# Francis Bashforth's books

Francis Bashforth wrote a number of books some of which are only around 30 pages. Many contain information about his thinking and the difficulties he had in pursuing his research. We give some whole Prefaces, some extracts from Prefaces, or some extracts from the Introduction.

Click on a link below to go to the information about that book.

General Table for facilitating the Calculation of Earthworks, for Railways, Canals, etc with a Table of Proportionate Parts (E & F N Spon, London, 1855).

Description of a Chronograph adapted for measuring the varying velocity of a body in motion through the air and for other purposes (Bell and Daldy, London, 1866).

A mathematical treatise on the motion of Projectiles founded chiefly on the results of experiments made with the author's chronograph (Asher and Company, London, 1873)

An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops (Cambridge University Press, Cambridge, 1883) with John Couch Adams

Revised account of the experiments made with the Bashforth Chronograph, to find the resistance of the air to the motion of projectiles, with the application of the results to the calculation of trajectories according to J Bernoulli's method (Cambridge University Press, Cambridge, 1890).

Supplement to a revised account of the experiments made with the Bashforth Chronograph, to find the resistance of the air to the motion of projectiles, with the application of the results to the calculation of trajectories and a historical sketch of the progress of ballistic experiments connected with the advanced class, Woolwich, 1864-1890 (Cambridge University Press, Cambridge, 1895).

A Historical Sketch of the Experimental Determination of the Resistance of the Air to the Motion of Projectiles (Cambridge University Press, Cambridge, 1903).

Ballistic experiments from 1864 to 1880 (Cambridge University Press, Cambridge, 1907).

1. Francis BashforthGeneral Table for facilitating the Calculation of Earthworks, for Railways, Canals, etc with a Table of Proportionate Parts (E & F N Spon, London, 1855).
1.1 Preface.

It appeared to the Author that a Table for facilitating the calculation of Earthwork might be so formed as to unite the great requirements of accuracy, generality, and facility of use. The accompanying Table is applicable to all combinations of "slopes" and widths of "formation level," (even when the slopes on the two sides are different,) and requires the extraction of only one number for each prismoid. It can be used with expedition for parliamentary estimates and rough comparisons, and is also capable of showing the contents of prismoids with great accuracy, by the simple addition of a few numbers when the heights are given to the $\large\frac{1}{10}\normalsize$th or $\large\frac{1}{100}\normalsize$th of a foot. This is effected by a method of proportional parts similar to that employed in Tables of Logarithms.

Numerous, and in some cases extensive works have of late appeared on this subject, - the two best known being those of Sir J Macneill and Mr Bidder. The defect of the former of these seems to be its great extent accompanied by a want of generality, whilst that of Mr Bidder, being perfectly general, involves twice the labour of the other. The present is an attempt to unite the excellencies of the two, and to extend them to fractional parts of a foot. If the heights be measured in feet alone, the three tables will give the same result; for the formulae from which they have been derived are essentially the same.

It has been deemed proper to give a full account of the mode of formation of the Table in order that those who are so disposed may have an opportunity of examining it, and that others may derive some confidence from its publication. In every case, the nearest integer has been inserted in the Table. As each term of the formula from which the tabular numbers have been derived is of two dimensions in $a$ and $b$, the heights of the two ends of the prismoid, a ready method is obtained of extending the Table to any imaginable heights.

Examples of the application of the Table will be found at Articles (7), (10), (12), (19).

Note. $\{a, b\}$ is used to denote the content in yards of the prismoid a chain long, the heights at the two ends being $a, b$ feet; or it is the tabular number at the intersection of the columns $a, b$. Thus {58, 63} = 8952.
2. Francis  Bashforth, Description of a Chronograph adapted for measuring the varying velocity of a body in motion through the air and for other purposes (Bell and Daldy, London, 1866).
2.1. Description of a Chronograph.

The resistance of fluids to the motion of solids is a problem of great practical and theoretical interest, because sometimes this resistance becomes a source of power, whilst at other times it is a large consumer of power which would otherwise be usefully applied. Thus the action of the wind upon sails is made to drive a mill or a ship. The wind acting upon the sails tends to drive the ship, and the resistance of the water is opposed to the progress of the ship. The resistance of the water to the motion of the oar, the paddle, or the screw, enables the rowers or the steam engine to drive the vessel. A very large part of the power developed in the locomotive is employed in overcoming the resistance of the air to the motion of the train. And quite recently it has been found that the friction of the tidal wave is probably slowly diminishing the velocity of the earth's rotation about its axis. We are dependent upon the resistance of fluids for our power to cross the ocean, for without that property we should not be able to use sail or oar, paddle or screw. Still little is known with accuracy respecting the laws of the resistance of fluids. It is extremely difficult to make satisfactory experiments on account of the great disturbance produced in the surrounding fluid, and as the mathematician knows neither the nature of this disturbance, nor the amount of resistance to be accounted for in particular cases, he is not able to find ground on which to base a satisfactory theory. For a history of what has been done, it will suffice to refer to the articles, "Ballistik" and "Widerstand," in Gehler's Physikalisches Wörterbuch, 1825 and 1842.

