A History of Invention in Mathematics

Ekkehard Kopp adopts a chronological framework to demonstrate that changes in our understanding of numbers have o� en relied on the breaking of long-held conven� ons, making way for new inven� ons that provide greater clarity and widen mathema� cal horizons. Viewed from this historical perspec� ve, mathema� cal abstrac� on emerges as neither mysterious nor immutable, but as a con� ngent, developing human ac� vity.

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Coordinates and Complex Numbers
If a man will begin with certainties, he shall end in doubts, but if he will be content to begin with doubts, he shall end in certainties.
Sir Francis Bacon, The Advancement of Learning, 1605

Summary
The next crucial steps that took mathematicians towards a fuller understanding of the structure of polynomial equations-leading, almost incidentally, to a more inclusive approach to the concept of number-were taken, apparently independently, by two Frenchmen: the lawyer Pierre de Fermat (1601-1665) and the philosopher René Descartes (1596 -1650). Both took initial steps toward the introduction of what we call a coordinate system, a term coined somewhat later by the German philosopher Gottfried Wilhelm Leibniz (1646Leibniz ( -1718. This enabled them systematically to reformulate, solve and generalise many geometric problems inherited from Ancient Greece by using the algebraic formalism developed by Arabian mathematicians and their Renaissance counterparts in Europe. This chapter highlights the innovations in Descartes' revolutionary contribution and to widespread use of algebraic notation, paving the way for acceptance of the system ofreal numbers (the 'number line') and 1 to what is now known as analytic geometry. His work inspired his successors, including Isaac Newton, to remove suspicions about negative numbers, and to a significant extent, to accept irrationals as numbers. However, the nature of square roots of negative numbers was clarified only in the nineteenth century, principally through the influence of Gauss and by Hamilton's abstract definition of the complex number system. This led to what became known as the Fundamental Theorem of Algebra.
1 Our focus will be on Descartes because Fermat did not publish his short treatise Introduction to Plane and Solid Loci in his lifetime. After his death it was published by his son in 1679, by which time Descartes' methods had already been widely disseminated and developed further by others.

Descartes' analytic geometry
1.1. Discours de la Méthode. René Descartes is known today principally as a philosopher: his Discours begins with an account of his rigorous search for indubitable truth, which led him to his dictum 'I think, therefore I am' ('Je pense, donc je suis' -later more widely known in its Latin translation cogito ergo sum). His reasoning was essentially that doubt, as an act of thought, cannot occur unless there is a thinker. He continued his search for further propositions that he should accept as certain, arguing that 'we ought never to allow ourselves to be persuaded of the truth of anything unless on the evidence of our reason.' Descartes' principal mathematical text appeared in 1637 as his (now famous) La géométrie, published as one of three annexes to his highly influential Discours de la Méthode. I will focus on how Descartes' new methodology led to the coordinate geometry nowadays routinely taught at school levela development which the English philosopher John Stuart Mill (1806-1873) later described as 'the greatest single step ever made in the progess of the exact sciences'.
Descartes' methodology differs fundamentally from what we now regard as scientific method, since he does not accept that our understanding of natural phenomena should be based on empirical observation. Rather like Plato, he wishes it to be based on metaphysical principles and on 'basic facts' of physical reality which in turn should be derived from the 'indubitable truths' supplied by 'pure reason'. In a letter to his friend, the priest Marin Mersenne, he chides Galileo, who 'without having considered the first causes of nature, has merely looked for the explanations of a few particular effects, and he has thereby built without foundations'.

Coordinate systems.
In Descartes' mathematical researches, however, the foundations were already in place, due largely to the work of Viète (see Chapter 3, Section 3.1). In La géométrie, his main concern was to bring order to geometric constructions ('the analysis of the Greeks') by combining them with modern algebraic notation ('the algebra of the moderns'). The synthesis he proposed would free geometry from its reliance on diagrams and, at the same time, give concrete meaning to algebraic operations through their geometric interpretation.
Descartes' goal remains a geometric construction, but his method of achieving this makes full use of algebraic operations. The crucial link between the two is created by choosing a point O in the plane, and a pair of reference lines that meet in O. Today, the reference lines are called axes, typically labelled as X and Y respectively. The position of any point P in the plane can then be described algebraically by means of a pair (x, y) of numbers that measure the distances needed to get from O to P by moving only in the directions of the two axes: from O we first move a distance x along Figure 19. Rectangular and polar coordinates the X-axis, and, from there, a distance y on a line parallel to the Y -axis, to reach P (or, alternatively, move distances y along the Y -axis and x on a line parallel to the X-axis). The path traced out by a curve or a geometric figure can then be described via an equation that relates these x and y coordinates by setting out in algebraic terms the geometric conditions that any point P = (x, y) on the curve (or figure) must satisfy.
