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This book offers a general introduction to the geometrical studies of Gottfried Wilhelm Leibniz (1646-1716) and his mathematical epistemology. In particular, it focuses on his theory of parallel lines and his attempts to prove the famous Parallel Postulate. Furthermore it explains the role that Leibniz’s work played in the development of non-Euclidean geometry. The first part is an overview of his epistemology of geometry and a few of his geometrical findings, which puts them in the context of the seventeenth-century studies on the foundations of geometry. It also provides a detailed mathematical and philosophical commentary on his writings on the theory of parallels, and discusses how they were received in the eighteenth century as well as their relevance for the non-Euclidean revolution in mathematics. The second part offers a collection of Leibniz’s essays on the theory of parallels and an English translation of them. While a few of these papers have already been published (in Latin) in the standard Leibniz editions, most of them are transcribed from Leibniz’s manuscripts written in Hannover, and published here for the first time. The book provides new material on the history of non-Euclidean geometry, stressing the previously neglected role of Leibniz in these developments.

This volume will be of interest to historians in mathematics, philosophy or logic, as well as mathematicians interested in non-Euclidean geometry.



Leibniz on the Parallel Postulate and the Foundations of Geometry


1. Introduction

Among the many contributions of Gottfried Wilhelm Leibniz (1646–1716) in mathematics and philosophy, his work on the foundations of geometry is especially relevant. In Leibniz’ times, the text of Euclid’s Elements still represented the starting point for any advanced mathematical theory, including Leibniz’ most celebrated discovery, the Calculus. The Greek treatise, on the other hand, was also the main model for deductive reasoning, and the touchstone of logical analysis and epistemology in general. In the seventeenth and eighteenth centuries, the debate on the Elements was extensive, and philosophers, philologists and mathematicians contributed, with dozens of emended and commented editions of the text, to a better understanding of Euclid’s intentions and a deeper insight into the nature of geometry itself. Given Leibniz’ great interest in logic, his involvement in foundational discussions about the new infinitesimal techniques, his wide erudition in the history of mathematics, and his didactical preoccupations with scientific education, it comes as no surprise that throughout his entire life he devoted a considerable part of his time to investigating the essence of geometrical reasoning or the system of principles needed to ground the whole of mathematics.
Vincenzo De Risi

2. The Theory of Parallel Lines in the Age of Leibniz

It may be useful to give a picture of the discussions on the Parallel Postulate in the age of Leibniz, as well as a list of his mathematical sources on the topic. The definition of parallel lines, in Euclid’s wording, is that of “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”. The Parallel Postulate appears as Postulate Five in the First Book of the Elements, and states that “if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which are the angles less than the two right angles”. In Early Modern editions of Euclid, however, the Postulate was normally arranged among the axioms.
Vincenzo De Risi

3. Leibniz’ Epistemology of Geometry and the Parallel Postulate

The seventeenth-century mathematicians whom we have mentioned so far did not share a common epistemology, and each of them had his own opinions about the nature, object and aims of geometry, as well as on the meaning of geometrical principles or the standards of rigor needed in a proof. This notwithstanding, there was some kind of consensus at least on this point: that the axioms of geometry were provable. Contrary to our modern understanding of the role of an axiom as an indemonstrable statement, they claimed that geometrical principles were self-evident truths that required no proof to be accepted, but nevertheless were not incapable of proof. The dispute was more about the opportunity to prove them than the possibility of doing this.
Vincenzo De Risi

4. Leibniz’ Attempts to Prove the Parallel Postulate

After a sketch of Leibniz’ epistemological views, we are now able to briefly discuss the properly mathematical texts that Leibniz devoted to the theory of parallels and the proof of the Parallel Postulate. Given the epistemological relevance of the system of (real) definitions in Leibniz’ geometry, however, it may be of some use to mention the most important characterizations that Leibniz employed in defining the basic geometrical terms.
Vincenzo De Risi

5. Reception and Legacy

Leibniz did not publish anything of his new geometry, the analysis situs, and nothing of his researches on the Parallel Postulate. His influence on the following developments of Euclidean and non-Euclidean geometry was almost inexistent. Yet, we may find a few episodes in the history of mathematics that show at least an indirect (and perhaps misconceived) reception of Leibniz’ endeavors.
Vincenzo De Risi

Leibniz’ Texts on Parallel Lines


6. Leibniz’ Texts on Parallel Lines

If, from three points on the same straight line, of which the extremes stand apart from the middle equally, three parallels meet another straight line, the sum of the extreme lines will be equal to twice the middle line.
Vincenzo De Risi


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