Charles Hutton, 1737–1823: The prototypical woolwich professor of mathematics

Journal of Mechanical Working Technology, 18 (1989) 195-230

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Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

C H A R L E S H U T T O N , 1 7 3 7 - 1 8 2 3 : THE P R O T O T Y P I C A L WOOLWICH P R O F E S S O R OF M A T H E M A T I C S

W. JOHNSON*

Schools of Engineering, Purdue University, West Lafayette, IN 47907-0499 (U.S.A.) (Received December 30, 1987; accepted January 10, 1988)

Summary Many details of Hutton's life are described, especially those passed professionally at Woolwich Royal Military Academy. He is most renowned for his experimental work on ballistics and for continuing the early work of Benjamin Robins. He corrected some of Robins' notions and illuminated and extended others. Besides work in ballistics he cooperated in conducting notable historical tests and calculations for finding the Earth's density. Hutton's professorial work included also publishing text books, from the simple to the relatively advanced, acting in a secretarial capacity for the Royal Society, consulting and cooperating with the ordnance authorities and advising on the strength of bridges. This paper is closely related to and indeed is intended to be read in conjunction with, The Woolwich Professors of Mathematics, 1741-1900 (see the previous article in this issue).

Introduction

Charles Hutton was appointed to a Chair in Mathematics at the Royal Military Academy, Woolwich near London, some 25 years after the death of Benjamin Robins of ballistic pendulum fame. The quickening of industrial development and expansion into overseas territories by European countries and Britain in particular, the growth in confidence in the usefulness of the natural laws of Physics (mainly as stated and reinforced by Newton (and Keill I ) and with the development of new methods of mathematical analysis) led to their being able to exploit successfully the growing natural physical sciences. One respect in which this was so relates to the performance of cannon balls and guns. Robins had led the way in showing how the speed of a ball could be deter*Emeritus Professor of Mechanics, Engineering Department, University of Cambridge, U.K. 1John Keill (1671-1721), Savilian Professor of Geometry, Oxford, and "a keen disciple of Newton". With the Principia available only in Latin and "a very difficult book to read", he was a key figure in helping its contents spread through Europe via's Gravesande (of the University of Leyden) and Desaguliers [4].

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mined, why its trajectory on earth was not parabolic and he had discussed the incurvations of the path of the ball [2]. H u t t o n carried on Robins' work and gave depth, further assurance and some generality to his methods and findings using mathematical methods. It was to be expected that this work would be carried out in military surroundings. Of course there are many more scientific respects in which H u t t o n contributed to late 18th century science and whilst these will be outlined below, the principal reason for giving attention to H u t t o n is as stated. It will become clear as the life and activities of H u t t o n unfold that we are seeing the growth of a forerunner of the modern professor of engineering mechanics. H u t t o n was a teacher of mathematics at various levels and of a scientific Strength of Materials appropriate to his time, he was a writer of textbooks and research tracts, he engaged in systematic ekperimentation and analysed the results, he displayed an interest in physics showing a wider awareness of its place in science than just ballistic engineering; he gave of his time and energy to help run the Royal Society (to encounter the not unfamiliar hazards of powerful interests different from, if not opposed to, his own) and was generous with help when it came to bringing along the generation of engineering scientists who would replace or succeed him. This essay is intended to be read, to a degree, as part of Ref. [1] to learn about H u t t o n ' s specific achievements. However it is hoped that the paper will also convey insight into the times in which he lived. H u t t o n will be seen to have been one of a short line of very successful professors who emerged outside of and independent of the universities. One has only to contrast the intellectual brilliance of one of his contemporaries, Lucasian 2 Professor Edward Waring (refer to p. 228) to see what un-useful results may flow from a world such as Cambridge then was - one far from the work-a-day world of the nation. As H u t t o n continued Robins' work, so have there now been many others. We may even recognise it in a recent obituary on Sir William Cook (The Independent, 19 Sept. 1987), which also involves Woolwich, in the following: "Bill Cook, as he was known to his closer colleagues, spent over 40 years in government science joining Woolwich Arsenal in 1928. One of his earliest successes was the development of a special camera to measure simultaneously the muzzle-velocity of the guns in the triple turrets of the Newcastle class cruisers, a new system which was showing unexplained errors in trial firings..."

H u t t o n : His life

A major source for the detail of H u t t o n ' s life is the relatively long article which Margaret E. Baron - a biographer of H u t t o n in the D.N.B. (Dictionary 2This was the Chair t h a t Isaac Newton had filled, 1669-1701.

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of National Biography) - has called an eulogy3 entitled A Brief Memoir on the Life and Writings of Charles Hutton, LL.D., in the Imperial Magazine for March 1823; it is in fact a glowing obituary written by Professor Olinthus Gregory, see Ref. [314 and Fig. 1. It contains a detailed commentary on most or all of Hutton’s papers and is an attempt to appreciate the man and his work. The article is composed of two parts, the first and longer one of 7000 words was initially commissioned by a Mon. Hachette of Paris and covers Hutton’s life up to 1818; it was intended to precede a translation of Hutton’s Tracts on Gunnery. However it was not then published but was completed on 1 February, THE

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of, these memoirs,. was born at Newcas- i tie-upor:-Tync, on the 14th of August, J. i737, Fig. 1. The Imperial Magazine. (Note the aphorism, line 5.) “D.E. Smith in History of Mathematics, Vol. II (Dover 1958, p. 458), writes that Charles Hutton “owed his prominence more to his perseverance than to his scientific ability and was chiefly known for his mathematical Tables and his dictionary.” The reason for this judgement is not elaborated on but readers may judge of its appropriatness for themselves, by reference to the details of this account. 4There is inconsistency between Refs. [ 3 ] and [ 41 concerning Hutton’s early life. Gregory gives no hint that he is writing about Hutton’s stepfather or indeed that he even had one. Similarly, there is later no indication that Hutton was twice married.

