The Impact of Ballistics on Mathematics



On 23 August 1774, the economist and politician Anne-Robert-Jacques Turgot (1727-1781) wrote to Louis XVI:

The famous Leonhard Euler, one of the greatest mathematicians of Europe, has written two works which could be very useful to the schools of the Navy and the Artillery. One is a Treatise on the Construction and Manoeuver of Vessels; the other is a commentary on the principles of artillery of Robins . . . I propose that your Majesty order these to be printed; . . . [Quoted, in English, by Clifford Truesdell, An Idiot's Fugitive Essays on Science (1984), p. 337]

Turgot had reason to know about these books, for at the time he was Ministre de la Marine. He was aware of the impropriety of translating a work without the author's permission so urged the king to compensate Euler with 5,000 francs from the secret accounts of the Navy.


Works by Benjamin Robins


1726:    "A Demonstration of the 11th Proposition of Sir Isaac Newton's Treatise of Quadratures. By Mr. Benjamin Robins Benjamin Robins, Isaac Newton," Philosophical Transactions (1683-1775), Vol. 34, 1726 - 1727 (1726 - 1727), pp. 230-236.


1735:    A discourse concerning the nature and certainty of Sir Isaac Newton’s methods of fluxions, and of prime and ultimate ratios. By Benjamin Robins. London, Printed for W. Innys and R. Manby, 1735. ii, 78 p. 20 cm. Columbia: SMITH 517 1735 R55


1739:    Remarks on Mr. Euler’s Treatise of motion, Dr. Smith’s Compleat system of opticks, and Dr. Jurin’s Essay upon distinct and indistinct vision. By Benjamin Robins, F.R.S.  London : printed for J. Nourse, 1739. vi,112p. : ill. Columbia's Butler Microforms, 401B: Eighteenth century ; reel 9682, no. 06. F a4254 reel 9682, no.06.

1740:    "A full confutation of Dr. Jurin's Reply to the remarks on his Essay upon distinct and indistinct vision. By Benjamin Robins, F.R.S.," printed for J. Nourse, 1740.


1742:    New principles of gunnery: containing the determination of the force of gun-powder, and an investigation of the difference in the resisting power of the air to swift and slow motions.  London, J. Nourse, 1742. Copy at NYPL.

1743: Paper in Phil Trans

1747:    A proposal for increasing the strength of the British navy, by changing all the guns, from the eighteen pounders downwards, into others of equal weight but of a greater bore. By Benjamin Robins, ... London : printed for J. Nourse, 1747. vii,[1],24p. Columbia's Butler Microforms, 401B: Eighteenth century ; reel 7626, no. 03. F a4254 reel 7626, no. 03.

1749:    "Observations on the Height to Which Rockets Ascend; By Mr. Benjamin Robins F. R. S." Philosophical Transactions (1683-1775), Vol. 46, 1749 - 1750 (1749 - 1750), pp. 131-133.

1761:    Mathematical tracts of the late Benjamin Robins Published by James Wilson, M.D. London, J. Nourse, 1761. 2 v. 3 pl. (1 fold.) diagrs. 21 cm. USMA Special Collections:  WPT 510.4 R553. Not in electronic catalog.

1783:    Nouveaux principes d’artillerie / de M. Benjamin Robins ; commentés par M. Léonard Euler ; traduits de l’allemand, avec des notes, par M. Lombard ... , A Dijon : Chez L.N. Frantin ... ; Et se trouve a Paris ; Chez Jombert fils aîní ..., 1783. Off-site at NYPL.

1805:    New principles of gunnery: containing the determination of the force of gun-powder, and an investigation of the difference in the resisting power of the air to swift and slow motions, with several other tracts on the improvement of practical gunnery / by Benjamin Robins... with an account of his life and writings by James Wilson. London : F. Wingrave, 1805. New ed. / corrected, and enlarged with the addition of several notes, by Charles Hutton. xlvi, [2], 341 p., [2] folded leaves of plates : ill. ; 24 cm. Signatures: a-c8A-X8Y4. USMA Special Collections: UF144 .R6 1805.

The following is probably an error:

Benjamin Robins, “New Principles in Gunnery”, London (1807). The only copy available in the United States is the 2nd Edition (1842) at the United States Military Academy, West Point, NY.



Ackroyd, J. A. D., "Sir George Cayley, the father of aeronautics. Part 1. The invention of the aeroplane," Notes and Records of the Royal Society, Volume 56, Number 2 / May 22, 2002, pp.167 - 181.

This paper, in two parts, seeks to describe and assess Cayley's technical development of his invention, the fixed-wing aeroplane. Particular attention is paid to such scientific results as were known to Cayley at that time. Part 1 deals with the invention of the aeroplane itself, this being prefaced by a biographical sketch outlining his wide-ranging interests, both scientific and humanitarian; part 2 will show how Cayley was not only able to develop his invention but through this was also able to demonstrate many of the basic principles of fixed-wing flight. The paper is based on the author's Cayley Lecture to the Royal Aeronautical Society in April 2000.

Allibone, T. E., "Edmond Halley and the Clubs of the Royal Society," Notes and Records of the Royal Society of London, Vol. 28, No. 2 (Apr., 1974), pp. 195-205.

