Foundations and the Threshold of Modernity: From Euclidean Geometry to the Symbolic Algebra of the Sixteenth Century

The mathematics of Leonhard Euler did not arise from nothing. Beneath the surface of his unified notations, his seemingly effortless generalizations, and his encyclopedic syntheses lies a long historical preparation in which the very meaning of mathematical activity was redefined. To understand how the Eulerian synthesis became possible, one must trace the transformation of the mathematician's working objects: from figures drawn upon sand or papyrus to letters manipulated on paper, from geometric construction to algebraic identity, from the contemplation of magnitude to the formal manipulation of symbols. The central question of this chapter is therefore a methodological one: through what specific historical and conceptual episodes did the criteria of mathematical legitimacy shift from geometric proof, anchored in the visualization of magnitudes, to symbolic manipulation governed by operational rules?

This question admits of no simple answer, for the transition was neither sudden nor unanimous. It spanned more than two millennia and unfolded across three civilizational matrices: the Greek world that produced the deductive paradigm, the Arabic-Islamic world that preserved, extended, and transmitted it, and the Latin-Christian world of the late Renaissance that transformed it. The argument proceeds through five principal movements. The first analyzes the deductive structure of the *Elements* and the function of the axiomatic method as both an instrument of proof and a constraint upon what counts as a legitimate mathematical object. The second examines the systematic transmission of Greek mathematics through the Islamic world and the emergence, in the work of al-Khwārizmī and his successors, of an algebraic discipline that was to remain rhetorical in expression but already operational in conception. The third movement treats the Italian Renaissance solution of the cubic equation and, in particular, the case of Rafael Bombelli, in which algebraic necessity forced mathematicians to accept objects—square roots of negative numbers—that had no geometric counterpart. The fourth examines François Viète's program of *logistice speciosa*, in which letters came to denote not merely unknown quantities but entire classes of magnitudes, thereby elevating algebra to a general science of formal relations. The fifth movement takes up the Cartesian identification of equations and curves, by which geometry itself was subordinated to algebraic representation. The chapter concludes by showing how these developments together constructed the symbolic apparatus within which Euler was to operate.

The Euclidean Inheritance and the Architectonic of Proof

Any genealogy of modern mathematics must begin with the *Stoicheia* of Euclid, a work whose authority over mathematical practice is difficult to exaggerate. Composed in Alexandria sometime during the early third century BCE and preserved through Byzantine manuscript traditions, the *Elements* established a paradigm of mathematical reasoning that would dominate European mathematics for more than two thousand years. Its central architectural achievement was the rigorous articulation of the axiomatic-deductive method: a body of propositions derived from a small number of self-evident postulates and common notions, with each subsequent proposition demonstrated solely on the basis of those already established.

The formal structure of the *Elements* has been the object of intensive philosophical analysis since at least the nineteenth century, when the discovery of non-Euclidean geometries revealed that the famous fifth postulate—the so-called parallel postulate, which Euclid himself had been reluctant to assert as self-evident—was independent of the others. Yet for the historical lineage that concerns us here, the deeper significance lies not in the question of whether Euclid's axioms are logically adequate by modern standards, but in the model of mathematical reasoning that the *Elements* promulgated. This model had three characteristic features that would shape the subsequent development of algebra.

First, the method proceeds from definitions. Before any property can be proved, the object to which it pertains must be defined. Euclid opens Book I with twenty-three definitions of the basic geometric objects: point, line, straight line, surface, plane, plane angle, right angle, obtuse angle, acute angle, and so on. These definitions are nominal: they tell us what the words mean, not what the objects essentially are. The points and lines of Euclid are not abstract entities in a modern set-theoretic sense; they are geometric primitives, immediately grasped through the activity of drawing.

Second, the method rests upon a small number of postulates (*aitēmata*) and common notions (*koinai ennoiai*). The five postulates of Book I are specific to geometry: that a straight line can be drawn between any two points; that any finite straight line can be extended continuously; that a circle can be constructed with any center and radius; that all right angles are equal; and, the controversial fifth, that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. The common notions are general logical principles: things equal to the same thing are equal to each other; if equals be added to equals the wholes are equal; if equals be subtracted from equals the remainders are equal; things that coincide with one another are equal to one another; the whole is greater than the part.

Third, the method employs diagrammatic reasoning. In the proofs of the *Elements*, the reader is invited to draw a figure, often described in detail, and to inspect it visually. The letters affixed to the diagram are notational devices that connect the figure to the prose argument. In the very first proposition—the construction of an equilateral triangle on a given finite straight line AB—the reader is instructed: "With center A and radius AB let the circle BCD be described; with center B and radius BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined." The proof depends for its intelligibility upon the actual or imagined drawing of the figure.

It is this third feature that most concerns our inquiry. For the *Elements* imposed a geometric constraint upon what could count as a legitimate mathematical object. A proposition about numbers could only be admitted as geometric if it could be exhibited in the form of a construction involving line segments and areas. The theory of ratios, treated in Book V, requires the most subtle geometric apparatus, precisely because Euclid wished to avoid the assumption that any two magnitudes are commensurable. Even arithmetic, when it appears in Books VII through IX, is conducted through the geometry of line segments: a "number" is conceived as a collection of units, and the operations of arithmetic are justified by reference to geometric configurations. The now-familiar formula $A = \pi r^2$, were it to appear in the *Elements*, would have to be demonstrated by an actual comparison of areas, with the constant $\pi$ itself reconstructed geometrically rather than supplied numerically.

This geometric constraint had two opposed consequences. On the one hand, it guaranteed a level of rigor and conceptual clarity that, by the standards of ancient mathematics, was unparalleled. The proofs of the *Elements* compel assent in a way that rhetorical argumentation cannot. On the other hand, the constraint imposed severe limitations upon what could be expressed. Operations that would later be considered natural—the manipulation of irrational quantities through formal rules, the combination of negative and positive terms, the handling of relations involving more than