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1823 and with a second part of some 2700 words the whole was published about two months after H u t t o n ' s decease. Charles H u t t o n was born in Newcastle-on-Tyne, a coal mining centre of England, on 14 August, 1737. His only claims to learned connections appear in the branch of his family which moved into Lincolnshire and a female who became the aunt of Isaac Newton by marriage and in being a first cousin of the renowned geologist J a m e s Hutton. His father was a colliery labourer, who died when his youngest son, Charles, was five years old. His stepfather was evidently a man of uncommon intelligence because, after being a pit foreman, he became a "viewer of mines" and later a land-steward to a local peer. H u t t o n was a hewer in the pit for a short period under his stepfather but after displaying some mathematical ability and suffering injury to one of his arms, the stepfather had him instructed by a local clergyman ~, a Mr. Ivison, in English, Latin and the leading principles of mathematics. At about the age of 18, he started up at Jesmond, near Newcastle, (replacing his late schoolmaster) as a teacher of mathematics with such success that apparently he was able to move into Newcastle in 1760 to take as pupils local landowners, and their offspring; these leading citizens included John Scott (Earl of Eldon) and H u t t o n ' s own future first wife, Elizabeth. He was often able to claim "double the terms that had previously been charged in that quarter of the kingdom". Aware also of his own mathematical deficiencies he read widely and, it is said, attended evening classes in Newcastle and much improved himself. (It is difficult to imagine who may have given such a course and in what institution. ) A gentleman named Shafto employed H u t t o n in the evenings to tutor his family, and lent him some advanced mathematical works. He also seems to have branched out so far as to be able to be consulted as a surveyor and in 1770 he was employed to draw an accurate map of the city. Further, before he was 23 he had contributed problems and solutions "of difficulty (and) utility" to the much respected Ladies' Diary (See Ref. [ 1 ], p. 179 ), The Gentleman's Diary and Martin's Magazine. Hutton's first book was a treatise on Arithmetic and Book Keeping, for use in schools, published in 1764 and many times thereafter. Next followed A Treatise on Mensuration in Theory and Practice in 1770, after its appearance in periodical numbers, 1767-70; an improved edition appeared later, with illustrations by T h o m a s Berwick in 1780 and two more editions later still. Gregory considered that after this the subject was largely exhausted, mere abridgements following. "Practice" in the title included land surveying, so it is probable that with knowledge of practical techniques this book was a prerequisite for the study of the determination of the mass of the Earth (see below). In 1779, H u t t o n then at the age of 42, received the degree of LL.D. from the mAt this time such educated mathematical clergymen came principally from the mediaeval university foundations of Oxford, Cambridge and Dublin.

199 University of Edinburgh 6 where Dr. M a t t h e w Stewart 7 and Mr. Dugald Stewart were joint professors of mathematics. H u t t o n retired in 1807, suffering from "a pulmonary complaint". He had been in post at Woolwich for 34 years and to recognise all he had achieved, the Board of Ordnance awarded him a pension of £500 per annum. According to Gregory however, the professor had already "acquired a very handsome fortune by the profits upon his laudable exertions". Also, "he then fixed his abode in Bedford Row, London, where he ... enjoyed his otium cum dignitate, heightened by the sweets of domestic intercourse ..."! He continued to do some examining at Woolwich for a period. For health reasons H u t t o n had been allowed to live, while in post, on Shooters Hill instead of in the Royal Arsenal where the Royal Military Academy (R.M.A.) then stood. Apparently, he greatly appreciated the pleasure of living at such an elevation and judging the area around him to have considerable economic future, he "bought land, employed brick-layers, directed the manufactory of his own bricks and planned and erected a series of genteel houses ''s. Thus he made the first and most important step occur in the urban development of Woolwich Common! W h e n the R.M.A. moved from the Royal Arsenal to Woolwich Common in 1806, the Board of Ordnance bought three of Hutton's houses for official residences of Field-officers of the Artillery~ and Professors of the Academy! In his last years, H u t t o n ' s principal companion was his eldest daughter 1°, his wife having died in 1817.11 His son, Lieut. General George Henry H u t t o n (d. 1827), moved to London, where with his family he helped tend his father in his closing days. Gregory [3, column 221 ] refers to a manuscript journal kept by H u t t o n but, disappointingly, only refers to its record in a visit to H u t t o n by an old friend of 50 years standing, one Dr. Trail, Professor Robert Simson's biographer. During his last years he issued, with the help of Professor Leybourn of the R.M.C., new editions of some of his works - a 15th edition of his Arithmetic, the 8th of his Compendious Measurer and the 6th of his Mathematical Tables. 6Honorary? 7The underlining henceforth indicates a short statement in Note 1 of the individual's achievements; it is someonewho usuallyplayeda part in Hutton's scientificlife. 8The entepreneurial ability of the people of this period is truly remarkable. Hutton's is only exceeded by that of James Nasmyth - supposed inventor of the steam hammer and vee anvil - who in 1836 openeda factoryat Patricroft, near Manchester,the buildinghavingbeen made from clay dug up from the original land, which had been burned into bricks. See p. 196 of James Nasmyth, Edited by Samuel Smiles,John Murray, 1885. 9ABritish Field-officeris an officerof the rank of major and above. ~°Huttonmarriedtwice, had two daughters and a son [4 ]. The seconddaughterwith her husband, Henry Vignoles, Captain of the 43rd regiment, died (as prisoners of war) of yellow fever in Guadeloupe. 11Hutton himselfwas buried in the familyvault at Charlton, Kent.

