Lectures on Elementary Mathematics

"Algebra is a science almost entirely due to the moderns... for we have one treatise from the Greeks, that of Diophantus... the only one which we owe to the ancients in this branch of mathematics. ...I speak of the Greeks only, for the Romans have left nothing in the sciences, and to all appearances did nothing."

"His [Diophantus'] work contains the first elements of this science [algebra]. He employed to express the unknown quantity a Greek letter which corresponds to our st and which has been replaced in the translations by N. To express the known quantities he employed numbers solely, for algebra was long destined to be restricted entirely to the solution of numerical problems."

"[H]e uses the known and the unknown quantities alike. And herein consists virtually the essence of algebra, which is to employ unknown quantities, to calculate with them as we do with known quantities, and to form from them one or several equations from which the value of the unknown quantities can be determined."

"Although the work of Diophantus contains indeterminate problems almost exclusively, the solution of which he seeks in rational numbers,— problems which have been designated after him Diophantine problems, —we nevertheless find in his work the solution of a number of determinate problems of the first degree, and even of such as involve several unknown quantities. In the latter case, however, the author invariably has recourse to... reducing the problem to a single unknown quantity, —which is not difficult."

"He [Diophantus] gives, also, the solution of equations of the second degree, but is careful so to arrange them that they never assume the affected form containing the square and the first power of the unknown quantity. ...he always arrives at an equation in which he has only to extract a square root to reach the solution..."

"Diophantus... does not proceed beyond equations of the second degree, and we do not know if he or any of his successors... ever pushed... beyond this point."

"Diophantus regarded the rule of the signs [the principle that in multiplication, + and -, give -; and - and -, give +] as a self-evident principle not in need of demonstration. ...[H]e is... likely to have considered it as an axiom, as did Euclid some of the principles of geometry."

"Diophantus was not known in Europe until the end of the sixteenth century, the first translation having been a wretched one by Xylander made in 1575. Bachet de Méziriac... a tolerably good mathematician for his time, subsequently published (1621) a new translation... accompanied by lengthy commentaries, now superfluous. Bachet's translation was afterwards reprinted with observations and notes by Fermat [1670]."

"Prior to the discovery and publication of Diophantus... algebra had already found its way into Europe. Towards the end of the fifteenth century [1494] there appeared in Venice a work by... Lucas Paciolus on arithmetic and geometry in which the elementary rules of algebra were stated."

"[T]he Europeans, having received algebra from the Arabs, were in possession of it one hundred years before the work of Diophantus was known to them. They made, however, no progress beyond equations of the first and second degree."

"In the work of Paciolus... the general resolution of equations of the second degree... was not given. We find in this work simply rules, expressed in bad Latin verses, for resolving each particular case according to the different combinations of the signs of the terms of equation, and even these rules applied only to the case where the roots were real and positive. Negative roots were still regarded as meaningless and superfluous."

"It was geometry really that suggested to us the use of negative quantities, and herein consists one of the greatest advantages that have resulted from the application of algebra to geometry, —a step which we owe to Descartes."

"In the subsequent period the resolution of equations of the third degree was investigated and the discovery for a particular case ultimately made by... Scipio Ferreus (1515). ...Tartaglia and Cardan subsequently perfected the solution of Ferreus and rendered it general for all equations of the third degree."

"At this period, Italy, which was the cradle of algebra in Europe, was still almost the sole cultivator of the science, and it was not until about the middle of the sixteenth century that treatises on algebra began to appear in France, Germany, and other countries."

"The works of Peletier [1554 ] and Buteo [i.e., Jean Borrel, who published the algebraic text, Logistica (1559)] were the first which France produced in this science..."

"Tartaglia expounded his solution in bad Italian verses in a work treating of divers questions and inventions printed in 1546, a work which enjoys the distinction of being one of the first to treat of modern fortifications by s."

