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. a 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 * 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. b Vieta, the great improver of almost every branch of the Mathematics 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 «vaλuw, resolvo,) in its general sense, INVESTIGATION; it is also named the SOLUTION, OF RESOLUTION of the problem. S. When the values of the unknown quantities are found and expressed in known terms, the substituting these values, each for its respective unknown quantity in the given equations; that is, by reasoning in an order the converse of analysis, and thereby ultimately proving that the quantities thus assumed have the properties described in the problem, is called the SYNTHESIS, Or SYNTHETICAL DEMONSTRATION of the problem, and frequently the COMPOSITION. 9. When the value of any quantity, which was at first unknown, is found and expressed in known terms, the translating of this value out of algebraic into common language, wherein the relation of the quantities concerned is simply declared, is called deducing a THEOREM '; but if the translation be exhibited in the form of a precept, it is called a CANON ", or rule. implies the resolving of any thing which is compounded, into its constituent simple elements: thus in algebra, several quantities, known and unknown, being compounded together, analysis is the disentangling of them; by its operation, each of the quantities included in the composition is disengaged from the rest, and its value found in terms of the known quantities concerned. This being the proper business of algebra, the science itself on that account is frequently termed analysis, which name however implies other branches besides algebra. * Synthesis (from the Greek avdeσıç, compositio) is the converse of analysis. By analysis, as we have shewn, compound quantities are decompounded; by synthesis, the quantities disentangled and brought out by the analysis, are again compounded, by which operation the original compound quantity is reproduced; hence synthesis is called the method of demonstration, and analysis the method of investigation. f A theorem (from the Greek twgenua, a speculation) is a proposition terminating in theory, in which something is simply affirmed or denied. Theorems, as we have observed before, are investigated or discovered by analysis, and their truth demonstrated by synthesis. 8 A canon (from the Greek xavw) or rule (from the Latin regula) is a system of precepts directing what operations must be performed, in order to produce any proposed result; such are the rules of common arithmetic. It is noticed above, that a theorem, and a canon, are of nearly the same import, differing only in the form of words in which they are laid down the distinc tion may appear trifling, but it is observed by writers, whose skill and judgment are unquestionable, and on that account we thought proper not entirely to omit it. 10. A COROLLARY is a truth obtained intermediately, and by the bye; an additional truth, over and above what the problem proposed to search out, or prove. 11. A SCHOLIUM is a remark or explanatory observation, intended to illustrate something preceding. 12. To make what has been delivered perfectly plain, to the analytical investigation of several of the following problems, is added the synthetical demonstration; instances are given of deducing theorems and of deriving canons or rules from the analysis; examples are likewise proposed, where necessary, to shew the method of applying the general conclusions to particular cases; and finally, the manner of converting any particular numerical problem into a general form, and of substituting and deriving expressions for the unknown quantities, in a great variety of ways, are shewn and explained. PROBLEM 1'. Given the sum and difference of two magnitudes, to find the magnitudes. ANALYSIS. Let x=the greater magnitude, y=the less, s= the given sum, d=the given difference. Then by the problem x+y=s. And x-y=d. The term corollary is derived from the Latin corollis, something given over and above; and scholium from exodov, a short comment. iSeveral of the problems here given, with others of the kind, may be found in Saunderson's Elements of Algebra, 2 vol. 4to. 1740. in the Abridgment of the same, and in Ludlam's Rudiments of Mathematics. * In the technical language of the mathematicians, Q. E. I. denotes, quod erat investigandum, which was to be investigated; Q. E. D. quod erat demonstrandum, which was to be demonstrated; and Q. E. F. quod erat faciendum, which was to be done. The first is subjoined to analytical investigations, the second to synthetical demonstrations, and the third to the proof that a proposed practical operation is actually performed and done. We have adopted the distinctions of analysis, synthesis, theorem, canon, &c. and likewise the above abbreviations in a few instances, to assist the learner in a know, ledge of their use, when any book containing them may happen to fall into bis bands, SYNTHESIS. Because by the problem x+y=s, and x-y=d, if the values found by the analysis be really equivalent to x and y respectively, then those values being substituted for x and y in the given equations, and the latter value added to the former in the first équation, and subtracted from it in the second, the results will be s and d. Let us make the experiment. s+d s-d First +. namely that x+y=s, =s, which answers the first condition, 2 2 dition, namely that x-y=d; wherefore the values of x and y found by the analysis, are those which the problem requires. Q. E. D. THEOREM 1. If the difference of any two magnitudes be added to their sum, half the result will be the greater magnitude; but if the difference be subtracted from the sum, half the result will be the less. SCHOLIUM. The form of any general algebraic expression may be changed at pleasure, provided its value be not altered thereby by this means a theorem may sometimes be laid down in a more convenient form than that derived immediately from the analysis. The value of r found above, viz. S d expressed, +; and the value of y, viz. s+d -may be thus hence we obtain the above theorem in a more convenient form, viz. THEOREM 2. Half the difference of two magnitudes being added to half their sum, the result will be the greater; and half the difference being subtracted from half the sum, the result will be the less. COROLLARY. Hence it appears, that theorems and canons may be derived from any general algebraic investigation, which will solve every particular case subject to the same conditions with the general problem, to which that investigation belongs. CANON 1. (From theorem 1.) Add the difference of any two magnitudes to their sum, and divide the result by 2, the quotient |