The resistance of the air to spherical balls, moving with high velocities, has been a subject of special interest for more than a century, because it was of practical importance to the science of gunnery, and because it offered a simple yet striking instance of the great resistance which a very rare medium would offer to a solid moving in it with a high velocity. Thus Robins states that Dr Halley thought it reasonable to believe that the opposition of the air to large metal shot is scarcely discernible, although in small and light shot, he acknowledged that it ought and must be accounted for. Robins further states, that a musket-ball, fired with a charge of half its own weight of powder, would leave the gun with a velocity of 1700 f.s., and that, with an elevation of 45°, its range would be 17 miles in a vacuum ; whilst practical writers on the subject say that the range in air is short of half a mile. This resistance of the air $p$, depends upon the velocity $v$, upon the form, and upon the size of the moving body. When a body is at rest in air, the horizontal pressure tending to move it one way is just equal to the pressure tending to move it in the opposite direction. If the body be put in motion by any force, the pressure of the air tending to prevent motion is greater than it was before in that direction, and the pressure in the direction of the motion is less than it was before. The difference of the pressures of the air before and behind the body is called the resistance of the air to that particular form of solid moving with the given velocity: but as the pressure of the air in direction of motion decreases rapidly as the velocity increases, it is commonly neglected.
3. Francis  Bashforth, A mathematical treatise on the motion of Projectiles founded chiefly on the results of experiments made with the author's chronograph (Asher and Company, London, 1873)
3.1. Preface.

Early in the seventeenth century Galileo determined the path of a projectile on the supposition that the horizontal motion would continue the same as if there was no vertical motion, and that the vertical motion would continue the same ns if there was no horizontal motion. But he afterwards explained that a projectile would not actually move in the parabolic path thus found, partly because he had neglected to take account of the resistance of the air, and partly because gravity did not really act in parallel lines. Mathematicians appear to have been content with this solution until Newton gave a mathematical investigation of the path described by a projectile moving under the action of gravity and the resistance of the air supposed to vary as the velocity, upon which, however, he remarked, "Caeterum corpora resisti in ratione velocitatis Hypothesis est magis Mathematica quam Naturalis" [Whereas the body's resistance ratio is more a mathematical hypothesis than a physical one]. As Newton did not give a solution in the case when the resistance of the air was supposed to vary as the square of the velocity, which he believed to be the law of nature, John Bernoulli inferred that he was unable to solve this problem. Although Bernoulli gave a solution when the resistance of the air was supposed to vary as any power of the velocity, that solution was of no practical use in the state in which it was left by its author. Moreover, as we now know that the resistance of the air does not vary even approximately, within practical limits, according to any single power of the velocity, this solution could not, under any circumstances, have led to a correct knowledge of the laws of resistance of the air to the motion of projectiles. Numerous mathematicians, as Euler, Legendre, Lambert, Français and Poisson have attempted to find the path of a projectile when the resistance of the air was supposed to vary as the square of the velocity. These solutions did not, however, lead to any definite conclusion respecting the law and amount of the resistance of the air. The real advance made in the science of ballistics was due to Robins, who first devised means for measuring, in a satisfactory manner, the velocities of projectiles. Afterwards Hutton, improving upon what Robins had done, obtained very good results. Numerous experiments have been made since his time, the most notable of which were those executed at Metz, which formed the basis of M Didion's Traité de Balistique. But the projectiles used in those experiments were spherical shot fired from smooth-bore guns, and the results obtained did little more than confirm the previous conclusions of Hutton.

On the institution of the Advanced Class for Officers of the Royal Artillery in 1861, there was no satisfactory work on ballistics, and no experiments made with elongated shot could be found which were consistent among themselves, and therefore deserving of confidence. The Navez electro-ballistic instrument, at that time in use in this country, was one of the most unphilosophical of philosophical instruments. Feeling that the satisfactory solution of any question in gunnery depended upon the construction of a trustworthy chronograph, it therefore became my duty to recommend that a proper instrument should be procured, and that a systematic course of experiments should be undertaken to determine, in the first instance, the resistance of the air to the motion of projectiles. Although I had the hearty support of the Council of Military Education in these recommendations, I met with no encouragement from the Ordnance Select Committee and its associates, who avowed the most perfect confidence in their instrument and the results it afforded, which were, however, of no use to me. Under these circumstances I undertook the construction of my own chronograph, because it was desirable for me to retain complete control over my own invention, on account of the numerous modifications and improvements which the first example of such an instrument might naturally be expected to require. However, this roughly constructed instrument far exceeded my expectations on its first trial in November, 1865, with ten equidistant screens.

The chronograph, after having thus passed its first trial with success, was next used to determine the resistance of the air to elongated projectiles of the same diameter, but provided with various forms of heads. The results of these experiments showed that it would not be necessary, for practical purposes, to employ more than one form of head in future experiments.

In the next place, an extended series of experiments was made with a view to find according to what function of the velocity the resistance of the air varied, and also to determine whether the resistance opposed to the motion of shot did accurately vary according to the square of the diameter. The form of head of shot selected for these experiments was the ogival struck with a radius of one diameter and a half. The diameters of the shot varied from about three to nine inches. Also in order to obtain great variations of velocity, the charges of powder used in firing each kind of shot were made to vary greatly. Afterwards experiments were made with spherical shot fired from the guns which had been used to fire the elongated shot. In this way the resistance of the air to service projectiles was found, in a conclusive manner, for all attainable velocities above 900 feet per second. As, however, the determination of the law of resistance for velocities below 900 feet per second was of little practical importance, and as new arrangements of the screens would have been required for that purpose, it was not thought desirable to enter on that branch of the enquiry without special authority for so doing. These experiments were concluded early in 1869.

