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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, we should say that in case X̅ and Ȳ are parallels (under this definition), and if the first observer moves from X1 to X2 with uniform motion, and the second from Y1 to Y2 again uniformly (and thus the lengths of their trajectories are equal, X1X2=Y1Y2), then we have: X1.Y1.X̅.Ȳ≃X2.Y2.X̅.Ȳ. This last formula should express the peculiar property of symmetry that we find in parallel lines, stating that the reciprocal situations of the two observers among themselves and in relations to the two whole lines that they are tracing (their trajectories) are indiscernibles, and thus congruent (≃). Starting with this key-formula derived by his consideration about indiscernibles, Leibniz goes on to state a common axiom of his characteristica geometrica, that similar determinants produce similar determinates. Then he goes on, claiming that the above formula can be reduced to X1.Y1≃X2.Y2, through a kind of simplification of equal situational relations (eliminating X̅.Ȳ from both sides, thanks to the axiom), and the congruence itself implies the equality of the lengths of the segments: X1Y1=X2Y2; which means that the lines are equidistant. Thus, starting from the definition of parallel straight lines as lines that have everywhere the same reciprocal situation, passing through some phenomenological considerations about the indiscernibles and a couple of combinatorial passages, Leibniz proved that two such lines are in fact equidistant. This amounts (as we know) to a kind of proof of the Parallel Postulate, applied to straight lines identically situated (instead of to non-intersecting lines). In fact, Leibniz immediately deduces (again through combinatorics) the parallelism of the two equal transversals X1Y1 and X2Y2 and the identity of the internal and external angles of these with X1X2 (Elements I, 29, equivalent to the Parallel Postulate), the existence of rectangles (a statement again equivalent to the Postulate) and Playfair’s Axiom on the uniqueness of the parallel line through a point. He concludes with a restatement of the formal definition of parallel lines, saying that if X̅ is a straight line and A a point outside it, then if A.X̅≃Y.X̅, the set Ȳ is the straight parallel line passing through A. Leibniz has proved that two straight lines which are parallel by his new definition are also equidistant; and thus the Parallel Postulate can be deduced from their definition (as it should be, Leibniz thinks). Once again, Leibniz should prove (as in the case of the definition through equidistance) that straight lines enjoying this new property of parallelism are possible and real. It should be clear, however, that Leibniz’ final goal is now closer, as he has a definition of parallel lines which is half-mathematical (or kinematical) and half-perceptual, and it fits very well with his plans of proving the Postulate through a property of space itself. In other words, the internal possibility of straight lines having those (still a bit vague) properties of uniformity needed to ground the Parallel Postulate, seems to follow from a property of symmetry of space itself, or perhaps (which is not too different for Leibniz) from a feature of our perceptual abilities. It is interesting to remark, even though Leibniz does not further develop this connection with physics and phenomenology, that the symmetry of space hinted at by his definition of parallel lines is that two observers in inertial motion (e.g. uniform and straight motion) may have such trajectories (e.g. parallel lines) as to be reciprocally undistinguishable.

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