An old idiom, employed by many writers over the years and popularly attributed to American polymath Benjamin Franklin, claims that nothing is certain except death and taxes. Franklin was not the first to use this phrase: versions of it appear in Daniel Defoe's 1726 book *The Political History of the Devil* [Defoe 1726, p. 269] and Christopher Bullock's 1716 play, *The Cobler of Preston* [Bullock 1716]. As it happened, Franklin came to use this idiom in a letter to his friend, French physicist Jean-Baptiste Le Roy, on 13 November 1789:

Our new Constitution is now established, and has an appearance that promises permanency; but in this world nothing can be said to be certain except death and taxes! [Franklin 1818, p. 259]

Six years prior, in 1783, two particular events occurred, one undeniably certain and one profoundly uncertain. On 19 September 1783, Le Roy himself was among the many witnesses present as a sheep, a duck, and a rooster became the first passengers aboard a hot air balloon constructed by the brothers Joseph-Michel and Jacques-Étienne Montgolfier, who had spent the past several months experimenting with their flying device. The balloon, dubbed the *Aérostat Réveillon* after wallpaper-maker Jean-Baptiste Réveillon, lifted off the grounds of royal palace at Versailles and flew for about 8-10 minutes and to an altitude of 1,500 feet over a crowd that included King Louis XVI and Queen Marie Antoinette [Kotar and Gessler 2003, p. 12].

**Figure 1.** The *Aérostat Réveillon *lifts off from the grounds of the

Palace of Versailles on 19 September 1783. The Wellcome Collection.

The night before this remarkable flight, famed Swiss mathematician Leonhard Euler succumbed to a brain hemorrhage at his home in St. Petersburg, Russia. The death of such a luminary as Euler was, if not unexpected, a profound loss for the mathematical community. Over his 56 year career, Euler contributed to such disparate fields as astronomy, number theory, mechanics, calculus, graph theory, and music theory. His prowess at mental calculation was widely known. In one amusing anecdote, related to us by Nicolas Fuss, Euler calculated the first six powers of all numbers under 20 when unable to sleep one night, and was able to recite them all from memory a few days later [Fuss 1783, p. 208].

In the Marquis de Condorcet's eulogy, Euler's death occurred suddenly in the bucolic setting of his St. Petersburg estate:

He had his grandson come and play with him and took a few cups of tea, when all of a sudden the pipe that he was smoking slipped from his hand and he ceased to calculate and live [Condorcet 1783, p. 67, translated by John S. D. Glaus].

Euler's influence would continue long after his death, with many of his works not published until 1862. All told, he wrote over 800 papers in mathematics and science, as well as some 30 books. His work revolutionized the mathematical sciences. However, Euler's work had a more specific and immediate impact after his death. Just before reporting the death of Euler in his eulogy, Condorcet wrote on Euler's particular interest in the weeks leading up to his death:

On [18] September 1783, after having enjoyed some calculations on his blackboard concerning the laws of ascending motion for aerostatic machines for which the recent discovery was the rage of Europe, he dined with Mr. Lexell and his family, spoke of Herschel’s planet and the mathematics concerning its orbit [Condorcet 1783, pp. 67–68, translated by John S. D. Glaus].

As we see here, Euler was engaged in the study of the Montgolfier brothers' novel “aerostatic machines” on the very day of his death. The calculations of the balloons’ motion remained on the blackboard even as Euler himself “ceased to calculate and live.” The following year, a short piece appeared in the *Memoires *of the Paris Academy, titled “Calculs sur les Ballons aérostatiques faits par feu M. Léonard Euler, tels qu’on les a trouvés sur son ardoise, après sa mort arrivée le 7 Septembre 1783” (“Calculations on aerostatic balloons made by the late Mr. Leonhard Euler, as they were found on his blackboard, after his death on 7 September 1783”) [Euler 1784, numbered E579]. Euler’s son, Johann-Albrecht, had forwarded a copy of these calculations to the Paris Academy, thus preserving the work in the public record.

In this article, we examine Euler’s valedictory and consider what conclusions can be drawn from it. We begin with some history of the study of balloon flights at the time, as context for his interest in the mechanics of balloon flight. Following our analysis of Euler’s calculation, we also present a classroom capsule with suggestions for how it can be used in a contemporary differential equations or physics course.