Today we regard Descartes and Fermat as the originators of analytic (or coordinate) geometry, in which we choose two perpendicular axes X and Y which, as shown in Figure 19, define the Cartesian plane (named in honour of Descartes). Any point P in this plane is then described by the distances (x, y) from O to the projections of P onto the (X, Y )-axes. This definition has the advantage that the length x 2 + y 2 of the line segment OP is given immediately by Pythagoras' theorem: for example, the collection of points P = (x, y) at distance r from O is (by definition) the circle with centre O and radius r, and the points x, y are related by x 2 + y 2 = r 2 , as the triangle in Figure 19 shows. However, neither Descartes nor Fermat actually used such a rectangular coordinate system. In fact, as we will indicate below, Descartes' system involved positive coordinates: in our terms, when using rectangular coordinates, we would call this the positive quadrant. The development of a full coordinate system was undertaken by his successors. Nevertheless, the key breakthrough -linking the apparently unrelated areas of geometry and algebra -had been made, making simplifications such as rectangular coordinates and extensions to more than two dimensions much easier to achieve.
Descartes further elaborated the algebraic symbolism he had inherited from Viète, and his algebraic expressions are essentially those in use today. Crucially, he also dispensed with the need to give all terms in a polynomial (such as a cubic or quartic) equation the same 'dimension', as Omar Khyayam had done. Unlike modern usage, and more in keeping with Greek tradition, he regarded the known parameters and the unknowns in his equations as lengths of lines, rather than as numbers. Yet he broke with Greek usage by considering x 2 , x 3 as lines, not as areas or volumes. For example, instead of treating the product of two line segments as a rectangle, he defined their product by using similarity of figures and proportionals.
It is the systematic introduction of a unit length in any geometric problem that provides the key to Descartes' approach. His geometric construction of the product ab of two line segments was extremely simple, as shown in Figure 20: letting two given line segments BC and BD meet at B (at an arbitrary angle), he chose an arbitrary distance BA along BD as the unit (extending the segment if necessary) and drew the line segment AC. Then he drew the line DE parallel to AC, to meet BC (extended) at E. The triangles ABC and DBE are obviously similar, which ensures that the ratios of corresponding sides are in proportion. Hence BC : BA :: BE : BD. Choosing BA as the unit length, while BC = a and BD = b, this shows that BE becomes the product BC.BD = ab. Using the same figure (and unit) the other way round one obtains BC as the quotient of BE by BD. Both product and quotient are therefore realised as line segments.
The construction of a mean proportional (Chapter 3, Figure 15) demonstrates how Descartes could interpret the square root similarly as a line segment: to find the square root of a line segment OA, extend it by a chosen unit AB to OAB with AB = 1, and from its midpoint draw a circle through O. The perpendicular AC from its circumference provides the desired square root of OA .
Descartes' notation when describing geometric problems was essentially algebraic. He used and extended Viète's new algebraic symbolism: 'it is sufficent to designate each [line segment] by a single letter. Thus, to add the lines BD and GH, I call one a and the other b, and write a + b. Then a − b will indicate that b is subtracted from a; ab that a is multiplied by b; a/b that a is divided by b; ...'.
As he points out, early in La géométrie, dimensional homogeneity in an equation can always be restored by multiplying or dividing a term sufficiently often by the chosen unit! For example: 'if it be required to extract the cube root of a 2 b 2 − b, we must consider the quantity a 2 b 2 divided once by the unit, and the quantity b multiplied twice by the unit.' (See [6].) Today this explanation may appear trivial and unnecessary, as the symbols are regarded as abstract entities, with no need to think of them in terms of geometric dimensions. Yet it is precisely this essential insight by Descartes that eased the way for his successors to regard the constants, coefficients and unknowns occurring in equations as symbols that can be represented by numbers. As Fauvel and Gray observe in [12], the simple step of eliminating dimensional considerations '...turns out to have lifted a weight off everyone's shoulders...' Descartes systematically used the initial letters of the alphabet for known quantities, leaving final letters such as x, y, z to denote the unknowns. He applied the exponential notation, by then commonly used, to the unknowns (although, while happy to write x 3 , x 4 , etc., he persisted in writing the square as xx rather than x 2 ), and he used the Germanic symbols +, − for for addition and subtraction. He did not use Robert Recorde's symbol = for equality; the symbol he employed, possibly drawn from the abreviation 'ae' of the Latin 'aequalis', remained popular on the Continent for some time.
