2.2. IMPROVEMENTS IN NOTATION 2.2 IMPROVEMENTS IN NOTATION Good mathematical notation serves more than one purpose. First, it displays mathematics in a clear and concise form, often encapsulating in a few symbols ideas that cannot be at all easily or briefly expressed in words; once one know the rules, mathematics written symbolically is more easily visualized, communicated, and understood. This in turn helps to generate new ideas, sometimes because the symbols themselves can be manipulated to show new connections, but at a deeper level because clear exposition almost invariably opens the way to further and faster progress in the mind of both reader and writer. New symbols usually lag some way behind the concepts they are required to express, and can take some time to settle into a standard form. Mathematical notation continues to evolve today, but widespread publication and rapid communication have made the process very much faster than it was four hundred years ago. Modern notation did not begin to appear until the late fifteenth century, when + and — were first used in Germany. They were followed some time later by the - sign, which first appeared in Robert Recorded Whetstone ofwitte in 1557 (and was described by Recorde as a paire of paralleles, or Gemowe [twin] lines of one lengthe, thus: ====== bicause noe .2. thynges, can be moare equalle'). It was many years, however, before these signs or any others became standard; many sixteenth-century writers continued to use p. for plus and m. for minus (or other inventions of their own), and continental writers followed René Descartes in using a version of oo as an equality sign well into the seventeenth century. Girolamo Cardano's great book, the Ars magna, of 1545 (see 12.1.1), was written entirely verbally, with some useful abbreviations but no genuine symbolic notation. Rafael Bombelli, trying to present Cardano's work more clearly, devised the notation ^, and so on, for squares, cubes, and higher powers, an idea that was taken up with slight modification by Stevin, who greatly admired Bombelli. Francois Viěte, writing in the 1590s, retained the verbal forms quadratus and cubits for 'squared' and 'cubed', but contributed the idea of using vowels A, E, I, ... for unknowns, and consonants 5, C, D,.,. for known or given quantities, which meant that equations could be expressed entirely in letters or, as he called them, 'species'. Viěte still used words, however, for the linking operations ('multiplied by', 'equals', and so on), so that his text remained primarily verbal rather than symbolic. CHAPTER 2. FRESH IDEAS 2.2.1 Harriot's notation, c. 1600 The earliest mathematical notation that appears to a modern reader both familiar and easy to read is that of Thomas Harriot, whose gift for devising lucid symbolism is apparent in all aspects of his mathematical and scientific work from the early 1590s onwards. Harriot's mathematical notation was based on Viete's insofar as it used vowels a, e, ... for unknown quantities, and consonants b, c, d, ... for known quantities or coefficients, but now in lower case rather than as capitals. He also introduced the convention of writing ah for a x b, with aa, aaa, and so on, for squares, cubes, and higher powers. His equals sign incorporated two short cross strokes to distinguish it from Recorde's simple parallels (since the latter were sometimes used by Viete to indicate subtraction) but these were abandoned in the printed versions. He also invented the now standard inequality signs < and >; in manuscript they, like his equals sign, included two vertical cross strokes, but these too were dropped as soon as the signs went into print. Also to be found in Harriot's manuscripts is the three-dot 'therefore' sign, designed to suggest that two propositions imply a third. Below is a single page from Harriot's posthumously published Artis analyticae praxis {The practice of the analytic art) of 1631, one of the few extracts in this present book that needs almost no translation. 2.2. IMPROVEMENTS IN NOTATION 45 Harriot's notation from Harriot, Artis analyticae praxis, 1631,10 10 SECT 1 O PRIMA. Comfarattomi fentt in [tqucnttbtis vfiirpanda. Mitatitii'it Aliaariuiis h ftgnifi{(l& mimrfmqtumb. FmftionartdHcibiUi reJuflitijf fuk Ktptatx. 