Fermat and Descartes in Light of Premodern Algebra and Viète

Abstract

There were two radically different algebras practiced in Europe in the time of Fermat and Descartes. One was the traditional cossic algebra grounded in arithmetic whose roots stretch back to al-Khwārizmī and Diophantus, and the other was the new geometrical algebra of François Viète. Both Fermat and Descartes chose to work in the latter. To understand their choice, and to determine how they understood their algebraic terms, we first outline the conceptual foundations of the two algebras, one that regards a monomial as a premodern number in which the coefficient is a multitude and the power is a kind, and the other that regards a monomial as the product of two magnitudes, one known and the other unknown. We find that cossic algebra could not have been modified to include undetermined coefficients, and thus would have been inadequate for the purposes of Fermat and Descartes. Further, both authors, when solving problems in geometry, worked with precisely the same non-arithmetized algebra as Viète, without a unit measure, respecting dimension, and using magnitudes of arbitrarily high dimension for which no justification is provided.

1 Introduction

Algebra played a limited role in mathematics prior to the seventeenth century. Along the path from Greek to Arabic to European algebra, it had remained a numerical technique for finding unknown numbers in problem-solving. Algebraic language and notation were not used to state arithmetical identities like (a + b)² = a² + b² + 2ab, formulas like A = πr², curves like y = x² + 1, or physical laws like $d=\frac{1}{2}{gt}^2$. For those purposes, arithmetical or geometrical language was used. Apart from rare applications in proofs in some Arabic authors (Oaks 2019), algebra concerned itself solely with the setting up and solving of equations to find the values of unknowns. While several authors such as al-Karajī and Bombelli had studied different aspects of algebra from a more theoretical viewpoint, these investigations remained internal to algebra as a problem-solving technique. This started to change in the first half of the seventeenth century, when Pierre de Fermat and René Descartes independently began to study curves through equations. Each of them discovered independently how curves can be represented by equations in two unknowns, and they both found ways to apply algebra to problems of determining normal and tangent lines to curves. These were among the first steps in what would eventually lead to the wholesale algebraization of mathematics, and by association of science as well.

In this chapter, I examine the ontology of the knowns and unknowns in the algebra practiced by Fermat and Descartes. The algebraic landscape of their time offered two basic alternatives: the traditional numerical algebra familiar to readers of such figures as Rafael Bombelli, Simon Stevin, and Christoph Clavius, and the new, geometrical algebra of François Viète and his followers. As the familiar story goes, Fermat and Descartes both built on the latter. To understand the nature of their algebraic terms and the rationale behind their choice, we should summarize the conceptual foundations of both kinds of algebra, especially because for each of them recent research has modified old views.

We begin with an overview of the numerical algebra that Europeans had acquired from the Arabic world. This premodern algebra was not simply an early version of the numerical algebra that we currently teach. It was grounded in a corresponding premodern notion of number, so that the monomials, polynomials, and equations we find in Diophantus, Abū Kāmil, Cardano, and Clavius were conceived in a radically different way from their modern counterparts. Very briefly, the core of the difference lies with the fact that the powers of the unknown were kinds of number, and not values as they are today. Our x is the value of an unknown, while Clavius’s cossic is a kind of number distinct from other kinds (powers). Only with a coefficient, like 1, does it become a value. Consequently, coefficients held a different relationship to the powers, and polynomials were aggregations, free of arithmetical operations. The different notations devised for working out problems in premodern algebra across all languages also reflect this conception, so that reading it as if it were some version of modern notation distorts its meaning.

Next we review Viète’s algebra nova, where knowns and unknowns, instead, are values deriving from geometry. More precisely, they are the non-arithmetized measures of magnitudes, possessing dimension but unfettered by the limitation to line, surface, and solid, and equations between them arise out of ratio and proportion. They are not, as many historians since Jacob Klein have maintained, uninterpreted symbols that transcend number and magnitude. Viète’s program was to develop propositions in geometry modeled on those in Euclid and other classical Greek authors, that is, in geometry without numerical measure, to serve as foundations for calculating the entries in astronomical tables. Ptolemy, after all, had based the calculations for his tables in Euclidean geometry, and Viète wished to respect this tradition. But Viète diverged from astronomers in the Ptolemaic tradition by reworking the “computational geometry” of the Greeks into a new algebra that would greatly facilitate calculation. Because his end goal was to use the geometric models for numerical calculation, he had no reason to restrict himself to three geometric dimensions – not just in his algebra, but even in his geometry. Instead of justifying these higher dimensions, he explained how they are useful. Not surprisingly, the notation Viète devised for his logistice speciosa, as he called his new algebra, is also a radical departure from that of contemporary algebraists.

When we finally turn to Fermat and Descartes, we find that in practice they both worked with the same non-arithmetized geometrical algebra as Viète, again with no justification for the higher-dimensional magnitudes that come with it. Fermat adopted Viète’s system without modification, and even kept the same notation. And although Descartes began his 1637 La Geometrie by describing an algebra of line segments that bypasses the problem of dimensionality, he did not apply it in his algebraic solutions to problems. Instead, he worked in the same non-arithmetized algebra as Viète and Fermat, in which the dimensions of the terms must be taken into account.

Last, we inquire what led Fermat and Descartes to choose Vietan algebra over premodern, cossic algebra. Two obstacles in particular prevented the latter option from being chosen. The most obvious is that it is a numerical art, while Fermat and Descartes worked in geometry. But even if the terms could have been reinterpreted geometrically, there was no natural way that premodern algebra could have been modified to accommodate undetermined knowns, which are necessary for the general solutions that Fermat and Descartes sought. The cause of this limitation lies with the conceptual foundation of premodern algebra in kinds, not values, and this foundation also made premodern algebra ill-equipped for other applications outside problem-solving. It was the shift from types to values, not the shift to abstract magnitudes, that allowed algebra to begin its path of overrunning the rest of mathematics.

2 Premodern Monomials, Polynomials, and Equations

European algebra in the time of Viète can be traced back via Italian and Latin texts to medieval Arabic algebra, and although we lack a textual connection, there is clearly a historical link between Arabic algebra and the algebra practiced in the Arithmetica of Diophantus of Alexandria (ca. third century CE) (Christianidis and Oaks 2013). It is the seemingly minor yet consistent differences in practice, wording, and notation between pre-Vietan algebra and our own elementary algebra that testify to a different way of conceiving of monomials, polynomials, and equations. Because my aim in this section is to provide a backdrop for comparison with the algebra of Viète, Fermat, and Descartes, I will describe the main features of premodern algebra using examples leaning toward their time. Many will be taken from Michael Stifel’s 1544 Arithmetica Integra, though just about any other sixteenth-century text could have been chosen. Stifel’s cossic notation was about as standard as one could find at the time, and it was the notation later adopted by Christoph Clavius, whose 1608 Algebra was studied by Descartes.

Stifel’s notation uses the standard cossic signs that had been common among German algebraists since the previous century. The names given to the powers of the unknown in this system were adapted from Latin terms which were themselves ultimately translations of the corresponding terms in Arabic. In the cossic system, the first degree unknown is called a radix, the Latin word for “root,” which was translated from the Arabic first-degree term jidhr (“root”). In notation its sign is . The square of a radix is called a zensus, deriving from the Latin census, translated from māl, the name of the second degree unknown in Arabic. Like the Latin census, the common meaning of māl is “sum of money” or “wealth.” The cossic sign for a zensus is . The cube of a radix is called a cubus, the Latin word translated from the Arabic kaʿb (“cube”), and its sign is . And just as in pre-Renaissance algebra, most higher powers are named with combinations of the terms for the second and third powers. The fourth power was called a zensizensus, which ultimately comes from the Arabic māl māl, and its sign is . The European powers diverge from the Arabic after this point. Stifel’s fifth power is called a surdesolidus (ß), his sixth is a zensicubus (), and higher powers follow without limit. The constant term in a polynomial, or what we call the zero-degree term, is called a numerus, and in Arabic, these numbers are usually counted in dirhams, a denomination of silver coin. Because algebra before Viète was founded in arithmetic, Stifel’s unknown radix is always found to be a number, and numbers in arithmetic from the earliest Arabic texts can take on any positive value, including fractions and irrational roots. In arithmetic, the cossic sign for square root is √, for cube root √, etc., so that for instance $\sqrt{72}$ was shown as √72.

In notation, this cossic algebra can look deceivingly modern. Take for example this polynomial from Stifel’s book (1544, f. 238b):

It is tempting to read this as being identical to 12x⁴ + 16x³ − 36x² − 32x + 24 and indeed if one follows Stifel’s calculations the modern reading makes sense. But there are many indications in the book that he is not thinking like a modern mathematician and these differences are shared with algebraists all the way back to al-Khwārazmī and still farther back to Diophantus

Before turning to the premodern arithmetical foundation of algebra, I will mention two differences between Stifel’s notation and ours. One is that when there is one of a term, he always writes the “1.” So where we write x² + 8x, Stifel shows “1 + 8” (1544, f. 277b.2), or where we write $x+\sqrt{205-{x}^2}$ Stifel has “1 + √.205 − 1.” (the periods serve as parentheses) (1544, f. 281a.5). This inclusion of the “1” is present in every premodern notation from Diophantus to Arabic algebra to Latin, Italian, and the various notations in Renaissance Europe. Where we would write simply x, Scheubel (1550) wrote “1 ra.”, Bombelli (1572) “1 ,” Stevin (1585) “1 ①,” and de Billy (1643) “1R” (Oaks 2017).

Another difference is that with a few exceptions beginning in Stifel’s century, premodern algebraists insisted on keeping their coefficients rational. For example, Stifel writes in one place, “I multiply √ $501\frac{9}{11}$ by √1, or √1, which makes √ $501\frac{9}{11}$ .” (“√ $501\frac{9}{11}$ multiplico per 1, seu per √1, facit √ $501\frac{9}{11}$ ”) (Stifel 1544, f. 292b.12). In modern notation, he multiplies $\sqrt{501\frac{9}{11}}$ by x to get $\sqrt{501\frac{9}{11}{x}^2}$, where we would write instead $\sqrt{501\frac{9}{11}}x$. In another instance he writes “I then multiply 1 by √864 − 1, which makes √864 − .”) (“Multiplico igitur 1 in √ 864 – 1, facit √864 − 1.”) (Stifel 1544, f. 288a.2). In modern notation, he multiplies x by $\sqrt[4]{864}-x$ to get $\sqrt[4]{864{x}^4}-{x}^2$, again keeping the power under the root. Algebraists starting with the earliest extant Arabic texts in the ninth century similarly ensured that their coefficients remained rational by taking the root of the coefficient and the power (Diophantus did not work with irrational numbers). We discuss the sixteenth-century exceptions in Sect. 3.2 below.