Some few experiments were subsequently made by me to determine the resistance of the air to Whitworth flat-headed projectiles in consequence of an application made to me from the Department of the Director of Artillery and Stores, Woolwich. Immediately after these experiments were completed my chronograph was removed from Shoeburyness, where it had been four years; for, having succeeded in perfecting my invention, and having trained three officers of the Royal Artillery in the use of my instrument and methods of reduction, there seemed to be no reason for me to push enquiries further, unless the subject was taken up officially and in earnest, as it was no part of my duty as Professor or Referee to invent and provide chronographs for the public service, or to make experiments, because it was distinctly stated at the first that the references themselves would always be "the application of analysis to some practical question." For evidence of the uniform success of my instrument and methods of experimenting I must refer to the collection of my Reports, published by the authority of the Secretary of State for War, which have been printed just as they were written at the specified dates.

The original chronograph, called for distinction the clock-chronograph, was constructed solely with a view to determine the resistance of the air to the motion of projectiles. The labour required to reduce each experiment would render this a most inconvenient instrument for use when it was desired to measure only the initial velocity of a shot, as in the proof of gunpowder, etc. But having received repeated applications from the War Office respecting a simplified form of my chronograph for common use, I succeeded, after considerable trouble, in perfecting the gravity chronograph, hereafter fully described. This instrument, constructed originally in a rough manner, has been often used in the trial of small arm ammunition, and has proved reliable and convenient to use.

To complete the subject, I have given an account of the way in which my chronograph may be used to measure extremely short intervals of time, as I wished to explain clearly to the Committee on Explosives the right application of my methods to such cases. I have also given a critical examination of the crude form of my instrument which was brought forward by Captain Andrew Noble in 1866, and used by the Committee on Explosives, although it must be confessed that such instruments, even in their best form, are not adapted to determine the pressure of fired gunpowder in the bore of a gun.

The consideration of the motion of a projectile naturally divides itself into three parts - first, its motion in the bore of the gun; second, its motion through the air; and third, its motion during its penetration into a solid substance. In the first and third of these cases I have had no opportunity of making any experiments whatever, and therefore I have nothing new to communicate on the subject. The little that is known respecting the laws of penetration of round shot into solid substances, is due to experiments made at Metz under the direction of competent mathematicians. The short chapter on the subject here given has been adapted from M Didion's Traité de Balistique. As to the pressures exerted by the gas of fired gunpowder, nothing for certain seems to be known. If we are to accept the dicta of the Committee on Explosives, the pressure of exploding pebble-powder is liable to extreme variation from round to round, even when fired under apparently the same conditions. If this be really the case, further experiments would be almost useless; but the methods of experimenting pursued by the committee are far from leading to conclusive results, as is explained in Chapter I.

In the remaining case, the whole of the data required for the calculation of the trajectories of spherical and elongated projectiles used in this treatise have been derived from recent experiments made with my clock-chronograph. These mathematical investigations are virtually the same as those of John Bernoulli, published a century and a half ago, but great labour has been required to render them practically useful. If the resistance of the air varied accurately as the cube of the velocity, and if the force of gravity acted in strictly parallel lines, it would now be easy to calculate with precision the trajectory of a shot fired at any inclination to the horizon. I mean that in no case is it necessary to have recourse to the supposition that $\large\frac{ds}{dx}\normalsize$ may be replaced by its mean value through a moderate arc of the trajectory, as is done by M Didion and M Helié and others. Whatever may, therefore, be the length of the arc through which the resistance can be supposed to vary as the cube of the velocity, through that arc the motion of the projectile may be accurately calculated by the methods and tables given in this treatise, excepting any deviation due to the rotation of the shot. It would be easy to take account of the variation of the density of the air corresponding to the height of any point in the trajectory. It is believed that the Tables given are sufficiently extended for all practical purposes; and when the resistance of the air to the motion of shot for velocities below 900 feet per second has been determined, it will only be necessary to furnish additional tables of coefficients corresponding to tables I. and II.

Just now the results of my eight years' labours have been arranged in a form adapted for the instruction of the Advanced Class of Royal Artillery Officers, the class itself is threatened with extinction, inasmuch as a sufficient number of candidates to form a new class did not, for some reason, offer themselves for examination last spring. In consequence of this a Committee was appointed to consider "whether the Advanced Class is of so much benefit to the Service as to justify its continuance, and whether any mode of educating the Royal Artillery Officers m the special branches of knowledge which the class was instituted to teach can be suggested in preference." The Committee drew up a circular letter containing four questions, copies of which were forwarded to thirty-four Officers of the Royal Artillery varying in rank from General to Lieutenant. The Committee in their report, presented to both Houses of Parliament, recognised "the great interest taken in the subject by the officers generally, and the importance which they (with hardly an exception) attach to the maintenance of some means of enabling officers of the corps to acquire that scientific knowledge which is so desirable in every respect." The Committee practically recommended the continuance of the class system, which provided at Woolwich instruction in Chemistry, Mechanics, Metallurgy, and Mathematics specially adapted to the wants of an Artillery Officer, at a total expense to government of about £1200 per annum. Several Officers of the Royal Artillery appearing to be distressed at seeing this large (?) sum of money ''swallowed up by the Instructional Staff," proposed that a few Artillery Officers should be selected by examination, and allowed to attend certain courses of lectures at the School of Mines, King's College, and University College, London. This formed the "alternative scheme" of the Committee, which, if adopted, would undoubtedly fail to lead to any result beneficial to the public service.

If now the system of instruction tried during the last eight years could have been pronounced a failure, there would have been some reason for its opponents to propose a change. But I maintain that the class system, instituted in 1864, has been eminently successful in training for the public service a sufficient number of Officers of great ability and unwearied industry. In support of this opinion, I refer to the official reports on the first, second, and third classes - to the text books pre- pared by Officers who have passed through the advanced class - to the valuable papers contributed to the Proceedings of the Royal Artillery Institution by former members of the advanced class; and lastly, to the satisfactory manner in which every Officer, who has received an appointment after leaving the advanced class, has discharged the duties of his office. There are certainly several other Officers of the Royal Artillery who have qualified by passing successfully the examinations of the advanced class, but have not hitherto received any appointment; while they have seen prizes, which have been held out to them as inducements to work, given to others whose scientific attainments had never been tested. And this goes far to explain the existing dissatisfaction, and the paucity of candidates for admission into the fifth advanced class. It, therefore, seems to me extremely opportune that public attention should have been so distinctly called at the present time to the higher scientific education of Officers of the Royal Artillery, and to some of the discouragements under which professors and their pupils have so long laboured.

As none of the civilian lecturers pretend to have even the most elementary knowledge of the regimental duties of an Artillery Officer, it is plain that their teaching is not likely to interfere with the prospects of those Officers who confine their attention to strictly professional duties. But just now great changes are taking place in the armaments of all nations, and it is a question who shall have the direction of the changes in this country. Shall selected Officers of the Royal Artillery receive, first, a high scientific training, and, when found qualified, he appointed to some subordinate posts, where they can carry on their education, and avail themselves of the opportunity to apply to practical purposes the instruction they have received in Chemistry, Mathematics, etc., so as to gradually fit themselves for the highest posts in the manufacturing departments? Or, shall the heads of departments be Officers who do not even pretend to any scientific knowledge, and who must, therefore, in deciding most important questions, be dependent for assistance upon subordinates, or inventors, or scientific men called in for the occasion. To me it appears that such duties would be best performed by Officers of the Royal Artillery, provided they possessed the necessary scientific knowledge, because they would be likely to be best acquainted with the wants of the service. And even if they did not feel sufficient confidence to deal single-handed with any important question, they would be able to consult distinguished scientific men, without being under any obligation to follow implicitly their advice.

The nation has suffered incredible losses of late years from the want of proper scientific training in its advisers. Take the Armstrong breech-loading system, a failure, upon which many millions sterling have been wasted. The "grip" in front of the seat of the shot was calculated to impede the initial motion of the shot, and therefore to cause the powder to exert its utmost destructive effect on the gun. It ought, therefore, to have been foreseen that such a contrivance must fail.

But in dealing with the muzzle-loaders, the very opposite course was taken. Some kind of increasing twist of rifling was adopted with a view to save the gun by facilitating the initial motion of the shot. On what principle, it may be asked, was the particular form of increasing twist chosen? The failure of several very heavy guns in time of peace seems to indicate that an error of a very grave kind has been committed in adopting the Woolwich system of rifling for those guns. Now the proper form of rifling depends upon the nature of the powder to be used. If the propelling force acting upon the shot could be rendered perfectly uniform, then the gun ought to be rifled with a uniform twist. Consequently, if the use of pebble-powder has a tendency to produce uniformity of pressure of the gas propelling the shot, the proper form of rifling must approximate to the uniform twist.

Quite recently we have heard much of the great power of a new 16-pounder field gun. When, however, the gun was tried with its proper carriage, the firing of a few rounds sufficed to break the axles of the gun-carriages. It need hardly be stated, either that the strength of axles should be perfectly understood, or that everything relating to the recoil of a gun, so far as it depends upon the weight and calibre of the gun, upon the weight of the shot, or upon the kind of powder. ought to be perfectly well known. The present difficulty may be got over by increasing the strength of the axles, and so adding to the weight of the gun, or by reducing the charge, and so diminishing the boasted powers of the gun, but in either case the gun ceases to be what it was intended to be.

The very numerous and costly experiments on the penetration of iron plates throw little light upon the subject, because they have been made on no system, by the use of shot which generally broke up in penetrating. Much valuable information might be derived from experiments carefully conducted on a small scale, by the help of a reliable chronograph, with shot made of the Whitworth metal, or some other metal which did not break up on penetrating.

If these and other questions of the same kind were made the subjects of careful experimental investigation under the direction of the best mathematicians of the day, most instructive lessons would be furnished to the Officers of the advanced class, and the conclusions arrived at would prove of the highest value to the nation. I here offer the results of my eight years' labours, voluntarily carried on under many discouragements and difficulties, as my contribution towards an improved state of things. The want of hearty official support has led to a great waste of labour, because I have had to do things for myself which ought to have been done for me. So long as the educational and the experimental departments do not work well together, it must be to the great disadvantage of the public service.

November, 1872.

3.2. Review in Nature.

The following review appears in Nature (16 October 1873), 503.

The Motion of Projectiles.

We are told in the Preface to this work that "the consideration of the motion of a projectile naturally divides itself into three parts - first, its motion in the bore of the gun; second, its motion through the air; and third, its motion during its penetration into a solid substance." The author directs his attention chiefly to the second of these parts. Galileo was the first person who determined with anything like accuracy the motion of a solid body moving through space under the action of gravity. Treating the vertical and horizontal motions as perfectly independent (which of course is in accordance with Newton's laws of motion), he showed that a particle moved in a parabola. In this theoretical investigation gravity is supposed to be constant, and to act in parallel directions, while the effect of the resistance of the air is totally disregarded, The parabolic motion is approximately true for bodies whose velocities arc small, but the greater the velocity of a projectile, the more does its path deviate from a parabola, and, in the present days of large guns and heavy charges, we can at once see the importance of solving with the greatest possible accuracy the problem of the motion of a projectile through the air, considering the air as a resisting medium materially affecting the motion of the shot. Newton solved the problem of the motion of a body through a medium whose resistance varies as the first power of the velocity, and John Bernoulli extended it to the case of resistance varying as any power of the velocity.

