Chapter 14
Origin of the Negative and the Imaginary. The Equation

98. The Equation in Egyptian Mathematics. While the irrational originated in geometry, the negative and the imaginary are of purely algebraic origin. They sprang directly from the algebraic equation.

The authentic history of the equation, like that of geometry and arithmetic, begins in the book of the old Egyptian scribe Ahmes. For Ahmes, quite after the present method, solves numerical problems which admit of statement in an equation of the first degree involving one unknown quantity.⁴³

99. In the Earlier Greek Mathematics. The equation was slow in arousing the interest of Greek mathematicians. They were absorbed in geometry, in a geometry whose methods were essentially non-algebraic.

To be sure, there are occasional signs of a concealed algebra under the closely drawn geometric cloak. Euclid solves three geometric problems which, stated algebraically, are but the three forms of the quadratic; $x^2+ax=b^2$, $x^2=ax+b^2$, $x^2+b^2=ax$.⁴⁴ And the Conics of Apollonius, so astonishing if regarded as a product of the pure geometric method used in its demonstrations, when stated in the language of algebra, as recently it has been stated by Zeuthen,⁴⁵ almost convicts its author of the use of algebra as his instrument of investigation.

100. Hero. But in the writings of Hero of Alexandria (120 b. c.) the equation first comes clearly into the light again. Hero was a man of practical genius whose aim was to make the rich pure geometry of his predecessors available for the surveyor. With him the rigor of the old geometric method is relaxed; proportions, even equations, among the measures of magnitudes are permitted where the earlier geometers allow only proportions among the magnitudes themselves; the theorems of geometry are stated metrically, in formulas; and more than all this, the equation becomes a recognized geometric instrument.

Hero gives for the diameter of a circle in terms of $s$, the sum of diameter, circumference, and area, the formula:⁴⁶

The area of a circle in terms of its diameter being $\frac{\pi d^2}{4}$, the length of its circumference $\pi d$, and $\pi$ according to Archimedes’ approximation $\frac{22}{7}$, we have the equation:

Clearing of fractions, multiplying by 11, and completing the square,

Except that he lacked an algebraic symbolism, therefore, Hero was an algebraist, an algebraist of power enough to solve an affected quadratic equation.

101. Diophantus. (300 a. d.?). The last of the Greek mathematicians, Diophantus of Alexandria, was a great algebraist.

The period between him and Hero was not rich in creative mathematicians, but it must have witnessed a gradual development of algebraic ideas and of an algebraic symbolism.

At all events, in the $\grave{\alpha }\rho \iota 𝜃\mu \eta \tau \iota \kappa \acute{\alpha }$ of Diophantus the algebraic equation has been supplied with a symbol for the unknown quantity, its powers and the powers of its reciprocal to the 6th, and a symbol for equality. Addition is represented by mere juxtaposition, but there is a special symbol, see Figure A, for subtraction. On the other hand, there are no general symbols for known quantities,—symbols to serve the purpose which the first letters of the alphabet are made to serve in elementary algebra nowadays,—therefore no literal coefficients and no general formulas.

With the symbolism had grown up many of the formal rules of algebraic reckoning also. Diophantus prefaces the $\alpha \rho \iota 𝜃\mu \eta \tau \iota \kappa \grave{\alpha }$ with rules for the addition, subtraction, and multiplication of polynomials. He states expressly that the product of two subtractive terms is additive.

The $\alpha \rho \iota 𝜃\mu \eta \tau \iota \kappa \grave{\alpha }$ itself is a collection of problems concerning numbers, some of which are solved by determinate algebraic equations, some by indeterminate.

Determinate equations are solved which have given positive integers as coefficients, and are of any of the forms $a{x}^m=b{x}^n$, $a{x}^2+bx=c$, $a{x}^2+c=bx$, $a{x}^2=bx+c$; also a single cubic equation, $a{x}^3+x=4x^2+4$. In reducing equations to these forms, equal quantities in opposite members are cancelled and subtractive terms in either member are rendered additive by transposition to the other member.