1.3. La géométrie. The first two books of La géométrie focus on applying algebraic techniques to geometry. Giving the 'unknown' a name, despite not knowing its value in advance, was an important step forward. It enabled him to treat known and unknown quantities on the same basis. His approach was to attack a geometric problem by converting it into an algebraic equation, simplifying the equation as far as possible and then to solve the equation geometrically. Given any geometric problem, his method for its solution is the following (the quotations are from [12]): '...we first suppose the solution already effected, and give names to all the lines that seem needful for its construction-to those that are unknown as well as those that are known. Then, making no distinction between known and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.' Descartes gives a simple example to show how an equation obtained in this way, once simplified as far as possible, can be solved geometrically. Figure 21 illustrates his argument. 2 a, that is, to half the other known quantity which was multiplied by z, which I supposed to be the unknown line. Then, prolonging M N, the hypotenuse of this triangle, to O, so that N O is equal to N L, the whole line OM is the required line z. This is expressed in the following way: He recognised that the degree of the algebraic equation expressing the curve in terms of the (X, Y )-coordinates would determine the means by which the geometric construction of its roots can be achieved: 'If it can be solved by ordinary geometry, that is by the use of straight lines and circles...there will remain at most only the square of an unknown quantity...' In other words, straightedge-and-compass constructions lead to linear or quadratic equations, and he showed how to solve these. Descartes knew that Viète had shown the duplication of the cube and angle trisection to lead to cubic equations, and he now asserted that these constructions could not be effected by straightedge and compass alone -although his attempt at a proof was defective. He argued, as had the Greek geometers, that all constructions should be effected by the simplest means possible; for him, this was determined by the degree of the equation. Quadratics could be handled by straightedge-and-compass, cubics and quartics required conic sections.
Descartes was encouraged by his success in dealing with a set of problems posed by the Alexandrian geometer Pappus (fourth century) which attracted much attention at the time. The simplest, the so-called three-line locus problem, assumed that one was given three lines in the plane. Pappus sought the locus of points such that the product of their distances from two of these lines is proportional to the square of the distance from the third line. 2 Descartes chose one of the three lines as AB and fixed the position of a typical point C of the locus he sought by marking AB as x and BC as y, so that these two became the reference lines (what we would call the axes today) for his system of coordinates. Expressing the other lines in terms of these two quantities he arrived at an equation (involving the variables x and y) that the point C must satisfy. He solved the problem by simplifying this equation as far as possible and effecting a geometric construction for the roots of the simplified equation.
In the second book he continued to extend this approach to a larger set of lines than Pappus had done and suggested a general classification of geometric problems: those leading to quadratic equations formed the first class and could be solved by straightedge and compass constructions, those leading to cubic or quartic equations he placed in the second class, since the roots could be constructed using conic sections. Quite generally, a problem of class n was associated with an equation of degree either 2n − 1 or 2n. He conjectured -incorrectly, as it turned out -that, since the solution of the quartic can be reduced to that of an associated cubic (as Ferrari had shown), a similar argument could always be found to solve an equation of degree 2n by reducing its solution to that of an equation of degree 2n−1. Although his classification did not hold water, his work encouraged the much freer use of higher plane curves in geometric constructions. A famous example is the folium of Descartes, which makes a loop in the first (i.e. 'north-east') quadrant, intersecting itself at the origin, while its two ends go off to infinity in the second ('north-west') and fourth ('south-east') quadrant of our rectangular (X, Y )-plane. Its equation has the form x 3 + y 3 − 3axy = 0.
He went on to classify as 'geometric curves' all those whose points can be determined by the intersection of two lines, each moving parallel to one of his axes (X or Y ) with commensurable velocities (whose ratio takes the form m n for some whole numbers m, n). His study of Pappus' three-and four-line problems led him to the general equation of a conic passing through the origin in the form: y 2 = ay − bxy + cx − dx 2 , where he identified the types that can occur for different choices of the constants a, b, c, d.
Modern terminology, following a more general classification introduced by Leibniz half a century later, defines an algebraic curve as the collection of points (x, y) in the plane satisfying a given polynomial equation in the variables x, y. In particular, the points (x, y) on a straight line satisfy the first-degree equation while the general quadratic equation in x, y, ax 2 + bxy + cy 2 + dx + ey + f = 0 will give rise to a conic section (see Figure 14 in Chapter 3). The simplest examples of conic sections occur when we have rectangular axes and centre the figure at the origin and symmetric to the axes. The circle of radius r is then given by the points (x, y) satisfying x 2 + y 2 = r 2 , an ellipse will have the form ( x a ) 2 + ( y b ) 2 = 1, a hyperbola is given by ( x a ) 2 − ( y b ) 2 = 1, a parabola typically by y 2 = 4ax.
Similarly, equations involving higher powers of the variables x, y have their degree specified by the highest power of these variables -so that the folium of Descartes has degree 3, for example. Leibniz called curves that cannot be specified by such a polynomial equation transcendental.
In his (earlier) classification Descartes in effect regarded as 'geometric' all curves we now describe as 'algebraic'. His primary objective was to establish a criterion that could be expressed in geometric terms, rather than our algebraic one, to describe such curves. His definitions implicitly assume that all geometric curves can be traced by a continuous motion. Thus the quadratrix, and other curves generally defined in terms of arc lengths, were to be excluded from his classification of geometric curves, since they arise from two simultaneous motions 'whose relation does not admit of precise determination'. For the quadratrix, one motion is a translation, the other a rotation, and, as we saw in Figure 16(a), the ratio of the lengths of the arc BED and the radius BA is π 2 , which is irrational; in other words, the two lines are incommensurable. Descarted stated this claim without proof -in effect, he claimed that the circle cannot be squared! He argued that such curves should be excluded from geometry: Geometry should not include lines (or curves) that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known...
