with clearness, to another, those truths which are completely understood by the instructer. The works of the ancient geometers, which have escaped the ravages of the barbarous ages, are mostly synthetical. The Elements of Euclid, so well known in our modern schools of geometry, furnish, probably, the most complete specimen of the ancient synthesis extant. And it is a remarkable fact in the history of science, that the geometry of Euclid, though written two thousand years ago, and some time anterior to any other mathematical tract that has reached us, is still one of the best elementary works on that subject, which has ever appeared. Analysis, or decomposition, on the other hand, assumes, as known, the proposition which is to be examined; or, as already effected the solution which is to be made, and thence proceeds to examine the consequences necessarily resulting from such supposition, until, in case of a theorem, a conclusion is attained, the truth or falsehood of which is already known, whence the correctness or absurdity of the supposition becomes established; or, in case of a problem, such relations are determined as prove the possibility or impossibility of the solution. In synthesis, observes Montucla, we proceed from the known to the unknown, from the trunk to the branches; in analysis, we proceed from the unknown to to the known, from the branches to the trunk. The analytical method is frequently indispensable, when new problems are to be solved, or new theorems investigated. No doubt, many propositions which the ancients have transmitted to us in the synthetical form, owe their discovery to analysis. Of this method of procedure, the mathematical collections of Pappus, and the work, De Sectiones Rationis, of Appollonius, furnish the principal specimens which the ancient geometers have left us. In the ancient geometry, the magnitudes under consideration, were mostly presented to the mind through the medium of representations, as similar as practicable to their antitypes. In some instances, however, this analogy was entirely abandoned, as in the fifth book of Euclid's Elements, where right lines are the only repre sentatives used, yet the reasoning is so conducted, as to be equally applicable to magnitudes of every kind, and even to abstract numbers. In this instance, we may discover a commencement of that species of generalization, which forms so conspicuous a feature in our modern mathematics. The ancient analysis furnished the germ of that branch of mathematics, which, in the hands of the moderns, has become the great master key to all the rest; but it does not appear to have assumed the character of a distinct science, till after the commencement of the Christian era. The earliest writer, in whose works the science of algebra is distinctly seen, was Diophantus, a mathematician of the Alexandrian school. The time in which he lived is not precisely known, but it was not later than the fourth century, as the daughter of Theon, the amiable and accomplished Hypathia, who died about the beginning of the fifth, wrote a commentary on his works. Whether Diophantus was the inventor, or only an improver of algebra, cannot now be known; the latter supposition, however, is the more probable, as the science, in his hands, exhibits a degree of maturity, which it can hardly be supposed to have attained in the first period of its existence. A part only of the original work of Diophantus is now to be found; in this he does not explain the first principles of the science, but teaches the solution of a great variety of difficult questions, in that branch of the subject, now called, from him, the Diophantine Algebra, or the indeterminate analysis, applied to equations of the higher orders. The various questions, if original, which he has formed, and the address with which he has conducted their solutions, necessarily inspire his readers with a high opinion of his invention and discernment. His work, written in the original Greek, was discovered in the Vatican Library, about the middle of the sixteenth century. Though the science of algebra appears to have originated among the Greeks of the Alexandrian school, the inhabitants of western Europe derived their knowledge of it from the Arabs, who are by some supposed to have been its inventors. Dr. Wallis observes, that they differ essentially from Diophantus in their manner of expressing the powers. The Greek analyst calls the 2d, 3d, 4th, 5th 6th, &c. powers, the square, cube, squared square, squared cube, cubed cube, &c.; each power being designated by the two inferior powers of which it is the product. But the Arabian algebraists denominate the 5th power the first sursolid, the 6th the squared cube; being the square of the cube, and not the product of a square and cube; the 7th the second sursolid, and so of the other powers. Hence, he infers, that the Greek and Arabian analyses were not derived from a common source. With due regard for the opinion of this eminent scholar, it appears quite as rational to suppose, that the Arabian mathematicians may have borrowed from their tutors, the Greeks, this branch, with the rest of the mutilated sciences, and