The principal goal of this article is to show Kepler's contributions to the development of calculus in a visual and interactive way.

During the century before Newton and Leibniz the works of Greek mathematicians were popular, especially the work of Archimedes. Infinitesimal techniques were developed for calculating areas and volumes, and Johannes Kepler (1571–1630), shown at left, contributed to these developments. His interest in calculating areas and volumes stemmed from an incident that occurred when he married for the second time in Linz, Austria, in 1613. Kepler had purchased a barrel of wine for the wedding and the wine merchant’s method of measuring the volume at first angered him. This inspired Kepler to study how to calculate areas and volumes and to write a book about the subject, Nova stereometria doliorum vinariorum (New solid geometry of wine barrels), which was his main contribution to the development of the integral calculus.

The wine barrel incident also led Kepler to take up a problem of differential calculus, the problem of maximums: What is the best design for a barrel in order to maximize its volume? Today, we solve this problem using derivatives because we know that, at a maximum (or minimum) value of a differentiable function, the derivative of the function is zero. Fermat was the first to relate maximum and minimum problems to tangents to curves: at a maximum or a minimum the slope of the tangent to the curve is zero. Kepler was able to show that, despite minor differences, the proportions of the Austrian wine merchant's barrels were such that the procedure used to calculate the volume actually would be quite accurate, after all.

Editor's note: A copy of this article, "Kepler: The Volume of a Wine Barrel," featuring four interactive applets is available at the website MatematicasVisuales. Three of those applets were created for this version of the article in July 2020.

Johannes Kepler (1571–1630) was a German mathematician, astronomer, and astrologer, and a key figure in the 17th-century scientific revolution. He lived after Copernicus and supported the heliocentric model of the universe. Kepler worked with Tycho Brahe and used Brahe's remarkable observational data to make his most famous discovery, the three laws of planetary motion now known as Kepler's Laws. Newton later showed that Kepler's laws could be deduced from Newton's laws of motion and universal gravitation law.

As a mathematician Kepler discovered two new regular polyhedra, worked on the problem of close packing of equal spheres, computed logarithms, and found volumes of solids of revolution. Our main purpose here is to understand Kepler's contributions to the development of the calculus.

Kepler lived before Newton and Leibniz. During the 16th and early 17th centuries, the Greek mathematical masterworks, including Euclid's Elements, the Conics of Apollonius, and the works of Archimedes, were studied seriously. Numerous mathematicians refined the method of exhaustion and applied it to a wide variety of new quadrature (area) and cubature (volume) problems. Another point of interest was the determination of centers of gravity of solids. (On the importance of centers of gravity for the development of the calculus, see Baron, p. 90.) Renaissance mathematicians were more interested in new results and methods of discovery than in rigorous proofs. They freely used intuitive concepts of the infinite to produce infinitesimal methods for the solution of area and volume problems. Kepler and Cavalieri were two key mathematicians who helped invent these infinitesimal methods.

Kepler's Era Copernicus Brahe Galileo Kepler Descartes Cavalieri Fermat Newton Leibniz

Nicolas Copernicus (1473-1543)

Copernicus published 'De Revolutionibus orbium coelestium' (1543) where he explained his heliocentric theory.

He gave several reasons why the sun would be at the center of the universe and not the Earth.

The heliocentric theory was defended by Kepler and Galileo and the theoretical evidence was provided by Newton's theory of universal gravitation.

Copernicus in Mac Tutor History of Math

Tycho Brahe (1546-1601)

Tycho Brahe was an astronomer who made very accurate observations with improved astronomical instruments.

Brahe did not accept Copernicus's theory that the Earch moved around the sun. He proposed an astronomical model with the Earth in the center and with the planets rotating around the sun. Then he avoided the problem of a moving Earth. Brahe's model was accepted by most astronomers.

Kepler used Brahe's accurate observations to deduce his three laws of planetary motion.

Brahe in Mac Tutor History of Math

Galileo Galilei (1564-1642)

Galileo was an Italian physicist, mathematician, and astronomer. He studied falling bodies and insisted on testing theories by conducting experiments. To make his astronomical observations he built a telescope.

For him, the laws of nature are mathematical. He wrote: "The universe ... It is written in the language of mathematics."