Consequently, Descartes called such curves mechanical.
In Figure 19 we also depict an alternative description of coordinates, first introduced by the Italian Bonaventura Cavalieri (1598-1647) in order to discuss the 'Archimedean spiral', which will feature in Chapter 5. This system of polar coordinates became widely used after Jacob Bernoulli (1654-1705) employed it more systematically. One chooses an origin O, or pole, as well as a directed line segment OX, which serves as the polar axis. The length OP then describes the radial distance r of a point P in the plane from the pole, and the angle XOP (denoted by φ, and taken anti-clockwise) determines the direction of OP relative to the chosen direction of the polar axis. Pythagoras' theorem and simple trigonometry show that the polar and rectangular coordinate systems are related by: Despite his dismissal of 'mechanical' curves from geometry, in 1638 Descartes was led (when considering the path of an object falling towards a rotating Earth) to a 'mechanical' curve that contradicted his assumption that no such curve could be rectified (this term means that we can construct a straight line whose length equals that of the curve). The curve in question is the logarithmic spiral, which is most simply defined in polar coordinates by r = ae bφ , where a, b are constants, and e is the base of 'natural' logarithms (which we will meet in Chapter 5). The Italian mathematican Torricelli (1608-1647) was the first to rectify this curve in 1645, by methods that foreshadowed Newton and Leibniz' invention of the Calculus over 20 years later.

Paving the way
The 'marriage' of geometry and algebra by Descartes and Fermat served to accelerate the acceptance of irrationals as genuine numbers. Descartes' classification of geometric curves included many different types of irrational roots which could be defined as distances from the origin O along the Xaxis, in exactly the same way as rational roots. In handling these quantities algebraically there was no need to distinguish between different sorts of lengths if one simply regarded them as points on a number line.

The number line.
The coordinate system provides a visual representation, not only of the curve being analysed, but also of solutions of the equation defining the curve. These appear as points on the X-axis, whether positive or negative, and can be treated as numbers, whether rational or irrational. As this provided a convenient visual (and practical) way of avoiding the inevitable philosophical question of what irrationals actually are, the more difficult problem of finding a viable arithmetical definition of irrationals was essentially shelved until the nineteenth century, by which time the previously unchallenged centrality of Euclidean geometry had become a serious issue for debate.
Throughout the eighteenth century, mathematicians were generally much too busy exploiting the rich rewards of the new techniques offered by Descartes' analytic geometry, and the competing formulations of the Calculus by Newton and Leibniz a few decades later, to concern themselves in detail with this philosophical issue. They seeemed content to regard as real numbers any points on the number line (positive or negative, rational or irrational).
Today it has become a commonplace to conceive of this 'number line', centred on some point O and extending indefinitely to left and right, as representing the real number system R upon which most common modern mathematical structures rest. For practical measurements, of course, we must always content ourselves with rational approximations (such as 3.14159 for π, 2.71828 for e, or 1.4142 for √ 2) since our mechanical or electronic instruments all have physical limitations. But today we nevertheless endow irrational numbers with as much 'reality' as we do 5 or 94 73 . This applies equally whether they are positive or negative, since negative numbers, represented by points lying to the left of the origin O on the number line, are regarded as just as 'real' as positive numbers. This perspective, due in good measure to the ubiquity of coordinate systems, lends weight to the argument that numbers may be seen as abstract entities. My personal preference (not shared universally among mathematicians, as we shall see later) is to regard them as human inventions that assist us collectively in making meaningful assertions about the world around us, rather than being or representing actual objects that 'exist' independently of us. This viewpoint differs markedly from the Platonic perception that numbers (and lines), as idealised abstractions, exist in some unseen 'World of Ideas'.
Descartes' analytic geometry may have had a significant impact on the gradual acceptance of a wider, more abstract, concept of number. Historically, however, it took the resolution of the further puzzle of finding a satisfactory visual representation of 'imaginary roots' (roots like √ −1, which occur in equations like x 2 + 1 = 0, and were given this name by Descartes) before our modern perspectives were fully accepted. These developments are outlined in the next section. Meanwhile, the nomenclature used even today (using the term real numbers for elements of the number line, while √ −1 is often called the imaginary unit), reflects this tortuous history and retains its potential for misconceptions.
Like Napier's invention of logarithms, Descartes' analytic geometry techniques, combining algebra and geometry, found acclaim upon their publication in 1637, and were developed and refined further by his contemporaries. This process involved considerable effort. Despite Descartes' insistence that one should always begin with the simplest possible constructions before moving on to more complex ones, his discussion of various geometric problems in La géométrie was hardly systematic, focussing instead on specific, often rather difficult, problems.