The indeterminate equations are of the form $y^2=a{x}^2+bx+c$, Diophantus regarding any pair of positive rational numbers (integers or fractions) as a solution which, substituted for $y$ and $x$, satisfy the equation.⁴⁷ These equations are handled with marvellous dexterity in the $\alpha \rho \iota 𝜃\mu \eta \tau \iota \kappa \grave{\alpha }$. No effort is made to develop general comprehensive methods, but each exercise is solved by some clever device suggested by its individual peculiarities. Moreover, the discussion is never exhaustive, one solution sufficing when the possible number is infinite. Yet until some trace of indeterminate equations earlier than the $\alpha \rho \iota 𝜃\mu \eta \tau \iota \kappa \grave{\alpha }$ is discovered, Diophantus must rank as the originator of this department of mathematics.

The determinate quadratic is solved by the method which we have already seen used by Hero. The equation is first multiplied throughout by a number which renders the coefficient of $x^2$ a perfect square, the “square is completed,” the square root of both members of the equation taken, and the value of $x$ reckoned out from the result. Thus from $a{x}^2+c=bx$ is derived first the equation

The solution is regarded as possible only when the number under the radical is a perfect square (it must, of course, be positive), and only one root—that belonging to the positive value of the radical—is ever recognized.

Thus the number system of Diophantus contained only the positive integer and fraction; the irrational is excluded; and as for the negative, there is no evidence that a Greek mathematician ever conceived of such a thing,—certainly not Diophantus with his three classes and one root of affected quadratics. The position of Diophantus is the more interesting in that in the $\alpha \rho \iota 𝜃\mu \eta \tau \iota \kappa \grave{\alpha }$ the Greek science of number culminates.

102. The Indian Mathematics. The pre-eminence in mathematics passed from the Greeks to the Indians. Three mathematicians of India stand out above the rest: Âryabhaṭṭa (born 476 a. d.), Brahmagupta (born 598 a. d.), Bhâskara (born 1114 a. d.) While all are in the first instance astronomers, their treatises also contain full expositions of the mathematics auxiliary to astronomy, their reckoning, algebra, geometry, and trigonometry.⁴⁸

An examination of the writings of these mathematicians and of the remaining mathematical literature of India leaves little room for doubt that the Indian geometry was taken bodily from Hero, and the algebra—whatever there may have been of it before Âryabhaṭṭa—at least powerfully affected by Diophantus. Nor is there occasion for surprise in this. Âryabhaṭṭa lived two centuries after Diophantus and six after Hero, and during those centuries the East had frequent communication with the West through various channels. In particular, from Trajan’s reign till later than 300 a. d.  an active commerce was kept up between India and the east coast of Egypt by way of the Indian Ocean.

Greek geometry and Greek algebra met very different fates in India. The Indians lacked the endowments of the geometer. So far from enriching the science with new discoveries, they seem with difficulty to have kept alive even a proper understanding of Hero’s metrical formulas. But algebra flourished among them wonderfully. Here the fine talent for reckoning which could create a perfect numeral notation, supported by a talent equally fine for symbolical reasoning, found a great opportunity and made great achievements. With Diophantus algebra is no more than an art by which disconnected numerical problems are solved; in India it rises to the dignity of a science, with general methods and concepts of its own.

103. Its Algebraic Symbolism. First of all, the Indians devised a complete, and in most respects adequate, symbolism. Addition was represented, as by Diophantus, by mere juxtaposition; subtraction, exactly as addition, except that a dot was written over the coefficient of the subtrahend. The syllable bha written after the factors indicated a product; the divisor written under the dividend, a quotient; a syllable, ka, written before a number, its (irrational) square root; one member of an equation placed over the other, their equality. The equation was also provided with symbols for any number of unknown quantities and their powers.

104. Its Invention of the Negative. The most note-worthy feature of this symbolism is its representation of subtraction. To remove the subtractive symbol from between minuend and subtrahend (where Diophantus had placed his symbol, see Figure A.) to attach it wholly to the subtrahend and then connect this modified subtrahend with the minuend additively, is, formally considered, to transform the subtraction of a positive quantity into the addition of the corresponding negative. It suggests what other evidence makes certain, that algebra owes to India the immensely useful concept of the absolute negative.

