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As Moreno and Waldegg mention: "Bolzano's work The Paradoxes of Infinity (1851) "officially" opened the discussion of the possibility of introducing infinity into mathematics as an object of study. ...the decisive step to be taken for this purpose was to conceive infinity as an attribute of a collection, and not as a noun or an adverb." Question: Is it true that Bolzano's work was the first attempt to mathematize infinity?Can we consider his work as a milestone to the conceptual development of infinity which distinguishes history of mathematics in,let's say "pre-Bolzano" and "meta-Bolzano" periods(with respect always to infinity)?

reference:Luis E. Moreno A., & Waldegg, G. (1991). The Conceptual Evolution of Actual Mathematical Infinity. Educational Studies in Mathematics, 22(3), 211-231.

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    See all **Ch.5 Paradoxes of the Infinite** of [Mancosu's book](https://books.google.it/books?id=60qaEePdqcoC&pg=PG118) in @Casper answer. – 2017-01-31

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Evangelista Torricelli (1608-1647) proved a theorem which shows that an infinitely long solid can have a finite volume: the trumpet-shaped figure to the right here has the same volume as the cylinder in the middle:

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With that theorem he challenged the Aristotelian dogma - which was almost universally accepted at his time - that there can be “no ratio between the finite and the infinite”.

Of course, whether this should be considered an earlier attempt than Bolzano's to "mathematize infinity" depends on what exactly is meant with "mathematize infinity".

References:

  • Evangelista Torricelli "Volume of an Infinite Solid", p. 227-231 in D. J. Struik "A Source Book in Mathematics 1200-1800", Harvard University Press, 1969.
  • Paolo Mancosu "Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century", Oxford University Press, 1996.