ERRAT A. PAGE 7 To the note at the bottom of the page add, "The sign ". denotes therefore." = 18 Last line, for the difference, read d=the difference. 44 Line 8, for n-1 read n- -1. Line 20, for n.n—1.n—2.n—3.n—4, read n.n-1.n-2.n-3.n-4. 64 Line 3, for Fo since, read For since ad. 66 Art. 68, after the word CONVERTENDO, add, Euclid pr. E. Book 5. 71 The note at the bottom is useless here, as it occurs in the latter part of The Properties of Numbers, pp. 108, 109. 97 Last line, for b=1.q, read b+1.q. 123 Dele the third and five following lines. 252 Art. 15. line 2, dele “or simple." 320 The three lines AG, BD, and EC in the figure, should intersect in the point Fon the circumference. Two or three of the figures in Part X. are very indifferently cut, but it is hoped that there is nothing which can possibly mislead, or affect the demonstrations. AN EASY INTRODUCTION TO THE MATHEMATICS, &c. PART IV. ALGEBRA. GENERAL PROBLEMS. ART. 1. ALGEBRA is divided into two kinds, numeral and literal, both depending on the same principles and employing the same operations. 2. Numeral algebra is that chiefly used in the solution of numeral problems, in which all the given quantities are expressed by numbers, the unknown quantities only being denoted by letters or other convenient symbols. This kind of algebra has been largely treated of in the preceding volume. 3. Literal or specious algebra is that in which all the quan b * Numeral algebra is that part of the science, which the Europeans received from the Arabs, about the middle of the 15th century. It does not appear that the latter people, or even Diophantus, (who is the only Greek writer on the subject at present known,) understood any thing of the general methods now in use; accordingly we find but little attempted beyond the solution of numerical problems, in the writings of Lucas de Burgo, Cardan, Diophantus, Tartalea, Bombelli, Peletarius, Stevinus, Recorde, or any other of the early authors who treated on algebra. Vieta, the great improver of almost every branch of the Mathematics B tities, both known and unknown, are represented by letters and other general characters. This general mode of designation is of the greatest use; as every conclusion, and indeed every step by which it is obtained, becomes an universal rule for performing every possible operation of the kind. 4. In literal algebra, the initial letters a, b, c, d, &c. are usually employed to represent known or given quantities, and the final letters x, y, z, w, v, &c. to represent unknown quantities, whose values are required to be found. 5. A general algebraic problem is that in which all the quantities concerned, both known and unknown, are represented by letters or other general characters. Not only such problems as have their conditions proposed in general terms, are here implied, every particular numeral problem may be made general, by substituting letters for the known quantities concerned in it: when this is done, the problem which was originally proposed in a particular form, is now become a general problem. 6. Every problem consists of two parts, the data, and the quæsita; the data include all the conditions and quantities given, and the quæsita the quantities sought. 7. The process by which the quæsita are obtained by means of the data, that is, by which the values of the unknown quantities are found, is called the ANALYSIS, or the ANALYTICAL known in his time, is considered as the first who introduced the literal notation of given quantities into general practice, about the year 1600. Cardan had indeed given specimens of such an improvement, in his algebra, as early as 1545; but as the advantages of a general mode of notation were then in all probability not sufficiently understood, the method was not adopted until about the time we have mentioned. The improvement of Vieta was further advanced and applied by Thomas Harriot, the father of modern algebra, about 1620; likewise by Oughtred in 1631, Des Cartes in 1637, and afterwards by Wallis, Newton, Leibnitz, the Bernoullis, Baker, Raphson, Sterling, Euler, &c. and is justly perferred by all modern algebraists, on account of the universality of its application. The letters of the alphabet are called by Vieta, species, whence algebra has been named arithmetica speciosa: reasoning in species, as applied to the solution of mathematical problems, appears to have been borrowed from the Civilians, who determine cases at law between imaginary persons, representing them abstractedly by A and B; these they call cases in species: this is the more probable, as Vieta himself was a lawyer. The word data means things given, and quæsita things sought. The word analysis, (from the Greek «v«λvw, resolvo,) in its general sense, |