When Galileo began publicly supporting the heliocentric views of Copernicus he was denounced to the Roman Inquisition.

Galileo in Mac Tutor History of Math

Johannes Kepler (1571-1630)

Kepler weas a German mathematician, astronomer, and astrologer. He supported the Copernican theory that the planets move around the sun and is best known for his laws of planetary motion.

Kepler studied the work of Archimedes and used infinitesimal techniques to calculate areas and volumes of bodies. He made contributions to the origin of integral calculus.

Kepler in Mac Tutor History of Math

Rene Descartes (1596-1650)

Descartes was a French philosopher, mathematician, and physicist.

As a mathematician he (and Fermat) invented the Cartesian coordinate system and founded analytic geometry, combining Geometry and Algebra. He identified each point of the plane with an ordered pair of real numbers. He expressed geometric properties using Algebra, then manipulated his algebraic expressions to get geometric results.

His work was crucial to the later developent of infinitesimal calculus by Newton and Leibniz.

Descartes in Mac Tutor History of Math

Bonaventura Cavalieri (1598-1647)

Cavalieri was an Italian mathematician.

He was one of the precursors of infinitesimal calculus. Cavalieri developed a "method of the indivisibles" which he used to determine areas and volumes.

We know his method as Cavalieri's principle: If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio.

Cavalieri in Mac Tutor History of Math

Pierre de Fermat (1601-1665)

Fermat was a French lawyer and mathematician.

He and Descartes were the founders of analytic geometry, which later played a major role in the development of infinitesimal calculus by Newton and Leibniz.

Fermat developed methods for determining maxima and minima for various curves using tangents to those curves.

Fermat is best remembered for his work in number theory, in particular for Fermat's Last Theorem.

Fermat in Mac Tutor History of Math

Isaac Newton (1643-1727)

The English mathematician and scientist Isaac Newton is considered to be one of the most influential mathematicians and scientists in history.

Using his theory of gravitation, he demonstrated Kepler's laws of planetary motion.

In mathematics, Newton shares credit with Gottfried Leibniz for the development of the differential and integral calculus.

Newton in Mac Tutor History of Math

Gottfried Leibniz (1646-1716)

Leibniz was a German philosopher and mathematician.

He invented infinitesimal calculus independently of Newton, and his notation has been in general use since then.

Leibniz in Mac Tutor History of Math

Figure 1. Mathematicians who influenced Kepler or were influenced by Kepler, from Copernicus to Leibniz, arranged by the centuries in which they lived. Read more about any or all of them at the MacTutor History of Mathematics Archive. In the copy of this article, "Kepler: The Volume of a Wine Barrel," at MatematicasVisuales, this diagram is interactive, allowing you to read a very short biography of each mathematician by clicking on his name.

Kepler had several children before his first wife died. In 1613, he married for the second time in a celebration in Linz, Austria. Kepler bought a barrel of wine for the wedding but questioned the method the wine merchant used to measure the volume of the barrel and thus determine the price.

In consequence, afterwards Kepler set out both to determine the correct volume of a wine barrel or cask, and to find the proportions that optimize the volume of such a barrel.

Figure 2. Wine barrels in the modern Spanish Pyrenees (left) and in Kepler's 1615 Nova stereometria doliorum vinariorum (p. 98, right). (Photo on left by the author; image on right used by permission of the Carnegie Mellon University Libraries.)

The drawing at right of a wine barrel lying on its side shows how the wine merchant determined the volume of the barrel, and thus the price of the wine. The merchant would insert a stick through the tap hole (at center top in the drawing) to the opposite edge of the lid of the barrel (at lower left or lower right). The length of stick that went into the barrel determined the price the merchant would charge!

To determine the volume of a wine barrel accurately, Kepler thought of the wine in a full barrel, or of any solid body, as made up of numerous thin sheets arranged in layers, and treated the volume as the sum of the volumes of these leaves. In the case of a wine barrel, each of these leaves was a cylinder (Klein, p. 209).

The animation below shows how Kepler could refine his approximation by using more and more cylinders. These cylindrical slices are taken perpendicular to the axis of the barrel. Each diameter is equal to that of the barrel at that particular height. The height of each cylinder can be made as small as we can imagine. These cylindrical slices are piled one on top of the other, with all the slices together constituting the barrel.