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Fig. 2 (a). Hutton in profile and his "logo".

In 1819 and 1820 Hutton had some correspondence with Laplace who apparently omitted to mention Hutton's work in his writing on the determination of the mean density of the Earth. It seems that a letter from Hutton to Laplace drawing attention to the omission went unanswered for some months and so he had it published in the PhilosophicalMagazine for February 1820 and in the Journal de Physique for April 1820. Laplace responded in the Connaissance des Temps for November 182012, and in direct personal correspondence, apologising and handsomely acknowledging Hutton's contribution; the reply also was printed in the PhilosophicalMagazine. On the same subject incidentally, Hut12Stated by Gregory to be for 1823.

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ton was convinced that Henry Cavendish's paper on the Earth's mean density contained many erroneous calculations. He found no one willing to go through the calculations de novo and hence, in his 84th year, did it himself. He did indeed find errors but they were "of trifling magnitude"; they are identified in the Philosophical Magazine for 1821. This was Hutton's last publication. Gregory records that a marble bust of Hutton by Sebastian Gahagan was executed from subscriptions contributed by his friends and presented to him personally in September, 1821. Hutton endeavoured to display his esteem for his friends by paying for the "striking off" of medals for the subscribers to the bust. On the obverse of the medal was the head of Dr. Hutton in profile, see Fig. 2 (a), and on the reverse, two emblems, one pertaining to his work on the density of the Earth and the other to that on the force or strength of gunpowder. Hutton, claimed Gregory, was the most popular of English mathematical writers, attributing this to simplicity of presentation and to the utility of his chosen subject matter. Gregory asserted that Hutton had an aversion to the "pedantry and parade of science" and observed that, curiously, in the matter of working habits, a practice of Hutton was to tabulate and classify memoranda on all subjects - "as though he had no memory". A portrait of Charles Hutton at age 75 appears as Fig. 2 (b). Hutton's appointment to the Professorship of Mathematics at the R.M.A.

In May 1773 Hutton became the fourth professor of mathematics to be appointed at the Royal Military Academy after the resignation of Mr. J.L. Cowley13who had occupied the post for the 12 years, 1761-1773. The Marquis George Townsend, then Master-General of the Ordnance, decided the appointment "would be made from the individual acquitting himself best in a public examination"! 14 It is noteworthy that the first five professors appointed had no formal training or education in mathematics at a university: they were all self- or privately-taught. It could have been therefore that one of the aims of examining them may have been that of ensuring that they were reliable and competent to teach across a significant part of the whole of the field of mathematics. There were ten competitors. Hutton was originally very diffident about seeking the post; he was evidently somewhat in awe of his predecessor, Thomas Simpson, who had been the second occupant of the Chair before him, 1743-61 [ 1, p. 163 ]. The first professor at the Academy is given as "-Derham 13, 1741-43" [1, Fig. 5(a)]. The competition was keen and the variety of attainments and claims of two l:~There is no entry about this man in the D.N.B. 14Colin Maclaurin had a ten-day competitive tiial ~efore appointment to a professorship at Marischal College, Aberdeen, in 1717; and Peter Barlow underwent "a severe c o m p e t i t i v e e x a m i n a t i o n ~' in 1801, see Note 2 and Ref. [I ].

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of the competitors justifies mentioning them briefly. These were, Benjamin Donn (or Donne ) (1729-1798), author of a book The Geometricians and other well known works on mathematics and book keeping; he had started a mathematical academy at Bristol. He was also Master of Mechanics to the King, then George III, [1, p. 184], had published maps of S.W. England, and charts of the Western Ocean. The second competitor was Hugh Brown who had translated and commented on Euler's Gunnery. The examination went on for several days and included the manner of communicating instruction, the best treatises to be employed in the different branches of science, the history of the principal departments of mathematics and the solving of a variety of problems. "More than one half of the candidates ... gave entire satisfaction to the examiners ... but the superiority of Mr. Hutton was in every respect so marked and decisive that they unanimously recommended him as peculiarly qualified to fill and adorn the situation ...".

The four examiners or assessors were: ( i ) Dr. Nevil MASKELYNE ( 1732-1811 ), then Astronomer Royal ( 1765 ) but a wrangler of Trinity College, Cambridge. He had established the Nautical Almanac in 1766, was the Copley medallist for 1775, awarded for Observations on the Attractions of Mountains. He was made a F.R.S. in 1758 and had observed the transit of Venus at St. Helena in 1761. (ii) Samuel HORSLEY ( 1 7 5 3 - 1 8 0 6 ) , Bishop of St. Asaph during 1802-6. He derived from Trinity Hall, Cambridge, graduating in 1758. He was made a F.R.S. in 1767. He published both mathematical and theological works and was secretary of the Royal Society in 1773 but left in 1783-4 after a dispute with Sir Joseph Banks. He edited Newton's works, 1779-85. (See also the entry re: Christie in Ref. [1], p. 167). (iii) Henry WATSON (1737-1786). Watson was an engineer colonel at Woolwich having been a captain in 1763, a field engineer in Bengal, 1764, and Chief Engineer, 1765. He had translated Euler's treatise on the Construction o/Ships. (iv) John LANDEN (1719-1790) was the author of Mathematical Lucubrations, 1755; he was elected F.R.S. in 1766. He propounded Landen's Theorem on an Hyperbolic Arc. (This states that the length of any hyperbolic arc is expressible in terms of two elliptic arcs. It was published in the Philosophical Transactions for 1775, and also in Mathematical Memoirs of 1780. ) His work is summarily assessed in the D.N.B. by, "he failed to develop and combine his discoveries". He is said to have written many profound mathematical treatises relating to rotatory motion and he was also the inventor of The Residual Analysis 15. l~Interestingly, C.B. Boyer, in The History of the Calculus, Dover Publications, 1959, p. 236/7, explains what Landen meant by The Residual Analysis. (1758), namely, "calculus ... without the aid of any foreign (principles) relating to an imaginary motion or incomprehensible infinitesimals". Boyer, remarking that Landen uncritically manipulates indeterminate forms, gives an example of his method.