Experiments, however, show that the resistance cannot be regarded as varying as any single power of the velocity, though, within certain limits, the third power gives pretty accurate results.

Mr Bashforth has applied himself to the task of throwing Bernoulli's solution into a practical shape, so that by means of copious tables, of which his book contains more than 100 pages, such problems as the following may be solved:- The 16-pounder muzzle-loading gun fires an ogival-headed shot 16 lb. in weight, and 3'54 inches in diameter. If the angle of projection be 2°, and the initial velocity 1,358 feet per second, find the trajectory and time of flight," "A Rodman shot weighing 452 lb. is fired with an initial velocity of 1,400 feet per second, at a target 500 yards off, find the striking velocity."

Experiments were made by Robins and Rumford last century to ascertain the pressure of fired gunpowder and several persons have attacked the problem during the present century. General Mayevski attempted to solve the problem by firing shot, into the back of which a rod was screwed, the rod running through an aperture in the breech of the gun, and carrying a knife edge which cut two thin wires at a given distance, the interval of time between the two breakages being measured as accurately as possible. Captain Rodman made use of the following arrangement:- A gun was mounted in a gun-pendulum, and a revolving cylinder was placed with its axis parallel to that of the gun. When the gun was fired, a tracing-point on the gun drew a curve on the revolving cylinder, the shape of which curve determined the whole motion of the gun's recoil. Mr Bashforth suggested that much greater exactness would be procured if the tracing-point were connected with the projectile. He managed to do this to some extent by firing a shot through a number of equidistant vertical screens, made of very thin metal wires. By an ingenious arrangement, the time of the shot breaking a wire in each screen was registered by means of an electric current on a revolving cylinder, special care being taken that all the registrations should be made under the same circumstances, so as to eliminate what we might call the personal error on the different registrations. This gave the times of transit of the shot over the successive intervals between the screens: from them, the velocities at the different screens can be calculated with great exactness, and also the resistance of the air on the shot. Mr Bashforth has made great numbers of experiments with shots of different shapes and sizes, fired with different charges of powder, and from them has with great labour calculated the tables above referred to, which are sufficient for the solution of the problems we have given above as examples of what Mr Bashforth has been able to accomplish.

The work is one which is too mathematical to do full justice to in our columns, but we have no hesitation in recommending it to such artillerists as are not unacquainted with mathematical analysis.
4. Francis Bashforth and John Couch AdamsAn attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops (Cambridge University Press, Cambridge, 1883).
4.1. From the Introduction.

Many years have elapsed since this work was commenced, and it is even now only partially completed. My object was to test the received theories of Capillary Action, and through them the assumed laws of molecular attraction, on which they are founded. To this end it was proposed to compare the actual forms of drops of fluid resting on horizontal planes they do not wet, with their theoretical forms.

After some trials a satisfactory micrometrical instrument was constructed for the measurement of the forms of drops of fluid, but my attempts to calculate their forms as surfaces of double curvature failed entirely, and my undertaking must have ended here, if I had depended upon my own resources. But at this point Professor J C Adams furnished me with a perfectly satisfactory method of calculating by quadratures the exact theoretical forms of drops of fluids from the Differential Equation of Laplace, an account of which he has now had the kindness to prepare for publication. After the calculation of a few forms, application was made to the Royal Society for assistance from the Government grant in making the needful calculations. The following extracts from the application (27 October 1855) will explain the objects of the undertaking. "I have carefully examined all the published experiments that I could meet with, but these have been generally made with capillary tubes, and in consequence of the difficulties inherent in this mode of observation they have not led to consistent and satisfactory results. It appeared to me that the best test of theory would be obtained by making careful measures of the forms assumed by drops of fluid resting on horizontal planes of various solids.
5. Francis BashforthRevised account of the experiments made with the Bashforth Chronograph, to find the resistance of the air to the motion of projectiles, with the application of the results to the calculation of trajectories according to J Bernoulli's method (Cambridge University Press, Cambridge, 1890).
5.1. Preface.

When my previous work on the Motion of Projectiles was published in 1873 the correct law of resistance of the air had been determined only for velocities between 900 and 1700 feet per second. The extensive experiments made at Shoeburyness in 1878, 1879 and 1880 with ogival-headed projectiles completed the law of resistance for velocities between 100 and 2800 feet per second, but it was not found possible to assign any simple expression for the law of resistance in terms of the velocity. The Newtonian and cubic laws may however be used, excepting perhaps a brief interval just below the velocity of sound.

The generous recognition of the practical value of my labours by the Marquis of Hartington, when Secretary of State for War in 1885, induced me to attempt to complete my labours by the calculation of tables of integrals for a resistance varying as the square of the velocity. So far as seemed necessary similar tables for the cubic law of resistance have been reprinted from my former work on the same subject.

The results of my experiments have been extensively used in government treatises on Ballistics since 1877. Also Captain Ingalls has given an extended and careful explanation of my results and method of experimenting in his Text-Book on Exterior Ballistics prepared for the use of officers under instruction at the United States Artillery School, 1886. And Major Wuich, Professor der Artillerielehre am k. k. höheren Artilleriekurse, Wien, has abridged my tables and presented them in a new form in his Aeussere Ballistik, 1886.

In order to furnish the reader with full information respecting the foundation on which my work rests, I have carefully revised all my original observations and given full particulars of the results finally adopted. This re-examination of every round has introduced trifling changes in the coefficients of resistance for both spherical and ogival-headed projectiles. I have therefore taken the trouble to recalculate my General Tables for both forms of projectile, in order to render my work consistent throughout. The whole has been adapted to the use of French as well as English measures.