Aerostatic balloon flight, like powered airplane flight, involved considerable experimentation and preparation. Like the Wright brothers, the Montgolfier brothers devoted their time, talent, and funds to realizing their dream. The first progress was made by Joseph Montgolfier in November 1782, when he set aloft a hollow, taffeta-lined box by heating the interior air. This early experiment was limited, occurring only briefly and indoors. But it demonstrated the soundness of the basic principles, leading to an outdoor experiment the next month, in which a larger hot-air ballon rose to a height of 70 feet [Kotar and Gessler 2003, p. 10].

The famous 1783 demonstration at Versailles was the first balloon flight by living creatures, but it was not the Montgolfiers’ first public success. That honor resides with a 4 June 1783 flight at Annonay, in which a hot-air balloon was flown to a height of 3,000 feet and an eventual distance of about 1.7 miles. This early creation was made up of silk sections with paper backing, attached to a mostly-spherical frame 30 feet in diameter [Holmes 2008]. Unfortunately, it caught fire after landing in a vineyard and was consumed, to be replaced some months later by the *Aérostat Réveillon*. However, the flight caught the attention of the wider public, and news of its success reached St. Petersburg in August of that year [Gillispie 1983, p. 32].

The Montgolfiers came to the science of lighter-than-air flight through their family's connection to the paper manufacturing business. Their father, Pierre, ran a successful paper company in Annonay, and he later turned over control of the business to the elder son, Joseph. The young businessman also had a longstanding interest in the behavior of gases, having relied on his cousin Matthieu Duret as early as 1777 for information on Joseph Priestley and Henry Cavendish’s experiments on oxygen (then called “dephlogisticated air”) and hydrogen (“inflammable air”), respectively [Gillispie 1983, p. 15]. As Joseph wrote in a letter to Duret, “All that you have taught me of chemistry only confirms me more fully in my ideas. I must make some experiments” [Gillispie 1983, p. 15].

**Figure 2.** A detailed sketch of the Montgolfier brothers' *Aérostat Réveillon*, including

information on its height, diameter, volume, and weight. Library of Congress.

These experiments began in earnest in late 1782, during which time Joseph and his younger brother Étienne attempted no fewer than three balloon flights, each with varying degrees of success [Kotar and Gessler 2003, p. 10]. While not adept at the physical mechanics of their balloons, the Montgolfiers did have a basic understanding of their functioning. As the *Journal de Paris* reported on the Montgolfiers' flight at Annonay,

He had a globe constructed, 35 feet in diameter, with canvas mounted on a framework of wood and wire. . . . According to the calculation of M. Mongolfier, the globe occupied a space for which a volume of air would weigh 2,156 pounds; but since the gas only weighed 1,078 pounds and the globe 500 pounds, there was an excess of 578 pounds for the force with which the globe tended to rise [Journal de Paris 1783, pp. 861–862, translated by the authors].

The overall principle—that heated air occupying a fixed volume is less dense and thus less heavy—appears to have been at the front of the Montgolfiers' minds throughout this time. As we saw above, the successful flight of the *Aérostat Réveillon* at Versailles occurred only one year after their first balloons took to the air. Le Roy and other members and officials of the Paris Academy of Sciences reported on this flight and several others by the Montgolfier brothers. They co-authored a memoir that was read to the Academy on 23 December 1783 and published the following year as a monograph. In it, they described the *Aérostat **Reveillon*'s descent as having occurred "so gently that it merely bent the tree branches on which it landed, and the animals that were suspended from it did not suffer in the slightest" [Le Roy et al. 1784, p. 15, translated by the authors].

Once the Mongolfiers’ balloons demonstrated the possibility of lighter-than-air travel, news traveled fast. Accounts of the various balloon flights circulated around the world, attesting to both the exhilaration of such a voyage and the scientific principles that made it possible.