Barkla, H. M., "Benjamin Robins and the resistance of air," Annals of Science, vol. 30, no. 1, March 1973, pp. 107 - 122.

Bektas, M. Yakup  and Crosland, Maurice, "The Copley Medal: The Establishment of a Reward System in the Royal Society, 1731-1839," Notes and Records of the Royal Society of London, Vol. 46, No. 1 (Jan., 1992), pp. 43-76.

Burckhardt, J. J., "Leonhard Euler, 1707-1783, Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 261-273.

Cajori, Florian, "Frederick the Great on Mathematics and Mathematicians." American Mathematical Monthly, Vol. 34, No. 3 (Mar., 1927), pp. 122-130.

Grattan-Guinness, Ivor, "The interest of G. H. Hardy, F.R.S., in the philosophy and the history of mathematics," Notes and Records of the Royal Society, Volume 55, Number 3 / September 22, 2001, 411 - 424.

Matthews, Michael R.; Gauld, Colin, and Stinner, Arthur, "The Pendulum: Its Place in Science, Culture and Pedagogy," Science & Education, Volume 13, Numbers 4-5 / July, 2004, pp. 261-277. Journal not at USMA.

Abstract  The study and utilisation of pendulum motion has had immense scientific, cultural, horological, philosophical, and educational impact. The International Pendulum Project (IPP) is a collaborative research effort examining this impact, and demonstrating how historical studies of pendulum motion can assist teachers to improve science education by developing enriched curricular material, and by showing connections between pendulum studies and other parts of the school programme especially mathematics, social studies and music. The Project involves about forty researchers in sixteen countries plus a large number of participating school teachers.The pendulum is a universal topic in university mechanics courses, high school science subjects, and elementary school programmes, thus an enriched approach to its study can result in deepened science literacy across the whole educational spectrum. Such literacy will be manifest in a better appreciation of the part played by science in the development of society and culture.

Radelet-de Grave, P., "La diatribe du Docteur Akakia, Médecin du pape,"   I think there is a very interesting story to be followed up here.

Saintsbury, George, "François Marie Arouet De Voltaire (1694-1778)," Encyclopedia  Britannica, 11th edition, vol 28, pp. 199-205. On line at . Very nice biography of Voltaire. Good connections with Emilie du Chatelet, Frederick, and Voltaire. Euler and Robins not mentioned.

Steele, Brett D., "Muskets and Pendulums: Benjamin Robins, Leonhard Euler, and the Ballistics Revolution," Technology and Culture, Vol. 35, No. 2 (Apr., 1994), pp. 348-382.

Steel, Brett D., "Military "Progress" and Newtonian Science in the age of Enlightenment," pp. 361-390 in The heirs of Archimedes : science and the art of war through the Age of Enlightenment, edited by Brett D. Steele and Tamera Dorland, Cambridge, Mass. : MIT Press, c2005.

Steel, Brett D., my article on Leonhard Euler's interior ballistics analysis and its influence on both military and engineering thinking in the late eighteenth century was published in  Gunpowder, Explosives, and the State: A Technological History, edited by Brenda Buchanan, Aldershot, England; Burlington, VT : Ashgate, c2006.

Swerdlow, N. M., "Montucla's Legacy: The History of the Exact Sciences," Journal of the History of Ideas, Vol. 54, No. 2 (Apr., 1993), pp. 299-328. Available in JSTOR.

Thomas, J. M., "Sir Benjamin Thompson, Count Rumford and the Royal Institution," Notes and Records of the Royal Society, Volume 53, Number 1 / January 22, 1999, 11 - 25.

On 19 September 1798, there arrived in London from Munich a 45-year-old, tall, blueeyed, handsome, American-born, opportunistic man of action. A former soldier and statesman, he was, by all accounts, ruthless and arrogant, callously cunning and devious, an unprincipled spy and a calculating womanizer; but he was also a philanthropist, a brilliantly effective social reformer, an ingenious inventor and an exceptionally innovative scientist. His name was Benjamin Thompson, better known as Count von Rumford of the Holy Roman Empire. He would found the Royal Institution.

Truesdell, C., review of “Exterior Ballistics” by E. J. McShane, J. L. Kelley, and F. V. Reno, Scripta Mathematica 20 (1954), 172-174. Not surprisingly, Truesdell is caustic (p. 56 "is the ugliest page I have ever seen). He is surprised that these distinguished authors did not do better. Yet he praises the 50 page historical appendix. He praises the book for this alone.  USMA: UF820 .M248.  R. A. Rankin has a long review in Mathematical Reviews (MR0060355) that predicts, correctly, that the book will become a classic.

Truesdell, C. A., "Published Works of Clifford Ambrose Truesdell III," Journal of Elasticity, Volume 72, Numbers 1-3 / July, 2003, 1-27.  Journal not at USMA.

Voltaire, François Marie Arouet de, "Traité de paix conclu entre M. le Président et M. le Professeur," dated 1 January 1753. On line copy at . Search for "Euler". Some of the clauses here are quoted from Robins's critique of Euler. Need to look into this. Would be nice to track down an English translation.

