EULER'S

Ian Bruce

This is a work with a long
history, which I will put in place here gradually, that you can read if you are
not familiar with it already; at present the original work of Robins is
presented, from his *Principles of Gunnery*,
together with the notes by Euler in his German edition for Frederic the Great,
and the first few propositions Prof. Lombard's equivalent notes in French
dating from around the year 1780. One should note here also that the idea of a
gun as a non-reciprocating heat engine lay in the future, and Euler labors to
present his ideas from the point of view of classical mechanics; this in itself
is illuminating, and shows the trouble one has with an incomplete theory.

References :

1. *Principles
of Gunnery*. Robins (1742);

2. *Neue
Gründsatze der Artellerie. *Tr. into German and considerably enlarged by
Leonhard Euler. (1753)

2. *True Principles of Gunnery*. Tr. from the
German by Hugh Brown (1777). Hard to locate and not available as pdf.

3. *Nouveaux Principes
D'artillerie De M. Benjamin Robins*. M.Lombard (1783)

4. *Collected Works of* *Benjamin Robins & Charles Hutton. W.Johnson. Phoenix *(2001)

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

Most of the early works are
available at the *e-rara* website ; the
augmented work by Charles Hutton (~1805) should not be confused with Robins
original work.

It is worthwhile to have a look at a small
paper from Euler's early days that never got published, coming from his *Opera Postuma, Vol. 2*., E853 : Click here .

Chapter I.

Click here for the Prop. I-VII of
** **The first 7 propositions are presented and examined critically by
Euler, according to his understanding of the physics of the day. Hardly any of
the theoretical results are correct, as the elastic fluid is neither isothermal
nor adiabatic, and a lot of the initial energy released is transferred into
heating the barrel of the gun, etc. etc.

Click
here for the Prop.
VIII of *To determine* *the Velocity, which any Ball moves with at any Distance
from the Piece, it is discharged from.*

The
business of measuring the speed of a bullet from a musket is undertaken here. A
useful form of the moment of inertia formula derived from experiment is used
for the ballistic pendulum invented by Robins. Euler makes for extended
comments on various aspects of Robins' calculations, including a refined method
dealing with bullet strikes away from the centre of percussion, calculating air
resistance, etc.

Click here for the Prop. IX
of *To compare the actual Velocities with which Bullets of different Kinds
are discharged from their respective Pieces, with their Velocities computed
from the* *theory.*

Euler
advances a number of ideas concerning the shape of the barrel and especially
the powder chamber, in order to obtain more spontaneous ignition, in his
concerns over Robins' data, which he considers to be too well-fitting to the
theory presented by Robins.

Click here for the Prop. X
of * **To assign the Changes in the Force of Powder, which
arises from the different State of the Atmosphere.*

This is a short proposition,
discussing the possibilities of measuring the water content of gunpowder from
the humidity of the atmosphere.

Click here for
the Prop. XI of * To investigate the Velocity which the Flame of* *the Gunpo**wder
acquires*, *by expanding **itself, supposing
it to be fired in a given Piece of Artillery without either a Bullet **or* *any **other Body before it.*

Here Euler has added an
extensive investigation into the nature of the firing of gunpowder in eight
extended remarks : time does not allow me to enumerate these in detail at
present, but apart from the obvious error in lacking a knowledge of the nature
of chemical reactions, the treatment is quite interesting. Several notes have
been added along the way by myself.

Click here for
the Prop. XII & XIII of *
To ascertain the Manner in which the Flame **of
Powder impels a
Ball, which is laid at a considerable Distance from the Charge; *

and of* *Prop. XII is :* To
enumerate the various Kinds of Powder, and to describe the properest Methods of
examining its Goodness.*

It is hard to assess the
validity of Euler's contributions to Robins essentially experimental work ;
some parts are obviously correct, and need no further comment; the idea of
replacing weights of exhaust gas and the ball or bullet itself by equivalent
air columns is unusual, while the use of the *Vis Viva* notion in vogue at
the time lends suspicion to the accuracy of the results. What one can say for
sure, is that in modern terms, around a third of the energy available in the
gunpowder ignition can be transformed into kinetic energy of the ball at most,
following the efficiency of an internal combustion heat engine; the adding of
an air space between ball and powder resulted in complete combustion, and the
setting up of shock waves, as Euler comments on. The complete lack of
understanding on Euler's part of the nature of chemical reactions and the lack
of reasoning based on thermodynamic principles, not yet discovered, leads me to think that the whole argument is
based on a false premise, and therefore the work in the first chapter can be
viewed only as an interesting off-shoot of little relevance to the modern
person, except perhaps for Euler's mathematics, brilliant as ever.