2.1 Premodern Numbers, Premodern Polynomials

These curiosities, along with other consistent differences in the wording of certain steps and in the overall approach to problem-solving, are reflections of a different way of understanding the relationship between the coefficient and the power, and this relationship derives from the way that numbers themselves were conceived. Today mathematicians typically think of numbers as being elements of some abstract set equipped with the binary operations of addition and multiplication that satisfy the ordered field axioms. Others might view numbers as being constructed from scratch, where zero is identified with the empty set. Many people with no university training in mathematics retain the idea that numbers are points on some imagined number line, or at least that the units in which one counts or measures (e.g., apples, pesos, meters, pounds) are not part of what constitutes a number.

Such was not the case with premodern numbers. I will first describe the setting in Arabic, from which European premodern algebra originated. (What I describe here applies also to numbers in ancient Greece (Klein 1968, 46ff; Mueller 1981, 59).) Numbers in Arabic, whether they arose in practical calculation or in the tradition of Aristotle and Euclid, were conceived as being numbers of something. A six was not simply a six, but must be six of some kind, like six geese, six men, six dirhams, six mithqals (a unit of weight) of gold, six sevenths ($\frac{6}{7}$, which in turn might be a number of some other unit, like mithqals), six degrees (6°), or six (abstract) units. In other words, numbers were not objects in themselves, but they necessarily modified – i.e., they counted or measured – a multitude of some material or intelligible unit. Some kinds of unit were divisible, such as units of weight and length, while others were indivisible, such as geese and the Euclidean unit. In an arithmetic book composed in 1301, the Moroccan polymath Ibn al-Bannāʾ wrote about the two aspects of a number, its “meaning” (maʿnan) and its “term” (lafẓ) (1994, 255.3). For “three men,” the meaning is “three” and the term is “men.” Of course, the term could be ignored when working out operations. And because numbers are amounts of some unit, they were necessarily positive, though sometimes “nothing” or “zero” entered into calculations.

Niccolò Tartaglia (1556, f. 2a-b), to pick a European example, frames these two aspects of number in Aristotelian terms while appealing to Euclid’s definitions of “unit” and “number.” Units, he tells us, come in two kinds, “Natural” (“Naturale”) and “Mathematical” (“Mathematico”). “The Natural [unit] considers the things according to their being, as according to reason conjoined with some sensible material.” (“Il Naturale considera le cose si secondo lesser, come secondo la ragione congionte con qualche materia sensibile” (Tartaglia 1556, f. 2a.29).) He gives many examples of Natural units, such as one gold ducat, one scudo, one fiorino (three denominations of currency), one yard of cloth, one pound of silk, one ounce of saffron, one pace, one yard, one degree, one minute (sexagesimal units), etc. Because they are rooted in the material world, such units are “infinitely divisible” (“divisibile in infinito”). “The Mathematical [unit], then, also considers the things conjoined, according to their being, with such sensible material (as with the Natural), but one takes them, or considers them as being abstracted from such sensible material according to reason.” (“Il Mathematico poi considera le cose pur congiunte secondo lessere, con tal materia sensibile, (si come fa anchora il Naturale) Ma le piglia, over considera poi si come astratte, da tal materia sensibile secondo la ragione” (Tartaglia 1556, f. 2a.45).) Unlike Natural units, Mathematical units are indivisible (“indivisibile”). After covering these two kinds of unit, he turns to the definition of “number”: “The number (as Euclid defined it in the second definition of the seventh [book]) is none other than a multitude composed of units,” (“Il Numero (come diffinisse Euclide nella seconda diffinitione dil settimo) non è altro, che una Moltitudine composta dalle unitade.” (Tartaglia 1556, f. 2b.14).) where these units can be either Natural or Mathematical. As in Aristotle and in medieval Arabic, numbers in Tartaglia are always multitudes of some unit, whether that unit is sensible or noetic. Even if some sixteenth-century mathematicians may have advocated for a different meaning of “number,” it is the common understanding exhibited in Ibn al-Bannāʾ and Tartaglia that informed the way algebraic monomials were conceived.

Premodern algebra was strictly an arithmetical art. As al-Khwārazmī wrote in the beginning of his seminal Book of Algebra (early ninth century), “When I considered what people need in calculation I found that it is always a number” ((al-Khwārazmī 2009, 97.1), my translation). Algebra could be used to solve problems in arithmetic, geometry, metrology, and even horology, as long as the parameters are given numerical measure. Algebra was not used, for instance, to find a geometric magnitude in itself, but only its numerical size.

The names of the powers in algebra are kinds of number in the same sense that geese, dirhams, and degrees are kinds of number. Again quoting al-Khwārazmī, “I found that the numbers which are necessary for calculation in algebra are of three kinds (ḍurūb), which are roots, māls, and a simple number unrelated to a root or a māl.” ((al-Khwārazmī 2009, 97.9), my translation. The plural of māl is amwāl, but we write it with the English suffix “s.”) So in addition to men, degrees, and mithqals, we can also count or measure roots, māls, and simple numbers. Al-Khwārazmī’s algebraic terms “eight roots” (2009, 147.1) and “four ninths of a māl” (2009, 181.11) are multitudes of different kinds just as “eight men” and “four ninths of a mithqal” are multitudes of different kinds. (The modern versions of these monomials are 8x and $\frac{4}{9}{x}^2$.) Sixteenth-century algebraists likewise often called these algebraic kinds “numbers” or “quantities,” like Tartaglia, who writes just before listing the names of the powers of the “different kinds of quantities considered in Algebra” (“diverse spetie de quantità, considerati in Algebra”) (Tartaglia 1560, f. 1a). Just five examples of the kinds as “numbers” are Stifel (1544, f. 228a), Scheubel (1550, 1ff), Peletier (1554, 5), Gosselin (1577, f. 54a), and Clavius (1608, 7). The algebraic kinds, or powers, were also called by words meaning “name” or “denomination” in every language from Greek to European vernaculars (Oaks 2018, 269).

The premodern concept of number just described was formed without consideration of irrational roots, most likely before irrational numbers entered into calculations. Although irrational numbers are abundantly common in Arabic arithmetic and algebra, they did not participate in this idea of a number as a multitude. One can have three loaves of bread or two and a half loaves of bread, but it makes no sense to have $\sqrt{10}$ loaves of bread. For this reason, the “number,” i.e., the coefficient, of a term in algebra was required to be rational. Stifel’s √ $501\frac{9}{11}$ $\left(\sqrt{501\frac{9}{11}{x}^2}\right)$ is acceptable because $501\frac{9}{11}$, the number of zensi, is rational, while the whole term is acceptable as a potentially irrational number (Oaks 2017). Words meaning “number” from al-Khwārazmī to Clavius were used in more than one sense. The whole term “five cubes” (5x³ in modern notation) could be called a “number,” and at the same time the “number” of the term (5 in this example) meant its multitude.

Let us compare Stifel’s term “16” from the polynomial shown above with its modern equivalent, 16x³. (The analysis would be no different if we were to choose in place of Stifel’s notation Cardano’s “16 cu,” Bombelli’s “16,” or any other pre-Vietan version of this term.) In our 16x³, the 16 and the x³ are both numbers, one known and the other unknown, that are understood to be multiplied together by the concatenation of their signs. But in Stifel’s 16, as in the rhetorical “sixteen cubes,” the “16” and the “” are two aspects of a single number, one telling the amount and the other the kind. It is like saying “16 bottles” or “16 euros.” With regard to this last example, “euros” is a kind of currency, while “one euro” is a value or an amount. If in place of “1 + 8” Stifel had written “ + 8,” it would have been read as “zensi and eight roots,” which leaves unanswered how many zensi there are, much like “euros and 8 dollars” leaves unanswered the number of euros. This is why, whenever there is one of a term in premodern algebra, the coefficient of “1” is always written in notation. In modern algebra, we can write simply x³ because the notation for the power already stands for a value. Here is another way of putting it: Imagine “sixteen euros” as a pile of sixteen one-euro coins. The “sixteen” and the “euros” are not two separate entities brought together to form the term, but are different aspects of a single amount. The same applies to Stifel’s “16”: the is the kind, and the 16 is how many there are. While we can today also interpret our monomials in this premodern manner, it only works for simple expressions. Reading modern notation as a number-kind pair breaks down when we stray just a little from simple monomials to terms like πr² or 3xy.

The addition of algebraic terms works like the addition of other kinds of number, which helps shed light on how polynomials were conceived. Adding, say, two cubes to three cubes gives five cubes, but when the kinds (powers) are different, the conjunction “and” or some equivalent is used. Sibṭ al-Māridīnī (fifteenth c.) gives the example “if you add two dirhams to three things, the answer is two dirhams and three things” (“thing” (shayʾ) was another name for the first power in Arabic algebra). Written in our notation, this does not seem to do anything: 2 + 3x = 2 + 3x. Unlike our equation, the Arabic is stated as an operation and its outcome: one adds the two terms to get the algebraic binomial “two dirhams and three things.” And where our 2 + 3x is written with the sign for “plus,” a word that functions as a quasi-preposition to indicate the arithmetical operation of addition, the Arabic “and” (wa) is simply the conjunction. There is no more addition in “two dirhams and three things” than there is in “two geese and three chickens.”

Some algebraists compared the names of the powers to denominations of coins to help explain the addition of the different kinds. In Europe, these include Luca Pacioli (1494, f. 112a), Michael Stifel (1544, f. 231a), and Marco Aurel (1552, f. 71a). To add three ducats to two crowns, Aurel explains, “I cannot said that they are five ducats, nor five crowns: in reality they are three ducats and two crowns more, or two crowns and three ducats more. In this same way, I want to sum 3 with 2: I cannot say that they are 5, nor 5: we are compelled to say that they are 3 and 2 more, or 2 and 3 more.” (“… no pueden dezir que son cinco ducados, ni menos cinco coronas: mas realmente pueden dezir que son tres ducados, y mas dos coronas, o dos coronas, y mas tres ducados. Assi mesmo quiero summar 3, con 2: no podemos dezir que son 5, ni 5: mas forçadamente, diremos que son 3, y mas 2: o 2, y mas 3” (Aurel 1552, f. 71a).)

Unlike English, Arabic has a word that serves as the negative counterpart to “and.” This word is illā, which I translate as “less,” though outside mathematics “except” is usually better. For example, one can say “all the students except (illā) Joan.” In arithmetic, the apotome that we write as $5-\sqrt{3}$ is stated by Ibn al-Bannāʾ as “five less a root of three.” (Ibn al-Bannāʾ 1994, 288.2), while he writes the corresponding binomial as “five and (wa) a root of three” (p. 287.21). As with the example sentence, the word illā indicates that what follows it is lacking from what precedes it. In algebra, the same “less” is used to state an amount in which one term has been removed from another term. Al-Karajī writes in one problem “If you cast away the thing from fourteen, it leaves fourteen less a thing” (Saidan 1986, 193.15). Like with the addition of different kinds, here the “fourteen less a thing” is the result of the operation. It is a deficient 14, specifically a 14 that is lacking a thing. The word illā does not mean the same as the English “minus,” which, like “plus,” is a quasi-preposition indicating the arithmetical operation. Incidentally, the word al-jabr (“restoration”), from which our word “algebra” originates, is used in the simplification of equations to “restore” such diminished amounts (Oaks and Alkhateeb 2007).