adopted, in the denomination of their powers, a mode of expression peculiar to themselves, as to believe, that while the other parts of Grecian science were sought with avidity, this was permitted to sleep amidst the dust of neglected libraries, till the same thing had been reinvented by a people much less advanced in scientific knowledge, and less remarkable for invention than their predecessors. In the history of scientific discoveries, there always appears a regular dependence in the successive stages. The discoveries of one age are the results of those made in the former. Logarithms were invented, when the discoveries in astronomy and trigonometry had rendered their use indispensable. The discoveries of Newton could not have been made, even by that gigantic genius, in the time of Copernicus; nor could Columbus have led his trembling companions across the Atlantic, before the invention of the mariners' compass. Whether the science of algebra was invented by the Arabs, or borrowed from the Greeks, the name is unquestionably of Arabic origin. The names given by the Arabs, for they used a plurality, were algebra v'almucabala. These words, according to Lucas de Burgo, sig nify restauratio et oppositio, restoration or rebuilding, and opposition. Golius defines the word gebera, or giabera by religavit consolidavit, it bound or consolidated; and mocabulat, by comparatio, oppositio, comparison, opposition. By these words they probably designed to indicate the general objects of the science. The quantity whose value is sought, is commonly interwoven with, or bound to other quantities, in such a manner as to form one or more equations, or comparisons of quantities set in opposition to each other. These equations are then transformed, or rebuilt, till the unknown quantity is brought out in opposition to a given or known one. The name almucabala was adopted by some of the Italian writers, and it is thus designated in some of the works of Cardan; but the term algebra appears now irrevocably fixed upon it. The most ancient authors on algebra among the Arabs, are Mohammed ben Musa, and Thebit ben Corah. The former is described by Cardan as the inventor of the method of solving equations of the second degree, a discovery in which he was certainly anticipated by the mathematicians of the Alexandrian school. From the title given to his book, it has been inferred, that he flourished during the reign of Almamon, or in the early part of the ninth century. Whether the Arabian algebraists proceeded beyond the solution of equations of the second degree, is an unsettled question. An accurate knowledge of the mathematical sciences, is seldom combined, in the same individual, with an extensive acquaintance with Arabic literature, and therefore, little is certainly known on this subject. The Bodleian Library in England, and that of the Escurial in Spain, are said to possess a great number of Arabic works on the subject of algebra. Leonard of Pisa, who lived near the beginning of the thirteenth century, impelled by a thirst for mathematical knowledge, travelled into Arabia and other parts of the east, and on his return, first communicated to his countrymen the science of algebra. I do not find that any of his writings on that subject have ever been published. In a treatise upon trigonometry, by Regiomontanus, of Franconia, written about the year 1464, some problems are solved by algebra, in which he refers to the rules, as though commonly known. But the earliest European author whose works, specially on this subject, have been published, was Lucas de Burgo, before mentioned. He was a franciscan, who travelled in the east, either in pursuit of knowledge, or for some purpose not now known, and after his return, taught mathematics at Naples, Venice and Milan. His work, in which the rules of algebra are laid down, was first printed in 1494. In this the science appears very far below our modern algebra. The rules for the solution of adfected quadratic equations, are given in semibarbarous Latin verse, and the different cases separately treated. His solutions do not rise to equations above the second degree. The solution of cubic equations appears to have been first effected by Scipo Ferrei, professor of mathematics at Bologna, about the beginning of the sixteenth century. This solution, however, included but one case, namely, that in which the first and third powers only, of the unknown quantity, were involved. Some questions, including this case, being afterwards proposed to Nicholas Tartaglia, an eminent mathematician of Brescia, he discovered a general solution of equations of the third degree. This discovery he communicated to Jerome Cardan, a physician of Milan, under an injunction of secrecy. Cardan, notwithstanding, having found the demonstration, published it in his work, De Arte Magna, in 1545. This is now commonly called the method of Cardan. Cardan first remarked the plurality of roots in quadratic equations, and their distinction into positive and negative. The solution of equations of the fourth degree was accomplished soon after the discovery of Tartaglia, by Lewis Ferrari, a pupil and coadjutor of Car dan. |