Chapter II.

*Of the resistance of the air, and of the track described
by the flight of shot and shells.*

Click
here for the Prop. I
of * To
describe the general principles of the resistance of fluids to solid bodies **moving* *in them. *Euler presents an intensive
analysis of the factors leading to the resistance of a body moving through a
fluid medium, starting from the basics of Newton's laws of motion applied to
rather non-realistic media, and leading on to a consideration of streamline
flow, and of the effect of shock waves for very fast motion in a medium.

Click
here for the Prop. II
of Ch. II. The original title of Prop. II is :* To determine the resistance of the air to projectiles by experiments. *Euler presents an intensive analysis into the linear projection of
rapidly moving cylinders and spheres through air. Here he compares his theory
for spheres with the experimental data obtained by

Click
here for the Prop. III
of Ch. II. The original title of Prop. III is :* To assign the
different augmentations of the resisting power of the air according to the
different **velocities of the resisted body. *Euler presents his own version of the formulas
relating the air resistance to the speed of the ball, and generates finally a
formula which he believes to be useful; as he notes finally, the complete story
is yet to be told.

Click
here for the Prop. IV
of Ch. II. The original title of Prop. IV is :*To determine the velocities with which musket and cannon-shot are
discharged from their respective pieces by their usual allotment of powder. *Euler presents an extended series of remarks here,
that may be summarized as comparing his calculations with those of Robins, as
Euler has included a number of causes omitted by the former. The calculations
are extended to cannon balls, though at this stage, only the speeds of musket
balls had been actually measured by Robins for various charges. Numerous tables
are given, and the general conclusion is that after a certain point, less
charge is needed in Euler's calculations than in those of Robins.

Click
here for the Prop. V
of Ch. II. The original title of Prop. V is : *When a cannon-ball of *24 *lb*. *weight, fired with a full charge
of **powder*, *first
issues * *from the piece, the resistance of the air on its surface
amounts to more than twenty times its gravity. *Euler first investigates the
scaling process by which Robins finds the resistance of a cannon ball from that
of a musket ball found experimentally; he then establishes the well-known
formulas for the motion of a projectile without air resistance.

Click
here for the Prop. VI
of Ch. II. The original title of Prop. VI is : *The track described by the flight of shot or shells is
neither a parabola, nor nearly a parabola, unless they are projected with small
velocities.*

Here Euler sets out an extensive investigation of the trajectory of a cannon ball involving the resistance of the air according to his formula, which depends both on the velocity of the ball squared and to the fourth power; he divides this into 3 sections, dealing in turn with horizontal motion, vertical motion, and finally the more general motion in two dimensions; he ignores the important result that Robins has discovered regarding the deviation in air of a spinning ball.

Click here for the Prop. VII & VIII of Ch. II.

The
original title of Prop. VII is : *Bullets
in their flight **are* *not* *only
depressed **beneath their original direction by the action of gravity, * *but are
also frequently driven to the right of left of that direction by the **action of some
other force.*

Euler
goes to considerable effort to show that the rotation of the ball in flight has
no bearing on the motion, which flies in the face of all experimental evidence
; it is unclear why he adopted this viewpoint. The problem is of course one of
great complexity, as has been indicated in the translation.

The
original title of Prop. VIII is : *If bullets
of the same diameter and density impinge on the same
solid surface with different velocities, they will penetrate that substance to
different depths, which will be in the duplicate ratio
of those velocities nearly. And the
resistance of solid substances to the penetration of bullets is uniform. *

Here Euler adopts his now familiar 'Boyle's Law' approach to solving this problem, by replacing all the forces by equivalent columns of air, and then allowing an infinitesimal change in the pressure, in order to derive the differential equation for the motion. This is the last chapter of this work.

Ian
Bruce. Jan. 28^{th}
, 2013 latest revision. Copyright : I reserve the right to publish this
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