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Hutton's p a p e r s a n d b o o k s

A paper which evidently aroused much attention at the time and appeared in the Phil. Trans. for 1776, was that entitled A New and General Method of Finding Simple and Quickly Converging Series. It was proposed as a "general (method) ... more universal than those of Machin, Euler and R. Simson, including their series...". However as this now has little interest we proceed to a paper in the Philosophical Transactions which is Hutton's classic, The Force of Gunpowder and the Velocity of Cannon Balls (projected from artillery). Gregory writes that the experiments for it commenced in 1773 but H u t t o n himself says he made his experiments in the summer of 1775 [3, p. 50]. The paper gives an account of the theory and equipment and the use of the Ballistic Pendulum, and remarks on the effects of friction, air resistance and the time of penetration of the wood block by a ball. The Royal Society awarded H u t t o n its Copley Medal 16 for the year in question and it was presented by the President, Sir John Pringle. The same year H u t t o n also presented to the Royal Society his Account of the Calculations made from the Survey and Measures taken at Mount ShichaUin (or SchiehaUion) in Perthshire, in order to ascertain the mean Density of the Earth. Dr. Maskelyne had explained the survey and the astronomical observations upon which the calculations were to be made, in a paper in the Phil. Trans. in 1775. However, "he declined (to do) them" as too laborious to carry out. As a result of the solicitations of the President and Council of the Royal Society, H u t t o n undertook the work and it took nearly a year of effort. Gregory remarks that they were more laborious than any undertaken by a single person since the preparation of the logarithmic tables! H u t t o n deduced 4.481 as the Earth's specific gravity and believed it to be too low. After the geological structure of the mountain had been re-examined by Professor Playfair of Edinburgh and a value of 2.6 established for its density, a new calculation was made and a value for the Earth established at 4.95; the latter result was published in 1808. This figure was confirmed, wholly independently, by Playfair in 1811. It is fair to observe that H u t t o n was the first to make a reasonably correct assessment of the mean density of the Earth. Unfortunately his name, for some unknown reason, is generally omitted from a list of those scientists who contributed seriously to the problem. A paper of 1779, entitled Calculations to determine at what point in the side of a Hill its attraction will be greatest, was generated by the immediately pre16This silver gilt medal (with, today, the sum of £2500 ), is the premier award of the Royal Society. It is given annually for outstanding achievement in research in any branch of science. It was a bequest made originally by Sir Godfrey Copley (d.1709), a baronet, a Member of Parliament and Controller of Army Accounts in 1691.

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ceding work. H u t t o n ' s next paper in 1780 was On Cubic Equations and Infinite Series, and for its time it was new. In 1783, he presented to the Royal Society, his Project for a New Division of the Quadrant. This was however soon abandoned due to mensuration changes in France. The Compendious Measurer followed in 1784 being a popular abridgment of his earlier Treatise on Mensuration. It was popular in as far as it omitted any demonstrations and confined itself to rules and examples. H u t t o n ' s Mathematical Tables of 1785 was a "heavier" piece of work; it contained among other items some circular functions to which we are now used and which are common, hyperbolic and logistic logarithms. An original History of Discoveries and Writing relating to the latter items, was also added. Apparently this latter book was needed because at the time many entries in the Tables were incorrect (actually thousands were). He also included some new Tables. The subjects of spherical and plane triangles were, at this time, also addressed. An history of early trigonometrical writings and tables was given and the contributions of such well known persons as Napier and Briggs, etc., are clearly and impartially identified. The text enjoyed five editions. In 1788 there was published a volume, Tracts, Mathematical and Philosophical. As well as work on series, roots of equations and properties of sections of a sphere and cone and certain geometrical properties of circles and ellipses, there is a large tract relating to the ballistic pendulum. Gregory writes that at this time of his life H u t t o n "seems to have thought every year lost in which he did not present to the public some new work!". Elements of Conic Sections with Select Exercises was published in 1787, seemingly intended for use at the R.M.A. Apart from other things considered, new problems discussed were ones of interest to engineers such as times for filling and emptying with water the ditches of fortifications. With George Shaw and Richard Pearson he edited an abridgment of the Philosophical Transactions for the years 1665-1680 in 18 thick volumes. It appeared in monthly "parts". H u t t o n was responsible for the "pure and mixed mathematics" in them and the theory of mechanics, hydro-dynamics, optics, astronomy, electricity and magnetism. He was responsible also for their general editorship, most of the biographical articles, translations and a number of his own notes. The two-volume, quarto work, Philosophical and Mathematical Dictionary, 1795 (6?), was said to be the best known of H u t t o n ' s work but was criticised for its unbalanced content and for being unduly cautious: but it was said to be a valuable source for later historians of mathematics. A new and enlarged edition with improvements appeared in 18151~. The first edition in two octavo volumes of H u t t o n ' s Course of Mathematics 17It is a work of two volumes, each about 750 pages and in the writer's opinion, a magnificent opus. See Fig. 2(c).