In Italian abacus texts of the fourteenth and fifteenth centuries, “less” was rendered as “meno” (“less”), while “and” became either “et/e” (“and”) or “più” (“more”). The words più and meno correspond to the Latin plus and minus, where they also mean “more” and “less.” (In my translation of the passage of Marco Aurel above, the “and two coronas more” is from “y mas dos coronas,” where “y mas” is the Spanish version of the Latin “et plus,” or the Italian “e più.”) These two words appear in arithmetical and algebraic expressions in several sixteenth-century books where they also take the same meaning as the Arabic “and” and “less.” In notation plus and minus are either abbreviated as “$\tilde{\mathrm{p}}$” and “$\tilde{\mathrm{m}}$,” or they are shown with the signs “+” and “−”. It is clear by the way authors before ca. 1600 described the terms and how they were applied in context that neither the Latin words nor the signs “+” and “−” were used in the modern sense. The words and the signs began to assume their modern meanings indicating the operations of addition and subtraction only with Viète (Oaks 2018, 253–4).

So Stifel’s polynomial 12 + 16 − 36 – 32 + 24 is not a linear combination of the powers constructed from operations. It is a 12, a 16, and a 24 of three different kinds that have been diminished by a 36 and a 32 of two other kinds. The signs , , , and represent kinds, not values like our powers of x, and the terms are joined together like we join any collection of multitudes of different kinds. (Stifel does not have a sign for units, though other cossic algebraists did.) There is no scalar multiplication, no addition, and no subtraction intended in the notation. Said another way, the polynomial is an aggregation, or a kind of inventory, listing how many of each kind are present, and how many are lacking. It is thus fundamentally different from our 12x⁴ + 16x³ − 36x² − 32x + 24.

Premodern polynomials were not thought of as functions or as elements of some abstract ring. They were ways of expressing unknown numbers using the same kind of language for saying known numbers. In Arabic one said “a hundred and seven and forty” (147), the same way one said “a māl and forty-nine dirhams and fourteen things” (in modern notation, x² + 49 + 14x). (The number and the polynomial are from (Abū Kāmil 2012, 449.9, 451.14).) Several Arabic algebraists called polynomials “composite numbers” to distinguish them from monomials, or “simple numbers.” In the sixteenth century polynomials were often referred to by terms like “quantities” or “cossic numbers.” Caspar Peucer writes about three kinds of cossic number (numerorum cossicorum): simple (like 20, 24, and 30), composite (like 1 +12, 1 + 1, and 2 + 8), and diminished (like 2 − 8 and 4 − 6) (Peucer 1556, f. Mviija). Then, on the next page, he gives the example of the “number” (numerum), 12 + 16 − 36 − 32 + 24, which he borrowed from Stifel.

Algebraic equations ideally expressed the equality of two cossic numbers, without any unresolved operations hanging over them. Jacques Peletier compared a two-term equation with an “equation” in currency (he writes “ç” for “,” and “” for “”):

Thus, an equation is an equality of value between numbers denominated differently, such as when we say, 1 écu is worth 46 sous: it is an equation between 1, with its denomination of écu, and 46, with its denomination of sous. So when we say, 1ç, equal to 4, it is an equation between 1 with its denomination of ç, and 4 with its denomination of .

(“Equacion donq, ét une equalite de valeur, antre nombres diversemant denommez. Comme quand nous disons, 1 Ecu valoèr 46 Souz: il y à Equacion antre 1 avec sa denominacion d’Ecu: e 46 avec sa denominacion de Souz. Einsi, quand nous disons, 1ç, egal a 4: il y à Equacion antre 1, avec sa denominacion de ç: e 4 avec sa denominacion de ” (Peletier 1554, 22).)

Because equations state the equality of numbers, however, represented, premodern algebraists routinely worked out all operations before stating equations. Thus, all equations should be polynomial equations, free of any unresolved operations. Four articles that deal with this topic for Arabic, Italian, Greek, and Renaissance European algebra, respectively are Oaks (2009), Oaks (2010), Christianidis and Oaks (2013), and Oaks (2017).

3 Modifications to Premodern Algebra

3.1 Divisions and Roots in Equations

Adding, subtracting, and multiplying polynomials results in another polynomial, but this does not hold for division and taking roots. When confronted with division by a polynomial, Arabic algebraists usually found some way of reasoning through the operations to construct expressions without unresolved divisions. The simplest way was to preemptively multiply all terms that would eventually form the equation by the divisor. Abū Kāmil was particularly adept at finding other ways around divisions, sometimes applying arithmetical identities, and sometimes giving names to the results of divisions and forming equations using them along with the names of the powers. Two names he and other algebraists gave are dīnār and a fals, which are denominations of gold and copper coin. In one problem, for example, he writes: “we divide a thing by ten less a thing, so it results in a dīnār, and we divide ten less a thing by a thing, so it results in a fals” (Abū Kāmil 2012, 397.2). He then works with these names along with dirhams, things, and māls, until multiplications can be performed to eliminate them. But some authors, notably al-Karajī, occasionally worked with terms of the form “A divided by B” in equations until the appropriate multiplications could be performed to cancel the division. Treating a phrase like “a thing divided by ten less a thing” (like our $\frac{x}{10-x}$) as if it were a term in a polynomial violated the idea that expressions should be free of operations, but it also made the solutions to many problems much simpler (Oaks 2009).

Resorting to terms of the form “A divided by B” never became common in Arabic algebra. But in medieval Italian abacus books, divisions occur more frequently because they were written in rhetorical solutions with the division bar, along with Arabic numerals, as in this example from Maestro Biagio as related by Benedetto da Firenze (1463): “$\frac{180\, \mathrm{chose}\, \mathrm{e}\, 120}{1\, \mathrm{censo}\, 3\, \mathrm{chose}}$, e questo è iguali a 10.” (In modern notation, $\frac{180x+120}{x^2+3x}=10$ (Biagio 1983, 117.7).) Divisions continued to be common in sixteenth-century Europe, still with the division bar. A simple example in Michael Stifel is “Aequantur igitur & 7, facit 1. $\frac{3}{7}$” (f. 266b) (“Equating, then, and 7, makes 1 $\frac{3}{7}$”).

Roots were treated similarly. As mentioned above, irrational roots did not fit the premodern way of conceiving of numbers as multitudes, yet they became an integral part of arithmetic anyway. Similarly, roots of monomials and polynomials were accepted in algebra.

The introduction of divisions and roots to equations did not open the door for other operations. This is true for addition and subtraction, but it is most evident for multiplication. No premodern text will admit an unresolved multiplication, such as “a root multiplied by ten less a root,” to an equation, since multiplications can always be performed beforehand. The two sides of an equation remained ideally polynomials, even if they may have become compromised in some solutions with divisions and roots.

3.2 Irrational Coefficients

Before ca. 1500, notation was primarily a tool for working out problems. To communicate this work in a book or to explain the general rules, a rhetorical version, sometimes with a little notation included, was composed. As long as the notation was somehow tied to speech, it had to make sense, so irrational coefficients were forbidden. After 1500, we find notation in books becoming increasingly independent of the spoken word. An early example is Christoff Rudolff’s 1525 Coss, where algebraic calculations are presented without the intervention of any verbalization of the equations. Rudolff was the first of a small group of algebraists of his century who saw an advantage in allowing irrational coefficients in notation. Rudolff used them hesitatingly, then they become more common in Girolamo Cardano (1539), Johann Scheubel (1550), Rafael Bombelli (1572), and Simon Stevin (1585), with some others following suit through copying these authors. But many more algebraists continued to respect the semantics of the expressions by not allowing for irrational coefficients, including Michael Stifel (1544) (who criticized Rudolff’s coefficients), Jacques Peletier (1554), and Niccolò Tartaglia (1560).

In the following century, irrational coefficients in premodern algebra became rarer. Benedetto Maghetti (1639) set off a controversy when he used them in his solutions, and authors like Christoph Clavius (1608), Jacques de Billy (1643), and Jacob R. Brasser (1663) followed tradition by ensuring that their coefficients remain rational. (See Oaks (2017, 161ff) for Maghetti’s story. De Billy, for example, multiplies “1R − 5 − 3” by “5 + 3” to get “5Q + 3R − 14 − 180” (He writes R and Q in place of and , and for √). In modern notation, he multiplies $x-\sqrt{5}-3$ by $\sqrt{5}+3$ to get $\sqrt{5{x}^2}+3x-14-\sqrt{180}$ (De Billy 1643, 269). Brasser, for example, squares “√1920–1” (our $\sqrt{1920}-x$) to get “1920 − √7680 + 1” ($1920-\sqrt{7680{x}^2}+{x}^2$) (Brasser 1663, 301.11).) Like the introduction of divisions to equations, the admission of irrational coefficients by a handful of authors was not indicative of any conceptual shift. All other characteristics that identify a piece of text as being premodern are still present in these authors. The irrational “numbers” of terms stems from a convenient preference for notational syntax over rhetorical semantics. (See Oaks (2017) for a full discussion. There is one significant typographical error in the published version of the article. On page 154, line 19, the polynomial is shown incorrectly as “1 + 2 + √2.” It should be “1 + 2 + √2.”)

3.3 Independent Unknowns

Different names for independent unknowns are attested already in the ninth century in Abū Kāmil’s Book of Birds, and by the sixteenth century, several European algebraists were working with independent unknowns in conjunction with different notations to express them. Christoff Rudolff (1525), Girolamo Cardano (1539), Jacques Peletier (1554), and Simon Stevin (1585) are just some of the algebraists whose independent unknowns are discussed in Heeffer (2010). We review here the notation of Stifel’s Arithmetica Integra, which was later adopted in the Algebra of Christoph Clavius.

Stifel introduces his unknowns on folio 251b. Recall that the signs for the powers of the (primary) unknown are , , , , etc. The powers of the second unknown are designated A, A, A, A, etc., for the third unknown B, B, B, B, etc. He writes that one can simply write “A” for “A,” “B” for “B,” etc. To multiply powers of different unknowns, one concatenates the signs, so, for example, the product of 2 by 4A is 8A. By this rule, 3A by 2 by 4B would be 24AB: the signs for the unknowns are arranged in order from to A, B, etc. This is a notation for multiplying kinds, not values. The traditional cossic names already employed this idea, where for example is the kind that results from multiplying zensi () by cubes (). If Stifel meant the notation to be the multiplication of values, then the result of 1 by 1A would be 11A instead of 1A. Here “A” is the kind that results from multiplying the kinds “” and “A.” Stifel, like other premodern algebraists, has no notation for the multiplication of values.

Simon Stevin also devised a notation for multiplying kinds. Where Stifel multiplied 3A by 2B to get 6AB, Stevin multiplies 3 sec.① by 2 ter.② to get 6 sec.① M ter.②. Here the kind “sec.①” is multiplied by the kind “ter.②” to get the new kind “sec.① M ter.②” (Stevin 1585, 259). Again, this is not the multiplication of values. The earliest use of concatenation for the multiplication of values I have found is in Vaulezard’s 1630 translations of Viète’s Isagoge and Zeteticorum. Concatenation appeared soon after in other algebraists, all of them indebted to Viète’s new algebra.

