Algebra

"Finally, all other properties of curves depend only on the angles which these curves make with other lines. But the angle formed by two intersecting curves can be as easily measured as the angle between two straight lines, provided that a straight line can be drawn making right angles with one of these curves at the point of intersection with the other. This is the reason for my believing that I shall have given here a sufficient introduction to the study of curves when I have given a general method of drawing a straight line making right angles with a curve at an arbitrarily chosen point upon it. And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know."

"[W]e have no possessed Treatise of it... ancienter than that of Diophantus, first published... by Xylander, and since... by Bachetus, with divers additions of his own; and Re-printed lately with some Additions of Monsier Fermat."

"That it was of ancient use also among the Arabs, we have reason to believe, (and perhaps sooner than amongst the Greeks;) which they are supposed to have received (not from the Greeks, but) from the Persians, and these from the Indians."

"From the Arabs (by means of the Saracens and Moors) it was brought into Spain, and thence to England (together with the use of the Numeral Figures, and other parts of Mathematical Learning, and particularly the Astronomical,) before Diophantus seems to have been known amongst us: And from those we have the name Algebra."

"The use of Numeral Figures (which... the Greeks had not) was a great advantage to the improvement of Algebra."

"The use of these Numeral Figures hath received two great Improvements. The one is the Decimal Parts, which seems... introduced by ', in his Trigonometrical Canons, about the year 1450; but much advance in the last present Century, by Simon Stevin, and Mr. Briggs, &c. And this is much to be preferred before Ptolemys Sexigesimal way..."

"The other improvement is that of Logarithms and other Trigometrical Calculations; introduced by Lord Neper, and perfected by Mr. Briggs... And these things, though they be not properly Parts of Algebra, are yet of great advantage in the practice of it."

"The first printed Author which treats of Algebra is Lucas Pacciolus, or Lucas de Burgo... printed in Venice in the year 1494, (soon after the first Invention of Printing,)... But he therein mentions Leonardus Pisanus, and divers others more ancient than himself, from whom he Learned it; but whose works are not now extant."

"This Fryer Lucas,in his Summa Arithmetica & Geometrica, (for he hath other Works extant) hath a very full Treatise of Arithmetick in all the parts of it; in Integers, Fractions, Surds, Binomials; Extraction of Roots, Quadratick, Cubick, &c. and the several Rules of Proportion, Fellowship, about Accompts, Alligation, and False Position, (so fully, that very little hath been thereunto added to this day:) And (after all this) of Algebra, with the Appurtenances thereunto, (as Surd Roots, Negative Quantities, Binomials, Roots Universal, the use of the Signs Plus, Minus, or + \; -, &c.) as far as Quadratick Equations reach, but no farther."

"And this he tells us was derived from the Arabs, (to whom we are beholden for this kind of Learning,) without taking notice of Diophantus (or any other Greek Author) who it seems was not known here in thoſe days."

"Afterwards Scipio Ferreus, Cardan, Tartalea, and others, proceeded to the Solution of (some) Cubick Equations."

"And Bombelli goes yet farther, and shews how to reduce a Biquadratick Equation (by the help of a Cubick) to two Quadraticks."

"And Nonnius or Nunnez... Ramus, Schonerus, [Bernardus or Bernhard] Salignacus, Clavius, and others... Record, Digs, and some others of our own... did (in the last Century) pursue the same Subject, in different ways; but (for the most part) proceeded no farther than Quadratick Equations."

"In the mean time, Diophantus... was made publick; whose method differs much from that of the Arabs (whom those others followed, ) and particularly in the order of denominating the Powers; as taking no notice of Sursolids, but using only the names of Square and Cube, with the Compounds of these. And hitherto no other than the unknown Quantities were wont to be denoted in Algebra by particular Notes or Symbols; but, the known Quantities, by the ordinary Numeral Figures."

"The next great step, for the improvement of Algebra, was that of Specious Arithmetick, first introduced by Vieta about the Year 1590. This Specious Arithmetick, which gives Notes or Symbols (which he calls Species) to Quantities both known and unknown, doth (without altering the manner of demonstration, as to the substance,) furnish us with a short and convenient way of Notation; whereby the whole process of many Operations is at once exposed to the Eye in a short Synopsis. By the help of this he makes many Discoveries, in the process of Algebra, not before taken notice of. He introduceth also his Numeral Exegesis, of affected Equations, extracting the Roots of these in Numbers."

"The method of Vieta is followed, and much improved, by Mr. Oughtred in his Clavis [Mathematicae] (...1631.) and other Treatises of his; and he doth, therein, in a brief compendious method, declare in short, what had before been the Subject of large Volums: And doth, in few small pieces of his, give us the Substance and Marrow of all (or most of) the Ancient Geometry."