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Fig. 3. The Whirling Machine. (From Howard Douglas's Naval Gunnery, 1855.)

for use by the "Gentleman Cadets in the Royal Military Academy", appeared in 1798. It was republished many times, a third volume being added in 1811. Gregory in his biography, [3, col.217], claims it was written "in conjunction" with him. However there is no acknowledgment of this on the title page of Vol. III. A three-volume collection of Tracts on Mathematical and Philosophical Subjects appeared in 1812, being written in the period of his retirement. It includes articles on the theory of bridges, on experiments on the force of gunpowder with applications to the modern practice of artillery, on the History of Writings and Investigations in Trigonometry and Logarithms, the Introduction to the Doctor's Mathematical tables and on the History of Discoveries and Investigations in Algebra. Some of the tracts describe experiments on gunnery and the whirling machine, see Fig. 3. The latter, introduced by Robins, is well described by Howard Douglas [5, p. 49 et seq. ], its purpose being to determine the resistance of spheres to motion at speeds of less than 300 ft/s. (Balls rebounded from the wooden block of the ballistic pendulum at this relatively low speed. ) A given falling weight W1 at M soon brings the angular speed of sphere P to a constant value w. If the weight W2 ( < W1 ) is next found which gives rise to the same o) for the arm without P, then from a knowledge of W1, W2 and the device dimensions, the resistance at P at a given speed is easily found. Hutton, appreciating the laws of resistance to motion, eventually was able to provide a table of the resistance experienced by a ball of, for instance, 2 in. diameter at speeds from 5 to 2000 ft/s. (A ballistic pendulum was used for obtaining data at higher speeds.) From the latter, results for resistance at different speeds could then be deduced for balls of various diameters and weights. A note on mathematical recreations and essays

Hutton translated Montucla's 4-volume edition of 1778 of Ozanam's 1694 work, Recreations in Mathematics and Natural Philosophy (London, 1803 ). He added notes and improvements to Montucla's modernisation which was entitled Mathematical and Philosophical Recreations of Ozanam.

207

Contemporary English-speaking interest in this subject is to be found in W.W. Rouse Ball's Mathematical Recreations and Essays, MacMillan & Co., 1949, pp. 418, which many of us have enjoyed since our school days. Rouse Ball, on pp. 2-3 of his book, explains that many of his elementary questions derive both from Bachet's Probl~mes Plaisans et Delectables, 1612, (which in turn uses problems from Alcuin and such Renaissance-Italian mathematicians as Tartaglia and Cardan) and from Ozanam's aforementioned book. The Royal Society quarrel Hutton was elected a Fellow in 1774 and served as an assistant to the secretaries in charge of foreign correspondence (which carried an honorarium of twenty pounds) from 1779 to 1783, in which year, due to dissension, he retired having been requested to do so by the then President of the Society, Sir Joseph Banks 18, on the grounds of his having failed to carry out his duties efficiently. It appears too that the Council of the Society required that the secretary should reside in London, when in fact Hutton lived at Woolwich. Dr. Maskelyne, Bishop Horsley, P.M. Maty and F. Maseres resigned from the Royal Society after having attacked Banks' management of the Society's affairs and believing that Hutton had been ill-treated. More about this is described elsewhere, see for instance Ref. [6], pp. 187/8. The speed of shot Charles Hutton made contributions to this subject in several respects, but principally, (i) In determining the speed of shot in flight after leaving a gun muzzle. (ii) On the effects of air resistance for different shot speeds. (iii) In studying to determine how the pressure of burning powder in a gun barrel varied with time. (iv) Examining the effects on ball speed of features such as the amount of linear windage. (v) In relating the variables of ball mass, velocity and charge weight in a given gun. These various topics were reported on by Hutton firstly in 1775 and secondly in 1778; and finally in about 1817 by Gregory. Some of the above topics are discussed in other parts of this paper but shortage of space has precluded addressing others. For this reason we refer the reader to Howard Douglas's trea~8In the s u m m a r y of his recent Sir J o s e p h B a n k s Memorial Lecture to the Royal Society (7 November 1988), Sir David S m i t h referred to his subject as "... a truly remarkable m a n ... whose personality was engagingly complex...", a n d continued with, "... B a n k s ... only published one serious scientific paper ...". (Yet) B a n k s was " P r e s i d e n t of the Royal Society for a record of 42 years."

208

tise [5], and in particular to a long Part II on The Theory and Practice of Gunnery, pp. 23-180, if he wishes to make up for the deficiencies of this section.