Figure 3. The animation illustrates how to estimate the volume of a wine barrel as a sum of volumes of cylinders. This interactive diagram also appears in the copy of this article, "Kepler: The Volume of a Wine Barrel," at MatematicasVisuales. (The applet here was re-created in Spring 2020 by Laura Turner.)

According to Otto Toeplitz (p. 83):

Working out finer approximations of various barrel shapes, [Kepler] consulted Archimedes and discovered that his own method of indivisibles had enabled him to obtain results in a far simpler and more general way than Archimedes, who had been struggling with cumbersome and difficult proofs. What he did not suspect was that Archimedes, too, had found his results by the same method of indivisibles (for the [Method] was lost until 1906!).

The reference to Archimedes' Method is to the work in which he described how he found many of his results on areas and volumes. After an old manuscript copy of this book was discovered in Constantinople in 1899, it was studied and deciphered by John Ludwig Heiberg from 1906 to 1908. The book then underwent restoration at The Walters Art Museum in Baltimore, and we can learn a lot about this fascinating work of Archimedes in the article, The Archimedes Palimpsest, in Convergence's Mathematical Treasures.

Archimedes' Method for Computing Areas and Volumes Article by Gabriela R. Sanchis in Loci: Convergence in which she explains Archimedes' method based on the Law of the Lever.

Kepler reported his results on wine barrels in his 1615 book, Nova stereometria doliorum vinariorum (New solid geometry of wine barrels). The word Stereometria is from the Ancient Greek stereos that means solid or three-dimensional and metron that means a measure or to measure. Stereometria then means the art of measuring volumes, or solid geometry. Doliometry is an old-fashioned word from the Latin dolium that means a large jar or barrel.

This book is a systematic work on the calculation of areas and volumes by infinitesimal techniques. Building on the results of Archimedes, it focuses on solids of revolution and includes calculations of exact or approximate volumes of over ninety such solids (Edwards, p. 102). Today we use integral calculus to solve these kinds of problems.

Figure 4. Title page of Kepler's 1615 Nova stereometria doliorum vinariorum. (Image used by permission of the Carnegie Mellon University Libraries)

The full text of the original Latin version of Kepler's Nova stereometria doliorum vinariorum (1615) can be viewed in electronic format, courtesy of the Posner Memorial Collection of the Carnegie Mellon University Libraries. Two pages of Kepler's own German version of his Nova Stereometria, published in 1616 and titled Auszug aus der uralten messekunst Archimedes, can be viewed in Convergence's Mathematical Treasures.

According to C.H. Edwards (p. 102):

Kepler's approach in [his Stereometria was] to dissect a given solid into an … infinite number of infinitesimal pieces, or solid 'indivisibles', of a size and shape convenient to the solution of the particular problem.

Kepler then added up the volumes of the component pieces to obtain the volume of the given solid. Kepler's infinitesimal elements had to possess the same dimensions as the body he wanted to measure. That is, if he wanted to calculate an area, he added up area elements and if he wanted to calculate a volume he considered solid infinitesimal elements with volume.

Kepler began his book with the simple problem of determining the area of the circle. He regarded the circle as a regular polygon with an infinite number of sides and therefore composed of infinitely many infinitesimal triangles of which the bases were the sides of the polygon, the heights or altitudes the radius of the circle, and the vertices opposite the bases the center of the circle (Boyer, p. 108). Since the area of a triangle is one-half the product of its base and height, the total area of the circle was then given by half the product of its perimeter (or circumference) and its radius.

Kepler in MatematicasVisuales - The Area of a Circle Kepler used an intuitive infinitesimal approach to calculate the area of a circle.

Similarly, by thinking of the sphere as being composed of an infinite number of infinitesimal cones whose vertices were the center of the sphere and whose bases made up the surface of the sphere, Kepler was able to compute the volume of the sphere. Since the volume of a cone is one-third the product of the area of its base and its height or altitude, the total volume of the sphere was then given by one-third the product of its surface area and its radius.

Kepler in MatematicasVisuales - Volume of a Sphere Kepler used an intuitive infinitesimal approach to determine the relationship between the volume of a sphere and its surface area.

