The Science of Ballistics: Mathematics Serving the Dark Side

1. The Science of Ballistics:Mathematics Serving the Dark Side William W. (Bill) Hackborn University of Alberta, Augustana Campus CSHPM/SCHPM Annual Meeting

2. Ballistics and its Context • Ballistics (coined by Mersenne, 1644) is physical science, technology, and a tool of war [Hall, 1952]. • Science consists of interior ballistics (inside the barrel) and exterior ballistics (after leaving the barrel). • Interior ballistics involves chemistry and physics, the thermodynamics of combustion and an expanding gas. Exterior ballistics involves the physics of a projectile moving through a resisting medium. • Tension between science, technology, and gunnery. • Affected by interrelations among scientists, engineers, industry, the military, and the state [Hall, 1952]. CSHPM/SCHPM Annual Meeting

3. Niccolò Fontana (Tartaglia) • Mathematical fame from priority dispute with G. Cardano over cubic equation (1547-48). • The New Science (1537) deals with ballistics. • Designed gunner’s quadrant. • Claimed maximum range at 45º. • Aristotelian and medieval baggage (violent and natural motion, impetus). • Had qualms about improving “such a damnable exercise”. CSHPM/SCHPM Annual Meeting

4. Galileo • Did experiments on motion, culminating in law of falling bodies (in a vacuum) and parabolic path of a projectile (ca. 1609). Published in Discourses on Two New Sciences (1638). • Professor in Pisa and Venice. Became “mathematician and philosopher” to Cosimo de Medici in 1611. • Recognized role of air resistance in causing “deformation in the [parabolic] path of a projectile”, but … • Thought parabolic theory still valid for low-velocity mortar ballistics, and included range tables in Discourses. CSHPM/SCHPM Annual Meeting

5. Toricelli • Galileo’s “last and favourite pupil” [Hall, 1952]. • Clarified Galileo’s results in Geometrical Works (1644). • Expressed range as r = R sin 2Φ, where R is maximum range; designed related instrument. • Dealt with cases where target is above/below gun and where gun is mounted on a fortification or carriage. • Corresponded with G. B. Renieri (1647) on unexpected point-blank vs. maximum range, etc. [Segre, 1983].  conflict of theory vs. practice CSHPM/SCHPM Annual Meeting

6. Huygens • Used period of a pendulum to determine gravitational acceleration, g = 981 cm/s2(1664). • Experiments on motion in a resisting medium (1669): • jet of water impinging on one side of a balance scale • block of wood pulled by weighted cord through water • air screens on two wheeled carts, one pulled at twice the speed • Concluded that resisting force at speed V is given by FR= kV2, analogous to Galileo’s law of falling bodies. • Abandoned attempt to determine trajectory of projectile subject to this square law of resistance. [Hall, 1952] • Found trajectory of projectile moving in a medium whose resistance varies as projectile’s velocity (as did Newton). CSHPM/SCHPM Annual Meeting

7. Newton • Principia (1687) has 40 propositions on motion in resisting mediums, investigated experimentally and mathematically. • Concluded that resistance associated with fluid density is FR= kV2, but resistance may have other components too. • Found projectile trajectory when resistance varies as the projectile’s speed: FR /m= f (V) = kV. • Partially analyzed trajectory whenf (V) = kV2. [Hall, 1952] CSHPM/SCHPM Annual Meeting

8. Johann Bernoulli • Solved ballistics problem for f (V)= kVn in response to a challenge from Oxford astronomer John Keill (1719) [Hall, 1952]. • Formulation of the problem: • Bernoulli’s 1721 solution [Routh, 1898]: Letting p = tan θ, where θ is the inclination angle, yields CSHPM/SCHPM Annual Meeting

9. How Significant is Air Resistance? • Consider a shot-put, terminal velocity 145 m/s [Long & Weiss, 1999], projected at 170 m/s at launch angle 45º. • Q denotes Quadratic Drag, i.e. f (V)= kV2. • The small inclination approximation [Hackborn, 2005] is CSHPM/SCHPM Annual Meeting

10. The Ballistics Revolution • Benjamin Robins wrote New Principles of Gunnery (1642). • Invented ballistics pendulum for measuring musket ball velocities. [Steele, 1994] • Did foundational work in interior ballistics. • Discovered Robins effect and sound barrier. • Euler translated and added commentary to New Principles, at request of Frederick the Great (1745). • Euler analyzed projectile trajectory subject to the square law of resistance, calculated range tables for one family (1753). • von Graevenitz published more extensive tables (1764); still sometimes used in World War II [McShane et al, 1953]. CSHPM/SCHPM Annual Meeting

11. Late 19th Century to World War I • Air resistance per unit mass described by where H(y) = e-.0003399y, air density ratio at height y feet, G(V) = kVn-1, Gâvre drag function, C = m/λd2, the ballistics coefficient, λ = form factor specific to projectile shape. • Gâvre function (named after French commission) found experimentally. Mayevski’s version (1883) [Bliss, 1944]: CSHPM/SCHPM Annual Meeting

12. Late 19th Century to World War I (continued) • The method of small arcs often used for trajectories. • F. Siacci, at Turin Military Academy, developed an approximate method for low trajectories with small inclinations, less than about 20º (ca. 1880) [Bliss, 1944]. • Siacci’s method adapted for use in U.S. by Col. J. Ingalls, resulting in Artillery Circular M (1893, 1918), still sometimes used in World War II [McShane et al, 1953]. • Siacci’s method accurate to O(Φ4), launch angle Φ. • Littlewood, 2nd Lt. in RGA, developed anti-aircraft method. Improved Siacci’s method to O(Φ6) and high trajectories, accurate to 20 feet in 60000 for Φ = 30 º [Littlewood, 1972]. CSHPM/SCHPM Annual Meeting

13. Roles of Governments and the Military • Extensive testing was done (e.g. Woolwich, Aberdeen). • Governments in England, Prussia, and France soon included work of Robins, Euler, etc. in military and university curricula (e.g. École Polytechnique). • Napoléon, a young artillery lieutenant, wrote a 12-page summary of Robins’ and Euler’s research in 1788. • Ballistics tables/tools used on battlefields [Steele, 1994]. • O. Veblen took command of office of experimental ballistics at new ($73 million) Aberdeen Proving Ground (Jan. 1918). • N. Wiener worked as a computer at Aberdeen, and later observed that the “the overwhelming majority of significant American mathematicians … had gone through the discipline of the Proving Ground” [Grier, 2001]. CSHPM/SCHPM Annual Meeting

14. Other Social Issues • The (mis)use of mathematical and human potential: • Time lost, opportunities missed, e.g. Ramanujan. • Time, talent wasted on “such a damnable exercise”. • ICBMs, ABMs, and SDI: • Government grants in the mathematical sciences. • Resistance to “Star Wars” in the Reagan years. • When Computers Were Human [Grier, 2005]: • Women in the mathematical work force. • Women in university mathematics and related professions. • ENIAC, silicon chips, and computing technology. CSHPM/SCHPM Annual Meeting