Augustin-Charles Piroux’s *L’Art de Voyager Dans les Airs* introduced the subject of air travel with a flourish: "The astonishment of the multitude joins the admiration of the scholars; Experiments multiply, intrepid men rise into the air, and the globes that carry them merge with the stars" [Piroux 1784, pp. 1–2, translated by the authors]. Benjamin Franklin (then serving as the newly-independent United States’ minister to France) described how, when a balloon was launched from the Tuileries Palace [on 1 December 1783], “all Eyes were gratified with seeing it rise majestically from among the Trees, and ascend gradually above the Buildings, a most beautiful Spectacle!” [Rotch 1907, p. 270]. The excitement is palpable in this conclusion from the *Journal de Paris*'s account: “M. Montgolfier was already known for the manufacture of the most beautiful papers that have been made in France. The experiment which we have just related announces a curiosity and enlightenment of another kind” [Journal de Paris 1783, p. 862, translated by the authors].

And balloon flights were not limited to the Montgolfier brothers—around the same time, the French scientist Jacques Charles was exploring the uses of hydrogen as a method for raising balloons. Charles, in collaboration with the engineers Anne-Jean and Nicolas-Louis Robert, oversaw the launch of a hydrogen balloon from Paris on 27 August 1783. It happened that Benjamin Franklin was also present at this flight, describing how they obtained hydrogen for the balloon by “pouring Oil of Vitriol upon Filings of Iron,” and reporting that “it was found to have a tendency upwards so strong as to be capable of lifting a Weight of 39 Pounds, exclusive of its own Weight which was 25 lbs. and the Weight of the Air contain’d” [Rotch 1907, pp. 260–261]. Indeed, the flight from the Tuileries that Franklin observed on 1 December was by another of Charles’s and the Robert brothers’ hydrogen balloons. Over the next several years, similar voyages were made repeatedly across France. Piroux, writing in 1784, noted that “Everyone wants to repeat the wonderful experiment of MM. Montgolfier, each according to his means. Aerostatic Globes are part of the show at all parties: the workers of Paris cannot supply the demands of the Province” [Piroux 1784, p. 29, translated by the authors].

**Figure 3.** An illustration of the hydrogen balloon flight

over the Tuileries Palace on 1 December 1783. Library of Congress.

Of equal interest to observers were the mechanical principles that made balloon flights possible. Many accounts combined the description of the event with an attempt at rigorous, scientific description of the forces at play. For the most part, these reports described the volume and weight of the balloon, with some stating the force exerted by the elevating agent (either heated air or hydrogen):

- Franklin described Charles’s balloon of 27 August as “A hollow Globe 12 feet Diameter was formed of what is called in England Oiled Silk,” and relayed that “some observers say, the Ball was 150 Seconds in rising, from the Cutting of the Cord till hid in the Clouds; that its height was then about 500 Toises” [Rotch 1907, pp. 260–262]. He further noted that “One of 38 feet Diameter is preparing by Mr. Montgolfier himself, at the Expence of the Academy, which is to go up in a few Days. I am told it is constructed of Linen & Paper, and is to be filled with a different Air” [Rotch 1907, p. 262].
- According to Piroux, “Montgolfier's machine of 19 September weighed seven or eight hundred pounds, contained forty thousand cubic feet of gas, and could, it is said, lift about twelve hundred pounds (its load was only six hundred)” [Piroux 1784, p. 15]. He also claimed that “the species of air employed by M. de Montgolfier offers an absolutely new phenomenon, since to obtain it, it is enough to simply burn wet straw and an animal substance such as wool. A very small quantity of these combustible materials, and ten minutes’ time, furnished the necessary 37,500 cubic feet of air, and such a volume of inflammable air [hydrogen] would not have been procured without perhaps eight or ten thousand francs, and five to six working days” [Piroux 1784, pp. 74–75, both quotations translated by the authors].
- In a report by Barthélemy Faujas de Saint-Fond, published under the auspices of the Royal Academy of Sciences in Paris, the balloon flown by the Montgolfiers at the Château de la Muette on 21 November “was 70 feet high, 46 feet in diameter: it contained 60,000 cubic feet [of air], and the weight it lifted was about sixteen to seventeen hundred pounds” [Faujas de Saint-Fond 1784, p. 22, translated by the authors]. (Notably, this was the first balloon flight crewed by humans.)

