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Euclid's Elements could tout to have the longest and most famed publishing history of any book ever written. First written in 300 B.C., Euclid's Elements became the standard text from which aristocracy and the common people learned mathematics well into the 1800s, giving it a little over 2000 years of use. For just about any metric one could create for a book, the Elements would consistently place first or second (only to the Bible).

Today, it is rarely seen or used. Synthetic Geometry itself hasn't gone away (although it doesn't get the same amount of spotlight as it did). Geometry is still taught today in the grade schools, but this is frequently taught out of more modern textbooks that only live to see a few editions (and I would scarcely call them more "readable").

Why, or how, did the Elements wane from usage?

A couple of suspicions I have are:

  1. Hilbert published his Grundlagen der Geometrie in 1899 in which he outlined different axioms which spotlighted the holes in Euclid's axiomatizations.

  2. The Enlightenment pushed Geometry away from the central role it had in mathematics for centuries in favor of the Calculus and subsequent branches of Analysis.

Although I'm sure these contribute in some manner, they don't really meld to create a story for the Elements decline. For such a work that had such great cultural significance to humanity, I feel like it deserves a far better eulogy.

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The contest between synthetic and analytic methods in geometry predates Hilbert and even calculus, one can trace its origins to Vieta's algebraic conversions of geometric problems that streamlined their solution, see Viète's Relevance and his Connection to Euler and their systematization in Descartes's analytic geometry. But the rise of calculus and eventually arithmetization and logicization of mathematics by Dedekind, Cantor, Frege, Pasch, Hilbert, etc., certainly sealed the deal. An early episode in the open rivalry between two styles of mathematical thinking, interesting due to its particular ferocity, is described by Mazzotti in Geometers of God. The confrontation took place over the first half of 19th century in Naples. Analytics tended to be more modern and liberal and emphacized the role of mathematics in sciences and practical matters. Synthetics were conservative traditionalists who saw analytics as (sic!) "morally depraved", "antiscientific", and "corrupting" the minds of young students. As Ventura put it in 1824:

"Among the sciences, the mathematical ones are those which have taken the more false and disastrous direction. They were the first to be included in the assault of the philosophers against Christianity... ; they have become deadly weapons in the hands of impiety and pride; they have broken every restraint; they have unchained all the passions; they have eroded the foundations of society and order".

Here is Mazzotti's more measured side by side comparison of the pros and cons of the two approaches:

"The synthetic method is specific: every problem to be solved calls for a different construction; thus the geometer requires skill, knowledge, and experience, which can be gained only through long training. This method relies on the intuition of the geometer, who must choose the construction most suitable for a particular problem. Although demanding, the synthetic method was held by its supporters to be the only natural and sound method of reasoning in mathematics; they presented analysis as an artificial tool whose operations are not only epistemologically subordinate to geometry but also much more complex and counterintuitive.

The analytic method is general: every problem can be put in the form of an equation, and then solved, by following the same steps. Through solving the "equation of the problem" we obtain, mechanically, all the possible solutions. Nothing is left to the intuition of the geometer. The analytics stressed that this method is "easy," "mechanical," and easily learned. The fact that it often required great ingenuity to formulate and solve the equation was not considered an essential problem. The point was to show that an extension of the "empire" of analysis to the field of geometry is possible, and indeed desirable, because it would render the problem-solving procedure entirely mechanical."

Despite the political headwinds, the popular triumph of the general and the mechanical over the old school ingenuity and subtlety is not surprising, nor is its replication elsewhere. The controversy was replayed, with less moralizing, between "synthesists" and "analysts" in the big leagues of mathematics in 19-th century, see Who are “analysts” and “synthesists” in mathematics? One of the key figures in the Neapolitan debate, Padula, offered arguments that one can imagine being made today

"Thanks to Lagrangian "algebraic analysis," Padula continues, "the mathematical sciences have finally been reduced to that unity of principles so sought by philosophers." Padula stresses the importance of enabling students to read works written in the "new language" as early as possible. For this purpose, he supports the use of French and North Italian textbooks of analytic geometry; these should replace more traditional synthetic textbooks on conic sections - which, he says, in the "civilized" countries have almost disappeared. The study of "geometrical abstractions," he claims, does not introduce the student to "the modem development of mathematics" and "to the enormous material advantages that society can obtain from the application of mathematics to the arts, manufactures, industries.""

As already mentioned, the stress on rigor at the end of 19th century only added to the dissatisfaction with synthetic methods. Recent studies of Euclid's method revealed, however, that it has resources (nondeductive use of diagrams, etc.) for maintaining reliability that are irreducible to Hilbert style axiomatic reasoning, and, in conjuction with heuristics, are far more effective for developing new results, as opposed to axiomatic organization of known ones. This makes it interesting as a model of mathematical practice and educational tool, aside from inspiring development of similarly minded approaches beyond classical geometry, see e.g. Rodin's Axiomatic Method and Category Theory. A recent classic in the study of Euclid's synthetic method is Manders's Euclidean Diagram (published in Mancosu edited volume, freely available). In a preface to it he addresses and evaluates perceived reasons for eschewing synthetics:

"One student, apparently the beneficiary of enthusiastic instruction in the virtues of the modern logical account, recently compared the study of Euclidean demonstration as mathematical justifications to the study of the Flat Earth. While myself not in a position to judge the contributions of the flat earth tradition to modern plate tectonics, I believe this expresses at least two complaints... that diagram-based reasoning in Euclid is unreliable and justificationally inadequate, and that the study of a tradition of argument so obsolete cannot benefit the philosophy of mathematics. Let me take these in turn.

Assessments based on diagrams are held to be unreliable on several grounds. Drawn diagrams are imperfect in that, say, lines are not perfectly straight; regardless, human assessments of straightness or equality of line segments would be imperfect. Moreover, geometrical figures are individual, or at least atypical compared to the generality of geometrical conclusions. Next, there are different forms of geometry, which differ in their conclusions, and so a single diagram-based form of reasoning cannot serve them all; indeed there are forms (such as plane coordinate geometry restricted to rational coordinates) in which the two circles would not have an intersection point. Finally, there are geometricals such as space-filling curves which utterly defeat diagram-based reasoning.

[...] In research profile, only a corner of modern geometry; but in terms of the footprint of mathematics in the modern world, that is most of modern mathematics. Euclid, and Apollonius and Archimedes, are virtually without error: their every result has a counterpart in modern mathematics... Nor should we fool ourselves that modern logical reconstructions by Hilbert (1899/1902) and Tarski (1959), however great their interest in other respects, give such an account [of their justification]: ancient geometers achieved their lasting, subtle and powerful results precisely by the means that philosophers dismiss so high-handedly today, without the benefit of modern logic and Hilbert’s refined control of coordinate domains... Euclidean diagram use forces us to confront mathematical demonstrative practice, in a much richer form than is implicit in the notions of mathematical theory and formal proof... and to confront rigorous demonstrative use of non-propositional representation."

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  • $\begingroup$ I find it funny that the emphasis of rigor that the Elements manifested contributed to its downfall. Just as there is no place in the world for ugly mathematics, the days of hand-wavy mathematics are numbered (even if it's only perceived as hand-wavy). Also, the Calculus did provide "a royal road to geometry" in the same breath as it explained physical phenomena. I'd much rather translate a geometrical problem into the Cartesian plane and use the Calculus than pile constructions upon each other. Thank you for this answer! $\endgroup$ – Robert Wolfe Mar 1 '18 at 2:31

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