The close agreement between calculated and experimental ranges and times of flight for high muzzle velocities and low elevations shows that my coefficients are well adapted for the best guns of the present day. But when projectiles are fired with high muzzle velocities at high elevations, the calculated ranges and times of flight are both generally less than those given in the range tables. This discrepancy, I have no doubt, is caused in a great measure by the vertical drift of the elongated projectile, which causes an increase of range and time of flight. In fact the explanation of lateral drift given by Magnus and others also accounts for a vertical drift which is really the origin of all drift

Recently some rounds have been fired from a wire gun at high elevations with a very high muzzle velocity, commonly spoken of as the Jubilee Rounds. But it unfortunately happened that the wind was more or less favourable to a long range in these experiments. And a moderate steady wind at the surface of the earth would become a very violent wind at a height of two or three miles, which would produce a marked effect on the motion of an elongated projectile exposed to its action for 50 or 60 seconds. I have calculated a complete range table for the case where there is no wind to disturb the motion of the projectile.

The statements and proceedings of some foreign writers on ballistics have rendered it incumbent on me to enter at some length into the history and progress of my work during the last twenty-six years. But I have confined these remarks chiefly to the conclusion of my work, so that the reader need not trouble himself unless he feels an interest in the matter.

In calculating trajectories it has of late become a common practice to reduce my coefficients, either arbitrarily, or so as to bring them into accord with those of Krupp. But I have not been able to find any satisfactory experimental authority for Krupp's tables issued in 1881. Certainly in the following year an "Annexe", consisting of 37 rounds, was put forward to support a foregone conclusion, but these experiments from their nature were not to be depended upon, and in no single case was the time of flight recorded. The specimen of the experiments made to determine the resistance of the air for velocities higher than 700 m.s. ought to establish the character of Meppen for ballistic experiments. In all cases the Krupp party were careful to follow and not to lead. An inspection of diagram will show how carefully they followed my law of resistance, merely reducing my coefficients, as is shown by line 3 compared with line 1 or 2.

In 1872 Mayevski combined my results published in 1868 with a few of his own experiments, from which he professed to have obtained "résultats russes et anglais," which however coincided with my previously published results. Consequently, so far as Mayevski's experiments had any value, they entirely supported my previous conclusions.

The method of calculating trajectories published by Siacci requires all the three tables previously used by Niven for that purpose. Ingalls has pointed out a grave defect in that Siacci has not found an analytical expression for a most important quantity... Turning to Niven's paper it will be found that the two values of this quantity required for distance and for time have been carefully determined, and still more so in a paper On certain Approximate Formulae for calculating the Trajectories of Shot, by Professor Adams (Nature, 16 January 1890). It must be plain that arbitrary coefficients of resistance, and empirical quantities are quite inadmissible in any calculations made to test the results of careful experiment. Krupp, Mayevski and Siacci use tables of the same kind as mine.

The reader will find in the following work a very full account of every round from which coefficients of resistance have been obtained by me for both spherical and ogival-headed projectiles. In consequence of the Krupp scare, special experiments were made in 1887 to test my coefficients on a long range, when they were found to be quite satisfactory. Still no notice seems to have been taken of this fact, or of Captain May's remarks, by calculators of trajectories.

My coefficients of resistance for low velocities have been tested by calculating a Range Table for the 6.3-inch Howitzer for elevations 5º to 35° with satisfactory results.

For high velocities I have used the Range Table for the 4-inch B.L. gun. The calculated ranges and times of flight for velocities 1900 to 960 f.s., and for elevations 1° to 4°, are quite satisfactory; and this conclusion is confirmed by the use of the General Tables. In the same manner the Range Table of Captain May, R.N., has been used to show the accuracy of my coefficients of resistance when the projectile moves nearly in the direction of its axis.

I therefore claim to have accomplished in a satisfactory manner all I undertook to do, namely, to find by experiment the law of resistance to spherical projectiles and also to elongated projectiles when they move approximately in the direction of their axes.

The tables and coefficients already given are sufficient for the calculation of trajectories of spherical projectiles and of elongated projectiles where there is no sensible drift. But in attempting to calculate the trajectories of elongated projectiles fired from rifled guns with high muzzle velocities and at considerable elevations, it will be well to recognise the truth of the statement of St-Robert - that the problem taken in all its generality presents great difficulties. I have endeavoured to explain the nature of the movement of such an elongated projectile, which is supposed to be projected with perfect steadiness from a rifled gun, according to the conclusions of St-Robert. It is evident that shortly after the elongated projectile leaves the gun it must be raised up bodily by the resistance of the air, so as to cause it to move as if it had been fired at a somewhat higher elevation than it really was. I have given the calculated ranges and times of flight for elevations of 1° to 15° for the 4-inch B.L. gun. As the elevation increases above 4° it appears that the calculated ranges and times of flight fall short more and more of those quantities respectively given by experiment. Suppose we reduced the coefficient's of resistance so as to obtain a calculated range equal to the experimental range for an elevation of 10°, we should find, as Captain May did, that these coefficients would not give a correct time of flight - and they would destroy the agreement actually obtained for low elevations. The reduction of the coefficients of resistance therefore cannot be the solution of the difficulty, as is commonly supposed. Some correction is required which will increase both the calculated range and time of flight.