"Mr. Harriot was contemporary with Mr. Oughtred (but elder...) and left many good things behind him in writing. Of which there is nothing hitherto made publick, but only his Algebra or Analytice... published by Mr. Warner... in... 1631. He alters the way of Notation, used by Vieta and Oughtred, for another more convenient. And he hath also made a strange improvement of Algebra, by discovering the true construction of Compound Equations, and how they be raised by a Multiplication of Simple Equations, and may therefore be resolved into such. ...In sum, He hath taught (in a manner) all that which hath since passed for the Cartesian method of Algebra; there being scarce any thing of (pure) Algebra in Des Cartes, which was not before in Harriot; from whom Des Cartes seems to have taken what he hath (that is purely Algebra) but without naming him. But the Application thereof to Geometry, or other particular Subjects, (which Des Cartes pursues,) is not the business of that Treatise of Harriot..."

"After this follows an account of Dr. Pell’s method, who hath a particular way of Notation, by keeping a Register (in the Margin) of the several Steps in his Demonstrations, with References from one to another."

"They who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation."

"There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted."

"Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established."

"Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world."

"Playfair... though he cannot condemn Cardan, seems to think Tartaglia rightly treated for concealing his discovery; and others have echoed this strain. Tartaglia himself says... that he meant to have divulged it ultimately; but, in that age, money as well as credit was to be got by keeping a secret: and those who censure him wholly forget that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility in the world."

"Ars Magna, published in 1545... contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases. Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta."

"Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists."

"Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna."

"If his [Diophantus'] works were not written in Greek, no one would think for a moment that they were the product of Greek mind. There is nothing in his works that reminds us of the classic period of Greek mathematics. His were almost entirely new ideas on a new subject. In the circle of Greek mathematicians he stands alone in his specialty. Except for him, we should be constrained to say that among the Greeks algebra was almost an unknown science."

"He [Diophantus] appears to be the first who could perform such operations as (x - 1) \times (x - 2) without reference to geometry. Such identities as (a + b)^2 = a^2 + 2ab + b^2, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation."

"In this work [' of Diophantus] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry."

"The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa."

"In the Greek geometry the idea of motion was wanting but with Descartes it became a very fruitful conception. ...This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"A great part of the progress of formal thought... has been due to the invention of what we may call stenophrenic, or short-mind, symbols. These... disengage the mind from the consideration of ponderous and circuitous mechanical operations and economise its energies for the performance of new and unaccomplished tasks of thought. And the advancement of those sciences has been most notable which have made the most extensive use of these... Here mathematics and chemistry stand pre-eminent. The ancient Greeks... even admitting that their powers were more visualistic than analytic, were yet so impeded by their lack of short-mind symbols as to have made scarcely any progress whatever in analysis. Their arithmetic was a species of geometry. They did not possess the sign for zero, and also did not make use of position as an indicator of value. ...The historical calculations of Archimedes, his approximation to the value of π, etc., owing to this lack of appropriate... symbols, entailed enormous and incredible labors, which, if they had been avoided, would... have led to [even] great[er] discoveries."

"Then followed the introduction of exponents by Descartes, the representation of geometrical magnitudes by algebraical symbols, the extension of the theory of exponents to fractional and negative numbers by Wallis... and other symbolic artifices, which rendered the language of analysis as economic, unequivocal, and appropriate as the needs of the science appeared to demand."

"For the development of science all such short-mind symbols are... paramount... and seem to carry within themselves the germ of a perpetual mental motion... for its unfoldment. Euler's well-known saying that his pencil seemed to surpass him in intelligence finds its explanation here, and will be understood by all who have experienced the uncanny feeling attending the rapid development of algebraical formulae, where the urned thought of centuries... rolls from one's finger's ends."

"[T]he mighty stenophrenic engine of which we here speak, like all machinery, affords us rather a mastery over nature than an insight into it; and for some, unfortunately, the higher symbols of mathematics are merely brambles that hide the living springs of reality."

"We have been following here, briefly and roughly, a line of progressive abstraction and generalisation... the process reached... its culmination and purest expression in Joseph Louis Lagrange... Lagrange's power over symbols has, perhaps, never been paralleled either before his day or since. ...His was a time when geometry, as he himself phrased it, had become a dead language, the abstractions of analysis were being pushed to their highest pitch, and he felt that with his achievements its possibilities within certain limits were being rapidly exhausted."

"When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the science was known about 1800 B.C., and probably still earlier."

"The first writer on algebra whose works have come down to us is . He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical."

"There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; , whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic... Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)... extends the work through quadratic equations."

"It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In 's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of notation. The rules given by Liu Hui form a kind of rhetorical algebra. The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving s. ...Sun-tzï solved such problems by analysis and was content with a single result... The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations. Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should... By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625). The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819)."

"With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra..."

"Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations. The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence. The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work... to appear in print was the General Trattato of Tartaglia..."

"The first noteworthy attempt to write an algebra in England was made by , whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde. The first Italian textbook to bear the title of algebra was Bombelli's work of 1572. By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675). So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century."

"Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due."

"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations."

"Greek algebra before Diophantus was essentially rhetorical."

"Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries."