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Symbolic Language in the Algebraization of Mathematics:

The Algebra of Pierre Hrigone (1580-1643)1

Abstract

The creation of a formal mathematical language was fundamental to the algebraization

of mathematics. A landmark in this process was the publication of In Artem Analyticen

Isagoge by Franois Vite (1540-1603) in 1591. This work was diffused through many other

algebra texts, such as the section Algebra in Cursus Mathematicus (Paris, 1634-1637, second

edition 1642) by Pierre Hrigone (1580-1643).

The aim of this paper is to analyze several features of Hrigones Algebra. Hrigone

was one of the first mathematicians to consider that a symbolic language could be used as a

universal language for dealing with pure and applied mathematics. Hrigones work was an

important means of diffusing Vites algebra. But we show that Hrigone did not make use

of Vites notation, presentation style and procedures. In addition, we emphasize how he

handled algebraic operations and geometrical procedures making use of some propositions

of Euclids Elements.

Keywords: Pierre Hrigone, Franois Vite, algebra, Cursus Mathematicus, algebraization,

symbolic language.

Introduction

The process of algebraization of mathematics took place from the beginning of the

seventeenth century to the beginning of the eighteenth century.2 It was mainly the result of

1 This article is an extended version of my communication Symbolic Language in the Algebraization of Mathematics, 1600-1660 for the symposium: Speciation in Science. Historical-Philosophical Studies on the Emergence and Consolidation of

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the introduction of algebraic procedures for solving geometrical problems. These

procedures enabled two fundamental advances in mathematics to occur: the creation of

what is now called analytic geometry and the emergence of infinitesimal calculus. These

disciplines became exceptionally powerful as connections between algebraic expressions

and curves and between algebraic operations and geometric constructions were established.

In this process of algebraization, the creation of a formal language for dealing with

algebraic equations, geometric constructions and curves was essential.

The publication in 1591 of In Artem Analyticen Isagoge by Franois Vite (1540-1603)

constituted an important step forward in the development of a symbolic language. He used

symbols to represent both known and unknown quantities3 and was thus able to investigate

equations in a completely general form. He solved equations geometrically using the

Euclidean idea of proportion: proportions can be converted into equations by setting the

product of the medians equal to the product of the extremes [Vite, 1970, 2].

Moreover, Vite introduced a new analytical art for solving problems. He divided his

new analytical procedures into three stages. The first stage consisted in transforming the

problem into one equation composed of known and unknown quantities (zetetics). In the

second, he tested the correctness of any stated theorem, which will be later used for solving

equations (poristics). The third stage, which was most important to him, consisted in

determining the value of the unknown quantity, thus solving the equation (exegetics). The

goal of his analytic art (also called algebra) was to solve all problems.4

Scientific Disciplines. This symposium took place at the XXII International Congress of the History of Science, which was held 24-30 July 2005 in Beijing. 2 There are many useful studies of this subject, including Mahoney, 1980, 141-156; Mancosu, 1996, 84-86; Pycior, 1997, 135-166; Bos, 1998, 291-317; and Panza, 2005, 1-45. 3 Nevertheless, Vite scarcely used symbols to represent operations. Further details on his notation are provided on page 21. 4 Vite sum up these ideas at the end of the Isagoge: Finally, the analytic art, endowed with its three forms of Zetetics, Poristics and Exegetics, claims for itself the greatest problem of all, which is TO LEAVE NO PROBLEM UNSOLVED, in

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Vite showed the usefulness of algebraic procedures for solving equations in

arithmetic, geometry and trigonometry [Vite, 1970; Giusti, 1992; Freguglia, 1999; Bos,

2001]. His symbolic language was the tool he used to develop this program. As Vite's

work came to prominence at the beginning of the seventeenth century, some

mathematicians began to consider the utility of algebraic procedures for solving all kind of

problems.5

In this paper, we outline the contribution of one of these mathematicians, Pierre

Hrigone (1580-1643). The originality of Hrigone is that he was one of the first

mathematicians who considered symbolic language a powerful tool for solving problems.

Hrigone wrote an encyclopedic textbook of five volumes (six volumes in the second

edition) entitled Cursus Mathematicus. The first four volumes were published in 1634 and

the fifth volume in 1637. A sixth volume appeared in 1642 as a supplement to the second

edition of the Cursus.

Hrigones aims in the Cursus were to introduce a symbolic language as a universal

language for dealing with both pure and applied mathematics using many new symbols,

citations and abbreviations.

In this paper, we analyze the section Algebra, in the second volume of the Cursus, and

two algebraic sections, entitled Supplement and Isagoge, in the sixth volume published with

the second edition of the Cursus. We will show that, though Hrigone dealt with equations

Latin, Denique fastuosum problema problematum ars Analytice, triplicem Zetetices, Poristices & Exegetices formam tandem induta, jure sibi adrogat, quod est, NULLUM NON PROBLEMA SOLVERE. [Vite, 1970, 12]. 5 Pierre de Fermat (1601-1665) was among the mathematicians who used algebraic analysis to solve geometric problems. He did not publish any of his work during his lifetime, although it circulated in the form of letters and manuscripts and was referred to in other publications. On Fermat see Fermat [1891-1922, 65-71 and 286-292] and Mahoney [1973, 229-232]. The most prominent figure in this process of algebraization was Ren Descartes (1596-1650), who published La Gomtrie in 1637. There are many useful studies on Descartes, including [Mancosu, 1996, 62-84; Giusti, 1987, 409-432; and Bos, 2001, 225-412]. With the dissemination of the works of Fermat, Vite and Descartes during the seventeenth century, symbolic language and algebraic procedures were applied in different ways to the resolution of problems and the process of algebraization was practically complete.

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and their solutions using Vites statements, his notation, presentation and procedures were

quite different. Indeed, Hrigone replaced Vites rhetorical explanations in geometrical

procedures by symbolic language in an original way, justifying it with the help of Euclids

Elements. We will also show how Hrigone stated and proved a theorem regarding

polynomial equations generalizing some examples in Vites algebra.

In Section 1, we examine several features of symbolic language at the time (1634) of

Hrigones publication of Algebra as part of the Cursus. In Section 2, after briefly

describing the different parts of Hrigones textbook, we analyze Algebra and describe its

notation, treatment of equations and geometric constructions. We also highlight how

Hrigone used Euclids Elements to establish the main results in his Algebra. In Section 3,

we discuss the influence of Hrigones Cursus on 17th-century mathematicians.

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1. Symbolic Language in the Early Seventeenth Century

For a better understanding of Hrigones contribution and in order to clarify several

points concerning Vites innovative use of symbols, we present an overview that begins

with the first manipulation of symbolic language. We focus on a number of significant

cases.

The language used before the seventeenth century was mainly rhetorical and then later

rhetorical with abbreviations [Cajori, 1928-29].6 For instance, in his treatise Al-kitab

almukhtasar fi hisb al-jabr wal-muqabala (c. 825), Al-Khwarizmi (780-850) describes

different kinds of equations using rhetorical explanations. His proofs are given in the form

of codified statements. There are no symbols in his work. In Robert of Chesters Latin

translation (1145) of Al-Khwarizmis work, the square power and unknown quantities are

translated by the word substantiae and radix or ra [Karpinski, 1915, 68]. Later, when

Leonardo de Pisa (1170-1240) (known as Fibonacci) expresses these Arabic rules in his

Liber Abaci (1202), he uses radix to represent the thing or unknown quantity (also

called res by other authors) and the word census or ce to represent the square power.

This rhetorical language continued to be used in several algebraic works in the early

Italian Renaissance, such as Summa de Arithmetica, Geometria, Proportioni e Proportionalit

(1494) by Luca Pacioli (1445-1514),7 Ars Magna Sive de Regulis Algebraicis (1545) by

Girolamo Cardano (1501-1576) and Quesiti et Invenzioni Diversa (1546) by Niccol

Tartaglia (1500-1557). Some works, such as Algebra (1572) by Rafael Bombelli (1526-

1573), feature an initial process of abbreviation of language as a tool for solving problems

6 In Diophantus Arithmetic (300 AD) we find symbols for unknowns, although he only mentions these in the introduction and rarely uses them in the text.

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[Parshall, 1988, 129-152].8 Typical features of the Italian school, in terms of symbolism, were

the use of the letters p and m to represent addition and subtraction and the letter R to

represent roots. To represent unknown quantities, cosa was abbreviated to co., the

square or census to ce., the cube to cu., etc.

The influence of German algebras, known as cossic algebras9, particularly texts such as

Coss (1525) by Christoff Rudolff (1499-1545), Arithmetica Integra (1543) by Michael Stifel

(1487-1567) and Algebrae Compendiosa Facilisque Descriptio (1551) by Johann Scheubel

(1494-1570), was also significant. In Germany, authors used the signs + and and the

symbol V for the square root. To represent unknown quantities they generally used a

different symbol for each power.10

In France, Jacques Peletier (1517-1582) was one of the influential mathematicians who

contributed to the development of algebra during the sixteenth century (before Vite) [Van

Egmond, 1988, 141; Cifoletti, 1996, 121-142]. He wrote LAlgbre (Lyon, 1554), which was

based on Stifels Arithmetica Integra and shows sign of Italian influence in its use of

symbolism (for example, he used the letters p and m to represent addition and subtraction)

[Peletier, 1554, 16]. To complete an overview of algebra texts, in France, we can quote, for

instance: Jean Borrel (Johannes Buteo, 1492-1572) that wrote Logistica Quae et Arithmetica

Vulgo Dicitur (Lyon, 1559) and used several Italian works as his main sources; Pierre de la

Rame (Petrus Ramus, 1515-1572) that wrote Algebra (Paris, 1560) using Scheubels Algebra

as a source; Simon Stevin (1548-1620) that wrote LArithmtique Contenant les

Computationsaussi lAlgbre (Leyden, 1585) using his own symbolism and terminology

7 Hoyrup provided an account of the innovations in Italian abacus algebra and referred to mid-fourteenth-century formal calculations of fractions [Hoyrup, 2000, p. 11]. 8 Bombelli used symbols to represent parentheses and powers, but algebraists in Renaissance Italy consistently used syncopations. 9 This name derives from the treatment of problems with an unknown quantity called cosa.

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except in the case of exponents, which he borrowed from Bombellis notation [Stedall, 2002,

49]. Consequently, there was no clear algebraists line in France, but rather many individual

contributions.

The notation and formalism of algebraic expressions evolved after the works by Vite

and Descartes (La Gomtrie, 1637) had been published (Vite, for instance, defined the

properties of operations between algebraic expressions). However, there were no unifying

criteria and so different notations were used in algebraic works for many years. If we consider

the symbolic language in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas

(1631) by Thomas Harriot (c.1560-1621) and in Clavis Mathematica (1631) by William

Oughtred (1573-1660), who publicized Vites work in England, we again find that there

were no standard criteria for symbols or mathematical terms in the seventeenth century

[Stedall, 2002, 55-125].

In fact, from 1637 to the middle of the eighteenth century, mathematics became

algebraized to a considerable degree. However, as different mathematicians used different

languages, new analytical methods derived from the algebraic analysis of Vite were not

considered a well-founded new science, even though they constituted a very powerful tool for

calculation in comparison with geometry. Therefore, this process of algebraization, which

involved a change from a predominantly geometrical way of thinking to a more algebraic or

analytical approach, was slow and irregular.11 Not all mathematicians adopted algebraic

procedures during this period: some considered these new techniques and symbolism as an

10 In sixteenth-century Spain, several algebra texts were published as part of works on arithmetic, but to our knowledge they did not influence European algebraic works. See Massa, in press. 11 In that period, classical mathematical thought was recovered through Latin translations of Greek texts. At the same time, algebraic procedures were introduced in mathematical works. These procedures were very productive. Their significance sometimes contradicted the understanding of classical techniques. According to Mahoney, this new algebraic thought had three features: the use of a new symbolism that abbreviated notation and acted as a powerful tool for operations; an emphasis on mathematical relations rather than mathematical objects; and abstraction, which meant freedom from mere physical representation [Mahoney, 1980, 141-155].

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"art" and tried to justify them according to a more "classical" form of mathematics;12 others

disregarded algebra because their research evolved along other paths. Eventually, a few

mathematicians accepted these new techniques as a complement to their mathematical

procedures.

Indeed, one of the key points in this process of algebraization of mathematics was the

establishment of a symbolic language as a formal language, so that the new language of

symbols and techniques could be used in operations to obtain new results. Therefore, in order

to reflect on this problem in a new way, we are going to consider Hrigones attempt to

make knowledge of mathematical science easier and more accessible by introducing his

own symbolic language as a universal language. As can be deduced from this overview, the

work of Hrigone represented an intermediate stage in the formation of a formal language.

2. Algebraic Procedures in Hrigones Cursus Mathematicus

Very little is known about Pierre Hrigones life. Per Stromholm claims that he was

from the Basque Country and that he taught mathematics in Paris [Stronholm, 1970-1991,

vol. 9, 299]. It is known that he, alongside tienne Pascal, Mydorge, Beaugrand and others,

took part in the committee appointed by Richelieu to judge whether Morins scheme for

determining longitude from the Moons motion was a practical one. In fact, on 23 July

1659, in a letter to Hartlib,13 Oldenburg14 states:

As for those inventors of the Longitude, I wish that they are not like the French Morinus,

who in 1634 published, that he had found it, but was contradicted in the truth of his

12 We can mention, for example, Thomas Hobbes (1588-1679), Bonaventura Cavalieri (1598-1647), Isaac Barrow (1630-1677), etc. More details on the controversies about the status of algebra can be seen in Hoyrup, 1996, 3-4; Massa, 2001, 708-710. 13 Samuel Hartlib (Poland, c.1600-England 1662) was an intelligencer in seventeenth-century Europe. An intelligencer was an agent for the dissemination of news, books and manuscripts. 14 Henry Oldenburg (1615-1677), secretary of the Royal Society of London, tried to established contact with other scientists abroad and to obtain copies of all scientific books published.

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invention by the ablest Mathematicians, that were then in France, Mrs Pascall, Mydorge,

Boulenger, Beaugrand and Herigon, deputed commissaries for to examine the invention.

[Oldenburg, 1975-1977, vol I, 289]

2.1 Hrigones Cursus Mathematicus15

Only one work by Hrigone is known to exist: Cursus Mathematicus.16 Published in

parallel French17 and Latin columns on the same page, the first edition was a five-volume

textbook entitled Cursus Mathematicus, nova, brevi et clara methodo demonstratus, per

notas reales & universales, Citra usum cuiuscunque idiomatis, intellectu faciles.18

In the (unnumbered) preface to the first volume, which bore the dedication To the

reader, Hrigone explained that he had invented a new method for making brief and

intelligible proofs that did not require the use of any other language. He claimed that

mathematics was a very obscure science, mainly due to the difficulty of understanding

proofs.

The best method for teaching science is to combine brevity and ease. However, this may not

always be as straightforward as it seems, particularly in the case of mathematics, which, as

Cicero pointed out, is a highly obscure science. Having tackled this matter myself, I have

come to the realization that the greatest difficulties lie in the proofs, which are dependent on

an all-embracing knowledge of mathematics. I have therefore devised a new method19

15 There are three other editions of Cursus, but according to Stromholm these consisted of nothing new. See Stromholm, [1970-1991, 299]. 16 Cifoletti mentioned other works on Euclid but Stromholm explained that they consisted of little more than the French portion of Vol. I of Cursus. See Cifoletti [1990, 146] and Stromholm [1970-1991, 299]. 17 When Hrigone published Cursus, there were just two other editions in French of Vites In Artem Analyticen Isagoge: Lalgbre nouvelle de M. Viette (1630) by A. Vasset and Introduction en lart analytic, ou nouvelle algbre de Franois Vite (1630) by J.L. de Vaulezard. Also, in 1630, M. Ghetaldi published De resolutione et compositione mathematica libri quinque opus posthumus, which dealt with Vites analytic art. 18 In French, Hrigone entitled this work COURS MATHEMATIQUES DEMONSTR D'UNE NOUVELLE BRIEEVE ET CLAIRE METHODE, Par NOTES reelles & universelles, qui peuvent estre entendues facilement sans l'usage d'aucune langue. 19 La meilleure methode denseigner les sciences est celle, en laquelle la briefvet se trouve conjointe avec la facilit : mais il nest pas ais de pouvoir obtenir lune & lautre, principalement aux Mathematiques, lesquelles comme temoigne Ciceron, sont grandement obscures. Ce que considerant en moy-mesme, & voyant que les plus grandes difficultez estoites aux demonstrations, de lintelligence desquelles dpend la cognoissance de toutes les parties des Mathematiques, jai invent une nouvelle methode Optimam methodum tradendi scientias, esse eam, in qua brevitas perspicuitati coniungitur, sed utramque assequi hoc opus hic labor est, praesertim in Mathematis disciplinis, quae teste Cicerone, in maxima versantur difficultate. Quae cum animo perpenderem, perspectumque haberem, difficultates quae in erudito Mathematicorum pulvere plus negotij facessunt, consistere in demonstrationibus, ex quarum intelligentia Mathematicarum disciplinarum omnis omnino

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Hrigone pointed out that his method was straightforward, clear and quick, and that it was

arranged in an order that made it easy to understand.

The first and second volumes of Cursus dealt with pure mathematics. The first

volume20contains Euclids Elements21 and Data, and Apolloniuss Coniques. The second

volume is devoted to arithmetic and algebra.22The third and fourth volumes deal with

applied mathematics, that is to say, with the mathematics used in the construction of

fortifications, navigation, etc.23The fifth and last volume of the first edition, published in 1637,

also deals with applied mathematics, including spherical trigonometry and music.24

As can be seen, this work covered all parts of mathematics and its applications. In the

Prolegomena, in the first volume, after the preface entitled On the Division of

Mathematics, he explains that Pythagoreans divided mathematics into four categories:

arithmetic, geometry, astronomy and music. Other scientists divided it into pure and

applied mathematics, specifying that in pure mathematics quantities were recognized as

being separate from matter. Thus, he considers that pure mathematics should be divided

according to the kind of quantity (either continuous or discrete) in geometry and arithmetic,

and that applied mathematics should be divided into optics, mechanics, astronomy and

pendet cognitio : excogitavi novam methodum [Hrigone, 1634, I, unnumbered]. All translations are the authors own unless otherwise stated. 20 Tome premier du Cours Mathematique, contenant les XV livres des Elements dEuclide. Un appendix de la geometrie des plans. Les dates dEuclide. Cinq livres dApollonius Pergeus du lieu resolu. La doctrine de la section des angles. Tomus primus, Continens Euclidis Elementorum Lib. XV. Appendicem Geometriae Planorum. Data Euclidis. Apollonii Pergei de loco resoluto, Lib. V. Doctrinam Angularium Sectionum. [Hrigone, 1634, I, unnumbered]. 21 Hrigone divided the thirteen books of Elements into four sections: the first included the six first books, the second the following three, the third the tenth and the final section the last five books. 22 Tome seconde contenant larithmetique pratique: le calcul ecclesiastique: & lalgebre, tant vulgaire que spcieuse, avec la mthode de composer & faire les dmonstrations par le retour ou rptition des vestiges de lanalyse. Tomus secundus. Continens arithmeticam practicam: Computum Ecclesiasticum: & Algebram, tum vulgarem tum Speciosam, un cum ratione componendi ac demonstrandi, per regressum sive repetitionem vestigiorum Analyseos. [Hrigone, 1634, II, unnumbered ]. 23 Le troisime, la construction et usage des tables des sinus et logarithmes, la gomtrie pratique, les fortifications, la milice et le mcanique. Tomus tertius. Continens constructionem tabularum sinuum, & Logarithmorum, un cum earum usu in Anatocismo, & triangulorum rectilineorum dimensione : Geometriam practicam : Artem muniendi : Militiam : & Mechanicas. [Hrigone, 1634, III, unnumbered]. La quatrime, la doctrine de la sphre du monde: la gographie et lart de naviguer Tomus Quartus. Continens Sphaerae mundi doctrinam: Geographiam tam veterem qum novam, gradibus & minutis longitudinum ac latitudinum designatam & Histiodromiam. [Hrigone, 1634, IV, unnumbered]. 24 La cinquime: la dioptrique, la perspective, (and) trois livres des sphriques de Thodose, avec un trait de la mesure des triangles sphriques: la thorie des plantes, la gnomonique et la musique. Tomus quintus ac ultimus. Continens Opticam,

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music. Hrigone stresses that all the parts in his volumes are separated in such a way that

readers may devote themselves only to those that are necessary to them. As regards the

order, he explains that in the first few volumes readers will find everything they need to

understand the volumes that follow. In addition, in each volume he groups related subjects

together. He changes the classical order of Quadrivium and begins with geometry. We

return to this idea below when we analyze the relationship between Hrigones Algebra and

Euclids Elements.

In the second edition (1642), he adds the sixth, final volume, which contains two parts

dealing with algebra (hereinafter, we call these two parts Supplement and Isagoge

respectively); it also deals with perspective and astronomy. He includes a chronological list

beginning in 2000 BC of the most important events in science and a list of outstanding

mathematicians, in which he mentions Archimedes, Aristarchus, Euclid, Plato, Pythagoras,

Ptolemy, Campanus, Bacon, Commandino, Kepler, Clavius, Descartes and others.25

2.2 Hrigones View of Algebra

In the second volume of Cursus Mathematicus, following the first part entitled

Arithmetic,26Hrigone includes a 296-page treatise entitled Algebra tum vulgaris tum

Catoptricam, Dioptricam, Perspectivam, Sphaericorum Trigonometriam, Theoricas Planetarum, tam secundum stantis, qum motae terrae hypothesim, Gnomonicam, & Musicam. [Hrigone, 1637, V, unnumbered]. 25 This supplement to Cursus Mathematicus is entitled Tome sixime et dernier, ou supplment du Cours Mathmatique, contenant les Affections Gomtriques des quations cubiques, pures & affectes. LIsagoge de lAlgbre. La mthode de mettre en perspective toutes sortes dobjets par le moyen du Compas de proportion. La Thorie des Plantes, distingue selon les hypothses de la terre immobile & mobile. LIntroduction en la Chronologie, avec une table des choses plus notables par ordre alphabtique: Et un Catalogue des meilleurs Auteurs des Mathmatiques. Tomus sextus ac ultimus, sive supplementum, continens Geometricas aequationum cubicarum purarum, atque affectarum Effectiones [Hrigone, 1642, VI, unmbered]. The supplement is divided up as follows: Supplement on Algebra (1-73), Isagoge (Introduction) to Algebra (74-98), On Perspective (99-116), Theory of the Planets (116-158) and In the Chronology (159-267). This book is written exclusively in French from page 74 onwards and was published in 1642 shortly after the publication, in 1637, of Ren Descartes Gomtrie. 26 Arithmetic is an 162-page treatise comprising 18 chapters. Hrigone defines numbers and operations using integer numbers, decimals numbers and fractions. He also defines rules for solving arithmetic problems, such as the rule of three, the false position rule, etc., and uses them to solve problems. See Hrigone [1634, II, 1-162].

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speciosa, un cum ratione componendi ac demonstrandi, per regressum sive repetitionem

vestigiorum Analyseos,27 of which there are twenty chapters.

1: several definitions and notations.

2, 3: operations involving simple and compound algebraic expressions.

4: operations involving ratios.

5: proofs of several theorems.28

6, 7: rules for dealing with equations, which are the same as those in Vites work.29

8: an examination of a theorem by poristics.30

9: rules of the rhetique or exegetic in equations up to the second degree.

10-13: solutions of several problems and geometric questions using his proofs (determined

by means of analysis).

14: solutions of several ambiguous equations.

15: solutions of problems concerning squares and cubes, referred to as Diophantus

problems.

16-19: calculation of irrational numbers.

20: several solutions of affected (negative sign) powers.

In the following paragraph we clarify the idea of algebra in the thought of Hrigone

through his explanations, definitions and uses.

27 In French, Hrigone entitled this treatise LAlgbre, tant vulgaire que spcieuse, avec la mthode de composer & faire les dmonstrations par le retour ou rptition des vestiges de lAnalyse. [Hrigone, 1634, II, unnumbered]. 28 According to Hrigone, these theorems by Vite are very useful for solving equations and particularly difficult problems. 29 These rules were the reduction of fractions to the same denominator (isomerie), the reduction of the coefficient of the highest degree (parabolisme), the depression of the degree (hypobibasme) and the transposition of terms (antithese). See Hrigone [1634, II, 83-89]. 30 As for the term poristics, Hrigone discussed four cases: 1) an equation becomes an identity; 2) an equation becomes impossible; 3) an equation becomes indeterminate and 4) an equation becomes a theorem. See Hrigone [1634, II, 92-94].

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His first explanation of algebra is found in the preface to this second volume. Hrigone

explains that Algebra was justly placed last in the book (after geometry and arithmetic), as

it is useful in all scientific fields:

In keeping with the order of doctrine, it is necessary for analysis (which is usually called

algebra) to be placed last in this course, because it works its magic on all of the books parts

().31

The usefulness of algebra in all branches of science was clear in the mind of Hrigone. He

emphasizes that analysis, usually called algebra, is not limited by the kinds of problems to

be solved: () the exposition of an art that does not recognize limits.32

In the first chapter of Algebra, entitled Definitions of Algebra, Hrigone calls algebra

the new analytic art, in the words of Vite, and defines it as follows:

Analytical doctrine or algebra -called cosa in Italy- is the art of finding the unknown

magnitude, by taking it as if it were known, and finding the equality between this and the other

given magnitudes.33

Like Vite, Hrigone presents this art as analysis, that is to say, he assumes that the unknown

quantity verifies the equality to be solved. This was not a new idea. In Spain, for example,

Marco Aurel [ca. 1520], in his Libro primero de Arithmetica Algebratica (Valencia, 1552),

put forward a similar idea:

I say that, by that rule, to ask a question you have to imagine that such a question has

already been asked, and answered, and that you now want to prove it.34

However, Aurel, like Antic Roca [ca. 1530] and others, only used this idea of analysis to

solve arithmetical problems, that is to say, to deal with questions that involved letters or

31 Pour observer lordre de doctrine, il falloit que lanalyse (quon appelle ordinairement Algebre) tint le dernier lieu en la distribution des parties de ce Cours, tant pour ce quelle opere ses merveilles sur tous les membres de ce corps . Doctrinae methodus exigebat, ut analysis (quam cum vulgo vocamus Algebram) partium omnium postrema traderetur [Hrigone, 1634, II, unnumbered]. 32 Lexposition dun art qui ne reconnat de bornes ut subsistentes in arte, quae terminos non agnoscit (Hrigone, 1634, II, unnumbered]. 33 La doctrine analytique, ou lAlgbre est lart de trouver la grandeur incognue, en la prenant comme si elle estoit cognue,& trouvant lgalit entre icelle & les grandeurs donnes. Doctrina analytica quae Algebra & Italico vocabulo cosa dicitur, est ars qua assumpta quaesita magnitudine tanquam nota, & constituta inter eam aliasque magnitudines datas aequalitate, invenitur ipsa magnitudo de qua quaeritur. [Hrigone, 1634, II, 1].

