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Ziieral, or Specious Algebra, <5r the K'wAlgeera, the Quantities, both known and unknown, by Symbols ot is that wherein the given or known Quantities, as well as Letters. — He alfo introduced an ingenious Method of cx- the unknown, are all expreffed or reprelentcd by their Spe- tracking the Roots of Equations, by Approximation; fi[ lcft cies, or Letters of the Alphabet. See Species, and Spe- much facilitated by Raphfon, in his Analyjis Jiiquationum 'eio-os. Vleta was follow'd by Oughtred, who in his Clavls Ala-
Thiseafes the Memory and Imagination of that vaftStrefs thematica, printed in 1631, improved Vieta's Method ■ and or Effort, requir'd to keep the feveral Matters neceffary for invented feveral compendious Characters, to fhew the Sums the Difcovery of the Truth in hand prefent to the Mind : Differences, Re&angles, Squares, Cubes, &e. For which Reafon this Art may be properly denominated Metaphyseal Geo?netry.
Specious Algebra, is not, like the Numeral^ connVd to certain Kinds "of Problems ; but ferves univerlally for the Inveftigation or Invention of Theorems, as well as the Solu- tion and Demonftration of all kinds of Problems, both Arith- metical, and Geometrical. See Theorem, t$c.
The Letters ufed in Algebra, do each feparately repre- fent either Lines or Numbers, as the Problem is Arithme- tical or Geometrical ; and together, they reprefent Planes, Solids and Powers more or lefs high, as the Letters are in a greater or lefs Number. — For inftance, if there be two
Mr. Harriot, anoiher Englijhman , cotemporary with Oughtred, left feveral Treatifes at his Death ; and among the reft, an Analyfis, or Algebra, which was printed in 11J31 5 where Vieta's Method is brought into a iHll more commodious form, being that which obtains to this Day.
r " 1S57, 2to Cartes publifh'd his Geometry, wherein
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he made ufe of the Literal Calculus and the Algebrakk Rules of Harriot ; and as Oughtred in his Cla-vis, and Ma- rin. Ghctaldus, in his Books of Mathematical Compofuion and Refolution publifti'd in 1630, applied Vieta's Arithme- tick to Elementary Geometry, and gave the Conftru&ions of Simple and Quadratick Equations ; fo Ties Cartes applied Letters, ab, they reprefent a Rectangle, whofe two Sides Harriot's Method to the Higher Geometry, explaining the are expreffed, one by the Letter a, and the other by b ; fo Nature of Curves by Equations, and adding the Contfruc- that by their mutual Multiplication, they produce the Plane t i Dns of Cubic, Biquadratic, and other higher Equations. a b. Where the fame Letter is repeated twice, as a a, they fQ es Cartcs's Rule for confronting Cubic and Biquadratic denote a Square. — Three Letters, a be, reprefent a Solid, Equations, was further improved by Tho. Baker, in his or a reftangled Parallelopiped, whofe three Dimenfions are Qavis Gcometrica Catholica, publifh'd in 1684; and the expretted by the three Letters a be; the Length by a, Foundation of fuch Conftruttions, with the Application of the Breadth by b, and the Depth by c : fo that by their Algebra to the Quadratures of Curves, Queftions de maxims mutual Multiplication they produce the Solid a be. an d minimis, the Centrobaryc Method of Gu!dinus } 8ic. was As the Multiplication of Dimenfions is expreffed by the gj ven by R. Sh/fius, in 166%; as alfo by Fermat, in his Multiplication of Letters, and as the Number of thofe may opera Mathematical Roberval, in the Mem. de Mathem. be fo great as to become incommodious; the Method is, only gef fo phyjique - and Barrow, in his Left. Geomet. Iq
to write down the Root, and on the right hand to write the Index of the Power, that is, the Number of Letters where- of the Power to be expreffed does confift $ as, a x , a*, a*, a s : the laft. of which fignifies as much as a multiplied five times into it felf 5 and fo of the reft. See Power, Root, Exponent, t£c.
For the Symbols, CharaBcrs, &c. ufed in Algebra, with
1708, Algebra was applied to the Laws of Chance and Gaming, by R. de Montmcrt ; and fince by de Moivre, and James Bernoulli.
Thus much for the Progrefs of Algebra. — The Elements of the Art were compiled and publifh'd by Kerfey in 1671 ; wherein the Specious Arithmetick, and the Nature of Equations are largely explain'd, and illuflrated by variety
their Application, &c. fee the Articles Character, Quan- f Examples : The'wholc Subftance of ' Diophantus is here
tity, &c. deliver'd ; and many Things added concerning Mathemati-
For the Method of performing the feveral Operations in AI- CA \ Composition and Refolution, from Ghetaldus. The like
gebra,y£<? Addition, Subtraction, Multiplication,^, has been fince done by Prefiet in 1^04.; and by Ozanam in
As to the Origin of this Art, we are much in the dark.— 1703.— But thefe Authors omit the Application of Algebra
The Invention is ufually attributed to Z)iopbantus, a Greek to Geometry ; which Defect is fupplied by Quifnee in a
Author, who wrote thirteen Books, tho only fix of 'em are ex- French Treatife exprcfly on the Subject, publifh'd in 1704*
tant, firft publifhed by Xylander, in 1575 ; and fince com- am i fHopital'm his Analytical Treatife of the Conic Seg- mented on and improved by Gafper Sachet, of the French Academy ; and fince by M. Fermat.