He also had a habit of leaving many details of his verifications to the reader. A frequent refrain in the text was 'it already wearies me to write so much about it'. He justified his often sketchy solutions by saying that he had omitted details 'in order to give others the pleasure' of discovering things for themselves! This exacerbated the difficulties that his initial audience found in understanding his methods, and gave impetus to extensive commentaries by other mathematicians. Prominent among these was a Latin edition of La géométrie by the Dutch mathematician Frans van Schooten (1615-1660), which appeared in 1649 and had seen four editions by 1700, firmly cementing Descartes' analytic geometry in the Continental mathematical tradition.

Wallis and Newton on numbers.
In Britain, the algebraic approach pioneered by Descartes was enthusiastically taken up by John Wallis (1616-1703), who held the Savilian Chair of Geometry at Oxford University from 1649. Ordained as a priest, he had been active in decoding Royalist messages for the Parliamentary side in the English Civil Wars, and had studied earlier mathematical works by William Oughtred and Thomas Harriot which had introduced him to the new methodologies developed on the Continent.
A highly original mathematician, he published Arithmetica Infinitorum in 1656, tackling many then prevalent problems-area and volume calculations, and finding tangents to various curves-by pioneering mainly arithmetical methods involving infinite sums and products. Prominent among these was his infinite product formula for π. 4 This allowed him to approximate π (the symbol ∞ n=1 means successive multiplication, n = 1, 2, 3, ...): Wallis treated a proportion simply as asserting the equality of two fractions, awarding ratios the same status as whole numbers. Although this marked a departure from long-held views, it seemed entirely natural to him. He remained less certain about the status of negative numbers, since in his view it was impossible for a quantity to be 'Less than Nothing, or any number fewer than None' [6]. But he recognised the usefulness of accepting negative numbers in calculations. Using analogy with movement to the right or left of a starting point, he argued that negative numbers could be represented by points to the left of a chosen origin on a number line. He went further, seeking to represent square roots of negative numbers geometrically, using a construction similar to that of the mean proportional ( Figure 15). 5 He argued that, while their status as numbers was uncertain, using them in calculations was 'not altogether absurd'. Wallis also maintained doubts about the status of irrationals as numbers, but nonetheless used them freely in his calculations, arguing, as Stevin had done, that they can be approximated arbitrarily closely by fractions.
Wallis' views were not universally accepted by English mathematicians. At Cambridge, Newton's mentor Isaac Barrow (1630-1677) was prominent in something of a backlash against algebraic methods for the solution of geometric problems, although even he used symbolic representations and numerical examples in his widely read 1655 edition of Euclid's Elements. He criticised Wallis' assertion that arithmetical equalities had a meaning independent of geometric interpretation, and argued that irrationals like √ 2 were best understood in terms of geometric ratios, rather than in terms of numbers and fractions. For him classical geometry, based upon axioms, provided a clarity of meaning that algebra had not yet achieved.
Isaac Newton (1642-1727) succeeded Barrow to the Lucasian Chair of Mathematics at Cambridge in 1669. An account of Newton's views on numbers can be found in his Arithmetica Universalis, first published in 1707 (an English translation, Universal Arithmetick, by Joseph Raphson-to which Newton refused to add his name-followed in 1720). This work is not, however, among his best-known today. The focus in the book is on the practice, rather than foundations, of the new algebraic techniques, and features many illustrative examples. Perhaps reflecting Barrow's influence, Newton stresses his preference for classical geometry in many of his comments in the text.
The origins of the text lie in drafts and lecture notes dating from the period between 1673 and 1683, when he had studied Cartesian methods closely and critically. The volume is not a carefully edited and polished publication in the spirit of his 1687 Principia Mathematica. Newton was only persuaded to agree, reluctantly, to the publication of Arithmetica Universalis (which was not overseen by him, but by his successor to the Lucasian Chair, William Whiston) when he needed to attract financial support from his academic colleagues for his campaign to enter Parliament as member for Cambridge in 1705. However, due to his towering stature in English science, the text was soon translated from its original Latin, was widely read and became influential in Britain and on the Continent.
Despite Newton's evident discontent with details of the publication, the definition of number given in this work provides a concise synthesis of earlier conceptions as follows: 'By Number we understand, not so much a Multitude of Unities, as the abstracted ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity. [Number] is threefold; integer, fracted, and surd, to which last Unity is incommensurable.' (Raphson's 1720 English translation, page 2.) 6 This is the clearest statement yet of an approach that defines numbers as abstract entities. They are not taken as quantities, but may represent either quantities or ratios of the same. The Greek distinction between multiples and magnitudes is no longer an issue, and both rational and irrational numbers appear on the same footing. The same applies to negative numbers, where Newton does not follow Wallis and others in worrying about the philosophical implications of being 'less than Nothing', but draws analogies with debts and shortfalls, and works directly with positive and negative outcomes of a calculation in the same vein. Moreover, by treating 'surds' as numbers, Newton's classification moves us closer to the modern concept of 'real number'.