Given the widespread interest in the mechanics of balloon flight, it is not surprising that Euler would become engaged in his own version of these calculations. In fact, Anders Lexell, Euler’s dining companion on the night of his death, recalled their dinner conversation a couple of weeks afterward (in a letter to J. H. de Magellan):

Talking afterwards upon the principles on which the aerostatic globes are constructed, he remarked that it was a curious mathematical problem to determine the motion of such a globe, from knowing the proportion between the density of the air contained in the globe and of the common air. He observed also, that supposing this proportion to be as one to two, the greatest velocity of the globe would be 41 feet in a second [Stén 2014, p. 234].

However, Euler’s goals were slightly different from those described above: Instead of reporting the specific size and weight of a hot-air balloon, or determining the expense required to obtain enough hydrogen for a flight, or even determining the force such a balloon could exert, he sought to use more general principles of mechanics to determine its maximum velocity. Of course, Euler’s death prevented us from reading the complete article he would likely have published on the matter. However, the preservation of his blackboard notes and the relatively rapid publication of E579 made his basic derivation available to the general public (and, eventually, to us).

For his final work in E579, Euler divided his analysis of the motion of a rising aerostatic balloon into three major parts. In the first part, he calculated air resistance and used it to find an expression for the velocity of the balloon at any altitude. Next, he determined the maximum altitude the balloon could attain and the time of ascent to this maximum altitude. Finally, he determined the maximum speed that the balloon would attain. The article ends with a sample calculation for a balloon of a particular size and weight.

To prepare for the calculation, Euler set \(a\) as the balloon’s radius, \(M\) as its weight, and \(N\) as the weight of the surrounding air displaced by the balloon. Of course, the mass of a fixed volume of surrounding air will diminish as altitude increases, so Euler would also need take atmospheric pressure into account. He accomplished this by introducing the approximate scale height (i.e., the height of an air column) of \(k=\) 24,000 feet. Using \(x = AM\) for the balloon’s altitude, air pressure at that altitude is modeled by \(e^{-x/k}\). Since Eulers analysis predated SI units, he did not have the luxury of measuring pressure in pascals; instead, he used a system in which air pressure at ground level is equal to 1.

**Figure 4. **Euler's setup for the calculation in E579, with a diagram in the left margin showing a column of air stretching

from \(A\) (on the ground) to \(H\) (at an altitude of 24,000 feet). The balloon is located at the point \(M\). The Euler Archive.

Then, in a turn that might be unexpected for a modern reader, Euler used \(g\) as the distance fallen in one second by an object subject only to gravity (i.e., \(g=16\) feet). This is another situation in which Euler’s units differed from a modern approach—today, we would instead use the acceleration due to gravity (say, \(g'\)), which equals \(2g\) feet per second per second. His resulting expression for air resistance—or, drag—at point \(M\) is

\[

\frac{vv}{4g}\cdot\frac{\pi aa}{2}\cdot\frac{N}{\frac{4}{3}\pi a^3}.

\]

With a little work, this can be made to align with the modern drag equation:

\[

\frac{1}{2}\left(\frac{v^2}{2}\right)\cdot\left(\pi a^2\right)\cdot\left(\frac{N/g'}{\frac{4}{3}\pi a^3}\right),

\] where the coefficient \(\frac{1}{2}\) approximates the drag coefficient for a sphere, the second factor corresponds to the maximal cross-sectional area of the balloon, and the third factor gives the density of displaced air at ground level. (As we know, Euler would eventually need to account for the decrease in atmospheric density as the balloon ascends.)

For now, he simplified his drag equation as \(\frac{3N}{8a}\cdot\frac{vv}{4g}\) and invoked the “principle of mechanics” (known today as Newton’s second law): \[2v \partial v=\frac{4g \partial x}{M} \cdot P,\] where \(P=\frac{M}{2g}\cdot v \frac{\partial v}{\partial x}\) was Euler’s way of expressing the relation Force = Mass \(\cdot\) Acceleration. (Here, the differential of \(v^2\) is written as \(2v\partial v\).) Euler’s goal was to equate \(P\) with the total of forces acting on the balloon. The net force acting on a rising balloon is the difference between the buoyant force \(Ne^{-x/k}-M\) and the drag force \(\frac{3N}{8a}\cdot\frac{vv}{4g}\): \[P \;=\; N e^{-x/k}-M-\frac{3N}{8a} \cdot \frac{vv}{4g} \cdot e^{-x/k}.\] For the buoyant force, Euler used Archimedes’ principle implicitly by subtracting the balloon’s weight from the weight of the displaced air (now accounting for the atmospheric pressure at altitude \(x\)).