Thus one of these dotted numbers is allowed to stand by itself as a member of an equation. Bhâskara recognizes the double sign of the square root, as well as the impossibility of the square root of a negative number (which is very interesting, as being the first dictum regarding the imaginary), and no longer ignores either root of the quadratic. More than this, recourse is had to the same expedients for interpreting the negative, for attaching a concrete physical idea to it, as are in common use to-day. The primary meaning of the very name given the negative was debt, as that given the positive was means. The opposition between the two was also pictured by lines described in opposite directions.

105. Its Use of Zero. But the contributions of the Indians to the fund of algebraic concepts did not stop with the absolute negative.

They made a number of 0, and though some of their reckonings with it are childish, Bhâskara, at least, had sufficient understanding of the nature of the “quotient” $\frac{a}{0}$ (infinity) to say “it suffers no change, however much it is increased or diminished.” He associates it with Deity.

106. Its Use of Irrational Numbers. Again, the Indians were the first to reckon with irrational square roots as with numbers; Bhâskara extracting square roots of binomial surds and rationalizing irrational denominators of fractions even when these are polynomial. Of course they were as little able rigorously to justify such a procedure as the Greeks; less able, in fact, since they had no equivalent of the method of exhaustions. But it probably never occurred to them that justification was necessary; they seem to have been unconscious of the gulf fixed between the discrete and continuous. And here, as in the case of 0 and the negative, with the confidence of apt and successful reckoners, they were ready to pass immediately from numerical to purely symbolical reasoning, ready to trust their processes even where formal demonstration of the right to apply them ceased to be attainable. Their skill was too great, their instinct too true, to allow them to go far wrong.

107. Determinate and Indeterminate Equations in Indian Algebra. As regards equations—the only changes which the Indian algebraists made in the treatment of determinate equations were such as grew out of the use of the negative. This brought the triple classification of the quadratic to an end and secured recognition for both roots of the quadratic.

Brahmagupta solves the quadratic by the rule of Hero and Diophantus, of which he gives an explicit and general statement. Çrîdhara, a mathematician of some distinction belonging to the period between Brahmagupta and Bhâskara, made the improvement of this method which consists in first multiplying the equation throughout by four times the coefficient of the square of the unknown quantity and so preventing the occurrence of fractions under the radical sign.⁴⁹

Bhâskara also solves a few cubic and biquadratic equations by special devices.

The theory of indeterminate equations, on the other hand, made great progress in India. The achievements of the Indian mathematicians in this beautiful but difficult department of the science are as brilliant as those of the Greeks in geometry. They created the doctrine of the indeterminate equation of the first degree, $ax+by=c$, which they treated for integral solutions by the method of continued fractions in use to-day. They worked also with equations of the second degree of the forms $a{x}^2+b=c{y}^2$, $xy=ax+by+c$, originating general and comprehensive methods where Diophantus had been content with clever devices.

108. The Arabian Mathematics. The Arabians were the instructors of modern Europe in the ancient mathematics. The service which they rendered in the case of the numeral notation and reckoning of India they rendered also in the case of the geometry, algebra, and astronomy of the Greeks and Indians. Their own contributions to mathematics are unimportant. Their receptiveness for mathematical ideas was extraordinary, but they had little originality.

The history of Arabian mathematics begins with the reign of Almanṣûr (754–775),⁵⁰ the second of the Abbasid caliphs.

It is related (by Ibn-al-Adamî, about 900) that in this reign, in the year 773, an Indian brought to Bagdad certain astronomical writings of his country, which contained a method called “Sindhind,” for computing the motions of the stars,—probably portions of the Siddhânta of Brahmagupta,—and that Alfazârî was commissioned by the caliph to translate them into Arabic.⁵¹ Inasmuch as the Indian astronomers put full expositions of their reckoning, algebra, and geometry into their treatises, Alfazârî’s translation laid open to his countrymen a rich treasure of mathematical ideas and methods.

