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According to Boyer, Salviati introduces the idea of a higher order infinitesimal on the “third day” in Galileo’s Two Chief Systems of 1632. They are introduced in order to counter Simplicio’s argument that an object on a rotating earth should be thrown off tangentially. Boyer includes a diagram intended to illustrate the role of the different orders of infinitesimals in the argument. Obviously it is impossible to draw an infinitesimal. However, one would naturally think that the different orders of infinitesimals would be represented in different proportion to one another, but they are show in equal proportion.

Boyer includes a quote in which Galileo appears to identify infinitesimals as indivisibles.

On another occasion, however, Galileo has Salviati assert that infinites and indivisibles “transcend our finite understanding, the former on account of their magnitude, the latter on account of their smallness; Imagine what they are when combined.”

If Galileo intends that an infinitesimal is an indivisible, then it is hard to imagine what is meant by higher orders of infinitesimals.

Boyer also notes that Galileo’s work implies that in an equation involving infinitesimals, those of higher order can be ignored since they have no effect on the final result.

If find all of this very confusing. How can indivisibility accommodate higher orders? Surely a higher order infinitesimal is not an infinitesimal portion of an infinitesimal measure.

Question : Is there a generally accepted description of what Galileo meant by higher order infinitesimals?

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  • $\begingroup$ It is the last part of the Second Day. $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 19:18
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    $\begingroup$ Galileo discuss the problem of the horn angle; see also Euclid, Book I, Def.8 and Book III, Prop.16. $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 19:30
  • $\begingroup$ On the "confused" ideas of Galileo on indivisibles, see Ch.5. Indivisibles in the Work of Galileo, by Vincent Jullien, of Vincent Jullien (editor), Seventeenth-Century Indivisibles Revisited (2015). $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 19:34
  • $\begingroup$ @Mauro Thanks for your very helpful comments. This is new territory for me, so my exposure is very limited. The Jullien work looks very useful and it is nice to see that, in this case, Google Translate appears to be very effective. $\endgroup$ – Nick R May 27 '16 at 19:47
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    $\begingroup$ You are welcome :-) On the "horn angle" (or Angle of Contingence) see also page 339-on. $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 19:49
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Galileo used the term non-quanta for infinitesimals (and quanta for ordinary quantities). A recent study on Galileo's infinitesimals is

Bascelli, Tiziana Galileo's quanti: understanding infinitesimal magnitudes. Arch. Hist. Exact Sci. 68 (2014), no. 2, 121–136.

Cavalieri's indivisibles were different from Galileo's infinitesimals. In particular, it seems that it would be difficult to express the idea of second-order infinitesimal in Cavalieri's scheme of things as far as I can tell, but perhaps he did.

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    $\begingroup$ That's a good point. Subsequently, I came to learn that G and C had different conceptions of infinitesimals. When I first read about this, I was confused by the "diagram" Boyer had included, which illustrated the G's two different types as appearing to be one and the same. $\endgroup$ – Nick R Aug 16 '16 at 17:12

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