Kepler also rotated a circle about a line external to the circle, and calculated, by infinitesimal methods, the volume of the torus thus generated. He then extended his work to solids not treated by the ancients. Some of his summations are remarkable anticipations of results found later by integral calculus (Boyer, pp. 108-109).

Kepler in MathWorld: Lemon Kepler in MathWorld: Apple In Wolfram MathWorld, we can see solids of revolution defined by Kepler.

The method the wine merchant had used to measure the volume of the barrel that had so shocked Kepler was to insert a stick through the tap hole (at \(S\)) in the diagram below) to the opposite edge of the lid of the barrel (at \(D\)).

Figure 5. Austrian merchants set the price of a full wine barrel according to the length \(SD\). (Underlying image from p. 98 of Kepler's 1615 Nova stereometria used by permission of the Carnegie Mellon University Libraries)

According to Toeplitz (pp. 82-83):

Then he read off the length \(SD = d\) and set the price accordingly. This outraged Kepler who saw that a narrow, high barrel might have the same \(SD\) as a wide one and would indicate the same wine price, though its volume would be ever so much smaller.

Figure 6. Two cylinders with the same measurement \(SD\) may have very different volumes. (After an image from Toeplitz, p. 82)

In consequence, Kepler tried to determine the best proportions for a wine barrel in order to maximize the volume. This led him to consider a number of problems on maxima and minima, which proved to be a very interesting contribution to the development of the differential calculus. For example, Kepler was able to establish that the cube is the largest parallelepiped that can be inscribed in a sphere (Baron, p. 115).

Of course, the problem of finding maxima and minima was not new. For example, Euclid had proven that, among all rectangles of equal perimeter, the square has the largest area (Toeplitz, 80). Pappus of Alexandria around 300 CE showed that, for equilateral and equiangular plane figures having an equal perimeter, the figure with the greater number of angles has the greater area and the largest such area is that of the circle with the same perimeter.

	The Sagacity of Circles: A History of the Isoperimetric Problem This Convergence article by Jennifer Wiegert gives a summary of the history of the problem of finding the region of greatest area bounded by a given perimeter.
	Thinking Outside the Box - or Maybe Just About the Box In an article in Loci by David Meel and Thomas Hern, the authors study a more realistic and therefore complex box problem than the traditional one and show how it can be used to investigate optimization.

To optimize the volume of the wine barrel, Kepler simplified the problem. He approximated the barrel by a cylinder with the diagonal measurement \(d=SD,\) radius of the base \(r,\) and height of the cylinder \(h.\) In modern notation, the formula for the volume \(V\) of a cylinder is: \[V=\pi r^2h.\]

Using the Pythagorean Theorem, we can write an equation relating \(d, h,\) and \(r:\) \[d^2 = \left(\frac{h}{2}\right)^2 + (2r)^2.\]

Figure 7. For a fixed value of \(d,\) the volume \(V\) is a function of height \(h.\) The graph at left shows how, for a fixed value of \(d,\) changes in height \(h\) result in changes in volume \(V.\) The figure at right shows how, for a fixed value of \(d,\) changing the height \(h\) changes the radius \(r\) and the shape of the cylindrical barrel. This interactive diagram also appears in the copy of this article, "Kepler: The Volume of a Wine Barrel," at MatematicasVisuales. (The applet here was re-created in Spring 2020 by Laura Turner.)

Next, Kepler asked: "If \(d\) is fixed, what value of \(h\) gives the largest volume \(V\)?" (Toeplitz, p. 83). After making some calculations and comparing them, he decided the answer was: \[h=\frac{2d}{\sqrt{3}}.\]

According to Toeplitz (p. 83):

That defined a barrel of definite proportions. Kepler noticed that in his Rhenish homeland barrels were narrower and higher than in Austria, where their shape was peculiarly close to that having a maximum volume for a fixed \(d\) - so close, indeed, that Kepler could not believe this to be accidental. So he imagined that centuries ago somebody had calculated barrel shapes, as he himself was doing, and had taught the Austrians to construct their barrels in this particular fashion - a very practical one, indeed. Kepler showed that if a barrel did not satisfy the exact mathematical specification \[3h^2 = 4d^2,\] but deviated somewhat from it, this would have but little effect on the volume, because near its maximum a function changes only slowly.