It is impossible to set a date to the entrance of Greek ideas. They must have made themselves felt at Damascus, the residence of the later Omayyad caliphs, for that city had numerous inhabitants of Greek origin and culture. But the first translations of Greek mathematical writings were made in the reign of Hârûn Arraschîd (786–809), when Euclid’s Elements and Ptolemy’s Almagest were put into Arabic. Later on, translations were made of Archimedes, Apollonius, Hero, and last of all, of Diophantus (by Abû’l Wafâ, 940–998).

The earliest mathematical author of the Arabians is Alkhwarizmî, who flourished in the first quarter of the 9th century. Besides astronomical tables, he wrote a treatise on algebra and one on reckoning (elementary arithmetic). The latter has already been mentioned. It is an exposition of the positional reckoning of India, the reckoning which mediæval Europe named after him Algorithm.

The treatise on algebra bears a title in which the word Algebra appears for the first time: viz., Aldjebr walmukâbala. Aldjebr (i. e. reduction) signifies the making of all terms of an equation positive by transferring negative terms to the opposite member of the equation; almukâbala (i. e. opposition), the cancelling of equal terms in opposite members of an equation.

Alkhwarizmî’s classification of equations of the 1st and 2d degrees is that to which these processes would naturally lead, viz.:

These equations he solves separately, following up the solution in each case with a geometric demonstration of its correctness. He recognizes both roots of the quadratic when they are positive. In this respect he is Indian; in all others—the avoidance of negatives, the use of geometric demonstration—he is Greek.

Besides Alkhwarizmî, the most famous algebraists of the Arabians were Alkarchî and Alchayyâmî, both of whom lived in the 11th century.

Alkarchî gave the solution of equations of the forms:

He also reckoned with irrationals, the equations

being pretty just illustrations of his success in this field.

Alchayyâmî was the first mathematician to make a systematic investigation of the cubic equation. He classified the various forms which this equation takes when all its terms are positive, and solved each form geometrically—by the intersections of conics.⁵² A pure algebraic solution of the cubic he believed impossible.

Like Alkhwarizmî, Alkarchî and Alchayyâmî were Eastern Arabians. But early in the 8th century the Arabians conquered a great part of Spain. An Arabian realm was established there which became independent of the Bagdad caliphate in 747, and endured for 300 years. The intercourse of these Western Arabians with the East was not frequent enough to exercise a controlling influence on their æsthetic or scientific development. Their mathematical productions are of a later date than those of the East and almost exclusively arithmetico-algebraic. They constructed a formal algebraic notation which went over into the Latin translations of their writings and rendered the path of the Europeans to a knowledge of the doctrine of equations easier than it would have been, had the Arabians of the East been their only instructors. The best known of their mathematicians are Ibn Aflah (end of 11th century), Ibn Albannâ (end of 13th century), Alkasâdî (15th century).

109. Arabian Algebra Greek rather than Indian. Thus, of the three greater departments of the Arabian mathematics, the Indian influence gained the mastery in reckoning only.

The Arabian geometry is Greek through and through.

While the algebra contains both elements, the Greek predominates. Indeed, except that both roots of the quadratic are recognized, the doctrine of the determinate equation is altogether Greek. It avoids the negative almost as carefully as Diophantus does; and in its use of the geometric method of demonstration it is actuated by a spirit less modern still—the spirit in which Euclid may have conceived of algebra when he solved his geometric quadratics.

The theory of indeterminate equations seldom goes beyond Diophantus; where it does, it is Indian.

The Arabian trigonometry is based on Ptolemy’s, but is its superior in two important particulars. It employs the sine where Ptolemy employs the chord (being in this respect Indian), and has an algebraic instead of a geometric form. Some of the methods of approximation used in reckoning out trigonometric tables show great cleverness. Indeed, the Arabians make some amends for their ill-advised return to geometric algebra by this excellent achievement in algebraic geometry.

The preference of the Arabians for Greek algebra was especially unfortunate in respect to the negative, which was in consequence forced to repeat in Europe the fight for recognition which it had already won in India.

110. Mathematics in Europe before the Twelfth Century. The Arabian mathematics found entrance to Christian Europe in the 12th century. During this century and the first half of the next a good part of its literature was translated into Latin.

