Mathematics goes ballistic: Benjamin Robins, Leonhard Euler, and the mathematical education of military engineers

Abstract

Efforts to understand the trajectory of cannonballs are an interesting example of the tensions between practical and theoretical knowledge. Although Galileo's 1638 parabolic trajectory was an important theoretical step forward, field gunnery practice was guided by Tartaglia's 1537 ‘mixed motion’ model until the eighteenth century. In 1742, Benjamin Robins published New principles of gunnery, and revolutionized the study of ballistics by suggesting the projectile's initial velocity—not its range—was the appropriate parameter to consider in accounting for air resistance. In 1745, Leonard Euler produced a German translation of New principles, adding his own extensive commentary. Euler's annotated translation quickly became a standard text—Napoléon Bonaparte studied ballistics from the French version—thereby influencing the education of artillery officers and, eventually, of all engineers. This paper surveys the contributions of Robins and Euler to mathematical ballistics theory, examines the influence of these developments on the education of eighteenth-century military engineers, and considers the extent to which the history of ballistics theory supports the thesis that the drive to reconcile practical knowledge with theoretical knowledge can be a critical element in shaping mathematical theory. We close with comments concerning the use of this history in today's classroom.

Notes

¹ Another of Galileo's students, Evangelista Torricelli (1608–74) attempted to make Galileo's theory more useful to gunners (Swetz 1995, 97–100).

² For a discussion of Thomas Harriot's work in this area, see Büttner 2003, 21–23, and Schemmel 2008.

³ Hall 1969b, 42, suggests this was due in part to the greater precision of Galileo's new mechanics, which allowed for tests of a greater order of accuracy than previously.

⁴ The exact date of Robins' birth is unknown. Although not well known among mathematicians, Robins has received a fair amount of attention from engineers in recent years for his work in ballistics and its accompanying contributions to aerodynamics and experimental fluid mechanics. This is due in part to a series of biographical articles published in engineering journals by William Johnson, FRS. A partial list of Johnson's articles appears in the references.

⁵ Robins also developed an instrument called a whirling arm to measure air resistance at velocities too low for the ballistics pendulum; see Johnson 1992c, 305–306.

⁶ Steele argues that Robins' resolution of this problem represents an early connection between nineteenth-century engineering thermodynamics and seventeenth-century mechanics (Steele 2005, 86).

⁷ This debate is treated at length in Alder 1997; for a summary of its history, see Steele 2005, 295, or Steele 1994, 207–212.

⁸ This effect is known today as the Magnus effect or Robins–Magnus effect; for further detail about Robins' contribution, see Johnson 1992c, 313–324, Steele 1994, 115–116, Segupta 2004.

⁹ Steele 1994 quotes Wilson to this effect on page 102, and discusses Robins' derivation of his gunnery table on pages 109–110.

¹⁰ The first published occurrence of the symbol e appears in Euler's 1736 Mechanica, the work which was the subject of Robins' 1739 critique. A partial translation is found in Smith 1959, 95–96.

¹¹ Secondary sources typically report that Euler translated the work in response to an inquiry from Frederick regarding the best available artillery text. The source of this report appears to be the Euler eulogy read at the Imperial Academy of Sciences of Saint Petersburg by Nicolaus Fuss on 23 October 1783. In 1744, Euler wrote to Frederick requesting permission to complete a translation of New principles which he had already begun. Frederick's written response to this letter is missing, but one surmises that permission was granted.

¹² All English quotations from Neue Grundsätze in this paper are taken from Brown 1777.

¹³ Although Alder agrees with Steele that Robins and Euler together ‘put ballistics on a new basis’ (Alder 1997, 104), he is also critical of certain of Steele's historiographical assumptions (Alder 1997, 91–92). Note especially Alder's argument that Steele's assumptions lead him to overestimate Robins' scientific contributions.

¹⁴ By 1772, for example, Newtonian fluxions had become part of the mathematics curriculum at the Woolwich Academy. Woolwich mathematics professor Hugh Brown published an English translation of Euler's annotated translation of Robins in 1777. An interesting feature of his translation is its use of Newtonian fluxion notation to represent Euler's analysis.

¹⁵ See Swetz 1995 for specific suggestions in this regard.

¹⁶ Hall's view concerning the motivation of theorists is a rejection of an alternative view which he describes as follows (Hall 1983, 115): ‘The common supposition has been – and perhaps among the less critical still is – that these men were naively eager to solve “useful” (or at least utilitarian) problems, not so much with the object of doing good (for even in the seventeenth century to increase the accuracy of shooting could hardly be regarded as clearly an act of kindly benevolence, and we have evidence that warlike inventions were then already regarded with horror), but rather in order to prove the general utility of the scientific approach to practical problems, and to demonstrate the particular powers of the individual writer.’