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abbreviations representing numbers.35 In contrast, Hrigone, like Vite, used the algebraic

method of analysis to solve all kinds of problems and also to prepare the proofs.

Indeed, Hrigone explicitly distinguishes vulgar algebra, which deals with numbers,

from specious algebra, which deals with species:

One may distinguish between vulgar and specious [Vites] algebra. Vulgar or numerical

algebra is that which is practiced by numbers. Specious algebra is that which exerts its logic by

species or by the forms of the things designated by the letters of the alphabet. Vulgar algebra

is only used to find the solution of arithmetic problems without proofs. But specious algebra is

not limited by any kind of problem, and is no less useful for inventing all kinds of theorems as

it is for finding solutions to and proofs of problems.36

The word specious has multiple interpretations: Wallis reflects his familiarity with the civil

law where the word is used to designate unknown or indefinite defendants [Wallis, 1685, 66].

Samuel Jeake claimed that Species are Quantities or Magnitudes, denoted by Letters,

signifying Numbers, Lines, Linear, Geometrical Figures, etc. [Vite, 1983, 13, fn 8]. In our

view, species is the name used to designate the letters that represent any magnitude, whether

discrete or continuous.

This idea of extending numerical algebra to solve any kind of problem was Vites.

Hrigone accepts Vites view that the symbols of analytic art (or algebra) can be used to

represent numbers and values of any abstract magnitudes. Thus, Hrigone calls vulgar or

34 Y digo que para hacer una demanda, por la dicha regla, has de imaginar que tal cuenta o demanda ya es hecha, y respondido, y tu agora la quieres provar. [Aurel, 1552, 77]. 35 We can read similar definitions in Prez de Moyas Arithmetica prctica y especulativa (1562) and in Antic Rocas Arithmtica compilacin de todas las obras que se han publicado hasta agora (1564) [Massa, in press]. 36 Elle se distingue en la vulgaire & en la specieuse. LAlgbre vulgaire ou nombreuse est celle qui se pratique par nombres. LAlgbre specieuse est celle qui exerce sa logique par les espces ou formes des choses designes par lettres de lalphabet. LAlgbre vulgaire sert seulement trouver les solutions des problmes Arithmetiques sans demonstrations. Mais lAlgbre Specieuse nest pas limite par aucun genre de problme, & nest pas moins utile inventer toutes sortes de thormes, qu trouver les solutions & dmonstrations des problmes. Distinguitur in vulgarem sive numerosam, & in vietaeam sive speciosam. Algebra vulgaris seu numerosa est quae numeris exhibetur. Algebra speciosa est, que per species sive rerum formas litteris alphabeti designatas, suam exercet logicam. Algebra vulgaris solutiones rpoblematum arithmeticorum tantum exhibet absque demonstrationibus. Algebra ver speciosa nullo genere problematum coercetur, nec minus utilis est ad invenienda omnis generis theoremata, qum solutiones & demonstrationes problematum. [Hrigone, 1634, II, 1]. Later, Ozanam [1640-1717] wrote similar definitions in his Dictionnaire: LAlgbre vulgaire ou nombreuse est celle qui se pratique par nombres. LAlgbre spcieuse est celle qui exerce ses raisonnements par les espces ou formes des choses dsignes par lettres de lalphabet. LAlgbre vulgaire sert seulement trouver les solutions des problmes Arithmtiques sans dmonstrations. Mais lAlgbre

15

numerical (vulgaire ou nombreuse) algebra that which deals only with numbers and he calls

specious (specieuse) algebra that which deals with any kind of magnitude.37 In addition, he

specifies that vulgar algebra is limited to arithmetic problems without demonstrations and that

specious algebra is useful for solving any kind of problem, for inventing all kinds of theorems

and for proving the solutions to problems.

Hrigone continues with this idea and presents the well-known classification of geometrical

problems: plane, solid and line-like.38

The Ancients determined three kinds of (geometrical) problems, which were called plane, solid

or line-like. Plane problems are those that depend on the first three postulates in Elements and

the affections derived from them. Solid problems are those that were solved using lines of

conical sections and cylinders. Line-like problems are those that were solved using lines

generated from the intersection of two straight lines moving in a plane, such as spirals,

quadratics, conchoids and cissoids.39

He explains that only the first kinds of problems can be solved by geometry, while the other

two kinds cannot be solved thus because they cannot be described geometrically. In this

second volume, there are no solutions for solid or line-like problems. Later on, in Supplement,

Spcieuse nest pas limite par un certain genre de problme, & nest pas moins utile inventer toutes sortes de thormes, qu trouver les solutions & dmonstrations des problmes [Ozanam, 1691, 61-62].37 Vite enounces a similar idea when he defines his numerical and specious logistic: numerical logistic is dealt with by numbers. Specious logistic is shown by species or forms of things designed by letters or the alphabet. Logistice numerosa est quae per numeros, Speciosa quae per species seu rerum formas exhibetur, ut pote per Alphabetica elementa. [Vite, 1970, 4]. Vite distinguishes between specious and numerical logistic. Klein [1968-1992, 11] considers the word logistic to mean the art of calculation. Hrigone, in 1642, in the sixth volume of Cursus, specifies: Specious logistic consists more in the explanation by letters of operations and discrete and continuous quantities for which one must find the unknown (in numbers or lines) than in the calculation. [Hrigone, 1642, VI, 76]. 38 Pappus explains in the Mathematica Collection: Ancients have admitted that problems belong to three kinds in geometry: ones are called plane, others solid and others line-like. [Pappus dAlexandrie, 1982, 38-39]. Bos said that this classification was usual at that time: Pappus distinction between plane, solid and line-like problems quickly became known after 1588; from the early 1600s mathematicians referred to it as a matter of course [Bos, 2001, 219]. 39 Or les anciens ont constitu trois genres de problmes, desquels les uns ont t nommez plans, les autres solides, & les autres linaires. On attribue aux plans tous (les) problmes qui dpendent de trois postulats du premier des lments, & des effectionis drives diceux. Les problmes solides sont ceux qui se ressoudent par le moyen des lignes des sections des cnes & cylindres. Les problmes linaires sont ceux qui se ressoudent par le moyen des lignes qui sengendrent de lintersection de deux lignes droites qui se meuvent sur un plan : comme sont les lignes spirales, quadratrices, conchodes, & cissodes. Veteres autem tria genera problematum geometricorum esse statuerunt, & eorum alia quidem plana appellari, alia solida, alia linearia. Ad plana referuntur omnia problemata quae tribus postulatis primi elementorum & effectionibus, inde derivatis pendent. Solida sunt, quae lineis corporum solidorum, ut pote, coni & cylindri sectione ortis, constant. Linearia sunt quae ex duarum linearum in una superficie sese intersecantium motu, & communis sectionis vestigio delineantur quales sunt lineae spirales, quadratrices, conchodes, & cissodes. [Hrigone, 1634, ll, 2].

16

he deals with the geometric constructions found in Vites Supplementum Geometriae

(1593) and there he solves solid problems using, as Vite did, a neusis postulate.40

In order to clarify Hrigones ideas on specious algebra we cite the explanation he

gives in the Isagoge. He claims that it is essential to repeat the principles of algebra, because

the beginning is always the most difficult part of a discipline.41

Specious algebra is the name given to describe the letters of alphabet which do have not any

particular meaning in a discrete quantity, which is in terms of actual numbers, or in continuous

quantities, but are only valid in terms of the attributed significance to them. For example, if we

attribute a value of 12 to the letter B, the reasoning applied to using this letter B can transcend

the number 12 and be used to represent any other number, such as 15, 20, etc., and the letter B

will represent the nature of numbers and not their individual features. It must also be

understood that a continuous quantity (letters) may represent a line, a surface or any other

quantity that may wish to assign to it. Therefore, by using these letters one can devise universal

theorems in both continuous and discrete quantities.42

Thus, we see that Hrigone comes back to the idea of specious algebra and states explicitly

that the letters can signify discrete quantities or continuous quantities such as lines, surfaces,

etc. He also specifies that through these letters one could invent theorems in both discrete and

continuous quantities.

The algebraists listed by Hrigone in the sixth volume give us a clear idea of his views on

algebra. When he cites the term algebra together with the term practical arithmetic, he is

40 Furthermore, he includes the publication of Fermats maximum and minimum method. Hrigone did not mention Fermat in the Chronology but when he publishes Fermats method of maximum and minimum, he states that Fermat is an excellent geometer and analyst who has restored Apollonius geometric loci and has written Introduction to plane and solid loci. [Hrigone, 1642, VI, 59-73]. 41 Isagoge de lAlgbre. Ce que nous avons dit de lAlgbre en ce livre jusques ici, est son vraie complment, & ce quil lui manquait pour sa parfaite intelligence: Mais cause quen icelle, de mme quaux autres sciences, on trouve plus de difficult en son Isagoge & entre, quau reste de la science. Nous rpterons ici succinctement les principes de son Isagoge, qui se divise en cinq parties, qui sont la logistique, des quantits simples, contenant laddition, la soustraction, la multiplication, & la division: (74-78). Logistique des quantits composes(7 8-80), Des reductions des equations (80-83), Methode dextraire la racine quarre des quations quadratiques affectes. (83-85) Les questions ncessaires pour lintelligence de la pratique de ces quatre premires parties, & de linvention des quations. (85-98). [Hrigone, 1642, VI, 74]. 42 LAlgbre Spcieuse se nomme ainsi des lettres de lalphabet, qui nont aucune signification particulier, ny en la quantit discrte, qui soit les nombres, ny en la continue, sinon celle quon leur attribue. Par exemple, si on attribue la lettre B12 pour sa valeur, le raisonnement quon fera avec icelle lettre B, sans considrer le nombre 12, conviendra aussi tout autre nombre comme 15, 20, etc & par ainsi la lettre B signifiera lespce des nombres & non les individus & particuliers. Ce quil faut aussi entendre en la quantit continue, pouvant signifier une ligne, une superficie, ou autre quantit telle quon voudra,

17

referring to practical arithmetic and numerical algebra, as distinguished from Vites specious

algebra. He refers to nearly all known algebraists,43 although, surprisingly, there is no mention

of Al-Khwarizmi, Fibonacci or Descartes. However, in the catalogue (before the chronology)

of this sixth volume, in which Hrigone lists outstanding mathematical works, he quotes

Descartes and Vite. This reference is the longest in the catalogue and Hrigone cites all the

works of Vite, including those that were published after his death, and points out all of the

breakthroughs he made.

He is the first to observe that an algebraic equation can have more than two solutions, and more so

the higher the order of the equation. He also devises the universal method for calculating the root of

numbers of affected powers and is the first person to introduce the law of homogeneity in algebra.

He likewise restores, and perhaps one should go as far as to say that he is the author of analytic art,

which is now widely used. His method consists in using symbols (species) or letters of the

alphabet, of which it could be said, as opposed to analysis that does not use symbols, is an ability

acquired through extensive practice, intelligence and memory rather than an art.44

Hrigone repeats the idea that Vite was the restorer or author of the analytic art, which used

species or letters of the alphabet, and that when analysis is used without species it is not an art,

but simply a skill acquired through practice.

The Descartes citation is not as long. Hrigone explains that he has used algebra to solve

the Apollonius problem, now known as the Pappus problem, as it appears in Descartes

par le moyen des quelles lettres on invente des thormes universels tant en la quantit continue que discrte. [Hrigone, 1642, VI, 76]. 43 He cites, in this order, the following authors of (practical) arithmetic and (vulgar) algebra: Stifel, Cardano, Clavius and Buteo. In the catalogue, Hrigone mentions Stifel, Stevin, Buteo and Clavius as authors of texts on arithmetic and vulgar algebra, and Bombelli and Nuez as authors of texts on vulgar algebra. Claude Bachet (1581-1638) is mentioned in relation to Diophantus algebra.43 Vites Specious Algebra is cited, as is De resolutione and Compositione Mathematica by Marinus Ghetaldi (1566-1626). In the catalogue, Ghetaldi is also mentioned as the author of texts on analytic art that follow Vites algebras. Hrigone specifies that he had published some of Ghetaldis propositions at the end of the first volume of Cursus Mathematicus. Then he lists the main authors who had written their works in Italian: Frater Luca de Burgo (Pacioli), Tartaglia and Bombelli. He mentions that the Algebra of Petrus Nonius (1492-1577) was written in Spanish, and also points out that Peletiers Arithmetic and Algebra was in French and Latin, and that Stevins was in French. Hrigone also cites Petrus Bungus (ca. 1599), who wrote Des Mysteres des Nombres. See Hrigone [1642, VI, 256]. 44 Il est le premier qui a observ quune quation dAlgbre peut avoir plus de deux solutions, & autant plus que lquation monte haut. Il est aussi inventeur de la mthode universelle dextraire les racines des nombres des puissances affectes, & le premier qui a introduit en lAlgbre la loi des homognes ; & est aussi le restaurateur, ou plutt auteur de lart Analytique, qui est maintenant en usage, par le moyen des espces ou lettres de lalphabet, au respect duquel, lAnalyse qui nuse point despce, est plutt une facult, qui sacquiert par un long exercice, & bont desprit & de mmoire quun art. [Hrigone, 1642, VI, 240].

18

Gomtrie. Hrigone also refers to the fact that Descartes has shown how to solve algebraic

equations of degrees greater than three by means of conic sections.45

Hrigones ideas on algebra become clear through his definitions and explanations. He

follows Vites specious algebra. Till Vite, algebraists only used the idea of analysis to solve

arithmetical problems; Vite went one step further by also using this algebraic method of

analysis to solve any kind of problem and to make proofs, as Hrigone did in dealing with

specious algebra. Hrigone states that analysis by species is an art, in contrast to analysis by

numbers, which can be considered a skill. He was aware that the new specious algebra of

Vite was more powerful and productive than the numerical algebra used in cossiques

algebras. It seems to us that in Hrigones view specious algebra means that letters are valid

for any abstract magnitude, such as a figure, angle, line or number, and that these letters

represent unknowns and givens. However, as we show below (page 25), Hrigone always kept

in mind the fact that a different treatment was required depending on whether letters

represented numbers or continuous quantities.

2.3 Hrigones Notation

Hrigones project was to introduce a universal language that would make it easier for

mathematicians to work in all branches of mathematics. Thus, in this paragraph we compare

his notation with that of Vite and Descartes in order to show its originality.

Hrigone describes the notation and the terms he is going to use in his specious algebra in

pages 4 to 9 of the first chapter. There are certain differences between Hrigones signs and

45 Ren DesCartes, au livre quil a intitul De la Mthode, explique la Dioptrique, & les Mtores, par le moyen des nouveaux principes quil suppose: Et en sa Gomtrie il a trouv, par le moyen de lAlgbre, la solution du problme dApollonius Pergeus, dont Pappus fait mention au 7 livre, qui sappelle, locus ad tres, quatuor, vel plures lineas. Il a aussi enseign rsoudre, par le moyen des Sections coniques, les quations dAlgbre, qui montent au 3 & 4 degr parodique. [Hrigone, 1642, VI, 244].

19

those used by Vite and Descartes. For instance, Hrigone represents equality as 2/2,

whereas Vite uses an abbreviation of the word aequalis and Descartes uses the symbol

. To represent multiplication, Vite uses the word in, whereas Descartes and

Hrigone write one letter next to the other.

Explicatio notarum, ou des notes.

ab A in B, A multiplie par B.

a2b signif. A quadratum in B, le quarr dA multipli par B.

ab2 A in B quadratum , A multipli par B quarr.

ap A planum, A plan. [Hrigone, 1634, II, 4].

However, Hrigone does not use the term ap, and when he wishes to represent a product of

two lines, he uses a rectangle or a square and a dot followed by two letters.

He uses the symbol to express the ratio between two quantities, whereas Vite uses

the expression ad and Descartes . In his works, roots are represented by the letter V,

with the number of the root to the right of it (so the square root is represented by V2, the

cubic root by V3, and so on), whereas Vite writes VQ. to represent the square root,

VC. to represent the cubic root, and so on. Descartes uses present-day notation.

Significationes signorum radicalium.

Significations des signes radicaux.

V, U V2. radix quadra, la racine quarre.

VC, U V3. radix cubica, la racine cube [Hrigone, 1634, II, 4].

In order to use symbols to represent quantities, Hrigone, like Vite, distinguishes between

vowels and consonants. He uses vowels to represent unknown quantities and consonants to

represent data. He also uses capital letters in the figures and, unlike Vite, a lower case

letter in the text to represent unknown quantities, the first of which is denoted by the letter

a.46 To represent the powers, Vite retains the words A quadratus and A cubus and

46 Harriot in his Praxis (1631) also used the letter a to represent the unknown and wrote one letter next to the other to express the product. See Stedall [2002, 91].

20

abbreviations such as A quad., etc. Descartes writes the exponents as they are written

today, with one exception: he sometimes uses xx to represent the square. Hrigone writes

the exponents on the right side of the letter (so the square is represented by a2, the cube by

a3, and so on).

Significationes characterum cossicorum.

Significations des charactres cossiques

2 3 l a , latus seu radix, le cost ou racine.

4 9 q a2 , quadratum, le quarr.

8 27 c a3, cubus, le cube.

16 81 qq a4, quadrato-quadratum, le quarr-quarr.

32 243 qc a5, quadrato-cubus, le quarre-cube.

64 729 cc a6, cubo-cubus, le cube-cube.

128 2287 qqc a, quadrato-quadrato-cubus, le quarre-quarre-cube. [Hrigone, 1634,

II, 4]

Hrigone calls letters used to represent numbers cosic characters.

In specious algebra magnitudes which ascend or descend proportionally from one genre to another

are called scalars, but in vulgar (algebra) they are called cosic numbers.47

Hrigone also considers these numbers in continuous proportion.