Till then the plight of mathematics in Europe had been miserable enough. She had no better representatives than the Romans, the most deficient in the sense for mathematics of all cultured peoples, ancient or modern; no better literature than the collection of writings on surveying known as the Codex Arcerianus, and the childish arithmetic and geometry of Boetius.

Prior to the 10th century, however, Northern Europe had not sufficiently emerged from barbarism to call even this paltry mathematics into requisition. What learning there was was confined to the cloisters. Reckoning (computus) was needed for the Church calendar and was taught in the cloister schools established by Alcuin (735–804) under the patronage of Charlemagne. Reckoning was commonly done on the fingers. Not even was the multiplication table generally learned. Reference would be made to a written copy of it, as nowadays reference is made to a table of logarithms. The Church did not need geometry, and geometry in any proper sense did not exist.

111. Gerbert. But in the 10th century there lived a man of true scientific interests and gifts, Gerbert,⁵³ Bishop of Rheims, Archbishop of Ravenna, and finally Pope Sylvester II. In him are the first signs of a new life for mathematics. His achievements, it is true, do not extend beyond the revival of Roman mathematics, the authorship of a geometry based on the Codex Arcerianus, and a method for effecting division on the abacus with apices. Yet these achievements are enough to place him far above his contemporaries. His influence gave a strong impulse to mathematical studies where interest in them had long been dead. He is the forerunner of the intellectual activity ushered in by the translations from the Arabic, for he brought to life the feeling of the need for mathematics which these translations were made to satisfy.

112. Entrance of the Arabian Mathematics. Leonardo. It was the elementary branch of the Arabian mathematics which took root quickest in Christendom—reckoning with nine digits and 0.

Leonardo of Pisa—Fibonacci, as he was also called—did great service in the diffusion of the new learning through his Liber Abaci (1202 and 1228), a remarkable presentation of the arithmetic and algebra of the Arabians, which remained for centuries the fund from which reckoners and algebraists drew and is indeed the foundation of the modern science.

The four fundamental operations on integers and fractions are taught after the Arabian method; the extraction of the square root and the doctrine of irrationals are presented in their pure algebraic form; quadratic equations are solved and applied to quite complicated problems; negatives are accepted when they admit of interpretation as debt.

The last fact illustrates excellently the character of the Liber Abaci. It is not a mere translation, but an independent and masterly treatise in one department of the new mathematics.

Besides the Liber Abaci, Leonardo wrote the Practica Geometriae, which contains much that is best of Euclid, Archimedes, Hero, and the elements of trigonometry; also the Liber Quadratorum, a collection of original algebraic problems most skilfully handled.

113. Mathematics during the Age of Scholasticism. Leonardo was a great mathematician,⁵⁴ but fine as his work was, it bore no fruit until the end of the 15th century. In him there had been a brilliant response to the Arabian impulse. But the awakening was only momentary; it quickly yielded to the heavy lethargy of the “dark” ages.

The age of scholasticism, the age of devotion to the forms of thought, logic and dialectics, is the age of greatest dulness and confusion in mathematical thinking.⁵⁵ Algebra owes the entire period but a single contribution; the concept of the fractional power. Its author was Nicole Oresme (died 1382), who also gave a symbol for it and the rules by which reckoning with it is governed.

114. The Renaissance. Solution of the Cubic and Biquadratic Equations. The first achievement in algebra by the mathematicians of the Renaissance was the algebraic solution of the cubic equation: a fine beginning of a new era in the history of the science.

The cubic $x^3+mx=n$ was solved by Ferro of Bologna in 1505, and a second time and independently, in 1535, by Ferro’s countryman, Tartaglia, who by help of a transformation made his method apply to $x^3\pm m{x}^2=\pm n$ also. But Cardan of Milan was the first to publish the solution, in his Ars Magna,⁵⁶ 1545.

The Ars Magna records another brilliant discovery: the solution—after a general method—of the biquadratic $x^4+6x^2+36=60x$ by Ferrari, a pupil of Cardan.