4 François Viète’s Geometrical Algebra

4.1 Preliminary Remarks on Viète

This section is condensed from my article (Oaks 2018), where I investigate the nature of Viète’s species. In some places below I give the corresponding section of that article with a reference like “(§2.3).”

Below is a list of Viète’s works cited in the present chapter. All were reprinted in his collected works, which were published in 1646 by Frans van Schooten, titled Opera Mathematica. After each title, I give the date of the original printing and the page number in the 1646 Opera Mathematica.

  In Artem Analyticem Isagoge (1591, p. 1) (henceforth Isagoge)

Variorum de Rebus Mathematicis Responsorum, Liber VIII (1593, p. 347) (henceforth Variorum)

Effectionum Geometricarum Canonica Recensio (1593, p. 229) (henceforth Effectionum Geometricarum)

Zeteticorum (1593/1600, p. 42) (The first 16 folios were printed in Tours in 1593, and the remainder in Paris in 1600. Folio 16 ends in the middle of Zetetic IV.6 (Van Egmond 1985, 362))

Supplementum Geometriæ (1593, p. 240)

Ad Problema Quod omnibus Mathematicis totius orbis construendum proposuit Adrianus Romanus (1595, p. 305) (henceforth Ad Problema)

Pseudo-Mesolabum & Alia Quædam Adiuncta Capitula (consisting of the two works Pseudo-Mesolabum and Adiuncta Quædam Capitula. 1595, pp. 258, 275)

Ad Angularium Sectionum Analyticen (1615, p. 287)

De Æquationum Recognitione et Emendatione Tractatus Duo (consisting of the two works De Recognitione Aequationum and De Æquationum Emendatione. 1615, pp. 82, 127)

Ad Logisticem Speciosam, Notæ Priores (1631, p. 13) ((Freguglia 2008, p. 53) dates the composition to before 1593. I have not located any copy of the 1631 printing, so I refer only to the 1646 Opera Mathematica.)

J. Winfree Smith’s English translation of Viète’s Isagoge is printed in (Klein 1968), and T. Richard Witmer’s English translations of several of Viète’s works are published in (Viète 1983). I have taken or adapted translations of many passages from these sources. Translations without a reference are mine. Sometimes I give Witmer’s modern rendering of Viète’s notation to clarify for the reader how that notation functions.

4.2 Overview of Viète’s Program

There were two main currents in sixteenth-century mathematics, which were at once conflicting and entwined. One stemmed from the classical Greek texts of Euclid, Apollonius, Nicomachus, and others, in which the arithmetical unit is indivisible and geometric magnitudes are free of numerical measure. The other was the practical arithmetic, algebra, and geometry that was known to have come from medieval Arabic, and which in fact traces back to earlier Old World cultures. Numbers in this current can be partitioned indefinitely and were routinely applied to lines, surfaces, and solids.

The main interest of François Viète lay with Ptolemaic astronomy, in which these two currents met but did not mingle. Hipparchus of Bithynia (active 147–127 BC) is credited with being the first astronomer to apply Greek geometrical models of planetary motion to observational data. The models are cast in the non-arithmetized geometry of Euclidean magnitudes, while the observations consist of numerical measurements often recorded using physical instruments and arranged in tables. Hipparchus’s approach was adopted by Ptolemy (second century CE), whose authority guaranteed its continuation down to the early seventeenth century, when the models, based as they were in uniform circular motion, were made obsolete with the work of Johannes Kepler.

Viète was one of the last astronomers to work in this Greek tradition. From his first published book of 1579, covering geometric models and the trigonometric tables derived from them, to the draft on planetary astronomy that he was working on at his death in 1603, his mathematical interests lay squarely with Euclidean geometry in the service of astronomical calculation. His publications in the intervening years deal directly or indirectly with the geometry and arithmetical calculations that underlie Ptolemaic astronomy, including his theorems on triangles, continued proportions, angular sections, and the numerical solutions to equations. Throughout these works, Viète remained faithful to the style and precepts of Greek authors.

One Greek work particularly influential for Viète’s approach was Pappus of Alexandria’s Collection, which was finally published in Commandino’s Latin translation in 1588. Book VII covers the “domain of analysis,” a topic that is barely touched on in other extant Greek works. In it, Pappus lists several books on, or important for, analysis by different Greek authors, and he describes their contents, including some of the propositions they contained (Pappus 1986, 66ff, 84). Most of these books are lost, and Viète was perhaps the first to compose a restoration of one of them. Based on the description given by Pappus (1986, 90–94), he wrote his Apollonius Gallus as a reconstruction of the Tangencies (’E$\pi \alpha \phi \tilde{\omega}\nu,$sometimes translated as On Contacts) of Apollonius. But more importantly, Pappus’s Book VII likely played a role in inspiring Viète to transform the geometrical equalities prevalent in Greek works into a geometrical algebra that parallels the numerical algebra practiced in his time. He introduced his logistice speciosa in his 1591 Isagoge, and that book opens by situating the new algebra in the context of ancient analysis.

4.3 Viète’s Magnitudes and Numbers

The letters that constitute the objects of Viète’s algebra will not make sense until we first establish how he understood geometric magnitudes and numbers, and for that we need to begin with the views of his contemporaries. The term magnitudo in sixteenth-century mathematics meant any geometric magnitude of dimension 1, 2, or 3, to the exclusion of numbers, time, angles, and other quantities (§2.3). Greek authors stated and proved propositions about these magnitudes without recourse to numerical measure, and this is naturally respected in medieval and early modern translations. But commentators often assigned numbers to the magnitudes in diagrams to illustrate the relationships between them, and many of these numbers are fractions or irrational. For example, in his commentary to Proposition II.12 of Euclid’s Elements, Federico Commandino makes line AC 16 and line AB 20, and then approximates $\sqrt{231}$, the length of line AD, as $15\frac{1}{5}$, and with that he finds the area of triangle ABD to be approximately $98\frac{4}{5}$. In his commentary to Proposition XIII.1 he keeps the irrational roots, measuring lines with numbers like $5-\sqrt{5}$ and $\sqrt{20}+2$ (Euclid 1572, ff. 34b, 230a). The numbers that these mathematicians assigned to magnitudes were not intended as absolute measures. Rather, they illustrate the relative sizes of the magnitudes in a diagram, much like the numbers 1, 2, and $\sqrt{3}$ that we assign to the sides of the 30°–60°–90° triangle (§3.7).

Viète respected the ancient tradition of stating and proving propositions in non-arithmetized geometry, and like his contemporaries, he often assigned numbers to the lines after finishing his proof to illustrate their relative sizes. His numbers, too, are sometimes shown as approximations and sometimes as irrational roots. Figure 1 shows two examples, the first from Variorum in which he characteristically approximates the lengths of sides not with fractions, but with large integers. He gives three different arithmetizations, where each side in turn is set to be 100,000 and the other sides are calculated from it (Viète 1593a, f. 42a; 1646, 420). In the second, from Zeteticorum, he shows the sides of a triangle as $\frac{153}{61}$, $\frac{158}{61}$, and $\sqrt{13}$ (the latter as “l 13,” where “l” stands for latus (side)) (1593c, f. 16a; 1646, 67) (§3.8).

Fig. 1

Two of Viète’s triangles, one showing approximations, the other fractions and an irrational square root

In some settings in sixteenth-century mathematics, the word numerus took its discrete meaning after Aristotle and Euclid, so that numbers consist only of positive integers. In other settings, the word took its continuous meaning that was common in practical arithmetic, algebra, and astronomy, where numbers include fractions and irrational roots. In Viète’s works, all numbers are regarded as being the measures of magnitudes, so they are of the latter, continuous type. The underlying context of all of Viète’s mathematical works is geometry, and because there is no canonical geometric unit, the numbers with which he measures his lines express the sizes of the magnitudes relative to each other, like those in Commandino and other geometers (§3.8).

One major way that Viète’s geometry diverges from sixteenth-century practice is that he stepped beyond the third dimension. This occurs most clearly in Propositions XIIII and XV of Effectionum Geometricarum (1593), where he works with four-dimensional magnitudes based in the diagram shown in Fig. 2. He begins with right triangle ABC with right angle at C, and next to it he shows the same lines AB and BC laid out on a line that becomes the diameter if a semicircle. From the proportion AB : BD :: BD : BC, he gets AB² : BD² :: BD² : BC², and substituting AC² + BC² for AB², he arrives at (AC² + BC²) : BD² :: BD² : BC². He then converts the proportion into an equality, first in logistice speciosa, and then again in terms of the magnitudes in the diagram, which gives BC⁴ + BC² ⋅ AC² = BD⁴. This is expressed in purely geometric language and notation unrelated to any algebra:

Fig. 2

The diagram for Viète’s Proposition XIIII in Effectionum Geometricarum

Quadrato-quadratum ex BC plus planoplano sub Quadrato ex AC & Quadrato ex BC aequetur Quadrato-quadrato ex BD. (Viète 1593b, f. 4b; 1646, 235)

He does not address the issue of the existence of these magnitudes, nor of the four-dimensional magnitudes in Proposition XV (§3.3).

In another direction, Viète found himself deriving relationships between magnitudes of arbitrarily high dimension in the setting of the trigonometry of multiple angles. He published his theorems on angular sections in two works: Chapter 10 of Variorum and Chapter 9 of Ad Problema. After Viète’s death, Alexander Anderson added proofs and other theorems and published them in Ad Angularium Sectionum Analyticen (1615). The main theorem on angular sections is in some sense a corollary of the corollary stated by Ptolemy to what is called Ptolemy’s Theorem (Ptolemy 1998, 51ff). Converting Ptolemy’s corollary from chords of arcs to sines and cosines, we can write it as:

$$ {\displaystyle \begin{array}{l}\cos \left(a+b\right)=\cos (a)\cos (b)-\sin (a)\sin (b)\\ {}\sin \left(a+b\right)=\sin (a)\cos (b)+\cos (a)\sin (b).\end{array}} $$

Viète expressed these rules in terms of the ratios of the sides of right triangles, and by applying them repeatedly for angles θ + θ, 2θ + θ, 3θ + θ, etc., he derived rules for what we would write as cos(nθ) and sin(nθ) in terms of cos(θ) and sin(θ). These rules constitute Viète’s main theorem, which is Theorem III in Ad Problema and Ad Angularium Sectionum Analyticen, and Theorem I in Variorum. In all three books, he first states the rules verbally, then in terms of logistice speciosa, and this is followed with several numerical examples. In logistice speciosa, he calls the hypotenuse of the single-angle triangle Z, the base D, and the height B. If we call the base of the triangle with angle nθ D_n, its altitude B_n, and its hypotenuse Z_n, then his proportions can be written as

$$ {\displaystyle \begin{array}{l}{D}_2:{Z}_2::{D}^2-{B}^2:{Z}^2\\ {}{B}_2:{Z}_2::2 DB:{Z}^2\\ {}{D}_3:{Z}_3::{D}^3-3{DB}^2:{Z}^3\\ {}{B}_3:{Z}_3::3{D}^2B-{B}^3:{Z}^3\\ {}{D}_4:{Z}_4::{D}^4-6{D}^2{B}^2+{B}^4:{Z}^4\\ {}{B}_4:{Z}_4::4{D}^3B-4{DB}^3:{Z}^4\\ {}{D}_5:{Z}_5::{D}^5-10{D}^3{B}^2+5{DB}^4:{Z}^5\\ {}{B}_5:{Z}_5::5{D}^4B-10{D}^2{B}^3+{B}^5:{Z}^5.\end{array}} $$

The rules for D₂ : Z₂ and B₂ : Z₂ are his versions of the rules for cos(a + b) and sin(a + b) shown above. Because the succeeding calculations are an easy corollary to those known rules, Viète does not give a proof or show a diagram. I drew the diagram in Fig. 3 to conform to the diagrams he shows for his numerical calculations. In the diagram, Z = Z_n, but this is not required in the theorems.