Wicher, Enos, "Ballistics pendulum," American Journal of Physics, vol 45, no. 7, 681-682, July 1977.


The two notebooks of Benjamin Robins


Johnson, W., "The Woolwich professors of mathematics (1741-1900)," Journal of mechanical working technology, 18(2), 145, 1989. Journal available at SIBL branch of NYPL.

Johnson, W., "Charles Hutton, 1737-1823: The prototypical Woolwich Professor of Mathematics," Journal of mechanical working technology, 18(2), 195, 1989. Journal available at SIBL branch of NYPL.

Johnson, W., "The origin of the ballistic pendulum : the claims of Jacques Cassini (1677-1756) and Benjamin Robins (1707-1751)," International journal of mechanical sciences, 1990, vol. 32, no4, pp. 345-374 (22 ref.)

Johnson W., "In search of the end of the life, in India, of Benjamin Robins, F.R.S," International journal of impact engineering, 1991, vol. 11, no4, pp. 547-571.

Johnson W., "Benjamin Robins, F.R.S. (1707-1751): New Details of His Life," Notes and Records of the Royal Society of London, Vol. 46, No. 2 (Jul., 1992), pp. 235-252.

Johnson W., "Benjamin robins (18TH century founder of scientific ballistics) : some European dimensions and past and future perceptions," International journal of impact engineering, 1992, vol. 12, no2, pp. 293-323.

Johnson, W., "Encounters between Robins, and Euler and the Bernoullis ; artillery and related subjects," International journal of mechanical sciences, 1992, vol. 34, no8, pp. 651-679.

Johnson, W., "Benjamin Robins during 1739-1742: 'Called to a Publick Employment... A Very Honorable Post," Notes and Records of the Royal Society of London, Vol. 48, No. 1 (Jan., 1994), pp. 31-42.

Johnson W., "Congreve's Details of the rocket system and the artillery museum in the Rotunda at Woolwich, London," International journal of impact engineering, Vol. 15, no. 6, pp. 833-842. 1994

William Congreve, Jnr (1772-1828), in a 40-page booklet, describes the various applications of his rocket system. This is partially and briefly summarized below, mostly using some of Congreve's figures. The contents of the booklet, accompanied by an essay by Captain Boxer, also appear in a recent Canadian facsimile reproduction, a short summary being included in the Notes. The second portion of this paper is concerned with the artillery museum in the Rotunda, Woolwich, established about 170 years ago. We mention the inception of the new museum and recommend it for visiting by students of impact engineering. Our first two Notes suggest or show that the renowned 18th century British pair, Benjamin Robins and Charles Hutton (founders of early ballistic theory), were well acquainted with the widely known Henry Pemberton and Sir William Congreve Snr respectively.

Johnson W., "Edward Gibbon and George Grote: A Bicentenary in Common," Notes and Records of the Royal Society of London, Vol. 48, No. 2 (Jul., 1994), pp. 221-22.

Johnson W., "A note on James Wilson M.D (1690-1771): Biographer and editor of some of the works of Pemberton, Robins and Newton," International journal of impact engineering, Volume 21, Number 8, September 1998, pp. 707-710.

Johnson W., "Aspects of the live and works of Martin Folkes (1690-1754)," International Journal of Impact Engineering, Volume 21, Number 8, September 1998, pp. 695-705.

Johnson W., "Early bridge consultants, Benjamin Robins, FRS and Charles Hutton, FRS and mis-judged bridge …," International Journal of Mechanical Sciences, 1999

Johnson W., "Short unpublished manuscripts of Benjamin Robins," International Journal of Impact Engineering, Volume 24, Number 4, April 2000, pp. 417-428.

Johnson W., International Journal of Impact Engineering, Volume 25, Number 2, February 2001, pp. 193-210(18)



Johnson W., "Benjamin Robins, Tobias Smollett and fictional Roderick Random: expeditions to Portobello (1739)," International journal of impact engineering,


Johnson W., "The Watts Academy, 1715–1750: Robins’ probable involvements and opportunities," International journal of impact engineering,






[CITATION] All guns blazing. Collected works on Benjamin Robins and Charles Hutton, by W. Johnson
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How Galileo solved the problem of maximum projectile range without the calculus

Herman Erlichson 1998 Eur. J. Phys. 19 251-257   doi:10.1088/0143-0807/19/3/007

   PDF (83.5 KB) | References

Herman Erlichson
Department of Applied Sciences, The College of Staten Island, City University of New York, 2800 Victory Blvd, Staten Island, NY 10314, USA

Abstract. The maximum range of a projectile in a vacuum is easily derived from the range formula or the calculus, but Galileo solved this problem in the 1600s without knowledge of either. This paper decribes his method and provides the first detailed study and discussion in the literature (to this author's knowledge) of Galileo's work on the maximum range problem.

Robins-Magnus effect: A continuing saga - group of 2 »
TK Sengupta, SB Talla - Current Science, 2004 -
... only experimental device at that time and ballistic pendulum for ... In writ- ing the
book on ballistics 2 , he ... 4. Anderson, J. D., A History of Aerody- namics and ...