The exponents of cosic characters show which quantity is in proportion from a first proportional, as

a4 shows that 16 or 81 is the fourth proportional.48

As we show below, the conceptual link Hrigone used for working with both specious and

numerical algebra is, as in Vite, the Euclidian idea of proportion.

Hrigone represents operations using the following symbols: addition is represented by the

plus sign +, subtraction by the symbol ~, and the sign of equality in a difference by a point

on the left and an additional point on the right of the previous symbol, thus: .~ :.

47 En lAlgebre specieuse les grandeurs qui montent ou descendent proportionnellement, de genre en genre, sappellent scalaires, mais en la vulgaire se nomment nombres cossiques. In Algebra speciosa magnitudines quae ex genere ad genus, sua vi proportionaliter ascendunt vel descendunt vocantur scalares, in vulgari ver appellantur numeri cossici. [Hrigone, 1634, II, 3]. 48 Les exposants des caractres cossiques montrent la quatrime est une chacune des proportionnelles, depuis la premire proportionnelle : comme a4 montre que 16 ou 81 est la quatrime proportionnelle. Exponentes characterum cossicorum

21

Explicatio signorum affectionis.

Explication des signes daffection.

+ plus

~ signifi. minus, U moins.

.~ : differentiam, U difference. [Hrigone, 1634, II, 5]

He also uses 3/2 to represent greater than and 2/3 to represent less than, 0 to

represent zero and U to represent the expression or.

Furthermore, Hrigone provides alphabetically ordered abbreviation tables (which he

called explicatio notarum) and an explanation of the citations used in the text (explicatio

citationum) at the beginning of each of the volumes that make up Cursus. In the table of

abbreviations, he writes column. Columna, colomne., err. Error, erreur., expo. Exponens,

exposant., nr. numerus, nombre., etc. In the table of citations, he writes 3.a.I {Tertium

axioma primi. Troisiesme axiome du premier livre., c. 17. I {Corollarium decimae septimae

primi. Corollaire de la dix-septiesme du premier., etc. In all cases he is referring to

propositions in Euclids Elements or in his own book.

Thus, we have seen that Hrigone used unusual symbols and abbreviations to represent any

expression or number and also used this language in an original way with the help of Euclids

Elements, as we show below.

2.4 Hrigones Algebraic Expressions

In the initial chapters Hrigone shows his specious logistic operating between

algebraic expressions, first of all between simple expressions and then between compound

algebraic expressions. In products and divisions, he again distinguishes between numbers and

abstract magnitudes, that is to say, the product of two numbers is a number (he names, as an

ostendunt, quota sit unaquae proportionalium, prima proportionali : ut a4 ostendit 16 vel 81 esse quartam proportionalem.

22

example, the number cossici) and the product of two lines is a rectangle (he shows the product

by means of a figure). In this third chapter, he gives several examples of operations, such as

Addition: a3 a2b Subtraction: 8a23ab

2a3 + ab2 7ab5a2

3a3 a2b+ ab2 13a210ab [Hrigone, 1634, II, 13].

Later, in the Isagoge, Hrigone specifies:

ab in numbers means that the number A must be multiplied by the number B; but in lines,

ab means that a line must be found whose square is equal to a rectangle determined by lines

A and B.49

2.4.1 The Treatment of Identities

To highlight the differences between Vites and Hrigones procedures, let us consider the

proof of a theorem in the fifth chapter entitled Collection of Several Theorems. Hrigone

proves, just as Vite did, several algebraic identities that are useful for solving equations. In

the proof of Proposition XXI Hrigone states:

The cube of the sum of two sides, minus the triple of the solid constituted by the sum of the

sides and the triple of the rectangle, is equal to the sum of the cubes of the sides.50

He proves the equality, which in modern notation would be expressed

51( ) ( )3 3 33 ,a b ab a b a b+ + = + as follows:

Hypoth. [Modern notation: Hypothesis]

a& b snt quantit; D. [a and b are given quantities]

[Hrigone, 1634, II, 6]. 49 ab, en nombres, signifie quil faut multiplier le nombre A par le nombre B : mais en lignes, ab, signifie quil faut trouver une ligne dont le quarr soi gal au rectangle, contenu sous les lignes A & B. [Hrigone, 1642, VI, 78]. 50 Le cube de la somme de deux costez, moins le triple du solide contenu sous la somme des costez & le triple de leur rectangle, est egal la somme des cubes des costez. Cubus aggregati laterum minus triplo solido sub aggregato laterum in rectangulum sub lateribus, est aequalis aggregato cuborum lateribus. [Hrigone, 1634, II, 47-48].

23

Req. . Demonstr. [It is required to prove]

Cub.. a+b, ~ .3ab, a+b 2/2 a3 + b3. [ ( ) ( )3 3 33a b ab a b a b+ + = + ]

Demonstr. [Proof]

cub. a+b est a3 +3a2b+3ab2+b3 [(a+b)3 is a3 +3a2b+3ab2+b3 ]

.3ab,a+b est 3a2b +3ab2. [3ab (a+b) is 3a2b + 3a b2 ]

Concl. ~ est a3 + b3. [Conclusion - is a3 + b3]

Thus, when Hrigone multiplies two quantities, one of which is a sum, he puts a square and a

dot at the beginning and a comma between the quantities to express the parenthesis. To raise

an expression to a third power, he writes Cub and one or two dots. To express the identities,

he uses est rather than the symbol 2/2, which represents the equal sign.

If we compare this proof with proofs of similar identities in Vites Notae Priores (1631),

we can see that Vite gives rhetorical explanations and verbal descriptions, uses few symbols,

uses capital letters to represent quantities, leaves no margins and writes the words cube,

quadratum, etc. to express powers. In contrast, Hrigone presents proofs in the Euclidean

way: hypothesis (known and unknown quantities), explanation of requirement, equation or

proof and conclusion, all of which are expressed using symbolic expressions with references

in the margin to the propositions used. Hrigone also formulates all identities and properties in

specious language, with no rhetorical explanations or verbal descriptions.

2.4.2 Solutions of Equations up to the Second Degree

51 Later, in Question 1 in the thirteenth chapter, Hrigone used this identity to demonstrate: Given the product of two sides and the sum of their cube, find the sides. [Hrigone, 1634, II, 181].

24

There are a number of differences in the treatment of equations in algebraic works by

Hrigone and Vite. Even though both transformed equations into three proportional

quantities, Hrigone always distinguishes whether the required quantity is a number (a discrete

quantity) or a line (a continuous quantity) and, in the latter case, uses Euclids Elements to

justify his procedures.

Hrigone, like Vite, defines an equation as the comparison between known and unknown

quantities. He also states that, because equations involve equalities, it is homogeneous

quantities that should be compared. He calls the independent term in an equation

homogeneous by comparison.

He deals with solutions to all kinds of equations, up to the second degree, in the ninth

chapter entitled On the Work of the Rhetic or Exegetic. He states:

The Rhetic or Exegetic is the construction carried out in the case of ordinate equations to find

the required magnitude by numbers or by lines. If the problem posed is to discover a number,

the operation is done by numbers; if a problem in geometry is to be solved, it is done by

lines.52

In the case of the second-degree equation with two terms (x2 = bd), if the required quantity is a

number, the solution is the root, and if the required quantity is continuous, Hrigone reduces it

to the corresponding proportion b : x = x : d. He states that this proportion can be set up using

Proposition VI.17 of Euclids Elements, and solved by Proposition VI.13.

With reference to the second-degree equation with three terms, in the geometrical case he

again reduces the equation to the corresponding proportion, using many examples. To solve

the equation x2 - bx = d2, for instance, he stresses that the three quantities x-b, d and x are

proportional, according to Proposition VI.16 of Euclids Elements, and that the corresponding

52 La Rhetique ou Exegetique est la construction quon fait, lquation tant ordonne, pour avoir la grandeur requise par nombres ou par lignes. Par nombres, si la question propose trouver quelque nombre, & par lignes, si la question propose faire quelque opration Gomtrique. Rhetique est qua, ordinata aequatione, exhibetur magnitudo quaesita arithmeticae,

25

proportion is (x-b) : d = d : x.53 He adds that in the equation x2 - bx = d2, d is the proportional

mean and b is the difference between the extremes, both of which are given. He emphasizes

that with these data the proportion can be solved by the scholie of Proposition VI.29 in

Euclids Elements, whose statement reads: Given the mean of three proportional (straight

lines) and the difference between the extremes, find the extremes.54 Hrigone gives two

proofs, one by geometric construction and the other using numbers. When he calculates the

solution by numbers, in the margin he cites Proposition I.47 (that is, Pythagoras

theorem).55

When the unknown is a number, Hrigone states two rules: one for equation x2 - bx = d2

and another for equation bx - x2 = d2. In the first rule, the algorithm for solving the second-

degree equation is worded in the following way:

If the negation is not inverse, the square of half of the coefficient is added to the independent

term and the sum is subtracted from the root. The resulting root is then added or subtracted

