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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 17th-century mathematicians recognized the need of an axiomatization of continuity. Leibniz will bring this kind of reflections to a further level of refinement. The demonstration of Elements I, 1, moreover, had been at the center of the modern logical debate on the possibility of reducing Euclidean demonstrations to syllogistic chains. This was an important epistemological problem in the Renaissance, that touched the methodology of science and the possibility of reconciling logic and mathematics. Semi-formal proofs of Elements I, 1, through four syllogisms, had been given by Piccolomini, Herlinus and Clavius in the 16th century (drawing on Greek materials). In the 17th century, Jungius had extended syllogistic logic with a few non-syllogistic inferences, and had produced a remarkable analysis of the proof of this proposition, reshaping it into a chain of twenty-one elementary inferences. Building on these researches, Leibniz offers here a truly remarkable example of a detailed proof composed by forty inferential steps grounded on explicit logical and geometrical principles.