1 id AC Ad V + - - + C d c ^ Ac — ba-\-Ac | id A c & had. J c dt t Ac — db dd s ddt A eg — ddb T i icb ~b~ A ~~i I T Ad icAA ~*7 d A 4 AAA Id ^ 1 1 bbb b b$>^g /Equario- TRANSLATION Signs of comparison used in what follows. Equality = as a — b signifies a is equal to b, Greater > as a > b signifies a is greater than b. Less < as a < b signifies a is less than b. Reducible fractions reduced to their equivalents [•••] 46 CHAPTER 2. FRESH IDEAS 2.2.2 Descartes' notation, 1637 Because of its enormous influence, Descartes' La geometrie (1637) was the book that did more than any other to standardize modern algebraic notation. Like Harriot, Descartes used lower case letters, and he began by using a, , from the beginning of the alphabet, but later adopted the convention of using x, y, and z as unknown quantities. Also like Harriot, he wrote xy for x times y, and xx for x times x, but introduced superscript notation x3, x4, ... for higher powers (though for some reason the convention of writing xx for ^-squared lingered on well into the eighteenth century). Continental mathematicians took up Descartes' notation wholeheartedly. In England, Harriot's a rather than Descartes' x survived until at least the end of the seventeenth century, but then fell out of use. On the other hand, Descartes' oo for equality was eventually ousted by Recorde's =, and Descartes' *JC. by the more adaptable The extract below is a very short section from Descartes' La geometrie, illustrating not only his notation but also some of the difficulties of contemporary typesetting. Descartes' notation from Descartes, La geometrie, 1637, 326 Mais fappofons la icy eftrc poffible, & pour en abregerlesteraies, au lieu des quan- . (fglz.-- d e tax. r . . titcs -,—.....- eicnuons xm , & au lieu de ť t •- cgz.x, ---- eicnuons —; &. aioli nous au- rons in * b cfglx -bcfgxx (jont [a raci- y y mmy~ ~ xy------ ne eft n x -p/ xm n x nnxx*~h bcfglx ■■ íi/jjtí, yx>m°- ~-f- mm'--—-—«s~-^-^ -í-- e z." e t z, z. imn z. & dereche f pour abreger , au lieu de bcfgl .. , »» --bif" H--^—eicnuonsOj&au lieu de-^ ■ tx, ■ cgz, t e ■• cgx. z. efcriuons ~m. car ces quantises eftant toutes donnees, nons les pouuons noramer comme ii nous plaift. & ainfi nous auons ______ y nm-^x -+■ ^ mm -f- ox-~.v.v, qui doit eft re h loiigeur dcla ligne B C, en laiflaut A B,ou x indeter- rnince. 2.3. analytic geometry translation But let us suppose here that it is possible, and to shorten the terms, in place of the cflgz — dekzz , . t r ^ezz + cfSz ~~ bcgz quantities---let us write 2m, and in place of-r-let us write ez* — egzz ezi - cgzz 2 tt jLti bcfplx — bcfgxx —; and thus we will have yy — 2 my--xy H----of which the root is z z ez* — cgzz nx / 2mnx nnxx +bcfglx — bcfgxx y = m---h J mm---1---r-■. z V z zz ez3 — egzz r 2mn bcfgl . . And again to shorten it, in place of---1--r-let us write o, and m place z ez* — egzz nn — bcfg p of — [+] —5——— let us write —. For these quantities all being given, we can name zz ez5 — cgzz m n I p them as we please, and thus we have y — rn--x + ./ mm + ox--xx, which must z V m be the length of the line BC, leaving AB, or x undetermined. 2.3 ANALYTIC GEOMETRY The invention of analytic geometry is usually attributed to Descartes and Fermat, but the foundations were laid before either was born, by the French lawyer and mathematician Francois Viete in the early 1590s. In this section we look at some of the difficult but important ideas put forward by Viete, and how they were used, adapted, and eventually superseded by Fermat and Descartes. 2.3.1 Viete's introduction to the analytic art, 1591 Viete's most important contribution to mathematics was his recognition that geometric relationships could be expressed and explored through algebraic equations, leading to a powerful fusion of two previously distinct legacies: classical Greek geometry and Islamic algebra. The central technique of algebra, taught in many sixteenth-century texts as the 'Rule of Algebra', was that one should assign a symbol or letter to an unknown quantity and then, bearing in mind the requirements of the problem, manipulate it alongside known quantities to produce an equation. For Viete, never content with a simple idea unless he could clothe it in a Greek term, this was the classical method of 'analysis' in which, he claimed, one assumes that what one is seeking is somehow known and then sets up the relationships or equations it must satisfy. Thus Viete saw the application of algebra to geometric quantities as the restored art of analysis. From this there followed some important consequences. One was that, in solving a problem or proving a theorem, all the relevant geometric magnitudes, given or sought, had to be represented by letters or 'species'. Viete set up a scale of dimensions: length, chapter 2. fresh ideas square, cube, square-square, square-cube, and so on, and frequently introduced artificially contrived 'species' such as Apianum or ZS0Mum to keep his equations homogeneous. This made his notation almost impossible to generalize beyond three or four dimensions, but it led for the first time to equations in which all the quantities, known or unknown, were represented by letters rather than numbers. A second consequence of Viete's method was that the art of creating equations (which he called zetetics) and solving them (which he called exegetics) came to be seen as essential tools for analysing and solving geometric problems, and Viete wrote separate treatises on each. In particular he introduced a numerical method for solving equations that could not be handled algebraically, the first European mathematician to do so.4 Viete believed that his methods of analysis and equation-solving could not only help to restore the lost or incomplete work of the Greek geometers, but could also enable mathematicians to handle previously intractable problems, in particular the classical problems of doubling the cube and trisecting an angle. So inspired was he by these new possibilities that he ended his first treatise, Ad artem analyticem isagoge {Introduction to the analytic art) of 1591, with his hopes for the future written in capital letters: Nullum non problema solvere (To leave no problem unsolved). Viete s idiosyncratic blend of Greek terminology and awkward notation make him a difficult author to read. Understanding of his ideas grows only with time and repeated reading of his various tracts, both singly and in relation to each other. Nevertheless, the opening chapter of the Isagogeis given below because it contains the seeds of some vitally important ideas. It was here, for example, that the word 'analysis' first entered modern European mathematics; it has since evolved through several changes of meaning but has never disappeared from the mathematical lexicon. It was here too that Viete first claimed that geometric magnitudes could be discovered through setting up and solving equations, and argued that all of this could be done in symbols. These were profound ideas that were to lead eventually to the development of powerful general techniques. 4. The relevant treatises are Zeteticorum libri quinque (1593), De numerosapotestatum ad exegesin resolutione (1600), and De recognitione aequationum (1615). 2.3- ANALYTIC GEOMETRY Viete's vision of the 'analytic art' from Viěte, Isagoge, 1591, 4 IN ART EM ANALYTICEM I S A G O G E. T>e definic ione &partieionczs4nalyßos} & deijsqueiuutinl Zeteticm. Ca n t i. °^-^Sga2 ST veiicacis mquiteftdas viaqusdarn in Máríítírnáacis,quatit Platopn'ßltis -i'jpi 'v* inueniíTedicicur^áTheonenorainatá AníIyíÍ35Kabeodefn definita,Aclíum-ptio quKÍíci tanquam conceííi pec confequencia ad verum concelíitm. Vc ccncrä Synthefis, Adfamptio concefťi per coňféquCntiá äd quatfiti fmeíti & comprehenfionem. Et quanquam veteres duplicem tancům propofucrunt Analyíim Qrrmxh» á wojii5iKl*o,adq«as definkio Thcbnismax'ime pertiíicr/oníHrííi ramen ctiam lertiam fpecícm)qua: dicatur p;™^» S^ayvnxii» confentanw11"1 e^ > vc fic Zetctice qua inuenitur a;quaIicaspfoportioucmagniiudinisdequa quamtur cumijs qua; data funr. Poriítice, quadc.arqualitateYeLprQporti.oncotdinatiTh.eoremátisveritästfcíaminatur. Exegeticc, qtrl ěx órdinacá arquaíitate vcí proportioneipfadequa qušťituř éxibeťur magnicudo, Acqucadeó totaars Analyricetiipleír illud fibi veridicans officium definia-tur.dočtrina bene inuenicňcíi in Mathcmati.Gis, Ae quod adZeteticém qtliděitl áttinet, íťiířituitdřartetogicapér fyllogifmos &c cnthymernaca,quorum fitmamétáTuntea ipfa quibus arqualkates & propoiuoncs concluduncur fyrnbola ,cam ex communibüs dcrí-uaivia nocionibus,quäm ordinandis vt jpfius Analyfeos dicoremaris.Fdrfriá aiité Zetc^ íim ine«ndicxartepropríácflr,noniamin:nuiticris fuam-logicarn exeríčíite,quse fuic ofcitancia veterum Analyítarum.fcd per logiftiCerafubfpecie nouíteriridučěňdam , fe-liciorem ňluffoSí potiorem numérofáacícomparandum inrer fe tnagrlituditleSjpropo-fitá primům homo-geneorüm lege,£tindeconftitutä,vc fit, folcßini ttiáglíitudinumex genetead geíius vi fuá proportionaliteradfceadentiiim vel defcendcntiumíeťiefeij Cca* lij quá gradus čarundem & genera in comparationibus dcíigneríttit ac diíthigilahttlť. De Symbolu aqualitatum&projjortiomim. C A í v T n. SYmbola axjualitatum &proportianum notiora qux habcnrtlť iriElementisadfumic Analytice vcdemonítrata,qualia fui.itferc,. i Tocum íuis pattibus xquau. z Qiuceidem :cquanciir,intcr fe eíTea;qualia. 3 Si arqualia squaiibus addantur,tota eíTe ícquai/ai A iiij chapter 2. fresh ideas TRANSLATION INTRODUCTION TO THE ANALYTIC ART On the definition and parts of Analysis, and thence what is useful to Zetetics. Chapter I. There is a certain way of seeking the truth in mathematics, which Plato is said to have first discovered, called Analysis by Theon, and defined by him as: assuming what is sought as though given, from the consequences the truth is given. As opposed to Synthesis: assuming what is given, from the consequences one arrives at and understands what is sought. And although the ancients proposed only two kinds of analysis, zetetics and poristics, to which the definition of Theon wholly applies, I have nevertheless also added a third kind, which is called rhetics or exegetics, agreeing that it is zetetic by which are found equalities in the ratios of magnitudes, between that which is sought and those that are given. Poristic is that by which the truth of a stated theorem is examined from the equality or ratio. Exegetic is that by which the magnitude one seeks is discovered from that stated equality or ratio. And therefore the whole analytic art taking these three tasks to itself, may be defined as the doctrine of proper discovery in mathematics. And what particularly pertains to zetetics is founded on the art of reasoning with syllogisms and enfhymemes, of which the main features are those laws from which equalities and ratios may be deduced, to be derived as much from common notions as from the theorems stated on the strength of Analysis itself. Moreover the way of discovery by zetetics is by its own art, exercising its reasoning not now in numbers, which was the shortcoming of the ancient analysts; but by a newly introduced reasoning with symbols, much more fruitful and powerful than the numerical kind for comparing magnitudes, first by the proposed law of homogeneity, and then by setting up, as it were, of an ordered series or scale of magnitudes, ascending or descending proportionally from one kind to another by power, from which the degree of each and comparisons between them are denoted and distinguished. 2.3.2 Fermat and analytic geometry, 1636 Pierre de Fermat was introduced to the mathematics of Viete by his friend Etienne d'Espagnet while he was studying law in Bordeaux between 1626 and 1630; d'Espagnet's father had known Viete personally and owned copies of his books, which were otherwise not easy to obtain. Fermat was profoundly influenced by what he read, and Viete's style of thinking and writing are clearly discernible in Fermat's early work. Viete had firmly established the connection between algebra and geometry, but his constructions remained essentially fixed and static, whereas Fermat began to investigate the locus, or range of possible positions, of a point constrained by certain conditions. In 1636 he attempted a restoration of two lost books of Apollonius on plane loci, and shortly afterwards wrote a short treatise entitled 'Ad locos pianos et solidos isagoge' 2.3. analytic geometry ('An introduction to plane and solid loci'), where he showed that all second degree equations correspond to one or other of the conic sections. (Fermat referred to general second-degree equations as 'affected equations', that is to say, the square term is 'affected' by the addition or subtraction of terms of lower degree.) Fermat's treatment was based directly on that of Apollonius, where each conic section is distinguished by a particular relationship between the ordinates and the diameter (see 1.2.4 for the parabola). That is to say, each section has a natural co-ordinate representation in which the diameter is taken as one axis and any line parallel to the ordinates as the other. The axes are not external, but embedded into the curve itself, and need not be at right angles, and Fermat was able to demonstrate that simple variations in the equation correspond only to changes in the choice of axes. Fermat sent 'Ad locos' to Marin Mersenne in Paris in 1637, but by the end of the year it was overtaken by the publication of Descartes' La geometrie, and remained unpublished in Fermat's lifetime. After his death, his son Samuel published some of his papers and letters in a compilation entitled Varia opera mathematica (1679), and the extract below is taken from that edition. While editors of mathematical texts generally try to remain true to the words of an original, they sometimes have less compunction about 'improving' the notation or diagrams, and Samuel was no exception. Fermat had used Viete's notation A quadratus or Aq, but Samuel upgraded this to the more modern A2. It is likely that Fermat's diagrams were also redrawn for the printed edition, and we do not know how close they are to the originals. Later editions and translations have introduced yet further changes to notation and diagrams (compare, for example, the various extracts from Fermat in this book with those in Fauvel and Gray or other source books).