To put it all together, Euler set \(b = \frac{8a}{3 \lambda}\) and wrote his differential equation as

\[

2v \partial v + \left(\frac{1}{b} \cdot vve^{-x/k}\right) \partial x = 4g \left(\lambda e^{-x/k}-1\right)\partial x, \;\;\;\;\;\;\;\; (1)

\]

with \(\lambda = \frac{N}{M}\) being his way of measuring a sort of "buoyancy ratio." A higher value for \(\lambda\) corresponds to a more buoyant balloon, while \(\lambda = 1\) corresponds to a balloon filled with unheated air.

As it stands here, Euler had a problem: this equation is not solvable! However, he circumvented this problem with some creative approximations. Taking a linear approximation \(e^{-x/k} \approx 1-\frac{x}{k}\) on the right side of the equation and a coarser approximation of \(e^{-x/k} \approx 1\) on the left side, he obtained

\[

2v \partial v + \left(\frac{1}{b} \cdot vv\right) \partial x \;=\; 4g \left(\lambda-1-\frac{\lambda x}{k}\right)\partial x,

\]

which can be solved by using an integrating factor method. Specifically, multiply through by \(e^{x/b}\) to get

\[

\partial \left( v^2e^{x/b} \right) \;=\; 4g \left(\lambda-1-\frac{\lambda x}{k}\right)e^{x/b}\,\partial x.

\]

After integrating (implicitly over the interval \([0,x]\)), Euler obtained

\[

v^2e^{x/b} \;=\; \frac{4\lambda gb}{k} \left((b+f)(e^{x/b}-1)-xe^{x/b}\right),

\]

where \(f = \big(1-\frac{1}{\lambda}\big)k\). With a little adjustment, we arrive at the equation

\[

v^2 \;=\; \frac{4\lambda gb}{k} \left((b+f)(1-e^{-x/b}) - x\right),

\]

which Euler reminded us is “an expression which determines the speed of the sphere at any altitude.”

With an approximate solution in hand, Euler could then derive formulas for the the balloon’s maximum altitude, total ascent time, and maximum speed. First, he noted that the maximum altitude \(h\) is reached when the vertical velocity vanishes: \[ 0 \;=\; \frac{4\lambda gb}{k} \left((b+f)(1-e^{-h/b}) - h\right).

\]

Since \(h\) will be considerably larger than \(b\) an approximation of \(e^{-h/b} \approx 0\) is enough to approximate the maximum altitude, as \(h = b+f\). Euler could have (and likely had) approximated \(h\) by appealing directly to his differential equation: if we use a linear approximation for \(e^{-x/k}\), the equation (1) becomes

\[

2v \partial v + \left(\frac{1}{b} \cdot vv\left(1-\frac{x}{k}\right)\right) \partial x \;=\; 4g \left(\lambda \left(1-\frac{x}{k}\right)-1\right)\partial x.

\]

Since the maximum altitude \(h\) is reached when velocity and acceleration *both* vanish, the differential equation may be reduced to \(0 = 4g \left(\lambda \left(1-\frac{h}{k}\right)-1\right)\), in which case \(h = \big(1-\frac{1}{\lambda}\big)k = f\).

Regardless of the approximation used for maximum altitude, the total ascent time is now easily found. Since \(v\) is velocity, Euler used \(v \partial t = \partial x\) to integrate solution \(v\) as a separable differential equation. After integrating, Euler set \(x = h\) and again used the approximation \(e^{-h/b} \approx 0\) to get \(\sqrt{\frac{kh}{\lambda gb}}\).

Maximum speed is not much more difficult. Euler needed the differential of \(v^2\) to vanish, i.e.,

\[

\partial \left((b+f)(1-e^{-x/b})-x\right) \;=\; 0.