That the volume function changes very slowly near its maximum value is illustrated in the graph below.

Figure 8. For a fixed value of \(d,\) the graph of volume \(V\) as a function of height \(h\) illustrates that, near the maximum volume, small changes in height \(h\) result in small changes in volume \(V.\)

Kepler tabulated values obtained by calculation to reinforce his idea that the volumes of such cylinders with equal diagonals change very little in the neighbourhood of a maximum (Baron, p. 116).

Figure 9. Kepler's table of volumes of wine barrels of various heights (altitudes) from p. 66 of his 1615 Nova stereometria. View this page and the whole book at The Posner Memorial Collection: Kepler's Nova Stereometria. (Image used by permission of the Carnegie Mellon University Libraries)

But what about the barrel of wine Kepler had purchased for his wedding in Austria? Was it priced fairly? According to Toeplitz (p. 83):

Thus, while the Austrian method of price determination, if applied to Rhenish barrels, would be a clear fraud, it was quite legitimate for Austrian barrels. The Austrian shape had the advantage of permitting such a quick and simple method. So Kepler relaxed in this instance.

This, then, was the major contribution by Kepler: He noted that as the maximum volume was approached, the change in the volume for a given change in the dimensions became smaller. Some years later, Fermat would consider maximum and minimum problems of this kind from a similar point of view.

Derivatives, Tangents, and Slopes

Fermat (born in 1601) took a slightly different approach than Kepler: In modern terms, he was interested in the tangent to a curve and the relationship between this tangent and the maximum (or minimum) of the function represented by the curve. Fermat's algebraic approach can be seen today as equivalent to studying the slope of the tangent to the graph of the function. Despite Kepler's intuition in this direction, Fermat is considered to have been the first to solve maximum-minimum problems by taking into account the characteristic behavior of a function near its extreme values. Newton and Leibniz understood even more clearly that a maximum or minimum was associated with a horizontal tangent.

Using our modern terminology, this is the geometric interpretation of the derivative of a function. We can see intuitively that if \(f(x)\) is a maximum (or minimum) value of the differentiable function \(f,\) then the value of \(f\) changes very slowly near \(x.\) Moreover, at the highest and lowest points on the graph of \(f,\) the tangent is horizontal; that is, its slope is \(0\). The derivative will be zero at extrema.

Returning to Kepler's problem of the proportions of a wine barrel, if \(V\) is the volume of the barrel (as a cylinder) with a fixed value of \(d,\) then \(V\) is a polynomial in \(h;\) namely, \[V=\frac{\pi}{4}d^2h-\frac{\pi}{16}h^3.\] Hence the derivative is easy to calculate: \[V^{\prime}(h) = \frac{{\pi}d^2}{4}-\frac{3\pi}{16}h^2.\]

For \(V\) to be a maximum, \(V^{\prime}\) must equal zero; hence \[3h^2=4d^2\,\,\,\,{\rm{or}}\,\,\,\,h=\frac{2d}{\sqrt{3}}.\]

And this was the result that Kepler found.

Figure 10. This applet shows the graphs of \(V\) and \(V^{\prime}\) as functions of \(h,\) along with the tangent line to the graph of \(V\) at any value of \(h,\) Here, it is assumed once again that, for a fixed value of \(d,\) the volume \(V\) of the wine barrel is a function of the height \(h\). In the diagram, the blue curve is the graph of \(y = V(h)\) and the red curve is the graph of \(y = V^{\prime}(h)\) for a fixed value of \(d.\) At the maximum volume of the barrel, the green line tangent to the graph of \(V\) is horizontal and its slope, the derivative \(V^{\prime},\) is zero. (Note that, while the animation permits negative heights and volumes for the barrel, in real life this would be impossible.) This diagram also appears in the Kepler's Barrel: Derivative, Tangents, and Slopes section of the copy of this article at MatematicasVisuales. (The applet here was re-created in Spring 2020 by Laura Turner.)

Conclusion

Thus, the practical problem of measuring the volume of a wine barrel inspired Kepler to make important contributions to the development of both the integral and the differential calculus.