His attitude to square roots of negative numbers, on the other hand, seems ambiguous. He recognised that where such 'impossible' numbers appear as solutions of a polynomial equation, they should be accepted as genuine solutions, although he may have treated their occurrence in particular problems as having no clear real-world applicablity. In any event, they were not classed as numbers in the above definition, and their status was only resolved more than a century later.
In their different ways, decimal expansions, logarithms and analytic geometry involved ideas that were 'in the air' at the time of their invention -much the same is true of the Calculus. Those now credited with these achievements were usually the first to publish comprehensive results (although, as we saw in the cases of Jobst Bürgi and Pierre de Fermat respectively, others had simultaneously, or even earlier, developed similar concepts). The initial published results were developed and sometimes improved by their peers. Newton's famous comment (referring to the work of Descartes) in a letter to Hooke in 1676: 'If I have seen further it is by standing on the shoulders of giants' (Newton wrote 'sholders') appears to be fully justified in this context.

Imaginary roots and complex numbers
But let us step back a little and return to Descartes. In the final book of La géométrie he turned to general principles for solving algebraic equations. He recognised that a polynomial of degree n, p(x) = a n x n + a n−1 x n−1 + ... + a 1 x + a 0 , is divisible by (x − α) exactly when α is a root of the polynomial, that is, a solution of the equation p(x) = 0. He proceeded to argue that this implies that a polynomial of degree n has n roots (foreshadowing what we now call the Fundamental Theorem of Algebra): 'Every equation can have as many distinct roots (values of the unknown quantity) as the number of dimensions of the unknown quantity in the equation '. Exactly what sort of 'numbers' (or 'values') should be allowed to represent these roots is not made explicit, although Descartes was surely aware that equations such as x 2 + 1 = 0 have no roots among the rational or irrational numbers represented by points of the geometric 'number line'.

The rule of signs.
Descartes also expounded his rule of signs, which provides information on the number of positive roots of a polynomial (with multiply occurring roots counted by the number of times they appear-their multiplicity).
A variation in sign occurs in a polynomial a n x n + a n−1 x n−1 + ... + a 1 x + a 0 if two consecutive coefficients have opposite signs. For example, x 2 − 3x + 2 has two variations: reading from the left we go from +1 to −3 to +2.
In modern terminology, Descartes' rule of signs can be stated as follows: the number of positive roots of a polynomial, each counted as often as its multiplicity, either equals the number of variations in the signs of its coefficients or is less than this number by an even number.
For example: the polynomials x 3 − 4x 2 + 5x − 2 and x 3 − 3x 2 + x − 3 each have three variations in sign (+ is followed by −, then +, then −). The rule of signs states that the positive roots of these polynomials will number either 3 or 1. Factorising each polynomial by inspection (trying out x = 1, 2, 3, for example) it is easy to see that the first can be written as (x − 1) 2 (x − 2), so the positive roots are 1, 1, 2 (the repeated root is counted twice). The second, however, becomes (x 2 + 1)(x − 3), so 3 is the only positive root, since x 2 + 1 has no real root.
Descartes himself formulated his rule of signs rather less clearly than stated here, for which Wallis took him to task in his Algebra (published in 1685). Wallis stated that Descartes had claimed that the number of positive roots would always equal the variation in signs, and remarked pointedly that the rule fails in general: 'it must be taken with this caution, that is, that the roots are real, and not imaginary'. In Descartes' defence, we might observe that his claim was only that the number of positive roots can equal the variation in signs, and that his text appears to suggest that he was aware that there are fewer when imaginary roots occur.

Representation of imaginary numbers.
These quotations suggest that, a century after Bombelli's struggle to make sense of them, imaginary roots of polynomial equations were no longer simply disregarded, but had become a possible object of study. In a further example of calculation, working implicitly with the 'positive square root' of −1, Leibniz factorised the expression x 4 + a 4 as although he did not attempt to simplify the even more mysterious quantity √ −1 any further.
In another computation, Leibniz worked directly with the square root of −3 to obtain where, under the outer square root sign, he employed the familiar identity x 2 − y 2 = (x + y)(x − y) with x = 1, y = √ −3. Multiplying out the square = 2 + 2(2) = 6.
The identity √ 6 = 1 + √ −3 + 1 − √ −3 followed by taking square roots. This again showed that calculations with imaginary numbers could generate real numbers. Precisely what an imaginary number should signify remained obscure, although Leibniz and his successors encountered them ever more frequently in their studies of the Calculus and differential equations. Leibniz summed up the prevailing attitude to imaginary numbers in the 1680s as follows: 'From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised'. [10].