Fig. 3

Reconstructed diagram for the theorem on angular sections

Here is how Viète states the theorem for the base of the quadruple angle triangle in all three books:

In ratione quadruplâ, Hypotenusa secundi [Z₄] fit similis Quadrato-quadrato-Hypotenusæ primi [Z⁴]. Basis [D₄], Quadrato-quadrato basis primi, minùs plano-plano sexies sub quadrato perpendiculi primi & quadrato basis eiusdem, plus quadrato-quadrato perpendiculi [D⁴ − 6B²D² + B⁴]. (Viète 1593a, f. 15b; 1595a, f. 15b; 1615a, 12; 1646, 289, 316, 372)

The language here belongs to geometry, not to arithmetic or algebra. The meanings of the terms “hypotenusa” and “basis” are “hypotenuse” and “base,” respectively, and “plano-plano” indicates a four-dimensional geometric magnitude just as it does in the example from Effectionum Geometricarum shown above.

Figure 4 shows one of the sets of calculations that follow. Viète began by assigning the radius AC of a semicircle to be “100 000” and BD to be “196 000,” from which he calculated BE through BK. Here, too, it is only the relative sizes of the numbers that matter. Viète’s higher-dimensional magnitudes may have no perceptible existence, but the rules in which they appear give the correct proportions for the sides of triangles when converted to numbers (§3.5)

Fig. 4

Some numerical calculations with angular sections (Viète 1595, f. 11b; 1646, 319) Ad Problema

Just as Viète gave no explanation for his magnitudes in Effectionum Geometricarum, he gives none for these higher-dimensional magnitudes, either. But he does address magnitudes of dimension greater than three in points 26 and 27 of Chapter VIII in Isagoge:

26 Since all magnitudes are either lines or surfaces or solids, of what earthly use are proportions above triplicate or, at most, quadruplicate ratio except, perhaps, in sectioning angles, so that we may derive the angles of figures from their sides or the sides from their angles?

27 Hence the mystery of angular sections, perceived by no one up to the present either arithmetically or geometrically, is now clear, and shows

How to find the ratio of the sides, given the ratio of the angles;

How to construct one angle [in the same ratio] to another as one number is to another. ((Viète 1591, f. 9a; 1646, 12). Translation adapted from Viète (1983, 32).)

Instead of justifying the existence of higher dimensional magnitudes, Viète explains that they are useful, and the use to which he puts them is in numerical calculation. Just as Ptolemy applied his version of the corollary to construct his table of chords, Alexander Anderson later remarked that Viète’s theorems on angular sections can be applied to construct trigonometric tables in increments of 1^′ (Viète 1615, 47; 1646, 303–304) (§3.6).

To summarize, Viète’s geometric magnitudes, like those of other mathematicians of his era, conform to classical Greek works in that they do not possess any inherent numerical size. But they are metrizable, and the numbers frequently assigned to them illustrate only their relative sizes, and not any absolute size. Viète parts from his contemporaries by freely admitting magnitudes of dimension greater than three. He gives no explanation of what kind of existence these magnitudes might have, but that would not have mattered to him because his interest lay with numerical computation. Viète’s numbers are the continuous, practical numbers of the astronomers (and the algebraists) that include fractions and irrational roots, and their only purpose in his works lies with the measurements of geometric magnitudes. They have no independent existence.

In the sixteenth century, geometric magnitudes could be measured in two different ways. Classical Greek magnitudes are measured through ratio and proportion, and Viète and other Europeans of his time set up and manipulated ratios and proportions in their geometric propositions. But these same early modern European mathematicians also routinely assigned numerical measures to magnitudes, so one could also perform arithmetical calculations that parallel those non-arithmetized proportions. This gives magnitudes a dual nature when it comes to measurement, and these two ways correspond to the two algebras that Viète presents in his books.

4.4 Viète’s Two Algebras

Viète called his new algebra logistice speciosa (specious logistic) to distinguish it from traditional numerical algebra, which he called logistice numerosa (numerical logistic). He borrowed the notation for the latter from Xylander’s 1575 translation of Diophantus’s Arithmetica. Translating the terms from Diophantus, Xylander called the first degree unknown a “Numerus,” which he abbreviated as “N.” The second degree is a “Quadratum,” abbreviated as “Q,” and the third degree is a “Cubus,” abbreviated as “C.” Higher powers, still translated from Diophantus, are “Quadratoquadratum” (QQ), “Quadratocubus” (QC), and “Cubocubus” (CC). Here is an example equation in this notation from Viète’s De Emendatione Aequationum: “30Q. − 156 N. − 1C., æquatur 440.” ((1615b, 123.7; 1646, 155.30). The 1646 printing is identical except that it omits the periods.) Distorted into modern notation, it would be 30x² − 156x − x³ = 440. The recurring coefficient of 1, the absence of operations, and other features of logistice numerosa throughout Viète’s works confirm that it belongs to the premodern algebra of the sixteenth century. And because all numbers in Viète are the measures of lines, planes, solids, etc., his logistice numerosa functions as an algebra for arithmetized geometry (§§4.1, 5.6).

Logistice speciosa, by contrast, is the algebra that Viète developed for geometry in which the knowns and unknowns are the relative sizes of the magnitudes themselves, and not numbers that measure them. This truly geometrical algebra was inspired in part by the computational geometry he read in Greek authors. Some propositions in Greek geometry concern spatial relationships of the magnitudes, while others specify only metrical relationships. (See, for example, Marie (1884, 4) and Panza (2006, 279).) In the latter case, it often happens that some of these magnitudes cannot exist in position. For example, Apollonius could write in proposition I.37 of his Conics that the rectangle contained by ΔZ, ZE is equal to the sum of the rectangle contained by ΔE, EZ and the square on ZE, even if no such rectangles can be drawn because the points Δ, Z, and E are colinear (Apollonius 1891, 112.5). Impossible magnitudes like these enter into the arguments only as (abstracted) sizes relative to other magnitudes deriving from the diagram. (Others, too, have made this observation, such as Mueller (1981, 42).) Moreover, the spatial relationships of some magnitudes are not relevant at all to the proposition, and in those cases they are drawn apart from the other magnitudes and are labeled with single letters. Euclid’s Proposition VI.17, for example, deals with proportionality and shows four individual lines labeled A, B, Γ, and Δ. In other propositions, it is an area that is designated by a single letter, like regions Γ and Δ in Elements Proposition VI.29.

Viète reworked these geometrical equalities into an algebra, partly modeled on logistice numerosa. The knowns and unknowns cannot be based in any particular diagram showing spatial relationships, so he chose single capital letters to represent them, and he followed the order of the transliterated Greek alphabet, calling his “species” A, B, G, D, Z, etc. This notation would have made sense to mathematicians of the time, given the presence and purpose of single-letter magnitudes in Greek geometry (§§5.4–5).

Viète presented the outlines of his logistice speciosa in the short yet enigmatic 1591 book In Artem Analyticem Isagoge (Introduction to the Analytic Art). After three chapters covering preliminary stipulations and laws, he introduces the species:Verse

Verse On the rules of calculating with species. Chapter IIII. Numerical logistic employs numbers, specious logistic employs species or signs for things, as, say, the letters of the alphabet. There are four basic rules for calculating with species just as there are for calculating with numbers.                                                   RULE I.                                    To add a magnitude to a magnitude. Let there be two magnitudes A and B. One is to be added to the other. ((Viète 1591, f. 5a; 1646, 4). Translation adapted from (Viète 1983, 17–18).)

Viète calls the species A, B, etc. of his algebra “magnitudes” (magnitudines) here and throughout his books. Unknown magnitudes are designated with vowels, and known magnitudes with consonants. These species, or magnitudes, can be raised to different powers, and these powers possess dimension. A simple “A,” “B,” “G,” etc. is of the first dimension and is called a “side” (latus). (In rare cases when it does not designate the measure of a line, it is called a “root” (radix), such as we see in Proposition LII in Notæ Priores (Viète 1646, 37).) The two-dimensional squares of these sides are called “A quadratm,” “G quadratum,” etc. Their cubes are called “A cubus,” “G cubus,” etc., their fourth powers, which are naturally four-dimensional, are “A quadrato-quadratum,” “G quadrato-quadratum,” etc., and higher powers continue to follow the pattern for the names in Xylander, translated from Diophantus: “quadrato-cubus,” “cubo-cubus,” “quadrato-quadrato-cubus,” quadrato-cubo-cubus,” etc. There is another series of terms used to label magnitudes whose lower-dimensional versions do not enter into the problem. A two-dimensional magnitude whose side does not appear is called “A planum,” “B planum,” etc. A three-dimensional magnitude is called “A solidum,” “B solidum,” etc., continuing with the four-dimensional “plano-planum,” the five-dimensional “plano-solidum,” etc., following the same pattern as the powers (§4.1).

The sample equation in logistice numerosa given above is a numerical version of this equation in logistice speciosa, here reproduced as shown in the 1615 printing (Viète 1615b, 122):

$$ \left.\begin{array}{l}\, \mathrm{B}\ \mathrm{in}\ \mathrm{A}\ \mathrm{quad}\ 3\\ {}-\mathrm{D}\ \mathrm{plano}\ \mathrm{in}\ \mathrm{A}\\ {}-\mathrm{A}\ \mathrm{cubo}.\end{array}\right\}\, \ae \mathrm{quetur}\, \left\{\begin{array}{l}\, \mathrm{B}\ \mathrm{cubo}\ \mathrm{bis}.\\ {}-\mathrm{D}\ \mathrm{plano}\ \mathrm{in}\ \mathrm{B}.\end{array}\right. $$

In the 1646 Opera Mathematica, the same equation is written on one line: “B in A quadr. 3 − D plano in A − A cubo, æquetur B cubo 2 − D plano in B” (Viète 1646, 155.23). Witmer transforms it into more modern notation as 3BA² − D^PA − A³ = 2B³ − D^PB (Viète 1983, 303).

