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.
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.
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 .
Click here for the Prop. I-VII of
here for the Prop.
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
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
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
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
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.
Of the resistance of the air, and of the track described by the flight of shot and shells.
here for the Prop. I
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. 28th , 2013 latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or any part of this translation as an e-book. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. See note on the index page. Please feel free to contact me if you wish by clicking on my name here, especially if you have any relevant comments or concerns.