Abstract
Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geometrical and finitistic axiomatic foundation (like Euclid’s planar geometry) for a class of differential problems. In particular, after a brief historical introduction, a model of such machines (i.e., the suggested components) is presented. Then, we introduce some preliminary results about generable functions, an example of a “tractional” planar machine embodying the complex exponential function, and, finally, a didactic proposal for this kind of artifact.
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Notes
- 1.
Even though Euclid’s works never introduced the straightedge and compass, his axioms for planar geometry include the idea that we can draw a circle of any known radius at any known point and that we can extend any line indefinitely. These axioms, purely mathematical in nature, can also be interpreted physically by saying that the geometer has access to a compass and a straightedge, tools that were used for many purposes even before Euclid.
- 2.
In general, not every Descartes scholar would agree about this point. In fact, Descartes never solved the problem of classifying the admissible curves in an unambiguous and complete way. However, in accordance with (Bos 2001; Panza 2011), we are adopting the characterization of Descartes’ geometrical curves as those that can be traced by geometrical linkages, that is, articulated devices basically working as joint systems, allowing a certain degree of freedom in movements between the two links they connect.
- 3.
- 4.
Riccati shows that, adopting modern terminology, it is possible to integrate any differential equation y′ = f(x, y), but he does not explicitly specify anything about the set of admissible functions f. According to the equations of the time, it is reasonable to assume that the function has to be obtained using only a finite number of algebraic operations and quadratures.
- 5.
With regard to the plane, in general it can be substituted by any other surface in a space (as usually made in differential geometry), but the adopted surface has to be considered as given a priori (all we are going to construct with machines are transformations over a surface, not new surfaces). This is why we restrict ourselves to the basic case of the plane (at least for the moment).
- 6.
To constrain a rod r to pass through M and N, we first pivot r in M by a joint, then put a cart on r, and finally attach the cart with N by another joint.
- 7.
Concerning the construction of a rod s perpendicular in P to another rod r, we can obtain the perpendicularity by imposing the passage of s through the vertex of a right triangle with one leg on r (the right triangle can be constructed by the junction of a Pythagorean triple as segment lengths).
- 8.
Note that, differently from (t 0, 0), t may vary in \(\mathbb{R}\). The point (t + 1, 0) can be obtained by linking at (t, 0) one end of a unitary length rod, whose other extremity is forced by a cart to move along q.
- 9.
Specifically, we determine the points \((t + 1,y_{i} + p_{i})\) in function of t and the (still) free \(y_{1},\ldots,y_{n}\).
- 10.
To link (t, y i ) with \((t + 1,y_{i} + p_{i})\), we place a rod r i at (t, y i ), then we place a cart on r i and move it to \((t + 1,y_{i} + p_{i})\).
- 11.
We have to deepen the idea of representing z + f(z) instead of f(z). If, at first glance, it seems so different from the representation in the real case, the main condition behind both is that the motion of the output point has to be determined by that of the input point, so it is necessary that the input drags the output. Mathematically, this is implemented by a vector sum (input + output), both in the real and complex cases. In the Cartesian plane, we denote the axes’ unit vectors as \(\hat{\imath}\) and \(\hat{\jmath}\), and the graph is defined as \(x\hat{\imath} + y\hat{jmath}\) (so, as the domain and range axes are linearly independent, the graph of a real function “statically” represents all the information of the function). In the complex case, the domain and range have to be merged in the same planar coordinates, losing the property that any point of the plane identifies a single couple of input/output. In particular, given a real function f in x, the usual real representation on the Cartesian plane (x, f(x)) can be interpreted as x + i ⋅ f(x) on the complex plane (i is the imaginary unit).
- 12.
Even though we do not encounter them in this chapter, there are problems when \(p_{k} = k \cdot f(z) = -1\) (the tangent is not defined, because p k is not moving). To overcome this, we would need to construct not only \(p_{k_{1}}\) and \(p_{k_{2}}\) but also \(p_{k_{3}}\) (with k i different from each other), so that there would be at least two well-defined tangent conditions everywhere.
- 13.
As we shall evince, the tangent to p k depends on the argument of z′. Concerning the polar form (ρ, θ) of any complex value z, the argument θ is determined if and only if z ≠ 0.
- 14.
In particular, the fact that two functions, one transcendent and the other algebraic, can be constructed through similar devices of equal complexity is an epistemological point, in contrast with the Cartesian dualism between the different legitimization of geometrical (algebraic) and mechanical (transcendental) curves. Concerning this, we may mention the letter that Poleni wrote to Hermann in September 1728 (published in Poleni 1729), in which the author wondered about the nature of tractional curves. With a simple modification to the exponential tractional machine (just changing an angle, which is essentially the same thing we did, as shown in Fig. 1.11), the author had realized that tractional machines draw curves defined by differential equations in a uniform way, regardless of their algebraic or transcendental nature.
- 15.
This definition is solved by the square root only for the real values; it does not apply in the complex extension.
- 16.
TMMs (and real artifacts) are not only able to visualize some properties (like in dynamic geometry) but also prove them in a specific register. Unfortunately, this register is currently not autonomous, and we are still trying to define a suitable theory to highlight the primitive concepts for the construction and functioning of the machines. Meanwhile, as visible in Table 1.1, we had to use some analytic properties in the geometrical/mechanical register (properties about continuous or monotonic functions) to obtain some informal proofs. In addition, the geometrical and mechanical registers, even if different, have been summarized in the same column for simplicity.
- 17.
Fractional calculus is the study of an extension of derivatives and integrals to non-integer orders (for further reading, see, e.g., Ross 1975).
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Milici, P. (2015). A Geometrical Constructive Approach to Infinitesimal Analysis: Epistemological Potential and Boundaries of Tractional Motion. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_1
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