Abstract
The considerations of yesterday will serve as a basis for a discussion about the resistance of solid bodies to fracture. The strength of bodies depends on the cement that holds them together and attaches their parts, in such a way that only a strong pull can separate them. After searching what could be the cause of this cohesion, which in some solids is very strong, and examining if this cause could be the fear of the vacuum, we have started many digressions that kept us busy all day and turned us away from studying the initial subject.
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- 1.
Galileo could not take into account the deformation of the material and the consequent proportionality between tensions and deformations later established by Hooke (1635–1703) as a consequence of his famous law. The prism is deformed before it breaks allowing its different sections to rotate around its neutral (i.e., unstressed) axis. Calculations considering this fact were subsequently carried out by Navier (1785–1836) and the results differed from those of Galilei only for numerical factors but not for the dependence on the power of linear dimensions. As a consequence, for example, it is also true in the Navier approximation that, if two nails are fixed to a wall, the one that has double diameter can carry a weight 8 times that of the other (as Salviati will state).
- 2.
Galileo’s pupil Vincenzo Viviani wrote in a note on his copy of the book that in reality this statement does not follow from the previous demonstrations, which however make it plausible. Moreover, Galilei does not explicitly say that this proof neglects the weight of the rod.
- 3.
Notice that Galilei’s figure is inaccurately drawn: the tangent to the curve in B should be vertical.
- 4.
We reproduced the demonstration by Galilei instead of using calculus, unknown to him. Calculus-based demonstration is trivial: by calling a the length of the AP segment and b the length of the AC segment, the ABP area is:
$$\begin{aligned} \int _0^a \frac{b}{a^2} x^2 dx = \frac{ab}{3}. \end{aligned}$$.
- 5.
Although Galilei writes that he is reproducing a proof due to Archimedes, the first part is different: Archimedes decomposes the surface using triangles, while Galileo uses rectangles.
- 6.
Indeed the curve thus obtained is a catenary and not a parabola, but it approximates a parabola.
- 7.
Galilei designed this instrument capable of carrying out complex mathematical and geometric operations and had it built in Padua in 1597 by his worker Marcantonio Mazzoleni, brother of Mario, chair of natural philosophy at the University. The instrument is described in the booklet The operations of the geometric and military compass, published in Padua in 1606 and dedicated to Cosimo II Medici. The proportional compass was very successful and Galilei had it mass produced to sell it. Here Galilei has just advertised a commercial product of his.
- 8.
As explained in footnote (a) on this day, Galilei’s calculation neglects the elasticity of the material. In this case, unlike the one discussed in the previous note, the more accurate calculation made by Navier does not have the same simplicity and the same dimensional dependence as that of Galilei.
- 9.
Galilei probably wanted to add some material at this point. The traditional “entr’acte” at the end of the day is missing and the beginning of the next day is abrupt, with the sudden appearance of the treatise in Latin written by the Academician (which had only been briefly discussed).
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De Angelis, A. (2021). Day Two. In: Galileo Galilei’s “Two New Sciences” . History of Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-71952-4_2
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DOI: https://doi.org/10.1007/978-3-030-71952-4_2
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