Certain features suggest that this notation should be read differently than the premodern notations we described earlier. First, a “1” is not written when there is only one of a term. In this equation, the “A cubo” has no coefficient that could count or measure the term, and this applies to every example in Viète’s works. This means that the letters are not kinds, but values. Second, these values are sometimes multiplied together. In the first term, “B” is expressly multiplied by “A quad” by the presence of the preposition in (“by”). Similarly, “D plano in A” expresses the multiplication of the two-dimensional “D plano” by the one-dimensional “A.” Perhaps the most noticeable difference is that known values are designated with letters and not numbers, which makes them provisionally undetermined. This feature has been cited as a great advance by historians dating back to Montucla’s 1758 Histoire des Mathematiques (1758, 488). The “3” and the “bis” (“2”), on the other hand, function more like premodern “numbers” of terms to indicate how many of that term there are.

Proposition XVI of Supplementum Geometriae nicely illustrates the similarities and differences between three ways of comparing magnitudes. There a particular relationship is first written as a geometrical equality, then as an equation in species, and finally as a numerical equation. For the equality, A, B, C, and E are points in the diagram (Viète 1593d, f. 17b; 1646, 249):

cubus ex AC minus triplo solido sub AC & quadrato ex AB, æqualis est solido sub CE & quadrato ex AB

For the equation in species, the unknown line AC is called A, and lines CE and AB are taken to be equal in length and are called Z:

A cubus minus Z quadrato ter in A, æquetur Z cubo

For the numerical equation, Viète sets Z to be 1 and A to be 1N:

1C − 3N, æquatur 1

The equality is naturally stated using geometrical language, including the phrase we can translate as “triple the solid under CE & the square on AB.” In the equation in species, the language of arithmetical operations is substituted, where this becomes “thrice Z squared [multiplied] by A.”

We saw above how Viète produced proportions involving magnitudes of arbitrarily high dimension. It is by virtue of Viète’s “supreme stipulation,” in Chapter II of Isagoge, that these magnitudes can be freed from their ratios and enter into equations as independent objects:

• But the supreme stipulation of equations and proportions and which is all-important in analysis is:

• 15 If there be three or four magnitudes and the result of the multiplication of the extreme terms is equal to the result of the multiplication of the mean by itself or to the product of the means, then those magnitudes are proportional. And conversely,

• 16 If there be three or four magnitudes, and the first is to the second, so that second, or else some third, is to another, the product of the extremes will be equal to the product of the means.

And so, a proportion can be called the composition (constitutio) of an equation, an equation the resolution (resolutio) of a proportion. (Translation adapted from Klein (1968, 324))

By this stipulation, the proportion a : b :: c : d is equivalent to the equality ad = bc. It generalizes Euclid’s Elements Propositions VI.16 and XI.34 by removing any restriction on the dimension of the terms in the equality. It is what allowed Viète to resolve the proportion in Proposition XIIII from Effectionum Geometricarum into a geometric equality with four-dimensional magnitudes.

In numerical logistic, one can add 1Q to 3N to get 1Q + 3N because numbers are homogeneous. But in geometry, magnitudes of different dimension are heterogeneous, so one cannot add, say, A quadratum to A. Viète solved this problem by multiplying the lower degree terms by known magnitudes of such dimension that all terms become homogeneous. For some given B, one can add “A quadratum” to “B in A” because they are both now two-dimensional. Thus his “Law of Homogeneity,” laid out in Chapter III of Isagoge, states that only terms of the same dimension (power) can be added or subtracted.

Specious logistic was designed for solving problems in geometry. Given magnitudes are named with consonants, an unknown is named with a vowel, and then an equation or proportion is set up in terms of these species and simplified. The resulting equation or proportion can then be read as a formula from which one can either perform a geometric construction to show the unknown magnitude, or numbers can be assigned to the givens and the unknown can be calculated numerically (Viète 1591, f. 8a; 1646, 10). These two ways of exhibiting the solution are a reflection of the dual nature of geometric magnitudes. In his Zeteticorum, Viète solves 82 problems with his new algebra, giving only numerical solutions. Two of his successors, Jean-Louis Vaulezard and Marino Ghetaldi, later took up the task of producing answers by means of geometrical construction (Ghetaldi 1630; Viète 1630).

4.5 Summary of Viète’s logistice speciosa

In creating an algebra for the non-arithmetized sizes of geometric magnitudes, Viète caused a radical reconception of polynomials, and of algebraic expressions in general. Now coefficients (knowns) and unknowns are both distinct objects of the same kind – measures of geometric magnitudes – that are necessarily multiplied together. And because monomials in logistice speciosa are now often stated with multiplications, and the final equation or proportion in a solution serves as instructions for either performing a geometrical construction or an arithmetical calculation, the signs “+” and “−” should be reinterpreted as operations, too.

Given quantities can only be undetermined in such a system. While one can name specific numbers, like 6 or 4 $\frac{4}{5}$, one cannot name the measure of a specific magnitude that is distinct from other magnitudes. Thus, given magnitudes are necessarily undetermined and are shown as consonants. The simplified equation or proportion obtained in solving a problem now conveniently becomes a formula stated in terms of the givens, so one can see precisely how the solution is obtained from the original parameters.

Two other features of this geometrical algebra differentiate it from numerical algebra. Because there is no canonical unit, all terms possess dimension, and this dimension must be respected when forming expressions. One can add and subtract only homogeneous terms, and multiplication results in a term of higher dimension than the multipliers. Because of this, the minimum dimension of a term is 1, so there can be no zero degree terms or reciprocals of terms as there are in numerical algebra. Division is thus possible only of a higher degree term by a lower degree term, as Viète tells us in Chapter IIII, Rule IIII in Isagoge (Viète 1591, f. 6a; 1646, 6). (Thus the expression shown in (Klein 1936, 180; 1968, 173) is Klein’s fabrication, since it is a third degree term divided by a third degree term). Also, roots must have positive integer degree. While Abū Kāmil could write “a root of three cubes” (in modern notation, $\sqrt{3{x}^3}$) (2012, 449.8), Viète could not take the square root of “A cubus 3” because the dimension would be fractional.

The letters of Viète’s algebra are not symbolic entities that transcend arithmetic and geometry, as Jacob Klein argued (1936, 1968). Klein read Viète’s rules for operating on the species in Chapter IIII of Isagoge as being axioms governing the relationships between the species. These rules, according to Klein, are general enough to cover both arithmetical and geometrical calculation, so that the species A, B quad., etc. in an algebraic expression can represent either numbers or magnitudes. The species themselves “are, therefore, comprehensible only within the language of symbolic formalism” (Klein (1968, 175), his emphasis). By this view, one can solve either arithmetic or geometry problems by translating the knowns and unknowns into logistice speciosa, and once the equation or proportion has been simplified, the solution can be found by reinterpreting the letters in terms of the original numbers or magnitudes.

Klein was apparently unaware that Viète worked with higher dimensional geometric magnitudes, and he passed over the several passages in Viète’s works that unequivocally associate the species with magnitudes and not with numbers. His error was taken up by many historians, including even Michael Mahoney, who writes that Viète’s logistice speciosa “is, to use modern terms, a language of uninterpreted symbols” (1994, 39), and Henk Bos, who saw it as “a method of symbolic calculation concerning abstract magnitudes” (Bos 2001, 147). I address Klein’s account more thoroughly in 2018, §§6.2–3.

While readers of Viète in the past century have had a difficult time understanding the nature of the species, his immediate successors seem to have had little problem. Viète is quite explicit that his letters are magnitudes, and contemporary geometers all understood that magnitudes can enter into calculations via ratio and proportion, and they can also be assigned numerical values. It is not the species, but magnitudes themselves that can be measured in two ways.

5 Fermat’s Algebra

5.1 Fermat, Student of Viète

When Pierre de Fermat (ca. 1607–1665) was in his early 20s, he traveled to Bordeaux, where he met students of Viète and studied Viète’s works. As Mahoney observes, Viète “shaped Fermat’s mathematical career as decisively as if the two men had stood in a living master-student relationship,” and that “everything that Fermat did in mathematics, however original, bears Viète’s imprint” (Mahoney 1994, 26). In Bordeaux, Fermat quickly and wholeheartedly embraced Viète’s new algebra, which, like Viète, he saw as a tool with which to restore the analysis practiced by ancient geometers. Fermat appropriated precisely Viète’s notation, too, which he continued to use to the end of his career even after the notation of Descartes had eclipsed it. This notation is already the main tool of analysis in Fermat’s Method for Determining Maxima and Minima and Tangents to Curved Lines, completed by 1629. (Methodus ad Disquirendam Maximam et Minimam, published in Fermat (1891, 133–179). The title and its translation are taken from Mahoney (1994, 55).) For example, to solve the problem of finding the cone of maximum surface that can be inscribed in a given sphere, he draws the semicircle shown in Fig. 5, calling line AB B and line BC A. (Following the edition of his works (Fermat 1891), we write Fermat’s species in italic font, and points in diagrams in regular font.) From there, he determines that he needs to maximize the fourth-degree amount “B cub. in A + B quad. in A quad. − B in A cub. − A quad.quad.” (Fermat 1891, 157).

Fig. 5

From Fermat (1891, 156)

It is also with this algebra that Fermat developed his analytic geometry . Like Viète, he undertook the restoration of a lost work of Apollonius described by Pappus in Book VII of the Collection. Fermat chose Apollonius’s Plane Loci, and it was during his struggle to prove Proposition II.5 of his restoration, and through his subsequent investigations of loci generally, that he hit upon the idea of transforming the symptômata of conic sections, that is, geometrical equalities that characterize the curves, into Vietan equations in two unknowns (Mahoney 1994, 112ff). Fermat completed his restoration of Plane Loci around 1635, and the following year he outlined a classification of equations and their corresponding curves in Introduction to Plane and Solid Loci. (Ad Locos Planos et Solidos Isagoge, published in Fermat (1891, 91–110).)

Where Viète had been concerned with developing geometrical models for astronomical calculation, Fermat’s focus was on the geometry itself. So unlike Viète, Fermat never assigns numerical measures to the species, and in work after work he relates the algebraic expressions and equations to the magnitudes in geometric diagrams.