Links

	Kepler in MacTutor History of Mathematics Archive A biography of Johannes Kepler (1571-1630)
	Original Latin version of Kepler's Nova stereometria (Posner Memorial Collection, Carnegie Mellon University Libraries) Kepler's German version of his Nova stereometria (two pages, Loci: Convergence Mathematical Treasures)
	Kepler in MatematicasVisuales - The Area of a Circle Kepler's intuitive infinitesimal approach to determine the area of a circle
	Kepler in MatematicasVisuales - The Volume of a Sphere Kepler's intuitive infinitesimal approach to determine the volume of a sphere
	Kepler in MathWorld: Lemon Kepler in MathWorld: Apple Two solids of revolution defined by Kepler
	Archimedes' Method for Computing Areas and Volumes Article by Gabriela Sanchis in Loci: Convergence explaining Archimedes' Method based on the Law of the Lever
	Archimedes in MatematicasVisuales - Area of a Parabolic Segment Archimedes used infinitesimal methods to discover areas and volumes 1800 years before Kepler and Cavalieri.
	The Sagacity of Circles: A History of the Isoperimetric Problem Article by Jennifer Wiegert in Loci: Convergence summarizing the history of the problem of finding the region of greatest area bounded by a given perimeter
	Thinking Outside the Box - or Maybe Just About the Box Article by David Meel and Thomas Hern in Loci presenting a box optimization problem more realistic and hence more complex than the standard one.

References

Margaret E. Baron, The Origins of the Infinitesimal Calculus, Dover Publications, New York, 1987; originally published in 1969 by Pergamon Press, Oxford, England.

Carl B. Boyer, The History of the Calculus and its Conceptual Development, Dover Publications, New York, 1959, reprint of 1949 edition of Boyer's The Concepts of the Calculus: A Critical and Historical Discussion of the Derivative and the Integral, Hafner Publishing, originally published in 1939.

Roberto Cardil, Kepler: The Best Proportions for a Wine Barrel (interactive applet), MatematicasVisuales, posted 2009-2010.

Roberto Cardil, Kepler: The Volume of a Wine Barrel (interactive applet), MatematicasVisuales, posted 8 January 2010.

Roberto Cardil, "Kepler: The Volume of a Wine Barrel" (copy of this article with four interactive applets), MatematicasVisuales, posted 2011

Roberto Cardil, MatematicasVisuales: www.matematicasvisuales.com

C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, New York, 1979.

Kepler, Johannes, Nova stereometria doliorum vinariorum, Linz, 1615, held in the Posner Memorial Collection of the Carnegie Mellon University Libraries, Pittsburgh, Pennsylvania, call number 520 K38PN, available in electronic format: http://posner.library.cmu.edu/Posner/books/book.cgi?call=520_K38PN

Felix Klein, Elementary Mathematics from an Advanced Standpoint - Arithmetic - Algebra - Analysis (especially pp. 207-210), Dover Publications, New York, 2004; originally published in 1908 in Leipzig.

Dirk J. Struik (ed.), A Source Book in Mathematics, 1200-1800 (see pp. 192-197: "Kepler. Integration Methods"), Harvard University Press, Cambridge, MA, 1969.

Otto Toeplitz, The Calculus: A Genetic Approach (especially pp. 80-83), University Of Chicago Press, 1963.

About the Author

Roberto Cardil Ricol is a Secondary School teacher of Mathematics in Spain. He studied as a teacher (Huesca University, Spain), and earned Bachelor of History (Barcelona University, Spain) and Bachelor of Mathematics (Zaragoza University, Spain) degrees. He has taught Mathematics to students from 6 years old to the University level. He maintains the site www.matematicasvisuales.com and enjoys the beauty of Mathematics.
 

Acknowledgments

I would like to thank Janet Beery, Convergence editor, for her support and encouragement. Without her help this article would not have been published.

I thank the anonymous referees for suggestions that improved the article.

I am grateful to the Carnegie Mellon University Libraries for granting permission to use images from their copy of Kepler's Nova stereometria doliorum vinariorum, held in the Posner Memorial Collection, in this article.

Miguel Cardil helped me in the artistic design of the page and also designed all graphic elements in the mathlets.

Finally, I would like to thank Kathleen Killorin for her help with the English translation.