The Impact of Ballistics on Mathematics

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The Impact of Ballistics on Mathematics

• Shawn McMurran
• V. Frederick Rickey
• Patrick J. Sullivan

A very preliminary report
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Benjamin Robins 1707 - 1751

• Born 1707
• Autodictat
• Had an important advisor
• A clear writer

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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
http//www.princeton.edu/his291/Sprat.html
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Authors Robins Studied

• Apollonius
• Archimedes
• Fermat
• Huygens
• DeWitt
• Sluse

• Gregory
• Barrow
• Newton
• Taylor
• Cotes

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Mediation on experiments made recently on the
firing of cannon. Eulers first paper on
cannon, E853, written 1727, published 1862.
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• Robins wrote a polemic against Johann Bernoulli,
  1728

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A polemic against Berkeley, 1735

• Robins wrote in defense of Newton

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A polemic against Euler, 1739
Too algebraic Uses infinitesimals
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Called to a Public Employment A Very Honorable
Post

• Sir Robert Walpole was prime minister, 1721
  1742
• He was reluctant to attack Spain

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An interlude

• Robins wrote three anonymous pamphlets in favor
  of war
• And became secretary of a secret tribunal

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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 1741 -1743.

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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

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• 1742
• Preface
• 55 pages
• Ch I Internal ballistics
• 65 pages
• Ch2 External ballistics
• 30 pages
• Total 150 pp.

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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
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• In 1749 Robins becameEngineer General for the
  East India Company
• Traveled to India
• Contracted a fever in the summer of 1750
• On 29 July 1751, Robins died with his pen in his
  hand while drawing up a report for the board of
  directors.

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• Euler returns to gunnery in E217,
• presented 1752,
• published 1755
• Translated in 1777

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1761 Posthumous Reprint

• Reprint of New Principles of Gunnery
• 13 other papers on ballistics
• 8 papers on other subjects in volume 2

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English translation by 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. . .

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Eulers English translation, 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.

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Uncharacteristic comments from Euler

• Robins is unacquainted with several books on the
  theory of artillery . . .
• Huygens
• Keil
• Hermann
• Taylor
• Daniel Bernoulli
• De la Hire
• Johann Bernoulli
• Papin
• Bachus
• Or else he wants to exult the merit of his own
  discoveries.

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PROP. VI
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Translations of Eulers Observations upon the
new principles of gunnery translation by Hugh
Brown, p. 276 to p. 303 28 pages
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Editions of New Principles of Gunnery

• 1745 to 1777 is a triple translation
• German to English
• Differentials to Fluxions
• Leibnizian to Newtonian notation

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Mathematics at Woolwich, 1772

• The Elements of Euclid
• Trigonometry applied to Fortification, and the
  Mensuration of Superficies and Solids
• Conic Sections.
• Mechanics applied to the raising and transporting
  heavy bodies, together with the use of the lever
  pulley, wheel, wedge and screw, c.
• The Laws of Motion and Resistance, Projectiles,
  and Fluxions.
• Now some calculus!

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• 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

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• Question
• What was the impact of ballistics on mathematics?
• Answer
• Calculus makes
• its debut intothe curricula of engineers and
  artillerists.

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New Principles of Gunnery by Benjamin Robins, p.1
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New Principles of Gunnery by Benjamin Robins,
p.66
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A Very Brief History of Projectile Motion

• Aristotles Impetus Notion

Daniel Santbech, Problematum astronomicorum et
geometricorum sectiones septem (Basel 1561)
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• Da Vincis Arcs

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

• Nova scientia, 1537

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Tartaglias Trajectories
Projectile motion depicted in Nova Scientia
(1537) by Niccolò Tartaglia (c.1500-1557)
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Galileos Parabolic Paths
Folios 116v and 117r, vol. 72, Galilean
manuscripts, 1608
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In Art
"Judith Slaying Holofernes" (c. 1620), by
Artemisia Gentileschi (1593-1652)
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Newton
A Treatise of the System of the World (published
posthumously, 1729)
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New Principles of Gunnery by Benjamin Robins,
p.66
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• In Eulers remarks he discusses the ideas of
  fluid mechanics which lead to the generally
  accepted law of resistance
• if the same body move in the same fluid with
  different degrees of velocity, the resistance
  will be proportional to the square of the
  velocity.

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• He goes on to corroborate Robinss claim
• since the fluid becomes denser before the body,
  as its celerity increases, it is sufficiently
  evident that the law of resistance in very swift
  motions is greater than that which is commonly
  received, agreeable to Mr. Robinss assertion.

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• Euler
• The resistance of the air to a plane with area c2
  moving with some velocity perpendicular to the
  plane will be measured by the weight of a column
  of air whose base is c2 and whose height is v
  where v is the height the body would freely fall
  in order to reach its velocity.
• Velocity is equal to vv.

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Prop V

• Robinss Scholium to Prop V
• So far Robins has demonstrated the existence of
  air resistance, and proposed a measurement for it
• Intends to examine the trajectory of a body in
  air and show how it deviates in every
  Circumstance from what it ought to be on the
  generally received Principles.
• Poses 7 postulates on the motion of a projectile
  in a medium with no, or very little, resistance.

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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.

Parabola graph fromhttp//www.mcasco.com/p1intro.
html
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• If any one postulate fails for a projectile, then
  that projectile must deviate from a parabolic
  path
• Robins intends to demonstrate by experiment than
  none of the postulates hold for a projectile in
  air.

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Eulers Annotations

• Analytically derives the equations of motion from
  the fundamental principles of motion in a vacuum
• Confirms each of the 7 postulates for the
  trajectory of a projectile in a vacuum

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Prop VI

• Robins gives experimental evidence to confute
  the 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 according to the fifth postulate.
• Actual range

Less than half a mile
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• Perhaps a heavier shot whose resistance is much
  less in proportion to its weight may coincide
  with the hypothesis?
• In a parabolic path, an iron ball of 24 lb
  weight with an initial velocity of 1650 ft per
  second shot at an angle of 45º should have a
  horizontal range of about 16 miles.
• Actual range

Less than 3 miles
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• What about projectiles that move slowly enough
  to have their path traced by the eye?
• Curve is shorter and less inclined to the
  horizon than that in which they ascended
• Confutes Postulates 1 3
• Vertex of their flight is much closer to the
  place they fall on the ground than the place from
  which they were discharged
• Confutes Postulate 2

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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
• Attempts to 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)

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Remark I Horizontal Shot

• Defining the variables and parameters
• b is the height from which the body must fall
  to acquire its initial velocity
• c the diameter of the ball
• n the ratio of the density of the ball to the
  density of air
• t the time it takes for the shot to advance to
  M or P
• x EP
• y PM
• vv the velocity of the ball at M

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Remark II Vertical Shot

• Time of ascent
• Time of descent
• Where a is given by

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• To illustrate the accuracy of his
• formula, Euler chooses an example
• given by Daniel Bernoulli in the
• Petersburg Commentaries
• Flight time reported by Bernoulli
• 34 seconds
• Flight time predicted by Eulers formulae
• Ascent 13.75 seconds
• Descent 20.11 seconds
• Total flight time

33.87 seconds!
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Remark III Shot at an angle
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Time's Up...