"Cardan published [1545] his treatise Ars Magna, or Algebra... Cardan was the first to perceive that equations had several roots and to distinguish them into positive and negative. But he is particularly known for having first remarked the so-called irreducible case in which the expression of the real roots appears in an imaginary form. Cardan convinced himself from several special cases in which the equation had rational divisors that the imaginary form did not prevent the roots from having a real value. But it remained to be proved that not only were the roots real in the irreducible case, but that it was impossible for all three together to be real except in that case. This proof was afterwards supplied by Vieta, and particularly by Albert Girard, from considerations touching the trisection of an angle."

"[T]he irreducible case of equations of the third degree... presents a new form of algebraical expressions which have found extensive application in analysis... it is constantly giving rise to unprofitable inquiries with a view to reducing the imaginary form to a real form and... it thus presents in algebra a problem which may be placed upon the same footing with the famous problems of the duplication of the cube and the squaring of the circle in geometry."

"The mathematicians of the period under discussion were wont to propound to one another problems for solution. These... were... public challenges and served to excite and to maintain that fermentation which is necessary for the pursuit of science. The challenges... were continued down to the beginning of the eighteenth century [in] Europe, and really did not cease until the rise of the Academies which fulfilled the same end... partly by the union of the knowledge of their various members, partly by the intercourse which they maintained... and... by the publication of their memoirs, which served to disseminate the new discoveries and observations..."

"[T]he resolution of equations of the fourth degree... was propounded in the following problem. To find three numbers in continued proportion of which the sum is 10, and the product of the first two [is] 6. ...[W]e get finally. y^4 + by^2 - aby + b^2 = 0According to Bombelli... Louis Ferrari of Bologna resolved the problem... dividing the equation into two parts both of which permit of the extraction of the square root."

"The Algebra of Bombelli [1572, 1579] contains not only the discovery of Ferrari but also divers other important remarks on equations of the second and third degree and particularly on the theory of radicals by means of which the author succeeded in several cases in extracting the imaginary cube roots of the two binomials of the formula of the third degree in the irreducible case, so finding a perfectly real result... the most direct proof possible of the reality of this species of expressions."

"The solution of equations of the third and fourth degree was quickly accomplished. But the successive efforts of mathematicians for over two centuries have not succeeded in surmounting the difficulties of the equation of the fifth degree."

"Yet these efforts are far from having been in vain. They have given rise to the many beautiful theorems... on the formation of equations, on the character and signs of the roots, on the transformation of a given equation into others of which the roots may be formed at pleasure from the roots of the given equation, and finally, to the beautiful considerations concerning the metaphysics of the resolution of equations from which the most direct method of arriving at their solution, when possible, has resulted."

"Vieta and Descartes... Harriot... and Hudde... were the first after the Italians... to perfect the theory of equations, and since their time there is scarcely a mathematician of note that has not applied himself..."

"As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics."

"The method... for finding and demonstrating divers general properties of equations by considering the curves which represent them, is a species of application of geometry to algebra... [T]his method has extended applications, and is capable of readily solving problems whose direct solution would be extremely difficult or even impossible... [T]his subject... is not ordinarily found in elementary works on algebra."

"[A]n equation of any degree can be resolved by means of a curve, of which the abscissæ represent the unknown quantity of the equation, and the ordinates the values which the left-hand member assumes for every value of the unknown quantity. ...[T]his method can be applied generally to all equations, whatever their form, and... only requires them to be developed and arranged according to the different powers of the unknown quantity."

"It is simply necessary to bring all the terms of the equation to one side, so that the other side shall be equal to zero. Then taking... the function of the unknown quantity... which forms one side of the equation, for the ordinate y, the curve described by these co-ordinates x and y will give by its intersections with the axis those values of x which are the required roots of the equation."

"[S]ince most frequently it is not necessary to know all possible values of the unknown quantity but only such as solve the problem in hand, it will be sufficient to describe that portion of the curve which corresponds to these roots, thus saving much unnecessary calculation."

"We can even determine in this manner, from the shape of the curve itself, whether the problem has possible solutions satisfying the proposed conditions."