\]

Some elementary differential calculus shows that the maximum speed is attained at an altitude of \(x = b\cdot \ln\left(\frac{b+f}{b}\right)\). From here, the speed itself can be seen to equal \(\sqrt{\frac{4\lambda gb}{k}\left(f-b\ln \left(\frac{b+f}{b}\right)\right)}\). Euler again used an approximation, this time \(\frac{b}{f}\approx 0\), to simplify his speed formula to \(2b\sqrt{\frac{\lambda gf}{bk}}\).

As we now see, the notes from Euler’s blackboard gave a set of explicit formulas to calculate (at least approximately) a hot-air balloon’s maximum altitude, ascent time, and maximum speed. One only needs to know the balloon’s radius \(a\) and buoyancy ratio \(\lambda\). From there, the intermediate parameter \(b = \frac{8a}{3\lambda}\) and air column of height \(k = 24,\!000\) feet provide the key to understanding the formulas:

\[ \begin{cases} \text{Maximum altitude} & \;=\; h \;=\; b+f \;=\; b+\Big(1-\frac{1}{\lambda}\Big)k, \\ \text{Ascent time} & \;=\; \sqrt{\frac{kh}{\lambda gb}}, \\ \text{Altitude of maximum speed} & \;=\; b\cdot \ln\left(\frac{b+f}{b}\right) \;=\; b\cdot \ln\left(1+\Big(1-\frac{1}{\lambda}\Big)\frac{k}{b}\right), \\ \text{Maximum speed} & \;=\; 2b\sqrt{\frac{\lambda gf}{bk}} \;=\; 2\sqrt{ gb(\lambda-1)}. \end{cases} \] The text of the article ends with a brief example, in which \(a = 30\) feet and \(\lambda = 5\) (thus making \(b = 16\)). Euler reported a maximum altitude as 19,200 feet, a maximum velocity of 64 feet per second (at an altitude of 112 feet), and an ascent time of 10 minutes and 32 seconds. As Gillispie notes, Euler’s son Johann Albrecht Euler “copied off the equations and sent them to the Academy of Science in Paris. Its *Mémoires* thus had the privilege of inaugurating mathematical analysis of the flight of aircraft with the publication of Euler’s valedictory” [Gillispie 1983, p. 32].

In this reading of Euler’s valedictory, we have generally assumed that his calculation was accurately transcribed, and that the work was correct. However, a closer analysis of Euler’s example reveals some discrepancies that deserve further attention. First of all, the formula he stated for maximum altitude should give 19,216 feet, not 19,000 feet. This is easily explained by the alternative calculation for \(h\) described above, which does not require a solution to the differential equation; so Euler could justifiably have used \(h = \big(1-\frac{1}{\lambda}\big)k = 19,\!200\) feet, as reported here. Second, the maximum speed was reported to occur at an altitude of 112 feet, even though the formula above gives an altitude of 113.45 feet. While the formula will clearly require \(\ln (1201) \approx 7.0909\), approximating the logarithm as 7 would have been entirely reasonable for the time. Of course, this approximation aligns with the maximum altitude of 112 feet recorded in the paper.

A greater difficulty lies with the reported ascent time of 10 minutes and 32 seconds, since his formula appears to give an answer of 10 minutes and 0.25 seconds. What happened to the missing time? Using the alternative value for \(h\) does not help, since this reduces the ascent time to exactly 10 minutes. A partial explanation may lie in the choice of units Euler used in his work. Since his calculations relied on the observations of flights across France, it is likely that when Euler measured distance he was using French units. While we have used the English word *foot* until now, this is somewhat misleading: the French unit (*pied*) was not the same as the English Imperial unit of the same name. Specifically, a pied was defined as 1/6 of the *Toise de l’Académie*, making it about 1.07 English feet. Crucially, this would make Euler’s gravitational constant \(g\) smaller than the typical value for English Imperial feet—and thus would make the ascent time greater. For pieds, \(g\) is very near 15, which gives an ascent time of 10 minutes and 20 seconds—much closer to the reported value. While this value of \(g\) would also cause the maximum speed to change (to about 62 pieds per second), overall this more closely approximates Euler’s own calculations.