Just over a century later, the concept of geometrical representation of imaginary numbers was very much in the air, and a subject of some controversy, as illustrated by the critical reception in England of a paper by the French émigré clergyman, Adrien-Quentin Buée (1745-1825), who maintained that √ −1 should be seen as 'a purely geometric operation. It is a sign of perpendicularity '. 7 In fact, within a period of less than fifteen years, geometric representations intended to represent numbers of the form a + √ −1b, where a, b are real numbers, appeared independently in three different European countries. In 1797 the Norwegian surveyor Caspar Wessel  was exploring ways of representing directed line segments-segments of lines pointing in a given direction, defined by their length and their directionthat we call vectors today. Wessel was led by his considerations to represent √ −1 as a vertical line segment of unit length, starting at the origin, in the Cartesian plane. He recognised that his geometric definition of the sum of two directed line segments (see Figure 23(a)) applied if the axes of his coordinate system are taken to represent real and imaginary numbers respectively. Wessel's paper, published in Danish, apparently remained unknown to most mathematicians of his day and only came to wider attention once it was translated into French a century later. Another significant and more widely known advance was made by a Paris accountant and amateur mathematician by the name of Argand. Elementary textbooks still interchangably use the names complex plane or Argand diagram for the plane described by means of two perpendicular coordinates (as in Figure 23(b)), where the real numbers are on the horizontal axis and the imaginaries on the vertical one, containing i at one unit above the origin. However, very little is known reliably about the man himself-a somewhat dubious 1874 biography names him as Jean-Robert Argand (1768-1822) and places him as born in Geneva, but none of this has been verified by original sources.
Argand considered the imaginary unit in an essay published in 1806, regarding it as the result of a rotation in the plane through a right angle. He argued that 1 is transformed into −1 by rotating the plane through 180 • , and concluded that a rotation through half of this angle should lead to √ −1 instead.
In the Argand diagram a + ib is the point reached by moving a distance a from the origin along the real axis and then a distance b parallel to the imaginary axis. It was the third member of the trio (and its sole mathematician) Carl Friedrich Gauss, who, in notes and various publications dating from 1811 to 1831, pointed out that this represents the complex number a + ib as a point (a, b) in the resulting plane. The resulting rectangular coordinate system has therefore also become known as the Gaussian plane.
Gauss remarked that, by dividing the plane into a grid by parallel lines, horizontal and vertical and one unit apart, the vertices of the resulting squares would become natural reference points for integral distances in both directions, with each point having four immediate neighbours. 'If this subject has hitherto been considered from the wrong viewpoint and thus was found to be enveloped in mysterious darkness, it is largely an unsuitable terminology which should be blamed. If +1, −1, √ −1 had been described, verbally, not as 'positive, negative, imaginary' (or [the latter] even as 'impossible') but, for example, as 'direct, inverse, lateral' instead, there would have been no cause to refer to any such darkness'.
In a celebration commemorating the 50th anniversary of his Brunswick doctorate in 1849 Gauss was introduced with the phrase 'You have made the impossible possible' [36].

Hamilton's definition of complex numbers.
In the 1830s it was made explicit by the Irish mathematician William Rowan Hamilton (1805-1865) that a complex number could be described quite precisely as a pair (a, b) of real numbers, representing the Cartesian coordinates of a point in the plane. He argued that this should be its definition as a number. The geometric interpretation leads naturally to the definition of the sum of two complex numbers, which is found by adding coordinates separately. This allowed Hamilton to extend algebra to this set of numbers, although the product of two such numbers still required definition.
Hamilton approached number systems in a novel way: he sought to define complex numbers purely in terms of real numbers, without recourse to notions of 'imaginary' units or geometrical representations, but simply by setting down arithmetical rules for combining such numbers in a way that would be consistent with previously established properties.
For this, he assumed that the arithmetic of real numbers was unproblematic, and sought to recover known properties of complex numbers without trying to establish their 'true nature'. Instead, his arithmetical rules should determine the properties of the system.
As mentioned above, addition of two pairs of reals (a, b) and (c, d) could be defined simply as since this would reflect the sum (a + ib) + (c + id) = (a + c) + i(b + d) of two complex numbers. To define subtraction, one need only note that (0, 0) acts as a neutral element for addition (i.e. (a, b) + (0, 0) = (a, b)), so that the additive inverse of (a, b) is (−a, −b) since (a, b) + (−a, −b) = (0, 0)). More generally, the difference between two pairs is To introduce multiplication Hamilton needed to reflect the property i 2 = −1 of the 'imaginary unit' i. This yields (a + ib) × (c + id) = (ac − bd) + i(ad + bc).
The product of two pairs of real numbers was therefore defined by Hamilton as (a, b) × (c, d) = (ac − bd, ad + bc).