The Enlarged Ballistic Pendulum and the Gun Pendulum Howard Douglas noted "that Mr. Robins first invented the Ballistic Pendulum by which velocities can be measured with great precision" [ 5, p. 28 ]. He continues by observing that his work was limited in "having used only such shot as could be discharged from musket barrels and thus that experiments should be made on a much larger scale". Dr. H u t t o n was "funded" to do just this by the British Board of Ordnance. The first set of experiments were performed in 1775; the report of it and several other sets of experiments resulted in the paper which was awarded the Copley medal for 1778 and to which I have referred elsewhere. A second series of tests or experiments was carried out in 1783-5. Dr. Gregory, in 1817, with General Millar and Colonel Griffith supervising the testing equipment, repeated and continued testing in the same vein but using a pendulum of about 7000 lb. wt., see Fig. 4. Douglas records that to the same end but using a different set-up, Chevalier d'Antoni 19 carried out tests in 1764 in Turin. The results arrived at were little different from those "subsequently obtained" by Gregory. Figure 4 shows in two elevations a diagram of the ballistic pendulum, used by Gregory according to Douglas [5, p. 29 ]. Displacement of the pendulum was measured by a pointer P on the lower surface of the block tracing a line "on a bed of soft grease which filled a groove in the circular piece of wood AB". At L'Orient between 1842 and 1846 French engineers used a pendulum which was a suspended gun, see Fig. 5 [5, p.29], together with a suspended receiver which corresponded to the block of English pendulums. This is usually referred to as a gun pendulum 2°. In 1843 the U.S. had a series of experiments carried out under the direction of Brevet-Major Mordecai with apparatus working on the same principle as the French device. The device of Major Mordecai is described by Douglas as "... a gun (24 or 32 pounder) suspended between two piers by a horizontal axis turning on knife edges; and of a hollow frustum of a cone of cast iron suspended in a similar manner; the axes of the gun and the cone, when both were at rest, being in the same horizontal line. The hollow frustum was filled with sand, in bags, to receive the impact of the shot; its front being covered with a plate of iron having in the centre a perforation, which was covered with sheet lead, and through this the shot passed after being discharged by the suspended gun. The whole weight of the gun pendulum was 10500 lbs., and that of the receiver pendulum 9385 lbs. The distance of the centre of gravity in each, from the axis l'Elsewhere [7 ] I have pointed out that Cassini has been designated by Brodie and Brodie [ 8] as the first to determine the velocity of ball ejection, but their source is not referenced. 2°I hope soon to publish a detailed account of the history and development of these pendulums.

209

E

n

o

w

~q c) o

i

<

210

E,

;Z

°°...°"

•

;

- ' ' ' ' ° ' ,o...-''"

H Fig. 5. Portion of a gun pendulum.

of suspension, was about 141 feet; and the sensibility of the pendulums was such, that when set in motion in an arc of 23 degrees, the receiver pendulum continued to vibrate about 24 hours, and the gun pendulum about 30 hours" [5, p. 29/30]. Subsequently, an additional weight of 667 lb. was applied under the gun and block so that "the distance from the centre of oscillation to the axis of suspension was rendered equal to the distance of the line of fire and point of impact from that axis" [5, p. 34]. There is little point in repeating the analysis of Douglas as it pertains to the various ballistic pendulums, for reference can easily be made to t h e m [5, p.30 et seq. ]. It is noted [5, p.35] that the distance of the block of a ballistic pendulum from the muzzle of a gun for the size and speed of projectiles of interest here, should be 30 to 50 ft., so that the blast from the discharge should not impinge on the block and so influence its motion. However, at such distances air resistance will tend to have reduced the speed of projectiles whilst the motion of the gases and the unburnt charge as ejected from the gun muzzle will tend to have increased it. Shot speed was also early found by shooting directly into one end of a hollow cylinder uniformly rotating about its central axis (the ends of it were covered or closed). The shot was required to have a flat trajectory and to move parallel to the axis. If the speed of rotation of the cylinder is co, and 0 the angle the

211 distal end rotates with respect to the near end, whilst the shot traverses the cylinder length l, then the shot speed would be lw/O. This method is described by Douglas [5, p.35/6] but unattributed. My suspicion is that it was proposed by G. Cassini. The force of fired gun powder and the initial velocities of cannon balls, determined by experiments from which is also deduced the relation of the initial velocity to the weight of the shot and the quantity of powder

Such is the lengthy title in the Philosophical Transactions, 68 (1778), pp. 50-85, of Charles Hutton's Copley medal-winning paper. Since it was so highly rated by the Royal Society it is fitting that we summarise it in detail. It was read on 8 January, 1778 and is the most directly related of all his papers to our interest here. Hutton commences by remarking that the experiments were made in the summer of 1775. He acknowledged the help of Royal Artillery officers and states explicitly that the experiments were made following "the method invented by Mr. Robins and described in his treatise, New Principles of Gunnery", (see also Ref. [7]), of which an account appeared in the Philosophical Transactions, for 1743. Hutton notes that "Before the discoveries of that ingenious gentleman very little progress had been made in the true theory of military projectiles". After some general introduction, Hutton proceeds to describe his pendulum "machines", see Fig. 6; one type was used for the first three sets of experiments and the second for another two. As remarked elsewhere, the detailed approach using the pendulum will already be well known to readers or they can be read about in Refs. [5,7 ]. Figure 6 depicts the first block he used, which is indicated by A, a cube of 20 in. length, of dry elm, fastened to a "strong iron stem on the back part of it". The pendulum length from axis BC to the ribbon at D was 102 ½in. To discover the center (sic) of gravity and the center of oscillation of the pendulum was elementary - by simple balancing or taking moments. Some pages are devoted to assessing the possible accuracy of the experiments conducted and it is shown that v, the ball speed, is assessed to within 1/3000. Next, as a ball remains in a block after a test, for use in a general formula, the block weight needs to be re-determined and similarly the magnitudes pertaining to the centres of gravity and percussion. But also Hutton notes that because the momentum of the ball is not given to the block instantaneously (but during its period of penetration - here about 2 ms - because of the resistance of the air on the back of the pendulum and because there is friction at the axis), the magnitude of these factors needs to be discussed and assessed; this he does and shows they can be safely neglected. The brass gun used in the experiments admitted a cast-iron ball of about 1¼ lb., or a leaden one of about 1 ~ lb., or sometimes a cylindrical shot of nearly 3