(depending on the opposite of the sign of the coefficient), the sum or the difference will be the

solution.56

This is the rule for solving the second-degree equation x2 bx = d2. The statements are, first of

all, to add the value of d2 to the square of half the value of b, which means (b/2)2+d2;

secondly, to obtain the square root of this result; and then, to add to or subtract from this value

half the value of b. Therefore, x = b/2 + (b/2)2 +d2. He goes on to solve equations such

vel geometricae. Arithmeticae quidem si de magnitudine numero explicada quaestio est, Geometricae ver si magnitudinem ipsam exhiberi oporteat. [Hrigone, 1634, II, 95]. 53 Later, in the problems, he specifies that the passage from equation to proportion can be justified by Proposition VI.16 of Euclids Elements. 54 tant donne la moyenne de trois proportionnelles, & la diffrence des extrmes, trouver les extrmes. Data media trium proportionalium, & differentia extremarum, invenire extremas. [Hrigone, 1634, I, 296]. 55 However, in this scholie he works with proportional lines and does not identify these values with the coefficients of an equation. 56 Si la negation nest inverse, soit adjoust lhomogene le quarr de la moiti du coefficient, & de la somme soit extraite la racine quarre, puis la racine trouve soit adjouste ou soustraicte (suivant la signification contraire du signe du coefficient) la moiti du coefficient, la somme ou le reste sera le nombre du degr parodique. Si negatio non sit inversa, quadratum semissis coefficientis addatur homogeneo, & ex summa extrahatur radix quadrata, deinde radici inventae addatur vel subducatur (secundum contrariam signi coefficientis significationem) semissis coefficientis, summa vel residuum erit numerus gradus parodici. [Hrigone, 1634, II, 104].

26

as x2 - 6x = 27; x4 - 6x2 = 27, x6 - 6x3 = 994000 and x2 + 5x = 66. Later, when Hrigone uses

the aforementioned rule to solve problems, he first writes the second-degree equation and then

specifies the three proportional quantities; lastly, he cites 9.c.alg. in the margin and writes

the solution to the equation using the following algebraic expression:57

2 2

2 2

2 2 / 2 2 [ :( )[ ]

1/ 4 21 12 / 2 [ 1/ 4 ].22 2

a ab d Modern notation x bx dx b da b d a

d xb

a b x b b dd

=

{+ = +

{+

]=

+

Therefore, Hrigone uses three steps when he solves the equations in problems: equation,

proportion and solution. We return to this point below.

As we can see, Hrigone bases his procedures for solving equations on the proofs of

propositions in Euclids Elements. In contrast, Vite reduces equations to proportions and

gives the solutions to the three proportional without mentioning Euclid.

2.4.3 The Treatment of Problems

Before applying these rules, Hrigone explains how to set up the equation in a problem in

the tenth chapter. He solves twenty problems corresponding to first- or second-degree

equations with two terms whose statements can be found in several numerical algebra texts.

He classifies the problems by specifying three solution methods.58 The first method

consists in setting up the equation by finding one or more unknown quantities, which are equal

to a given or known quantity. He applies it in Questions 1, 20 and others in the tenth chapter.

57 To represent the parenthesis in the root, he used the symbol {.

27

In Question 1, for example, he solves the problem: Given the sum and the difference of two

sides, find the sides.59

As for the second method, he explains that the equation can be set up by finding four

proportional quantities, because there is an equality between the rectangle that is determined

by the extremes and the one determined by the middles. He applies it in Questions 2, 3 and

others in the tenth chapter. In Question 3, he solves the problem: Given two numbers, find

the proportional mean in music proportion.60

Finally, in the third method he explains that the equation can be set up by finding

unknown quantities that are equal to equal quantities and are therefore equal to each other.

One might see this method in Questions 4, 5 and others in the tenth chapter. In Question 13,

for example, he solves the problem: A captain assigning soldiers to an infantry square

finds he has 284 soldiers to spare, but if he adds one soldier to every row he finds he is

missing 25 soldiers. How many soldiers are there? 61

Hrigone describes how to solve arithmetic problems through the classification of

methods for solving equations or, in other words, by trying to generalize.

2.5 Hrigones Construction of Equations

The relationship between the roots and coefficients of an equation is one of the most

important achievements in Hrigones Algebra.

58 In his Annotations on Algebra (at the end of the volume) he explains these methods. He stresses that it is better to begin with arithmetic proofs rather than geometric proofs, because the former are easier. Here he gives some examples of resolution with numbers and in the tenth chapter he solves the problems with letters using the properties of equations. 59 Estant donnez laggreg des costez & leur difference trouver les costez. Dato aggregato laterum & differentia eorundem invenire latera. [Hrigone, 1634, II, 108]. 60 Entre deux nombres donnez trouver le moyen proportionel en proportion musique. Datis duobus numeris, medium harmonic proportionalem invenire. [Hrigone, 1634, II, 110]. 61 Un Capitaine rangeant ses soldats en un bataillon quarr trouve 284 soldats de reste mais pour augmenter chaque rang dun homme il luy manque 25 soldats, savoir combien de soldats il a? Dux quadratam aciem instruit, & milites reliquos habet 284, deinde in singulos ordines unum militem adjicit, sed 25 desunt ad quadratum explendum, quot igitur milites habet? [Hrigone, 1634, II, 120]. In Isagoge, he writes these three methods again and solves more problems of each type; some of them, as he says, can also be found in Peletiers Algbre.

28

Even though Hrigone hardly gives explanations, he provides a meaningful account of the

number of roots of any equation at the beginning of the fourteenth chapter entitled On

Ambiguous Equations. Ambiguous equations are, as in Vites work, complete equations in

which the signs of the expressions are alternately positive and negative. Hrigone states:

[An] ambiguous equation is one whose power can be explained by several roots. Moreover, for

any given magnitude one is able to find an ambiguous equation, whose power can be explained

by as many roots as there are unities contained in its exponent.62

This text reminds us of the first statements on the fundamental theorem of algebra, although

only for ambiguous equations.63 Having made the above statement, Hrigone puts forward

several examples using letters and numbers and constructs several ambiguous equations. He

reduces the coefficients of a complete equation to its roots. Therefore, he constructs the

equation by calculating its coefficients from its roots, and writes:

If x3- mx2+nx = c with m = f+g+l and n = fg + fl + gl, c is given and f and g are arbitrary,

then as one knows that c = fgl, so the solutions of equation are f or g or l. As shown in

Example 2 below, in which he tests the truth of this statement (he first writes it in letters and

then in numbers, in two cases):

C,30, F,2. G,3. H,6. L,5. M,10. N,31

F,2. G,4. H,8. L, 3 . M,9 N,30

c est magd. D. [Modern notation: c is a given magnitude]

f & g snt magd; arbitr; [f and g are arbitrary magnitudes]

h 2/2 . f,g [h = fg]

62 Lquation ambige est celle dont la puissance peut tre explique par diverses racines. Or quelconque grandeur donne on luy peut trouver une quation ambige, dont la puissance pourra tre explique par autant de racines, quil y aura dunits en son exposant. Aequatio cuius potestas pluribus radicibus potest explicari est ambigua. Datae autem cuicumque magnitudini potest inveniri aequatio ambigua, cuius potestas tot radicibus explicetur, quot unitatibus constat eius exponent. [Hrigone, 1634, II, 190].

29

h msur: c l.64 [c : h = l]

m 2/2 f+g+l, [m = f+g+l]

n 2/2 fg +fl +gl, [n = fg + fl + gl]

a3 ~ ma2 +na 2/2 c, [x3-mx2+nx = c]

a est 2/2 f, g, l. [x is equal to f or g or l]

In numeris, en nombres [In numbers]

a3~10a2 +31a 2/2 30, [x3 -10x2+31 x = 30]

a 2/2 2, 3, 5. [x = 2, or 3, or 5]

a3 ~9 a2 + 30 a 2/2 30, [or x3- 9 x2 + 30 x = 30]

a 2/2 2, 4, 3 . [x = 2, or 4, or 3 ]

After giving these examples, Hrigone states:

By the same method, in the case of higher-order equations, equations can be found that may

be explained by means of as many roots as there are unities in the exponent of the power.65

In contrast, Vite also gives some examples (in letters and numbers, but just for n = 25)

at the end of De Emendatione Aequationum, tractatus secundus (1615). He does not

provide proofs but just short explanations as to how he has dealt with the subject.66

63 Descartes would later state in his Gomtrie: 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 [Descartes, 1954, 159]. 64 The expression h msur: c l indicates that the division of c by h gives l. 65 Par la mme mthode, aux autres degrs plus hauts, on trouvera des quations qui se pourront expliquer par autant de racines quil y aura dunits en lexposant de la puissance. Eadem arte in altioribus gradibus invenientur aequationes tot radicibus explicabiles quot unitatibus constat potestatis exponens [Hrigone, 1634, II, 192]. 66 Vite states: I have dealt [elsewhere] at length and in other respects with the elegant reasoning behind this beautiful observation, [so] this must be the end and the crown [of this work]. Atque haec elegans & perpulchrae speculationis sylloge, tractavi alioquin effuso, finem aliquem & Coronida tandem imponito [Vite, 1970, 158].

