### In the Classroom

Solutions to area problems are some of the most fundamental and easy-to-grasp results in calculus. The Fundamental Theorem of Calculus is an amazing tool for solving these problems, but it isn't the only tool. Historically, the quest to solve general area problems can be traced to Archimedes' particular solution to the area problem for a parabola, and there is a rich history of diverse methods for solving this specific area problem which stretches from ancient Greece to the modern calculus. If a detailed discussion of Archimedes' solution to the area problem for the parabola is included in a history class, the *Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta* can also provide examples of the many different approaches to solving this particular problem, and so is an important original source in the history of calculus. It is also accessible to any student who is comfortable with Books V and VI of Euclid's *Elements* and has experience reading original sources up to the level of Archimedes. Here are some specific examples of how it might be used as an original source in an upper level history of mathematics class:

- For those students who have read Archimedes'
*Quadrature of the Parabola*, in particular the proof of the area formula given in Propositions 21-14, Torricelli's Lemmas 24-27 and Proposition 15 present a proof identical in spirit but without resort to the method of exhaustion. What is the purpose of Torricelli's painstaking attempt to justify his infinite summation in geometrical terms? How does this contrast with the method of exhaustion that Archimedes utilized? Does it lack anything in rigor? (See the section "Torricelli's Quadrature of the Parabola" above for a modern explanation of how Torricelli was attempting to wrestle with infinite summations.)
- As an extension of the previous exercise, Torricelli's Lemma 28 (with the accompanying "In Another Way" discussion) and Proposition 16 give a nearly identical proof of Archimedes' result using triangles that are tangent to the parabola instead of inscribed in it. What would an Archimedean proof using the method of exhaustion for this result look like?
- For those students who have read the first Proposition of Archimedes'
*The Method*, Torricelli's Lemmas 32-33 and Proposition 20 should be familiar, because (with some minor differences) Torricelli's proof is the same as the proof Archimedes' gives in the method. Has Torricelli indepedently rediscovered Archimedes' *Method* more than 250 years before it finally came to light again in 1906?
- Those students who have read the proof of Proposition 8 of Book II of Archimedes'
*On the Equilibrium of Planes* know that it depends on the formula for the area of a segment of a parabola given in *Quadrature of the Parabola*. Torricelli's Proposition 30 demonstrates (using the Law of the Lever as found in Book I of *On the Equilibrium of Planes*) that conversely one can find the area formula for a segment of a parabola once one knows the center of gravity of a parabola (a result which Torricelli proves quite easily using infinitesimals in Lemma 30 of his work). Do you think Archimedes was aware that the two results were equivalent?
- Perhaps the most important part of Torricelli's work is that it articulates Cavalieri's method of indivisibles better than Cavalieri himself did. Moreover, whereas Cavalieri restricted himself to comparing figures having equal altitudes, Torricelli used indivisibles much more freely. Those students who have read about Cavalieri's Principle at the level of, say, [Katz, p. 514-517], would benefit from reading Torricelli's most general expression of Cavalieri's Principle as it is found in Lemma 29. How does Torricelli's statement of Cavalieri's Principle compare to Cavalieri's own statement?
- Immediately following Lemma 29, Torricelli uses this version of Cavalieri's Principle to show (in Proposition 17) that Proposition 24 of Archimedes'
*On Spirals* can be used to find Archimedes' area formula for a parabolic segment. Was Archimedes aware that these two results were also equivalent?

One issue that students may have with our translation is that we have made no attempt to modernize the language of proportionality used in the text. Students may find it easier to read if they understand how to make the translation to modern fractional notation themselves. Thus, the proportion "DF will be to FB as CE is to DF" may be easier to understand when written as "DF/FB = CE/DF". For a model of how to construct modern interpretations such as this of some of Torricelli's arguments, see [Leahy, pp.178-183].

### Conclusion

Infinitesimal methods for solving area and volume problems were some of the most important developments in mathematics during the seventeenth century. Cavalieri's method of indivisibles was the most well known of these techniques, but his works were not widely understood. As Andersen noted, Torricelli is "an important link between Cavalieri's method and the general understanding of it," and his sole work, *Opera Geometrica*, in particular was "influential in spreading knowledge of the method of indivisibles" [Andersen, p. 356]. Among the various parts of this work, *Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta* stands out as one of the most clear, concise, and compelling presentations of Cavalieri's method that was available at the time. We hope that the translation presented here will aid in understanding the importance of this work.