(B) HISTORY OF ALGEBRA
(v) Lucas de Burgo (Lucas Paciolus) (1494)
The earliest printed book on algebra was composed by Lucas Paciolus, or Lucas de Burgo, a minorite friar. It was first printed in 1494, and again in 1523. The title is Summa de Arithmetica, Geometria, Proportioni, et Proportionalita.
This is a very complete treatise on arithmetic, algebra, and geometry, for the time in which it appeared. The author followed close on the steps of Leonardo; and, indeed, it is from this work that one of his lost treatises has been restored.
Lucas de Burgo's work is interesting, inasmuch as it shows the state of algebra in Europe about the year 1500; probably the state of the science was nearly the same in Arabia and Africa, from which it had been received.
The power of algebra as an instrument of research is in a very great degree derived from its notation, by which all the quantities under consideration are kept constantly in view; but in respect of convenience and brevity of expression, the algebraic analysis in the days of Lucas de Burgo was very imperfect: the only symbols employed were a few abbreviations of the words or names which occurred in the processes of calculation, a kind of short-hand, which formed a very imperfect substitute for that compactness of expression which has been attained by the modern notation.
The application of algebra was also at this period very limited; it was confined almost entirely to the resolution of certain questions of no great interest about numbers. No idea was then entertained of that extensive application which it has received in modern times.
The knowledge which the early algebraists had of their science was also circumscribed: it extended only to the resolution of equations of the first and second degrees; and they divided the last into cases, each of which was resolved by its own particular rule. The important analytical fact, that the resolution of all the cases of a problem may be comprehended in a single formula, which may be obtained from the solution of one of its cases, merely by a change of the sings, was not them known: indeed, it was long before this principle was fully comprehended. Dr. Halley expresses surprise, that a formula in optics which he had found, should by a mere change of the signs give the focus of both converging and diverging rays, whether reflected or refracted by convex or concave specula or lenses; and Molyneux speaks of the universality of Halley's formula as something that resembled magic.
The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in some cases the two sciences may serve to illustrate each other, there is not now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former. It was otherwise in former times. Lucas de Burgo found it to be convenient, after the example of Leonardo, to employ geometrical constructions to prove the truth of his rules for resolving quadratic equations, the nature of which he did not completely comprehend; and he was induced by the imperfect nature of his notation to express his rules in Latin verses, which will not now be read with the kind of satisfaction we receive from the perusal of the well-known poem, "The Loves of the Triangles."
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