In a section below the calculated ranges are arranged in a different manner. I have found from the Range Table the elevation and time of flight corresponding to each calculated range. It is evident that the corrections for elevation at once give the correct ranges and very approximate corrections for the times of flight. These latter corrections would have been still more satisfactory if the decrease in density of the air corresponding to the height of the shot had been taken into account in the calculation of the trajectories. For the reason stated this mode of correction will be only an approximation to the truth - but it will perhaps be found to be satisfactory. The law of the correction can only be obtained by the calculation of numerous trustworthy Range Tables, or by theoretical considerations.

I fear that the reader will meet with some repetitions in the following work, but it was impossible to avoid them entirely on account of the complicated nature of the various questions to be dealt with. Although it will not surprise me to find that what has been said produces little immediate effect, it will always be a satisfaction to me to have stated my case carefully and supported it by reference to, and specimens of, my early results and tables, in none of which have I found it necessary to introduce any important change.

The English Range Tables I have made use of appear to me surprising from their minute accuracy. I have derived much assistance from Captain Ingalls's excellent work on Exterior Ballistics, and the numerous references to that work will explain in what respect I am indebted to his labours.

Minting Vicarage,
March, 1890.
6. Francis BashforthSupplement to a revised account of the experiments made with the Bashforth Chronograph, to find the resistance of the air to the motion of projectiles, with the application of the results to the calculation of trajectories and a historical sketch of the progress of ballistic experiments connected with the advanced class, Woolwich, 1864-1890 (Cambridge University Press, Cambridge, 1895).
6.1. Preface.

The Advanced Class was established at Woolwich in 1864 to meet the wants of those Royal Artillery Officers who were desirous of pursuing the study of Artillery in its higher scientific branches. And it was also rightly considered that by this means the fittest men would be obtained for the management of the experimental and manufacturing branches of the profession. As Professor of Applied Mathematics I was anxious to determine the resistance of the air to projectiles for the benefit of my class, and for that of the service generally. But the Ordnance Select Committee were very much opposed to any experiments being made under my direction. They appeared to consider that such a proceeding was an encroachment on their province. But I was merely proposing to do for projectiles in general that which Robins and Hutton and Gregory, the latter two being Professors at the R M Academy, had. attempted to do with some success for round shot, and, so far as I know, without exciting any jealousy on the part of military men. The O S Committee condemned my results of 1866 without mercy. The Committee afterwards were bold enough to publish some of my results condemned by them the year before, as their own discovery, for which they were duly reproved. In consequence of such unworthy treatment I refrained from seeking permission to make any experiments after 1866. Government had been afforded a good opportunity of learning what I was able to do for them. Henceforth, if they required my assistance they had to ask for it.

In 1870 a Director of Artillery was appointed who was opposed to the scientific training of the Advanced Class as then constituted. When a new class of Officers should have been assembled in 1872, there was not a sufficient number of candidates to form a class. In part explanation of this state of things it was stated that appointments, which had been held out as incentives to Officers to pass through the course, had been conferred on other Officers, who had satisfied no test. The opponents of the Advanced Class were utterly defeated before the Committee of enquiry in 1872. The Class was revived in 1874, and I have not heard of any further opposition to it. Although I was at this time of age to retire on my pension, it had been officially intimated to me that I was at liberty to hold on my Professorship, But my opponents proposed as the reward of all the good work I had done to reduce my small stipend, which proposition was promptly met by my resignation. The worst stage in the history of the Advanced Class had now been reached.

But soon after, the prospect brightened. Abundance of good work had been done, which now began to produce fruit. Several P A C Officers had been appointed Captain Instructors, etc. who provided new Text Books for their various classes, making use of my Tables and Experiments. For many years after my retirement, I continued to advise and assist my former pupils. When the so-called "practical" Director of Artillery had ruled for his five years, his successor invited me to lend my Chronograph and return to complete my experiments at Shoeburyness, which was done, 1878-80. I had thus, in the end, every facility for making all the experiments I desired, 1865-80. If my results are not correct and trustworthy, it is my own fault, unless there was some defect in the shooting of the experimental guns, which I do not believe. I have thought it advisable to explain the precision of my observations, and point out the consistency that reigns among all my published results. In my concluding remarks I have also endeavoured to meet such objections as have been made to my results by the use of the range table of the 4-inch BL gun selected on the part of government for that purpose.

Sir G B Airy, Professor Stokes and Professor Adams very kindly accepted the reference of my experiments 1867-8. This reference was originally intended to be a very simple affair, such as is usually adopted by learned societies. But the submission to them of additional questions respecting other forms of Chronograph, which had no connection with my experiments, caused great delay, and very much increased the labour of the Referees. The Report of the Referees was duly published in 1870, and since then it has been practically ignored by the advisers of the Secretary of State for War. On this account my obligation is all the greater to those R A Officers, who have brought my results into general use in Text Books for the Royal Military Academy 1879, 1883 and 1887; for the Navy 1880; for the Army (Small Arms) 1884 and 1888; in the Handbook for Field Service 1878, etc.; so that probably more than 15,000 copies or my General Tables have been printed. Ballistic Tables founded on the results of my experiments have been in use during the last ten years in the U.S. America. Krupp, Mayevski and Siacci have made free use of my results in Germany, Russia and Italy. Lord Hartington, Secretary of State for War, when he conferred the government award of £2000, in 1885, fully recognised the value of the services rendered by me to the War department. These are very good reasons for my being so far perfectly satisfied with the general reception of my results.

But in the Postscript I have explained how the Chronograph might be used to still greater advantage, in measuring the steadiness of the projectiles fired from all new rifled guns and thus testing them thoroughly at the least possible expense end in the shortest possible time.