5.2 Fermat’s Interpretation of the Species

With about four decades separating them, and given their different interests, we might wonder if Fermat interpreted the species differently than their creator did. Reading over Fermat’s works, one finds that he rigorously observed the law of homogeneity and that he followed Viète in working with magnitudes of arbitrarily high dimension. At least for the mechanics of the algebra, there is no indication that Fermat’s understanding of the species diverged from that of Viète. Mahoney, however, argued that Fermat viewed his algebraic terms as all being homogeneous:

Although Fermat paid lip service to Viète’s Law of Homogeneity, his use of algebra in geometric situations clearly shows that he no longer attached dimension to the degree of an expression. Fermat operated as if all combinatory operations on geometrical magnitudes yielded homogeneous products. At no time, however, does he justify his behavior. (Mahoney 1994, 42)

Mahoney came to this conclusion based in part on this claim:

…every equation of Viète’s analytic art had dimension, and the dimension was directly linked to the degree of the equation. For example, the solution of a cubic equation corresponded to a solid construction in three-dimensional space … (Mahoney 1994, 42)

Viète never solves a problem in his Zeteticorum with a geometrical construction, so such solutions to equations will not be found among his worked-out problems. But he does give the constructions necessary for solving particular problems in two other works: Effectionum Geometricarum, which covers basic ruler-and-compass constructions, and Supplementum Geometriae, which covers propositions on the trisection of an angle. In 7 of the 45 propositions presented in these 2 books, Viète states algebraic equations that can be solved by the given construction, and of those 7, the degree of the equations in 4 of them do not match the dimension of the construction. (Propositions IX, X, XI, XIIII, and XV in Effectionum Geometricarum, and Propositions XVI and XXIIII in Supplementum Geometriae. Only the first three in the first work show a match between degree and dimension.)

In the introduction to Supplementum Geometriae, Viète postulates a neusis construction: “To draw a straight line from any point to any two given lines, the intercept between these being any possible predefined distance” (Viète (1593, f. 13a; 1646, 240), translated in Bos (2001, 168)). In Proposition IX, he applies this construction to the problem of trisecting a given angle in the context of a two-dimensional diagram. Then, in Proposition XVI, with essentially the same diagram, he gives a third-degree algebraic equation that corresponds to that solution: “A cubus minus Z quadrato ter in A, æquetur Z cubo.” Viète also arrives at cubic equations in Proposition XXIV, for the problem of inscribing a regular heptagon in a circle. In both propositions, the degree is 3 and the dimension is 2.

A similar situation arises in Propositions XIIII and XV in Effectionum Geometricarum. These are the propositions in which Viète works with four-dimensional magnitudes, discussed above in Sect. 4.3. The diagrams are again two-dimensional, and this time the equations are of the fourth degree. After showing the relationship between the lines as a proportion involving ratios of two-dimensional magnitudes, Viète gives a corollary showing the fourth-degree equation that this construction will solve. In Proposition XIIII, this is “A quadrato-quadratum, plus B quadrato in A quadratum, æquetur D quadrato-quadrato” (Viète 1593b, f. 5a). Given the equation, one can appeal to Theorem I in Chap. XII of De Recognitione Æquationum to make a substitution that transforms it into a quadratic equation, whose solution is constructible (Viète 1615b, 27–28; 1646, 99; 1983, 191). The equation in Proposition XV can be similarly reduced to a quadratic equation by Theorem III in the same chapter. There is in fact no rule in Viète linking the degree of an equation with the dimension of the solution. We can only infer from the few examples in these two works that it is the dimension of the original problem that corresponds to the dimension of the solution.

Mahoney also misunderstood Viète’s higher-dimensional magnitudes. To account for their geometrical meaning, Mahoney wrote: “their interpretation was possible, though it required the introduction of ‘mechanical’ procedures like the Pseudo-Mesolabe for generating fourth, fifth sixth, etc., proportionals to two given line segments” (Mahoney 1994, 42). This Pseudo-Mesolabe, described in Proposition II in Viète’s Pseudo-Mesolabum (1595b, f. 2b; 1646, 259), serves specifically to produce four lines in continued proportion. The propositions that follow likewise all deal with exactly four lines. There is no mention of producing five or more lines in continued proportion, so the mechanical procedure only gives lines whose relationship can be expressed with a cubic equation. Nor is there mention in any other work of Viète’s of a “mechanical” procedure to produce a series of five or more magnitudes in continued proportion (though Descartes later would do it with his Mesolabe (Descartes 1637, 317ff, 370)). Instead, Viète generates proportional series in terms of the higher powers of his species. Proportional series are introduced in Proposition II of Notæ Priores (1646, 13). Two magnitudes A and B are followed by “$\frac{B\, quadr.}{A}$,” “$\frac{B\, cubus}{A\, quadr.}$,” “$\frac{B\, quad. quadratum}{A\, cubo}$,” etc. He finds the proportionals between two given magnitudes A and B in Proposition V (1646, 15), with the example of quintuplicate ratio. He first makes the series of five-dimensional species in which the extreme terms are “A quadrato-cubus” and “B quadrato-cubus.” Witmer translates this series as A⁵, A⁴B, A³B², A²B³, AB⁴, B⁵ (1983, 37). Viète then takes the fifth roots of each term so that the extreme magnitudes are A and B. In Viète, higher dimensional magnitudes are used to produce the continued proportions, not the other way around. As I explained in the previous section, Viète did not give any geometric meaning to his magnitudes of dimension greater than three. They arise naturally in the theorem on sectioning angles, and he worked with them because they were useful for numerical calculation.

With the misunderstandings that the degree of the equation corresponds to the dimension of the problem and that Viète’s higher dimensions are artifices that are justified through proportional lines, Mahoney concluded that

both Fermat and Descartes saw clearly the difficulties that Viète’s translation of geometry into algebra caused if one tried to apply it to a realm of problems which themselves involved dimensionality, e.g. locus problems. If one wishes to determine the nature of a locus, which itself may be dimensionless or have a dimension depending on the conditions, one cannot use a vehicle of analysis that is already burdened by dimensionality. (Mahoney 1994, 42)

But when viewed against Viète’s actual practice, there is nothing about his species that would prevent them from being employed to solve locus problems, and consequently there is nothing in Fermat’s writings to suggest that he held a different view of the species than Viète. But one should wonder, then, how Fermat understood higher-dimensional terms like the four-dimensional expression quoted above, or the terms in an equation like “Aqq. æquale Zpl. in Aq. − Zs. in D” from Introduction to Plane and Solid Loci. ((Fermat 1891, 109). Fermat abbreviates quadratum, planum, and solidum as q, pl, and s.) Characteristically, he does not offer an answer. All I can suggest is that like Viète, he saw higher-dimensional magnitudes in his intermediate calculations as useful, since they give the correct results when the simplified equations are interpreted in two- or three-dimensional space.

6 Descartes’s Algebra

Descartes published his only mathematical treatise, La Geometrie, as the third appendix to his Discourse on Method. The book appeared in 1637, the same year that Fermat shared his Introduction to Plane and Solid Loci with the mathematicians in Paris (Mahoney 1994, 72). Descartes, like Fermat, was interested specifically in geometric problem-solving (Bos 1990; 2001, 229), and like Fermat, his analytic geometry was inspired at least in part by his solution to a locus problem originating in Book VII of Pappus’s Collection. This problem had been proposed to him in 1631 by Jacobus Golius, a professor at University of Leiden, and Descartes had solved it by January of the following year (Bos 2001, 271). It asks for a point C such that lines extending from C to given lines and forming given angles with them make certain products equal (Sasaki 2003, 206; Bos 2001, 313ff; Pappus 1588, f. 165a; 1986 I, 120). The solutions to this indeterminate problem form a curve, and Descartes’s solution to the general case of four lines, given in Books I and II of La Geometrie, illustrates his method of representing loci by equations in two unknowns. The method is equivalent to that of Fermat: one unknown is measured along a given line from a given point, and the other is measured at some angle from it. He sets up equations for other locus problems, too. One of these is the equation that describes the curve traced by what Bos calls the “turning ruler and moving curve” procedure (Bos 2001, 278), which Descartes employs later to trace the solutions to equations. He also forms an equation for the solution to a special case of the five-line Pappus problem, and algebraic expressions and equations are also instrumental in his solutions to the other problems in Book II, both for finding normal lines to curves and for constructing ovals satisfying particular optical properties. (The equations for the three loci are given at Descartes (1637, 322.5, 325.-1, 337.10).)

For his version of specious algebra, Descartes uses lowercase letters, usually with x and y as unknowns and other letters as knowns, and he shows multiplication by concatenation. He also introduces superscripts for exponents. What was “E cubo-cubus” in Viète becomes “y⁶” in Descartes, though for the square he preferred yy to y². Here are two sample equations, the first one describing a curve traced by the turning ruler and moving curve procedure from Book II, and the second an example provided for a rule for manipulating equations from Book III (Descartes 1637, 322, 379):

$$ yy= cy-\frac{cx}{b}y+ ay- ac $$

$$ {x}^3-\sqrt{3} xx+\frac{26}{27}x-\frac{8}{27\sqrt{3}}=0. $$

There was precedence for indicating multiplication by juxtaposing species. Jean-Louis Vaulezard did so in his 1630 French translations of Viète’s works, and we find it also in the arithmetical versions of logistice speciosa in Thomas Harriot (1631), William Oughtred (1631), and James Hume (1636). These last three authors had no need for the law of homogeneity, since their species are now numbers. Harriot, for example, could write an equation like ^“aaa − 127296. a = 85700000^” (Harriot 1631, 145), and Oughtred could divide “BA + A” by “A” to get “B + 1” (Oughtred 1631, 11).

In the beginning of his book, Descartes tells us how to interpret his letters. He famously outlines an algebra of line segments by “taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily …,” and then redefining multiplication, division, and root extraction so that the outcome is always another line segment. For multiplication, “having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity” (Descartes (1637, 297), translated in Descartes (1954, 2)). He defines division and square root similarly, and he gives ruler-and-compass constructions for these operations on the next page. As Henk Bos has stressed, “in Descartes’ interpretation the algebraic operations as applied in geometry did not concern numbers but geometric magnitudes, namely, the segments. Although he did introduce a unit line segment, he did not identify line segments with their numerically expressed lengths.” (Bos 2001, 296). By redefining multiplication, division, and square root, Descartes created an algebra of lines whose operations agree with the corresponding operations of numbers in practical algebra. With it he could make sense of powers greater than three, and he could add and subtract what would otherwise have been nonhomogeneous terms. (See Bos (2001, 293ff) for a discussion on Descartes’s redefinitions of the operations.) He was thus free to work with equations like the second example shown above.

But with this apparent abandonment of homogeneity, it is important to consider what he writes next:

It should be noted that all parts of a single line should always be expressed by the same number of dimensions, provided unity is not determined by the conditions of the problem. Thus, a³ contains as many dimensions as abb or b³, these being component parts of the line which I have called $\sqrt[3]{3{a}^3-{b}^3+ abb}$. (Descartes (1637, 299), translated in Descartes (1954, 6).)

So if no unit is declared, then the operations of multiplication, division, and root extraction necessarily revert to their traditional geometrical meanings, and the law of homogeneity must be obeyed. Indeed, apart from one particular instance of the Pappus problem, no unit is posited in any problem posed in Book II. All expressions and equations in those solutions are homogeneous, like the first equation shown above. This does not stop him from working with higher-degree equations, however. For example, one homogeneous equation in Book II is of the sixth degree, beginning “y⁶ − 2by⁵…” (Descartes 1637, 344).