212 t'higa.¢. T r ana" . Va l. L ~ -

.Tab .1".

a

b

* , ~ , . S¢,4~ . Fig. 6. Hutton's pendulums, as used in work reported in 1778, taken from his Copley Medal paper.

lb. wt. The gun bore was about 2.1 in. diameter and its length was about 20 calibres or about 42.5 in. The gun powder used was typical of t h a t used by government suppliers and

213 the charge weights were 2, 4 or 8 oz.; they were contained in a light flannel bag (for safety against fire) and r a m m e d as required, without a wad before it. The pendulum was located 30 ft. from the gun muzzle, being based on a fired 8 oz. charge (without a ball) which did not visibly move the pendulum. The distances of penetration of the balls into the wood were uncertain since all balls tended to strike the pendulum block near the same spot. However, a 2 oz. charge seemed to give about 3 in. of penetration into the wood, only. With a pendulum weight of about 328 lb., the distance of the centre of gravity to the axis was about 72 in., and the distance of the centre of percussion about 88 in. The first set of six tests with 2 oz. charges gave an average of 626 f t / s and two tests with 4 oz. charges, 915 ft/s; the tests were conducted with balls of about 1.97 in. diameter. The ratio 915/626= 1.46; for a doubling of a charge, other things being the same, the velocity of shot would be expected to change by about ~ or 1.42; so the two earlier ratios at 1.46 are sufficiently close. The second set of tests (June, 1775) used powder from the bottom of a barrel, the charges also being rammed "a little closer t h a n on the former day"; the ball diameter on this occasion was 2.08 in. The fourth and fifth shots were made using cylinders with hemispherical ends; the axial length was about twice the shot diameter. A British experiment with a ball of about 19 oz. wt. gave a speed of about 973 ft/s, and t h a t of a 46½ oz. shot, 749 ft/s. The speed ratio 973/749= 1.3 was expected to be as the ratio of the square root of the shot masses, i.e. (46.5/ 19 ) ~= 1.56. We expect that, all other things being equal, approximately, ½mlv = ~ m 2 v 22

so

v2/v 1 = (ml/m2)

1/2

There was clearly a significant discrepancy here - of about 20%. Most remarkable was the difference in average shot speeds on successive days: on the first day it was 626 f t / s and on the second, 973. This difference was assigned principally to the difference in windage on the two occasions: the much smaller a m o u n t on the second occasion had the balls closer to the gun TABLE 1 Charge (oz.): Shot velocity,v (ft/s) 4th test: lead 5th test: iron Vir,,,/Vlead

Shot weight, w (oz.) 4th test: lead 5th test: iron (Wlead/Wiron) 1/2

2

4

8

613 701 1.14

873 992 1.14

1162 1397 1.20

28~ 18~ 1.23

28½ 182 1.23

28~ 1.24

214

bore diameter. Secondary reasons were the taking of powder from the bottom of the barrel and the employment of a higher degree of ramming. A third set of tests with a nearly constant weight of ball but charges of 2 and 4 ounces, gave mean speeds of 738 and 1043 ft/s. The ratio of the latter is 1/1.41, i.e. is as the square root of the charge ratio precisely. The first pendulum slowly disintegrated and eventually a second one was substituted. Apart from some minor design improvements, the weight of the new pendulum was 552 lb., all other dimensions being more or less as previously. Lead balls (weight about 28½ oz. ) were used on the first day (fourth set) and iron ones on the next, the charges investigated being two, four and eight ounces. As three tests were conducted for each weight of charge they gave nearly uniformly mean speeds of 613, 873 and 1162 ft/s; the latter ratios are 1:1.42:1.90, which is nearly as the square roots of the charge weights. The discrepancy of 5% in the last figure of 1.90, rather than 2, calls for explanation. It is attributable to (i) the reduced length of barrel or cylinder through which the ball passes when the charge is large (and hence how long in the barrel) and the smaller length over which the cylinder gas pressure acts on it; (ii) that the quantity of powder blown out through the muzzle unfired is the greater the greater is the charge. The final set of tests was much the same as in the last test except that the balls fired were of iron of about 18½ ounces weight. The velocities achieved averaged 701,993 and 1397 ft/s or, as ratios, 1 : 1.42 : 1.99; the ratio of the square roots of the charges, again, were 1 : 1.42 : 2.00. Disparities are attributed to the same causes as before. The shot velocities for the last two tests of lead and iron balls are compared in Table 1. Again, disparities are attributable to the same two factors as before. The conclusions from these tests are fairly obvious. The principal recommendations new to persons untrained in naval artillery were, that by diminishing the amount of windage then permitted by the Naval authorities, a considerable saving in powder could be achieved - as much as 1/3 or more of that "presently used". T w o topics from Hutton's Course of Mathematics

The title page of Volume II121 of H u t t o n ' s Course of Mathematics is seen as Fig. 7 (a) and page 281 (on p. 216 herein, set as Fig. 7 (b) ) is interesting having the complicated-looking layout of a text-book page, typical of that year. The type-setting has improved to help the reader during the intervening one and one half centuries! Note the use in the text of the old Newtonian term fluent, ~lOriginally, I am grateful to Prof. A.G. Atkins for drawing my attention to these extracts. 'Recently Professor' rather than 'Late Professor' using to-day's literary convention, would have been a more accurate designation of him in 1813!

215

A

COURSE OF

MATHEMATICS. IN

THREE

VOLUMES.

COMPOSED

FOR

THE USE OF THE ROYAL MILITARY ACADEMY, BY

ORDKM

O'P H I S

LOMDSHI~

THE MASTER GENERAL OF THE ORDNANCE. |

BY

CHARLES

HUTTON, L L . D . F.R.S.