Minting Vicarage
18 July 1895
7. Francis BashforthA Historical Sketch of the Experimental Determination of the Resistance of the Air to the Motion of Projectiles (Cambridge University Press, Cambridge, 1903).
7.1. Resistance of the Air to the Motion of Projectiles.

Galileo and most early writers on ballistics assumed that the resistance of the air to projectiles produced little or no effect on their motion. Robins remarks that "if it were necessary, a large catalogue of geometers of note might be here produced, who have asserted in their works, that, in the operation of gunnery, the resistance of the air was too minute to merit attention. And I do not remember, that any author has formally or expressly contradicted this position; whatever may in general be concluded, from what some of them have at times advanced as to the laws observed by resisted bodies. Indeed I may venture to affirm, that it is now the almost universal opinion of those, who have studied the doctrine of projectiles from the treatises, which have hitherto been published thereon: that all shells and bullets in their flight do nearly describe the curve of a parabola ; and consequently, that the resistance of the air to the motion of these bodies is altogether inconsiderable." Tracts, 1. 176.

Robins appears to have carried out his ballistic experiments chiefly at his own expense. His first ballistic pendulum weighed 56 pounds, afterwards increased to 97 pounds. Robins arrived at the conclusion that for a velocity less than 1100 f.s. and greater than 1200 f.s., the resistance of the air to spherical projectiles will vary as the square of the velocity, but that the "absolute quantity of that resistance in these greater velocities will be near three times as great, as it should be by comparison with the smaller velocities." Tracts, 1. 182. My experiments shew that the law of resistance to spherical projectiles varies approximately as the square of the velocity for velocities less than 840 and greater than 1300 f.s., where the coefficient for high velocities is nearly double that for low velocities.

The Royal Society awarded their Copley Medal to Robins in 1746. His experiments and theories are said to have met with the greatest approbation from the best judges at home. His work was translated by Euler into High Dutch, 1745, and by Le Roy into French, 1751. He was appointed engineer-general to the East India Company, 1749, for which he was to receive £500 a year for life, provided he gave his services for five years in India.

Two years after Hutton arrived at Woolwich, 1775, he "in (1775) conjunction with some able officers of the Royal Regiment of Artillery, and other ingenious gentlemen, first instituted a course of experiments on fired gunpowder and cannon balls" for government. His account of them was presented to the Royal Society, who honoured it with the gift of the annual gold medal, and printed it in the Philosophical Transactions for the year 1778."

Hutton remarks "That part of Mr Robins's book has always been much admired, which relates to the experimental method of ascertaining the actual velocities of shot, and in imitation of which, but on a large scale, those experiments were made which were described in my paper." The projectiles used by Hutton varied from one to near three pounds in weight, afterwards increased to six pounds. Further experiments were carried on from 1783 to 1791, and the results obtained were considered satisfactory. Hutton at first made use of a pendulum weighing about 600 lbs. It was increased to 1014 lbs. in 1788, afterwards to 1655 lbs., and at last to 2099 lbs. Hutton's health began to fail in 1806, and when he retired the Board of Ordnance manifested their approbation of his long and meritorious services by granting him a pension for life of £500 per annum.

Gregory made some experiments at Woolwich in 181518 with a ballistic pendulum weighing 7046 lbs. chiefly on the effect of windage.

The first experiments made with the ballistic pendulum at [1839] Metz were carried out during 1839, 1840. The weights of the projectiles were about 9, 13, 26 and 51 lbs. The weight of the pendulum was 13,228 lbs. Other experiments were made at Metz 1856, 7 and 8.
8. Francis Bashforth, Ballistic experiments from 1864 to 1880 (Cambridge University Press, Cambridge, 1907).
8.1. Institution of the Advanced Class for Royal Artillery Officers.

When the Council of Military Education were preparing to establish the Advanced Class for Royal Artillery Officers, in 1863, they kindly invited me to become a candidate for the office of Professor of Applied Mathematics. But, as I was otherwise engaged, I felt obliged to decline the invitation. However, when they pressed me a second time, I had considered the matter, and found that it would be possible to determine the resistance of the air to the motion of projectiles by the use of a chronograph specially devised for that purpose. With proper assistance, it seemed to me that this work might be accomplished in about two years. In the Government application for temporary leave of non-residence for me it was stated that:
No other candidates could be found (although the Council pushed their enquiries in every direction) possessing to the same extent the requisite combination of attainments. Should Mr Bashforth's services, in consequence of the obstacle before mentioned, be not available, the Council are of the opinion that the loss to the public service will be very great ...
When I arrived at Woolwich, April 1864, I found the President and Vice-President of the Ordnance select Committee were very decidedly opposed to any new chronograph. They had made numerous experiments, and could furnish me with any quantity of results from their stores! Chronographs with rotating cylinders had been tried and had failed! etc. It was clear that the O S C knew nothing whatever about the application of mathematics to ballistics.

I soon came to the conclusion that if my proposed chronograph was to be used I must provide my own instrument. But still, I felt that it would be only prudent to ascertain whether, if I provided my own chronograph, facilities would be afforded me for trying it, and for experimenting with it. A favourable reply was promptly received from the Director of Ordnance, who concluded by stating:
I am to add that Earl de Grey and Ripon recognises with much pleasure the zealous and practical manner in which you have entered upon your duties.
17 May 1864.
The construction of the chronograph was commenced in my own workshop during our August Vacation, 1864, and it was reported ready for trial with ten equidistant screens in June 1865. But it was not till the following November when the Select Committee thought proper to afford me an opportunity to try my new invention in Plumstead Marshes.

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