The Pappus problem deserves special attention. Descartes also works with higher powers there, too, though his general equation is only of the second degree. He sets up this equation in unknowns x and y, and in knowns b, c, d, e, f, g, k, l, and a parameter z used in defining ratios:

In more modern format, it is this:

$$ {y}^2=\frac{\left( cfglz-{dekz}^2\right)y-\left({dez}^2+ cfgz- bcgz\right) xy+ bcfglx-{bcfgx}^2}{ez^3-{cgz}^2}. $$

After making some simplifying assignments, he solves the equation for y in terms of x. From there, he gives detailed instructions on how to construct solutions, and he explains based on whether different givens are added, subtracted, or “nothing” (nulle) whether the locus of solutions is a line, circle, ellipse, parabola, or hyperbola. All this is done with homogeneous expressions without any declared unit. Then, after completing his description of the constructions of solutions, Descartes writes:

And if one wants to show all the given quantities by numbers, by making, for example, EA = 3, AG = 5, AB = BR, BS = $\frac{1}{2}$ BE, GB = BT, CD = $\frac{1}{2}$ CR, CF = 2CS, CH = $\frac{2}{3}$ CT, and that angle ABR be of 60 degrees, and finally that the rectangle of the two, CB, & CF, be equal to the rectangle of the two others CD & CH … (Descartes 1637, 333; 1902, 405)

Making EA = 3 establishes the unit. Making AG = 5 is equivalent to saying that AG = $\frac{5}{3}$ EA, and making angle ABR 60^∘ serves to make EB = 2BS. It is from the last condition that the equation “yy = 2y − xy + 5x − xx” is established, which describes a circle (the axes are at a 60° angle). It was not necessary for the construction of the solution for Descartes to have assigned a unit. Had he written “AG = $\frac{5}{3}$ EA” in place of “EA = 3, AG = 5,” then since EA = k, the unitless, homogeneous equation would have been $yy=\frac{2}{3} ky- xy+\frac{5}{3} kx- xx$, and the construction of solutions would have been exactly the same. Descartes may have posited EA = 3 to illustrate his algebra of line segments, or perhaps to give meaning to the higher degree terms in the derivation of the general equation that preceded these assignments.

In Book III, where Descartes investigates the roots and transformations of equations and how to construct solutions, he usually works with examples of heterogeneous equations for which some unit has been chosen. These examples are not given in the context of any particular problem, however. Bos has noted that Descartes works with these two interpretations of the operations, one from his algebra of line segments and the other from traditional geometry that takes into account dimension. Descartes himself does not address the issue of the compatibility of these interpretations, which suggests to Bos “a certain carelessness about the matter” (Bos 2001, 301). (Bos notes that they are in fact compatible, as one early commentator showed.)

Perhaps Descartes’s statement that the unit “can in general be chosen arbitrarily” should be understood to mean that it is potentially present in problems where it is not declared. No unit is declared in the general solution to the Pappus problem, but afterward Descartes’s choice of a unit for one particular arrangement of the lines can be interpreted as retroactively giving meaning to all the terms of degree greater than three in the general solution leading to that specific setup. But that is all it would do, because the selection of a unit is unnecessary for constructing such particular solutions. Further, Descartes does it for only one case of one problem. No other solution in Book II is followed by a solution “by numbers.” We can infer, then, that Descartes preferred to find general solutions, after which a solution “by numbers” was merely an option. Because of this, he effectively worked in the same non-arithmetized geometry as Viète and Fermat. Every term in Descartes’s expressions arising from general solutions have positive integer degree, and the law of homogeneity is observed. As in Viète, magnitudes in the problems solved by Descartes are without any intrinsic numerical measure, but one can later assign numbers to them for whatever purpose one has in mind.

There is one minor formal development in Descartes’s notation for homogeneous expressions that may have been inspired by his geometry of line segments. Where Viète would have written $\frac{\mathrm{B}\, \mathrm{in}\, \mathrm{A}}{\mathrm{Z}}$, a dimension 2 term divided by a dimension 1 term, Descartes writes many homogeneous terms like $\frac{a}{z}x$ (1637, 328), where the letters are arranged so that it appears that the $\frac{a}{z}$ has some geometrical meaning by itself. But this way of writing, the term only functions as a notational device to separate the unknown x from the rest of the expression. The whole term is one-dimensional, like the corresponding term in Viète.

In solving problems in geometry, then, Descartes works with essentially the same algebra as Viète and Fermat. This is an algebra rooted in the non-arithmetized sizes of geometric magnitudes, which can, if one chooses, be given numerical measure (Fermat never takes this option, but it is prevalent in Viète). All terms have positive integer degree, only homogeneous terms can be added and subtracted, and powers greater than three are given no geometric meaning. Descartes’s algebra of line segments, which appears to be a creative solution to the problem of magnitudes of higher dimension, comes into play only when numbers are assigned to lines. Such arithmetizations can help illustrate the general solutions or they may be applied to real-life problems, but they are not a necessary part of geometrical problem-solving.

7 Synthesis

One important, but ultimately inconsequential, difference between premodern algebra and Viète’s logistice speciosa is that the unknowns of the former are numbers, while Viète’s species belong to geometry. We have seen that Harriot, Oughtred, and Hume all reinterpreted the species as numbers, and in the other direction, numerical algebra could be applied to problems in classical geometry by positing a unit line, as al-Khayyām had already done in the eleventh century. What definitively differentiates premodern algebra from Viète’s specious logistic is how expressions were conceived. Premodern algebra is based in multitudes of kinds of numbers and their aggregations, while Viète’s algebra is based in values assembled via operations. We can ask, then, what advantage Viète’s algebra had over premodern algebra for the work of Fermat and Descartes. How is it that they chose Viète’s species, and could some version of premodern algebra have sufficed?

Descartes learned cossic algebra from Clavius’s 1608 textbook, and it is that algebra that he employed from the late 1610s down to at least 1628 (Rabouin 2010). There was at least one time in this period that he could have benefitted from undetermined coefficients. In notes he took in 1619–1620 on cubic equations in relation to his mesolabe compass, he sometimes expressed a particular type of equation by putting a circle, or an “O,” in place of the coefficients, such as an “equation between O & O + ON” (“aequationem inter O & O + ON” (Descartes 1908, 244)). We would write this in modern notation as an equation between ax³ and bx + c. The fact that three copies of the same sign “O” stand in the places of three potentially different amounts indicates that they are more like circles in which one could fill in numbers than letters standing for undetermined values. Clavius expressed an equation type like this by simply not writing anything for the coefficients, like his generic equation “between  + , & N” (“inter  + , & N” (Clavius 1608, 52.6)) for what we would write as an equation between ax² + bx and c. This is a notational example of the rhetorical equation types that go back to al-Khwārazmī, who wrote this particular type as “[some] māls and [some] roots equal a number” before giving the specific example “a māl and ten roots equal thirty-nine dirhams” ((al-Khwārazmī 2009, 101.2), my translation).

No premodern algebraist is known to have devised either words or notation for undetermined coefficients. One could suggest that Stifel or Stevin were on the verge of such a notation, because they could have regarded their independent unknowns as being given, so that our ax² + bx = c, for example, could have been written by Stifel as “1A + 1B equals 1C.” But this construction would have been decidedly inconvenient, because the coefficients become embedded in the kinds, which obliterates the fundamental premodern distinction between the coefficient (i.e., the multitude) and the power (i.e., the kind). Multitudes and kinds cannot just be merged together into new, composite kinds that themselves are assigned a multitude. For one thing, one should be able to replace the letters with determinate numbers, but in this scheme that cannot happen without dismantling the types. I had suggested in Oaks (2018, 287) that independent unknowns could have served as undetermined givens, but now I see that it could not have worked.

Or one might wonder why they could not have just written something like “A + B equals C,” where A, B, and C now stand for the values of numbers such as we find in Book II of Pappus’s Collection or in De Numeris Datis of Jordanus de Nemore (early thirteenth c.). Such letters as values were known in Stifel’s time, though they served as labels in premodern arithmetic and are not found together with algebraic terms. If such a contrivance was ever considered, there would have been a problem as soon as there was a need to operate on the terms, since the letters, as labels, were not operated on to produce more complex expressions (§5.4). Xylander (1575) and Gosselin (1577) took small steps in introducing operations to letters that designate values, though they did not mix this notation with algebraic notation. Those steps might have eventually led to another modification of premodern algebra, but they came too late (Oaks 2018, §5.4).

With nine different known magnitudes in Descartes’s general solution to the Pappus problem, it would have been impossible for him to have written the equation in cossic algebra even if he had reinterpreted the kinds as being magnitudes. Descartes is silent about when he first saw examples of logistice speciosa, but it was likely to have occurred between 1628, when Beeckman recorded samples of cossic algebra from Descartes’s hand, and January 1632, when Descartes solved the Pappus problem, presumably with essentially the same algebra that he shows in La Geometrie. (Sasaki (2003, 235ff) reviews the evidence regarding when Descartes may have first become aware of Viète’s algebra.)

It is no surprise that Fermat chose to adopt Viète’s algebra and its notation, given that he studied both Viète and Pappus, and that he, too, sought to restore Greek analytic geometry. Numerical algebra would have held no appeal for him when he had at hand an algebra designed for the geometer. His work, too, necessarily required undetermined givens, so cossic algebra would have also been inadequate for him.

Premodern algebra had always been an excellent arithmetical technique for solving problems, but it was limited by the identification of expressions as numbers, which were by their nature amounts of some unit or units. Geometric magnitudes are not multitudes of units, so when Viète set about making an algebra for geometry, it was naturally based in values, and thus expressions were necessarily formed through operations, and givens could only be undetermined. These last two features of logistice speciosa were somehow accidental, but they offered critical advantages. We have seen that in Viète, solutions to equations in problem-solving become for the first time formulas expressed in terms of given parameters, and in Fermat and Descartes general locus problems and problems of finding tangent and normal lines, with unspecified givens, can be solved algebraically. These accidental features were also necessary for later applications of algebra. Try to write an identity like (a + b)³ = a³ + b³ + 3a²b + 3ab² in any premodern notation and you will understand why Cardano states the rule rhetorically (Cardano 1545, f. 16a). Likewise, how could A = πr², $f(x)=3{x}^3\sqrt{x^2+6}$, or $d=\frac{1}{2}{gt}^2$ be expressed if not with letters that stand for values that are joined through operations?

We can credit Book VII of Pappus’s Collection with playing a significant role in inspiring both Viète’s renovation of algebra and Fermat’s and Descartes’s application of that algebra to the study of curves. But even if these three authors worked in the sphere, or at least in the shadow, of classical Greek mathematics, it was they who paved the way for the eventual abandonment of geometry in favor of the new algebra for conducting original work in mathematics, and also for the slow demise of premodern algebra in favor of a numerical version of logistice speciosa for practitioners. Viète’s algebra was supposed to be a restoration of ancient analysis. Instead, ultimately it made obsolete both the geometrical analysis of Pappus and the arithmetical analysis (premodern algebra) of Diophantus.