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Who Developed Indivisibles? The Precursor to Infinitesimal Calculus in Mathematics

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

Calculus is a rather recent branch of mathematics when compared to central pillars like algebra and geometry, but its uses are far reaching (to underrepresent the situation). Like all fields of mathematics, it too has interesting origins, and one key aspect of calculus, the infinitesimal, had hints of it established as far back as Archimedes. But what additional steps did it take to become the tool we know of today?

Galileo Begins the Wheel

Oh yes, everyone’s favorite astronomer of Starry Messenger and major contributor to heliocentrism has a role to play here. But not as direct as things may seem. You see, after Galileo’s 1616 decree incident, Galileo’s student Cavalieri presented him with a math question in 1621. Cavalieri was pondering the relationship of a plane and a line, which can reside in a plane. If one had parallel lines to the original, Cavalieri noted that those lines would be “all the lines” with respect to the original. That is, he recognized the idea of a plane as being constructed from a series of parallel lines. He further extrapolated the idea to 3-D space, with a volume being made of “all the planes.” But Cavalieri wondered if a plane was made of infinite parallel lines, and likewise for a volume in terms of planes. Also, can you even compare “all the lines” and “all the planes” of two different figures? The issue he felt existed with both of these was the construction. If an infinite number of lines or planes would be needed, then the desired object would never be completed because we would always be constructing it. Plus, each piece would have a width of zero so therefore the shape made would have an area or volume of zero as well, which is clearly wrong (Amir 85-6, Anderson).

No known letter exists in response to Cavalieri’s original question, but subsequent correspondences and other writings hint at Galileo being aware of the matter and the troubling nature of infinite parts making up a whole thing. Two New Sciences, published in 1638, has one particular section of vacuums. At the time, Galileo felt they were the key to holding everything together (as opposed to the strong nuclear force as we know today) and that the individual pieces of matter were indivisibles, a term Cavalieri coined. You could build up, Galileo argued, but after a certain point of breaking matter apart you would find the indivisibles, an infinite amount of “small, empty spaces.” Galileo knew mother nature abhors a vacuum and so he felt it filled it with matter (Amir 87-8).

But our old buddy didn’t stop there. Galileo also talked about Aristotle’s Wheel in his Discourses, a shape constructed from concentric hexagons and a common center. As the Wheel spins, the line segments projected on the ground made from the contacting sides differ, with gaps appearing because of the concentric nature. The outer boundaries will line up nicely but the inner will have gaps, but the sum of the lengths of the gaps with the smaller pieces equals the outer line. See where this is going? Galileo implies that if you go beyond a six-sided shape, and say get closer and closer to infinite sides, we end up with something circular with smaller and smaller gaps. Galileo concluded then that a line is a collection of infinite points and infinite gaps. That folks is awfully close to calculus! (89-90)

Not everyone was excited about these results at the time, but a few did. Luca Valerio mentioned those indivisibles in De centro graviatis (1603) and Quadratura parabola (1606) in an effort to find the centers of gravity for different shapes. For the Jesuit Order, these indivisibles were not a good thing because they introduced disorder in the world of God. Their work wanted to show math as a unifying principle to help connect the world, and to them indivisibles were demolishing that work. They will be a constant player in this tale (91).



Cavalieri and the Indivisible

As for Galileo, he didn’t do much with indivisibles, but his student Cavalieri certainly did. To perhaps win over skeptical people, he used them to prove some common Euclidean properties. No big deal here. But before long, Cavalieri finally used them to explore the Archimedean Spiral, a shape made by a changing radius and a constant angular velocity. He wanted to show that if after a single rotation you then draw a circle to fit inside the spiral, that the ratio of the spiral area to the circles would be 1/3. This had been demonstrated by Archimedes but Cavalieri wanted to show the practicality of indivisibles here and win people over to them (99-101).

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As mentioned before, evidence points to Cavalieri developing the connection between area and volumes using indivisibles based off letters he sent to Galileo in the 1620s. But after seeing Galileo’s Inquisition, Cavalieri knew better than to try and cause ripples in the pond, hence his strive to extend Euclidean geometry rather than profess something someone might find offensive. It is partially why despite having his results ready in 1627 it would take 8 years for it to be published. In a letter to Galileo in 1639, Cavalieri thanked his former mentor for starting him on the path of indivisibles but made it clear that they weren’t real but merely a tool for analysis. He tried to make that clear in his Geometria indivisibilibus (Geometry by Way of Indivisibles) in 1635, where no new results were derived, just alternate ways to prove existing conjectures such as finding areas, volumes, and centers of gravity. Also, hints of the mean value theorem were present (Amir 101-3, Otero, Anderson).



Torricelli, the Successor of Galileo

While Galileo never went crazy with indivisibles, his eventual replacement would. Evangelista Torricelli was introduced to Galileo by an old student of his. By 1641, Torricelli was working as a secretary to Galileo in his finals days leading up to his death. With a natural math ability to his credit, Torricelli was appointed as Galileo’s successor to the Grand Duke of Tuscany as well as a professor of the University of Pisa, using both to boost his influence and let him accomplish some work in the indivisibles arena. In 1644, Torricelli published Opera geometrica, connecting physics to the area of parabolas via…you guessed it, indivisibles. And after finding the area of the parabola 21 different ways with the first 11 the traditional Euclidean ways, the slick indivisible method made itself known (Amir 104-7).

In this proof, the method of exhaustion as developed by Euxodus was used with circumscribed polygons. One finds a triangle to fit inside the parabola completely and another to fit outside of it. Fill in the gaps with different triangles and as the number grows, the difference between the areas goes to zero and voila! We have the area of the parabola. The issue at the time of Torricelli’s work was why this even worked and if it was a reflection of reality. It would take fore3ver to actually implement the idea, people of the time argued. Despite this resistance Torricelli had included 10 other proofs involving indivisibles, knowing full well the conflict it would cause him (Amir 108-110, Julien 112).

It didn’t help that he brought new focus onto him, for his indivisibles approach was different from Cavalieri’s. He took the big leap that Cavalieri wouldn’t, namely that “all the lines” and “all the planes” were the reality behind the math and implied a deep layer to everything. They even revealed paradoxes that Torricelli adored because they hinted as deeper truths to our world. For Cavalieri, creating initial conditions to negate the results of the paradoxes was paramount. But rather than waste his time on that, Torricelli went for the truth of the paradoxes and found a shocking result: different indivisibles can have different lengths! (Amir 111-113, Julien 119)

He came to this conclusion via ratios of the tangent lines to the solutions of ym=kxn otherwise known as the infinite parabola. The y=kx case is easy to see since that is a linear line and that the “semignomons” (region formed by the graphed line, and axis, and interval values) are proportional with respect to the slope. For the rest of the m and n cases, the “semignomons” are no longer equal to each other but are indeed proportional. To prove this, Torricelli used the method of exhaustion with small segments to show the proportion was a ratio, specifically m/n, when one considered a “semignomon” with an indivisible width. Torricelli was hinting at derivatives here, people. Cool stuff! (114-5).

Works Cited

Amir, Alexander. Infinitesimal. Scientific American: New York, 2014. Print. 85-91,99-115.

Anderson, Kirsti. “Cavalieri’s Method of Indivisibles.” Math.technico.ulisboa.pdf. 24 Feb. 1984. Web. 27 Feb. 2018.

Julien, Vincent. Seventeenth-Century Indivisibles Revisited. Print. 112, 119.

Otero, Daniel E. “Buonaventura Cavalieri.” 2000, Web. 27 Feb. 2018.

© 2018 Leonard Kelley

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