Foundations of Geometry

Overview of Ars Representatoria (1691)

The Ars Representatoria is a general and programmatic paper interested in problems of characteristics. Leibniz reaffirms the starting point and purpose of his analysis situs, meant to avoid the use of extrinsic algebraic proceedings in geometry, without however having to give up the advantages of the characteristica speciosa and go back to the elegant, but complicated and unproductive, synthetic geometry. It is a matter of reforming algebraic formalism and finding another one that may be proper to geometry. Although it can be blindly manipulated, such formalism must have this advantage on ordinary algebra: that, when interpreted, it immediately offers the geometrical object to our imagination. It is therefore a cogitatio caeca that promises a vast view. The discussion in the Ars Representatoria in many respects anticipates Leibniz’s late essays on the theory of knowledge and interlocks with the theory of the imaginatio distincta that a little earlier Leibniz began to envision as the solution to several... view

Overview of De curvis similibus et similiter positis et parallelis

The text discusses the notion of parallelism in relation to curves. This is the concluding text in a series of studies by Leibniz on parallelism, in which Leibniz had defined parallelism as equidistance. The problem Leibniz saw with this notion is that in general a curve equidistant to a given curve is not similar (in the sense of Euclidean similarity) to the original curve. This had been noted by Leibniz about conic sections, and had to do with the transformations by evolution that Leibniz (following Huygens) was studying in the 1680s. The present essay offers a full discussion of this topic.

The fundamental problem underlying these Leibnizian investigations, however, was the provability of the famous Parallel Postulate. Many attempts to prove this postulate, in fact, were based on a definition of parallelism through equidistance, and then in the assumption that the line equidistant to a straight line is also a straight line. This last assumption is false in non-Euclidean geometry, and if one assumes it, it actually becomes possible... view

Overview of Duae rectae parallelae sunt 

In this important essay, Leibniz tries to exploit a phenomenological characterization of parallel lines, in connection with the combinatorial techniques of his characteristica geometrica, to prove the Parallel Postulate. He starts with a new definition of parallels (“ubique eodem modo se habere”), and characterizes the definition perceptually by imagining that an observer moves along the straight lines X̅ looking at the line Ȳ, while another observer moves along Ȳ looking at X̅, the both of them being in uniform motion. He remarks that everything is the same: the two observers perceive no change while moving (each of them always look at the same straight line from equivalent points of view), nor are the perceptions of one observer different from those of the other. This should be the very meaning of the definition of parallels. The grounds of this indiscernibility may be found in the similarity of all straight lines to each other; but also their reciprocal position (situation), that we have to investigate. Thus, if we consider metric properties and distances... view

Overview of Leibniz’ Notes to Arnauld’s Nouveaux elemens de geometrie

The Nouveaux Éléments de Géométrie were published by Arnauld in 1667, and in a second edition in 1683. It is a work that made school in the study of the foundations of geometry, and had a vast influence on French and European geometry in general. Arnauld's approach to the foundations of geometry was, however, very different from Leibniz', and was based on the assumption of a large number of principles and axioms that Arnauld considered "self-evident" and therefore impossible to prove. Leibniz discussed this work many times during his lifetime, and frequently noted that Arnauld had assumed propositions that in his opinion needed to be proven in order to establish the foundations of geometry. These reading notes testify to the attention with which Leibniz read this fundamental text, and they are the origin of a large number of essays and subsequent observations on the foundations of geometry.

Overview of Scheda (mid–1690s)

This "Scheda" from the mid-1690s, which Leibniz himself characterized as "good" amidst many other writings of the same period, provides a good overview of the foundations of analysis situs in this period. In it Leibniz presents his own thinking in deductive form (which is uncommon, but not altogether rare in writings on geometry), thus showing which propositions he considers more elementary and which are derivable from previous ones.

In particular, it is relevant that Leibniz assumes here as the initial term of the whole essay the notion of an extensum, which is defined as a coexistent continuum (a definition that had already appeared in the first writings on Characteristica geometrica). The notion of a point, however, which in the 1680s had been coordinated as a second primitive to that of an extensum is seen here as derived from it.

Some fundamental ideas appear in the text that often recur in the writings of the time. Principal among these is probably that all points have situational relations to each other, and therefore they all... view

Overview of Specimen Analyseos Figuratae in Elementis Geometriae (1683)

The Specimen analyses figuratae offers a detailed analysis of the first proposition of Euclid’s Elements, teaching how to construct an equilateral triangle. This proposition had been the object of foundational analysis innumerable times before Leibniz, and since antiquity several issues were found in its demonstration. In particular, the early modern discussion on this proposition had concentrated on the implicit assumption of Euclid that the two circles drawn in the construction of the problem should have a point in common. This assumption seemed to require some discussion on the continuity of the circles themselves, as the ground of the existence of their point of intersection. Already in 1532, Oronce Fine had added an intersection axiom to Euclid’s original principles in order to bridge this demonstrative gap, and since then several 17^th-century mathematicians recognized the need of an axiomatization of continuity. Leibniz will bring this kind of... view