In spite of the discrepancies, in a roundabout way these equations allow us to view a sort of poetic coincidence. Let us return to the flight of 19 September 1783, in which a sheep, a duck, and a rooster lifted off from Versailles the day after Euler’s death. Piroux reported that their balloon had a volume of 40,000 cubic feet, which corresponds to a radius of approximately 21.216 feet. At dinner the night before, Euler remarked to Lexell that a value of \(\lambda = 2\) should correspond to a maximum speed of 41 feet per second. If we understand this balloon’s radius to be measured in pieds, and use a value of \(g = 15\) for gravitational acceleration, Euler’s formula produces a maximum velocity of . . . 41.2 pieds per second! So while Euler’s example suffers from some inaccuracies, there is little doubt that his analysis was close to the mark.

In this section we offer some suggestions on how Euler’s *opus moribundum* can be used in a contemporary mathematics or physics classroom. The most natural connection is to differential equations, with an emphasis on mathematical modeling and solving *unsolvable* differential equations through approximation. Other than the notion of the *integrating factor*, the background material needed to carry out the exercises discussed below is addressed in first-year calculus, and anyone who knows derivatives up to integration by parts can take on this work. Finally, this episode can also provide an historical flavor to an introductory physics course in mechanics, especially if the consideration of units (from the 18th century to SI) is included. Therefore, E579 (or its English translation) can be used as a basis for a special assignment or project in such courses.

Below is a suggested sequence of self-contained exercises that solve a differential equation for the velocity of the balloon and calculate its maximum height.

**Step 1.** Consider the differential equation developed by Euler, with buoyancy ratio \(\lambda = 5\) and scale height \(k = 24,\!000\) pieds (presented here in modern notation):

\begin{eqnarray}

2v \frac{dv}{dx} + \left(\frac{1}{16} \cdot v^{2}e^{-x/k}\right) = 2g\left(5 e^{-x/k}-1\right),

\end{eqnarray}

where \(v = v(x)\) is the vertical velocity of the balloon as a function of its altitude \(x\), and \(g\) denotes the acceleration due to gravity.

*Task*: Show that this equation is a first-order, linear differential equation in the variable \(y = v^2\).

**Step 2.** Given the air pressure function \(f(x) = e^{-x/k}\) and its approximation \(g(x) = 1-\frac{x}{k}\), we can measure the relative error of the approximation via

\[

\text{RE} \;=\; \frac{g(x) - f(x)}{f(x)}.

\]

*Tasks*: Calculate the relative error at 1000 feet and at 10,000 feet. Based on your results, provide a range of altitudes for which you would consider \(g(x)\) to be a reasonable approximation for air pressure.

**Step 3.** Now consider the differential equation obtained after approximating the air pressure:

\[

2v \frac{dv}{dx} + \frac{1}{16}v^2 \;=\; 4g \left(4-\frac{5x}{k}\right).

\]

*Task*: Again using the variable \(y = v^2\), solve this differential equation using the integrating factor method.

**Step 4.** Once we know the solution \(y(x)\) from Step 3, we can easily find the balloon's maximum velocity.

*Tasks*: Explain why the maximum velocity will occur when \(y'(x) = 0\). Then, using \(k = 24,\!000\) pieds, find the maximum velocity attained by the balloon.

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Michael P. Saclolo is a Professor of Mathematics at St. Edward’s University in Austin, Texas, where, along with courses in mathematics, he has also enjoyed teaching mathematically-themed first-year and honors seminars, literature, and French. His English translation of Euler's E579, the paper that inspired this article, is available on the Euler Archive. He also currently serves on the editorial board of the Euler Archive's companion journal, *Euleriana*.

Erik R. Tou is an Associate Professor of Mathematics at the University of Washington Tacoma, where he has taught since 2015. He has been involved with the Euler Archive since 2003, currently serves as its director, and is a managing editor of *Euleriana*. Some of his other research interests include the history of quadratic forms, the mathematics of juggling, and number theory over imaginary quadratic fields. He was a 2022 recipient of the MAA's Halmos-Ford Award for excellence in expository writing, for his article "A Prime Testing Algorithm from Leonhard Euler," alongside co-author Dominic Klyve.