood, 

 Manning, Frend, Bonnycaftle, Sec. &c. &c. 



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/Ihelra is a peculiar kind of Aritiimftic, 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 a train of de- 

 dutlion, till the quantity at firll only fuppofed to be known, 

 or at leall fome power of it, is found to be equal to fome 

 quantity or quantities which arc known, and confequently 

 itfclf is known. 



Algebi-a is of two kinds, numeral and literal. 



Algebra, numeral, or vulgar, is that which is chiefly con- 

 cerned in the refolution of arithmetical queftions. In this, 

 the quantity fought is reprefented by fome letter or charatler ; 

 but all the given quantities are exprefi'ed by numbers. Such 

 is the algebra of the more ancient authors, as Diophantus, 

 Paciolus, Stifelius, &c. This is thought by fome to have 

 been an introduftion to the art of keeping merchants' ac- 

 counts by double entry. 



Algebra fpecimis I or literal, or the nciv algebra, is that in 

 which all the quantities, known and unknown, are exprefled 

 or reprefented by their fpecies, or letters of the alphabet. 

 There are inftanccs of this method from Cardan and others 

 about his time ; but it was more generally introduced and 

 ufed by Vieta. Dr. Wallis (Algebra, p. 66.) apprehends, 

 that the name of fpecious arithmeticappliedto algebra is given 

 to it with a reference to the fenfe in which the Civilians ufe 

 the word fpecies. Thus, they ufe the names Titius, Sem- 

 pronius, Caius, and the hke, to reprefent indefinitely any 

 perfon in fuch circumftances ; and cafes fo propounded, they 

 call fpecies. Vieta, accuftomed to the language of the 

 civil law, gave, as Wallis fuppofes, the name of fpecies to 

 the letters A, B, C, &c. which he ufed to reprefent inde- 

 finitely any number or quantity, fo circumftanccd as the 

 occafion required. 



This mode of expreffion frees the memory and imagina- 

 tion from that ftrefs or effort, which is required to keep fe- 

 veral matters, neceffary for the difcovery of the truth in- 

 velligated, prefent to the mind ; for which reafon this art 

 may be properly denominated metaphyfical geometry. Spe- 

 cious algebra is not- like the numeral, confined to certain 

 kinds of problems ; but ferves univerfally for the inveftigation 

 or invention of theorems, as well as the folution and demon- 

 ftration of all kinds of problems, both arithmetical and 

 geometrical. 1'he letters ufed in algebra do each of them 

 fcparately reprefent cither lines or numbers, as the problem 

 is either arithmetical or geometrical ; and together, they 

 reprefent planes, folids, and powers more or lefs high, as 

 the letters are in a greater or lefs number. For inftance, if 

 there be two letters, a b, they reprefent a reftangle, whofe 

 two fides are exprefted, one by the letter a, and the other 

 by i ; fo that by their mutual multiplication they produce 

 the plane a b. Where the fame letter is repeated twice, 

 as a a, they denote a fquare. Three letters a b e, repre- 

 fent a folid or a reclangular parallelepiped, whofe three 

 dimenfions are exprellVd by the three letters a b c ; the 

 length by a, the breadth by b, and the depth by f ; fo that 

 by their mutual multiplication, they produce the fohd 

 a b c. 



As the multiplication of dimenfions is exprefled by the mul- 

 tiplication of letters, and as the number of thefe may be fo 

 great as to become incommodious, the method is only to write 

 down the root, and on the right hand to write the index of the 

 power, that is, the number of letters of which the quantity 

 to be exprefled confills ; as a', a', a*, &c. the laft of which 

 fignifies as much as a multiplied four times into itfelf ; and 

 fo of the reft. But as it is necefl'ary, before any progrefs 

 can be made in the fcience of algebra, to underftand the 

 method of notation, we (hall here give a general view of it. 



In 