And yet Algebra feems to have been not wholly unknown
to the antient Mathematicians, long before the Age of 2)io- ■phantus : We fee the Traces, the Effects of it in many Places ; tho, it looks as if they had defignedly concealed it.— Something of it there feems to be in Euclid^ or at leaft in "itheon upon Euclid, who obferves that 'Plato had begun to teach it. — And there are other Inftances of it in 'Pappus, and more in Archimedes and Apollonius.
But the Truth is, the Analyfis ufed by thofe Authors is rather Geometrical than Algebraical ; as appears by the Examples thereof which we find in their Works : So that we make no fcruple to fiiy, that e Diophantus is the firft, and only Author among the Greeks who has treated of Algebra profeffedly.
This Art, however, was in u(e among the Arabs much earlier than among the Greeks. And 'tis faid the Arabs too borrow'd it from the Perfians, and the Perflans from the Indians. — 'Tis added, that the Arabs carried it into Spain ; whence, fome are of opinion, it pafs'd into Eng- land, befor.e "Diophantus was known among us.
The firft who wrote on the Subject in this part of the World, was Lucas Pacciolus, or Lucas de Burgos, a Cor- delier ; whofe Book, in Italian, was printed at Venice in 14.94. — This Author makes mention of one Leonardus Pi- fanus, and fome others, of whom he had learnt the Art ; but we have none of their Writings. — He adds, that Algebra came originally from the Arabs j and never mentions Dio- phantus : which makes it probable, that that Author was not yet known in Europe. — Flis Algebra goes no further than Simple and Quadratick Equations. See Quadratic, He.
After Pacciolus appear 'd Stifclius, a good Author ; but neither did he advance any further.
After him, came Scipio Ferreus, Cardan* Tartalea, and fome others ; who reach'd as far as the Solution of fome Cu- bick Equations. — Bombelli follow'd thefe, and went himfelf a little further,— At laft came Nonnius, Ramus, Schoner, Saliguac, Clavius, Sic. who all of them took different Cour- fes, but none of them went beyond Quadraticks.
About the fame time, jOiophantus was firft madepublick j whofe Method is very different from that of the Arabs, which had been follow'd till then.
In 1590, Vleta enter'd on the Stage, and introdue'd what he calfd his Specious Arithmetick, which confifts in denoting
tions, in 1707. — The Rules of Algebra are alfo compendi- oufly deliver'd by Sir /. Newton, in his Arithmetica Uni- verfalis, firft publifh'd in 1707 5 which abounds in choice Examples, and contains feveral Rules and Methods invent- ed by the Author.
Algebra ^has been alfo applied to the Confideration and Calculus of Infinites; from whence a new and very extenfive Branch of Knowledge has arofe, call'd the UoElrine of Fluxions, wc Analyfis of Infinites, or the Calculus Differev.-
tialis. See Fluxions. .The Authors on this Subject, fee
under the Article Analysis.
ALGEBRAICAL, fomctbing that relates to Algebra. See Alceur a.
InthisSenfe, we fay, Algebraical Characters, or Symbols. See Character.
Algebraical Curve, is a Curve, wherein the Relation of the Abfciffes to the Semiordinates, may be defined by an Algebraical Equation. Sec Curve.
Thefe are alfo called Geometrical Lines. See Geome- trical Lines.
Algebraical Curves ftand contradiftinguifh'd to Mechani- cal or Tranfcendental ones. See Mechanical, and Tran- scendental.
Algebraical Solution. See Resolution.
ALGENEB, in Aftronomy, a Fixed Star of the fecond Magnitude, on the right fide of Perfeus— Its Longitude, La- titude, gefc. fee among the rcfl of the Conftcllation Perseus.
ALGOL, or Mcdufa\ Head, a Fixed Scar of the third
Magnitude, in the Conftellation Perfeus. Its Longitude,
Latitude, &c. fee under the Article Perseus.
ALGORISM, a Term ufed by fome Arabick Authors for the practical Operation of the feveral Parts of Specious A-
rithmetick, or Algebra. Ses Algebra.- Sometimes it
is alfo ufed for the Practice of common Arithmetick, by the ten numeral Figures. See Arithmetick.
ALGORITHM, an Arabic Term, which fome Authors, and especially the Spaniards, make ufe of to fignify uie Doctrine of Numbers. See Numeer.
Algorithm is properly the Art of numbering truly, w& readily; and comprehends the fix Rules of common Arith- metick.— It is fometimes called Ltgiflica Nunieralis. Sec Arithmetics, Rule, &c.
In this Senfe, we fay, the Algorithm of Integers, the Algorithm of Fractions, the Algorithm of Surds, i$c $™ Fraction, Surp, &c.
4 ALGUA-
