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

A Very Brief History of Projectile Motion

Aristotle’s “Impetus” Notion

Daniel Santbech, Problematum astronomicorum et geometricorum (Basel 1561)

Da Vinci’s Arcs

"Four Mortars Firing Stones into the Courtyard of a Fort" (c.1504)

Tartaglia

Nova scientia, 1537

The Mariners Magazine, or [Samuel] Sturmy’s Mathematicall and Practical Arts (1669)

1. Straight violent motion

2. “mixt or crooked motion”

3. Vertical descent of natural motion

• 150 yeas after Tartaglia

Tartaglia’s Trajectories

Projectile motion depicted in Nova Scientia

(1537)

by Niccolò Tartaglia (c.1500-1557)

Galileo’s Parabolic Paths

Folios 116v and 117r, vol. 72, Galilean manuscripts, 1608

In Art

"Judith Slaying Holofernes" (c. 1620), by Artemisia Gentileschi (1593-1652)

Benjamin Robins 1707 - 1751

• 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.”

Euler’s first paper on cannon, E853, written 1727,published 1862.

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 Robins’s previous attacks

• Mr. Derham became the first professor of mathematics at Woolwich, served 1741 -1743.

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 Robins’s results

• Euler adds annotations

2400

English translation of Euler, 1777

From Euler’s 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 Euler’s 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 45º 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

E xpress the velocity of the body a t E by b ,and resolve the motion into components:

dx

dt b sin

dy

dt b cos ,

provided the sine of 90o is 1 .

"W elche Vergle ichung integrirt w ird" (and taking the fluent) and, he obta ins:

x b sin t.

N ow gravity ac ts vertica lly and since veloc ity in this direc tion is dy/dt, the space descended to genera te this ve locity is dy dt2 . N ow if d t is constant, w e have

d

dtdy2

dt2

2dy ddy

dt2.

The R H S is to dy as the vertica l force on the body is to its w eight. B ut this force is the w eight, and gravity decreases ve locity , so

2dy ddy

dt2 dy 1 1,

or d2y dt2 1 2. Hmmm? So 2 dy/dt = c - t.

N ow dy/dt is the vertica l ve loc ity a t the end of the time and a t t = 0

it is b sin , so

c 2 b sin

and so

dy dt sin b t dt 2Integra ting w e have

y sin b t t2 4.This does not look quite right, but since E uler has g = 1/2, w e obta in

y sin b t g t2 2.Thus w e have the equations for projec tile motion 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 ¾ of an inch in diameter that has

an initial velocity of 1700 feet per second at an angle of 45º should have a horizontal range of about 17 miles.

Actual range: Less than half a mile

Euler’s “Remarks”OneDerives equations of motion for a shot in a horizontal line

(7 pages)

TwoDerives equations of motion for a vertical shot

(10 pages)

ThreeDerive equations of motion for a shot made under an oblique angle with the horizon and compare his results with the conclusions of Robins’s 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 Euler’s formulae: Ascent: 13.75 seconds Descent: 20.11 seconds

Total flight time: 33.87 seconds!

Mathematics at Woolwich, 1772

1. The Elements of Euclid2. Trigonometry applied to Fortification, and the

Mensuration of Superficies and Solids3. 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

Newton

A Treatise of the System of the World (published posthumously, 1729)

Robins wrote a polemic against Johann Bernoulli, 1728

A polemic against Berkeley, 1735• Robins wrote in

defense of Newton