30

Moreover, Hrigone, concludes the last chapter of Algebra entitled On the Numerical

Resolution of Affected Powers by stating a theorem that generalizes this result and thus

warrants further explanation:

If a positive power is affected by all the degrees and by the independent term, which are

alternately negative and positive, and the coefficient of the degree following the higher degree

being the addition of as many numbers as there are unities in the exponent of the power; the

coefficient of the following degree is the result of adding all plane numbers of those numbers;

the coefficient of the third following degree is the result of adding all solids, and so on until the

independent term is reached, which is the product of these numbers continuously multiplied;

the sum of all positive numbers will be equal to the sum of all negative numbers and

consequently if the independent term makes one part of the equation and the power with all

degrees the other part, the root of the power may be explained by each of the proposed

numbers.67

This theorem deals with finding equations with a given set of roots and Hrigone stresses that

it is very important. It can be stated in modern notation as:

1 21 2 1...

n n nn n ox a x a x a x a

+ + =

jn

p

where if p1, p2, p3, pn are n-roots of the equation, then the other terms represent

the sum of the roots p

01

;i n

ii

a p=

=

=

i combined, that is, , and so on. 1 21 1

;i n

n i n ii i j

a p a p=

= <

= =

This theorem can be found at the end of Hrigones volume on algebra (1634), after

calculations (similar to those of Vite) that consisted in finding the upper or lower bounds

67 Si une puissance affirme est affecte sous tous les degrs parodiques & sous lhomogne de comparaison, quils soient alternativement niez & affirmez, & que le coefficient du degr parodique prochain la puissance, soit lagrg dautant de nombres quil y aura dunits en lexposant de la puissance : le coefficient du second degr inferieur suivant, soit l agrg de tous les plans des mmes nombres : le coefficient du troisime degr, soit lagrg de tous les solides, & ainsi de suite jusques lhomogne de comparaison qui est le produit des dits nombres multipliez continment : la somme de tous les affirmez sera gale la somme de tous les niez, & par consquent si lhomogne de comparaison fait une partie de lquation, & la puissance avec tous ses degrs parodiques lautre partie, la racine de la puissance pourra tre explique par un chacun des nombres proposez. Si potestas affirmata, sit affecta sub omnibus gradibus parodicis, alternatim negatis & affirmatis, sitque coefficiens, primi gradus parodici potestate, aggregatum totidem numerorum, quot sunt unitates in exponente potestatis : coefficiens secundi gradus, aggregatum omnium planorum eorundem numerorum : coefficiens tertij gradus, aggregatum omnium solidorum, & ita deinceps usque ad homogeneum comparationis, quod gignitur ex continua multiplicatione eorundem numerorum : aggregatum omnium affirmatorum erit aequale aggregato omnium negatorum, ac

31

in the numerical solutions of ambiguous equations. As far as we know, this generalization

cannot be found in Vites work, and it suggests that Hrigones underlying idea is to

provide a universal theorem that would be useful in the case of a polynomial equation of

arbitrary degree.68 In fact, Hrigones theorem provides degree-independence for

constructing ambiguous equations and shows a way of establishing a general relationship

between roots and coefficients in polynomial equations.69

2.6 Hrigones Geometric Constructions

Vite and Hrigone differ substantially in their treatment of geometrical constructions.

Hrigones attempts at expressing all proofs in logical symbolic expressions for geometrical

constructions achieve the highest degree.

In order to explore the geometrical procedures used by Hrigone in more depth, we

present a proposition from the twelfth chapter of Algebra that deals with second-degree

equations and the geometric construction of their solutions. A comparison with Vites

solution and geometric constructions shows us once again that Hrigones reasoning and

proinde si homogeneum comparationis faciat unam aequationis partem, & potestas cum omnibus suis gradibus parodicis alteram, radix potestatis erit explicabilis de quolibet illorum numerorum. [Hrigone, 1644, 195-196]. 68 Albert Girard in his book Linvention nouvelle en lalgbre (1629) formulated this theorem, like Hrigone, for an equation of arbitrary degree. Harriot in his Praxis (1631) provides similar calculations. On Girard, see Struik [1969, 81-87] and on Harriot, see Stedall [2000, 491]. Harriots achievements are also explained in Chap. XXXVII, entitled The Composition of Coefficients, of Treatise on Algebra by Wallis. [Wallis, 1685, 142]. We do not believe, however, that Hrigone was aware of Harriots work, because it is very different in terms of the notation, procedures and even the statements.. 69 Later, in Supplementum, he deals with ambiguous equations in a different way. First, he repeats the idea, In each degree of scalar magnitudes, equations can be found and explained by as many roots as there are dimensions in the power. [Hrigone, 1642, VI, 18]. After reminding the reader of the theorem cited above, he specifies that he will then construct ambiguous equations in another way. He begins, like Descartes, with simpler equations and uses the product to construct ambiguous equations of higher degrees. For example, if x = 2, x-2 = 0; x = 3 and x-3 = 0, then (x-2).(x-3) = 0, the ambiguous equation is 6 = 5x x2, and its roots are 2 and 3. He carries out these operations until he obtains third-degree equations. Then he continues to deal with the roots of equations using Vites syncrisis, that is, he compares two correlated equations in order to discover their structure. This new way of constructing ambiguous equations using the product of factors would seem to indicate that Hrigone, in this second edition (1642), may have been inspired by Descartes Gomtrie, which was published in 1637. However, he does not mention this source of inspiration.

32

presentation are different to Vites. The statement of Vites equation, which can be found in

Zeteticorum libri quinque (1593),70 is as follows:

Zeteticum (III,1). Given the mean of three proportional straight lines and the difference

between the extremes, find the extremes.71

The equation A (A+B) = Z2 is considered by means of the proportion A : Z = Z : (A + B),

where A is the unknown quantity, Z is the given proportional mean and B is the given

difference between the extremes. Vite based the solution on Zeteticum II, 3, which stated:

given the area of a rectangle (Z2 ) and the difference between the sides (B), find the sides. In

order to solve this new proposition, he explained that the square of the difference between the

sides (B) added to four times the rectangle (the product, that is, Z2) would give the square of

the sum; in modern notation, [(A + B) A]2 + 4A(A+B) = [(A+B) + A]2. Vite then claimed

that this would give both values: the sum and the difference of the sides, and thus the sides, as

he had already proved in Zeteticum I, 1.

Later, Vite published Effectionum Geometricarum Canonica Recensio, in which he

geometrically constructed the solutions of second- and fourth-degree equations. Vites

construction corresponding to the previous example is set out below, and can be compared

with Hrigones construction, which is shown in Figure 2.72

Proposition XII. Given the mean of three proportional magnitudes and the difference between

the extremes, find the extremes.73

FIGURE 1. Vites construction74

[This involves] the geometrical solution of a square affected by a [plane based on a] root [x2 +

dx = b2].

70 This work constituted the exemplification and the application of the method proposed by Vite in Isagoge, a procedure which w s understood as a new form of calculation. a71 Data media trium proportionalium linearum rectarum, & differentiae extremarum, invenire extremas. [Vite, 1970, 56] 72 Vite had previously said: We were able to show this in the Zetetica and it is now demonstrated synthetically from a geometric figure. [Vite, 1983, 377]. 73 Propositio XII. Data media trium proportionalium & differentia extremarum, invenire extremas. [Vite, 1970, 233] 74 This figure is in Vites Effectionum geometricarum. Canonica recensio. [Vite, 1970, 234]

33

Let FD be the mean of three proportional and let GF be the difference between the extremes.

The extremes are to be found.

Let GF and FD stand at right angles and let GF be cut in half at A. Describe a circle around

the centre A at the distance AD and extend AG and AF to the circumference at the points B

and C.

I say that has been done that was to be done, for the extremes are found to be BF and FC,

between which FD is the mean proportional. Moreover, BF and FC differ by FG, since AF and

AG are equal by construction and AC and AB are also equal by construction. Thus, subtracting

the equals AG and AF from the equals AB and AC, there remain the equals BF and FC. GF, in

addition, is the difference between BF and BG or FC, as was to be demonstrated.75

Here (see Figure 1), Vite sets up the equation x2 +dx = b2 by means of a proportion, which

can be expressed, in modern notation, as (x + d) : b = b : x. Vite draws FD = b and GF = d,

making a right angle, and divides d by half AF = d/2. He describes a circle whose radius is

equal to AC, which we can identify with the hypotenuse of the triangle formed by d/2 and b,

AD = AC =(d/2)2 b2. The solutions are then the segments FC = AC - AF and BF = BA +

AF, which take BA = AC = radius.

In Question 3 in the twelfth chapter of Algebra entitled Questions about Second-Degree

Equations, Hrigone sets up a second-degree equation that is similar to Vites. However, he,

unlike his counterpart, immediately solves the equation arithmetically, by the rule for solving

second-degree equations, and geometrically, by constructing the solution, for which he

provides no rhetorical explanation.76 Instead, he uses a symbolic language to convey his

statement in a few lines, and refers repeatedly to Euclids Elements.

75 Mechanice quadrati adfecti sub latere. Sit data FD media trium proportionalium, data quoque GF differentia extremarum. Oportet inveni