To define division, he made use of the fact the quotient of two whole numbers, m n , is the product of m with the multiplicative inverse 1 n of n. So the important step was to find the inverse, the number x satisfying n × x = 1. The same idea can be applied to complex numbers, where the neutral element for multiplication is shown by the above definition to be (1, 0). So the inverse (x, y) of (c, d) should satisfy (1, 0) = (c, d) × (x, y) = ((cx − dy), (cy + dx)).
Thus cy + dx = 0, hence y = − d c x, and 1 = cx − dy = (c + d 2 c )x, so that the multiplicative inverse of (c, d) = (0, 0) becomes Hamilton could therefore define the quotient of the pair (a, b) by the pair (c, d) = (0, 0) as set out to provide the proof. The first proof to gain wider acceptance appeared in 1799 in Gauss' doctoral thesis, which immediately established him as a major figure in his subject.
In his thesis, Gauss reviewed the attempted proofs of his predecessors, pointing out that all had assumed that any polynomial must have roots, although they did not specify which number system would contain these roots! This overlooked the main issue, he maintained, which was to demonstrate that any polynomial will have at least one root in some well-defined number system. His claim was that this is true if the number system in question is the complex plane. Thus the key result needed was the following, which we will call
A polynomial with leading coefficient a n = 1 is called monic. Restricting attention to this case is no real restriction. If the degree of a polynomial c n z n +c n−1 z n−1 +...+c 1 z +c 0 is n, the coefficient c n must be nonzero, so that we can divide whole the polynomial by c n to obtain a monic polynomial.
With Gauss' theorem we can can show that any monic polynomial of degree n is a product of n linear factors:
This shows that (z − w) is a common factor of all the remaining terms in the expansion of p(z) = p(z) − p(w), so that we can re-arrange terms to write p(z) = (z − w)q(z) for some polynomial q of degree (n − 1). Write α 1 = w, and use Gauss' theorem to find a root w of the polynomial q. As before, (z − w ) is then a common factor of q(z) = q(z) − q(w ), so we can find a polynomial r of degree (n − 2) such that q(z) = (z − w )r(z). Write α 2 = w . We have shown that p(z) = (z − α 1 )(z − α 2 )r(z).
This completes the proof of the theorem.
This illustrates how what we have called Gauss' theorem is the key result required for understanding the structure of polynomials. A different proof of the Fundamental Theorem was published by Argand in 1814. His proof was not deemed rigorous and it was not widely accepted at the time -although it now thought that Argand's ideas provide the most direct approach to the problem. In fact, all the attempted proofs published in the early 1800s could only be made fully rigorous in the 1870s, when the crucial role played in these arguments of completeness of the real number systemwhich we discuss in Chapters 7 and 8 -was better understood.
Argand's ideas are based on a result first announced by the French mathematician Jean le Rond d'Alembert (1717-1783, who had also made two attempts (1746 and 1754) to prove the Fundamental Theorem.

d'Alembert's lemma:
If the complex polynomial p(z) is non-constant and p(z 0 ) = 0, then any neighbourhood of z 0 contains a point w with |p(w)| < |p(z 0 )| .
In his papers d'Alembert takes for granted that, as z varies across the complex numbers, the curve traced out by |p(z)| will be continuous. In his time, continuity was described in terms of 'infinitesimal change', so that a continuous curve was regarded as one that could be drawn without lifting the pencil, and the notion of 'neighbourhood' relied on geometric intuition. In the nineteenth century it became clear that these ideas do not provide adequate definitions.
There are many modern proofs of versions of Gauss' theorem. Those making direct use of d'Alembert's result are perhaps the most accessible. A relatively brief, but authoritative, summary of the history of this important result and its many different proofs is given in Chapter 4 of [10]. A short modern proof is given in MM.
Today the Fundamental Theorem of Algebra can arguably be regarded as mis-named, since it deals exclusively with polynomial equations, which are far removed from the myriad abstract algebraic structures that have been invented in the two centuries since Gauss' heyday. Moreover, its early proofs implicitly assumed deeper properties of the number line than could be made visible by eighteenth century algebra. The fact that Gauss returned to the theorem three times throughout his career testifies that he was aware that his original argument in 1799 contained a significant gap. It was only in 1920 that the Russian mathematician Ostrowski fully completed Gauss' original proof, in line with modern standards of rigour.
Gauss' theorem is a good example of the perspectives of modern mathematics which he did much to encourage. The focus is not on finding a construction of the actual roots, but on showing that, in general, roots of polynomials will always exist. It is proved that they are there to be found, without actually specifying how to find them. By shifting perspectives in this way, mathematicians found that much larger and previously intractable, even unimagined, areas of enquiry became available. The focus now shifted to analysing the structure of the objects (here, the collection of all polynomials) being investigated. From this point of view Gauss' theorem is fundamental, since it shows that, as long as we are flexible about the kinds of numbers we allow as roots, then the structure of any polynomial is fully understood once we know all its roots.