Thus in Italy, within fifty years of the new birth of algebra, after a pause of sixteen centuries at the quadratic, the limits of possible attainment in the algebraic solution of equations were reached; for the algebraic solution of the general equation of a degree higher than 4 is impossible, as was first demonstrated by Abel.⁵⁷

The general solution of higher equations proving an obstinate problem, nothing was left the searchers for the roots of equations but to devise a method of working them out approximately. In this the French mathematician Vieta (1540–1603) was successful, his method being essentially the same as that now known as Newton’s.

115. The Negative in the Algebra of this Period. First Appearance of the Imaginary. But the general equation presented other problems than the discovery of rules for obtaining its roots; the nature of these roots and the relations between them and the coefficients of the equation invited inquiry.

We witness another phase of the struggle of the negative for recognition. The imaginary is now ready to make common cause with it.

Already in the Ars Magna Cardan distinguishes between numeri veri—the positive integer, fraction, and irrational,—and numeri ficti, or falsi—the negative and the square root of the negative. Like Leonardo, he tolerates negative roots of equations when they admit of interpretation as “debitum,” not otherwise. While he has no thought of accepting imaginary roots, he shows that if $5+\sqrt{-15}$ be substituted for $x$ in , that equation is satisfied; which, of course, is all that is meant nowadays when $5+\sqrt{-15}$ is called a root. His declaration that $5\pm \sqrt{-15}$ are “vere sophistica” does not detract from the significance of this, the earliest recorded instance of reckoning with the imaginary. It ought perhaps to be added that Cardan is not always so successful in these reckonings; for in another place the sets

Following Cardan, Bombelli⁵⁸ reckoned with imaginaries to good purpose, explaining by their aid the irreducible case in Cardan’s solution of the cubic.

On the other hand, neither Vieta nor his distinguished follower, the Englishman Harriot (1560–1621), accept even negative roots; though Harriot does not hesitate to perform algebraic reckonings on negatives, and even allows a negative to constitute one member of an equation.

116. Algebraic Symbolism. Vieta and Harriot. Vieta and Harriot, however, did distinguished service in perfecting the symbolism of algebra; Vieta, by the systematic use of letters to represent known quantities,—algebra first became “literal” or “universal arithmetic” in his hands,⁵⁹—Harriot, by ridding algebraic statements of every non-symbolic element, of everything but the letters which represent quantities known as well as unknown, symbols of operation, and symbols of relation. Harriot’s Artis Analyticae Praxis (1631) has quite the appearance of a modern algebra.⁶⁰

117. Fundamental Theorem of Algebra. Harriot and Girard. Harriot has been credited with the discovery of the “fundamental theorem” of algebra—the theorem that the number of roots of an algebraic equation is the same as its degree. The Artis Analyticae Praxis contains no mention of this theorem—indeed, by ignoring negative and imaginary roots, leaves no place for it; yet Harriot develops systematically a method which, if carried far enough, leads to the discovery of this theorem as well as to the relations holding between the roots of an equation and its coefficients.

By multiplying together binomial factors which involve the unknown quantity, and setting their product equal to 0, he builds “canonical” equations, and shows that the roots of these equations—the only roots, he says—are the positive values of the unknown quantity which render these binomial factors 0. Thus he builds $aa-ba-ca=-bc$, in which $a$ is the unknown quantity, out of the factors $a-b,a+c$, and proves that $b$ is a root of this equation and the only root, the negative root $c$ being totally ignored.

While no attempt is made to show that if the terms of a “common” equation be collected in one member, this can be separated into binomial factors, the case of canonical equations raised a strong presumption for the soundness of this view of the structure of an equation.

The first statement of the fundamental theorem and of the relations between coefficients and roots occurs in a remarkably clever and modern little book, the Invention Nouvelle en l’Algebre, of Albert Girard, published in Amsterdam in 1629, two years earlier, therefore, than the Artis Anatyticae Praxis. Girard stands in no fear of imaginary roots, but rather insists on the wisdom of recognizing them. They never occur, he says, except when real roots are lacking, and then in number just sufficient to fill out the entire number of roots to equality with the degree of the equation.

Girard also anticipated Descartes in the geometrical interpretation of negatives. But the Invention Nouvelle does not seem to have attracted much notice, and the genius and authority of Descartes were needed to give the interpretation general currency.