The Impact of Ballistics on Mathematics The work of Robins and Euler in the eighteenth-century Shawn McMurran, Cal State San Bernardino V. Frederick Rickey, USMA 16th ARL/USMA Technical Symposium, 23 October 2008 A Very Brief History of Projectile Motion Aristotles Impetus Notion Daniel Santbech, Problematum astronomicorum et geometricorum (Basel 1561) Da Vincis Arcs "Four Mortars Firing Stones into the Courtyard of a Fort" (c.1504) Tartaglia Nova scientia, 1537 The Mariners Magazine, or [Samuel] Sturmys Mathematicall and Practical Arts (1669) 1.Straight violent motion 2.mixt or crooked motion 3.Vertical descent of natural motion 150 yeas after Tartaglia Tartaglias Trajectories Projectile motion depicted in Nova Scientia (1537) by Niccol Tartaglia (c ) Galileos Parabolic Paths Folios 116v and 117r, vol. 72, Galilean manuscripts, 1608 In Art "Judith Slaying Holofernes" (c. 1620), by Artemisia Gentileschi ( ) Newton A Treatise of the System of the World (published posthumously, 1729) Benjamin Robins Born 1707 Autodictat Had an important advisor A clear writer Biography of Robins Studied on his own Met Dr. Henry Pemberton Moved to London Studied more mathematics Traveled to the continent Elected FRS, age 21 Became a teacher Frontispiece to Sprat's History of the Royal Society Authors Robins Studied Apollonius Archimedes Fermat Huygens DeWitt Sluse Gregory Barrow Newton Taylor Cotes Mediation on experiments made recently on the firing of cannon. Eulers first paper on cannon, E853, written 1727, published 1862. Robins wrote a polemic against Johann Bernoulli, 1728 A polemic against Berkeley, 1735 Robins wrote in defense of Newton A polemic against Euler, 1739 Too algebraic Uses infinitesimals Called to a Public Employment A Very Honorable Post Sir Robert Walpole was prime minister, 1721 1742 He was reluctant to attack Spain Robins wrote three anonymous pamphlets in favor of war And became secretary of a secret tribunal An interlude: From Teacher to Professor ? Robins hoped to be the first professor of mathematics at Woolwich Planned a course on fortifications and gunnery Walpole displayed his displeasure with Robinss previous attacks Mr. Derham became the first professor of mathematics at Woolwich, served Mathematics at Woolwich, 1741 That the second Master shall teach the Science of Arithmetic, together with the principles of Algebra and the Elements of Geometry, under the direction of the Chief Master. That the chief Master shall further instruct the hearers in Trigonometry and the Elements of the Conick Sections. To which he shall add the Principles of Practical Geometry and Mechanics, applied to raising and transporting great Burthens; With the Knowledge of Mensuration, and Levelling, and its Application to the bringing of water and the draining of Morasses; And lastly, shall teach Fortification in all its parts. But no calculus Preface 55 pages Ch I: Internal ballistics 65 pages Ch2: External ballistics 30 pages Total: 150 pp Published 1742 Euler 1745 Frederick the Great asks about the best book on gunnery Euler magnanimously recognizes Robins Euler starts researching Robinss results Euler adds annotations 2400 English translation of Euler, 1777 From Eulers Preface Some are of the opinion that fluxions are applicable only in such subtle speculations as can be of no practical use... But what has been just now said of artillery is sufficient to remove this prejudice. English translation of Euler, 1777 More from Eulers Preface It may be affirmed, that things which depend on mathematics cannot be explained in all their circumstances without the help of fluxions, and even this sublime part of mathematics has met with difficulties which it has not fully mastered. Postulates for motion of a projectile in a vacuum Postulate 2: If the Parabola, in which the Body moves, be terminated on a horizontal Plain, then the Vertex of the Parabola will be equally distant from its two Extremities. Postulate 4: If a Body be projected in different Angles, but with the same Velocity, then its greatest horizontal Range will be, when it is projected in an Angle of 45 with the horizon. Although these properties are demonstrated in many books, we shall here investigate them from the first principles of motion, partly to give clear insight into this matter; partly, and chiefly, the better to enable us to determine the real track described in the air. Euler, p. 271 Remark III: Shot in a vacuum Historical Questions Did anyone, before Euler, give an analytic (calculus based) derivation of the equations of projectile motion? When was this derivation cleaned up and presented in textbooks? Robins gives experimental evidence to confute the 7 postulates posed in Proposition V. For example, according to postulate 5 in Prop V: A musket ball of an inch in diameter that has an initial velocity of 1700 feet per second at an angle of 45 should have a horizontal range of about 17 miles. Actual range: Less than half a mile Eulers Remarks One Derives equations of motion for a shot in a horizontal line (7 pages) Two Derives equations of motion for a vertical shot (10 pages) Three Derive equations of motion for a shot made under an oblique angle with the horizon and compare his results with the conclusions of Robinss experiments (9 pages) Remark II: Vertical Shot Time of ascent: Time of descent: Where a is given by: To illustrate the accuracy of his formula, Euler chooses an example given by Daniel Bernoulli in the Petersburg Comentarii: Flight time reported by Bernoulli: 34 seconds Flight time predicted by Eulers formulae: Ascent: seconds Descent: seconds Total flight time: seconds! Mathematics at Woolwich, The Elements of Euclid 2.Trigonometry applied to Fortification, and the Mensuration of Superficies and Solids 3.Conic Sections. 4.Mechanics applied to the raising and transporting heavy bodies, together with the use of the lever pulley, wheel, wedge and screw, &c. 5.The Laws of Motion and Resistance, Projectiles, and Fluxions. Now some calculus! The Impact of Ballistics on Mathematics: Calculus was taught in artillery schools Piedmont-Savoy in 1750s Royal Artillery and Military Academy, Turin Prussian Artillery Corps French regimental school at Auxonne Australian Artillery Academy Ecole Polytechnique, 1794 West Point, 1807, 1813, 1823 to date. Bonaparte read Robins / Euler in French. Bonaparte rightly said that many of the decisions faced by the commander-in-chief resemble mathematical problems worthy of the gifts of a Newton or an Euler. Carl von Clausewitz, Vom Kriege, 1832 I have recently received two treatises that you kindly sent me, one on cannons and the force of powder, the other containing the theory of the motion of planets and comets. For this double gift I am deeply grateful. I have now nearly finished reading the first of these books, but I have relied completely on the correctness of your computations and haven't verified them myself, for many of them seem too complicated to me. Johann Bernoulli to Euler, 23 September 1745