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?