LATE PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, ira',,

i

VOL.

III.

LONDON: lbKINTZD F 0 1 7 . C. A I D J. I I ¥ . I N 6 T O N } 01, WILKIV. AND $. IOBINSON~ J, V A L K l i ; R, B A L D W I N ; J, M U R K A y ; LACE-" XNOTONj ALLXNp AND CO. j LONQMAN, HUKST, RKKS, OKMmp AND BROWN ~ CADZLL AND DATIES; J. CUTHELL; B.CRoSBY AND CO.; J . RICHARE~0N; 3. M. RICHAKD~0N; RLACR, PARRY, AND CO.; O. APD 8. ROBINSON; OAL]~I CURTIS, AND FENNER~ AND $. JOHXSO,',r AND CO. 18 I$, Fig. 7 (a). Title page from H u t t o n ' s C o u r s e of M a t h e m a t i c s .

216 OF

GUNNERY~

prob. 2, will be t h e resistance o f the air against the ball in avolt:dupols pounds; to which if the weight o f the ball be added, then ( m y ~ - n v ) d z J- ~¢, will be the whole resistance to the bali's motion ; this divided by w, the weight o f t h e ball in motion, gives ('0 . . . . . ),t,+,, _ , , ~ , - ~ , d , + 1 = . / t h e tV

tll

retard-

ing force. H e n c e the general formula v v - - o-gfk (theor. 10 pa. 342 vol. o, edit. 6 ) b e c o m e s - v ' v -" 2 g k × (- , , u- ' --, v )- , t ,p+ w to

m a k i n g ~ negative because v is decreasing, where g = 16 i t . i and hence rm ,'~ ,.~, •~ ------- ~ X (,,~,_t,o)u,+w - - ~g,~a'---~,× n to ,

_

- - t o

tJ 3 - -

v +

- -

mtt ~

N o w , for the easier finding t h e fluent o f this, assume " "; -"z ,o tJ - 2.~-- = * then v - - z + ~m' and v z '~ + ,n +'-~-~m" and vv = z~ + ~,,,~, '--'-~ a n d C - ,~,

=

v+

~=z

", and v 2

- - ~-~t ; these being substituted in t h e above value

o f ~-, it becomes .~ = •

tt

•

n putting p -- ~ ,

~2 and

-- - -

- p', or p" +

9' -- *'

T h e n the ~eneral fluents, taken b y t h e 8th and 11 th forrm vol. ~ pa. 307, give .~ = N - ' ~ x [-~ log. ( P + f~) + ~to

to tad. ~'. and tan. ~] - - ~ - ~

x are ~

× [" log. (v~ - - "vm + ~--~)+

~, x arc to tad. q and tang. v - - p ] .

But~ at the beginning

o f the motion, when the first velocity is v for instance, and the space x is = 0, this fluent becomes

o -- ~-~-~-, x E~- log. (v' -

~'v+ ,~

'0 + ~ - X arc radius ~, ,,,¢,)

tan. v - p]. H e n c e b y subtraction, and t a k i n g v = 0 fo~ t h e end o f the motion, the correct fluent becomes t, '~v ~' x - - ~$,~, x []- log. (v'* -- ,,* + ~-~,) --~- ~log ~t, +

---~ x

C

(arc tan. v - p - arc tan. -- p to tad. e)]. But as part of this fluentt denoted by -~ x the dif. o f tho two arcs to t ; ~ . v -- p arid - p~ is always very small in comparison Fig. 7 (b). A typical page from Hutton's Course of M a t h e m a t i c s .

217

the very easy steps in derivations but the drawn out manner of mathematical writing (or the want of tighter notation), in the last two lines. Much of the work in these three volumes - and to be adumbrated upon below - had already been dealt with in Hutton's Tracts (especially XXXVII ) of 1812.

(i) Ballistics: internal and external In Chapter III on the Theory and Practice of Gunnery, Hutton begins in the first paragraph by remarking that the parabolic theory of solid (spherical) projectile flight can only be useful "in the slower kind of motion, not above the velocities 2, 3 or 400 ft/sec." On p. 269 he discusses the "elasticity of the air resulting from fired gunpowder"; and then, allowing Hutton to use his own words, he writese2' "These are circumstances which have never before been determined with any precision; Mr. Robins, and other authors it may be said, have only guessed at, rather than determined them. That ingenious philosopher, by simple experiment, truly showed that by the firing of a parcel of gunpowder, a quantity of elastic air was disengaged, which, when confined in the space only occupied by the powder before it was fired, was found to be near 250 times stronger than the weight or elasticity of the common atmospheric air. He then heated the same parcel of air to the degree of red hot iron, and found it in that temperature to be about 4 times as strong as before, when he inferred, that the strength of the "inflamed fluid", must be nearly 1000 times the pressure of the atmosphere. But this was merely guessing at the degree of heat in the inflamed fluid, and consequently of its first strength, both which in fact are found to be much greater. It is true that this assumed degree of strength accorded pretty well with that author's experiments; but this seeming agreement, it may easily be shown, could only be owing to the inaccuracy of his own further experiments; and, in fact, with far better opportunities than jell to the lot of Mr. Robins, we have shown that inflamed gunpowder is about double the strength that he has assigned to it, and that it expands itself with the velocity of about 5000 feet per second."

These last remarks, criticisms or shortcomings may be held in mind when reading Robins' own work, described in Ref. [7], though the italicised words are generous indeed. On p. 274 Hutton comes to the penetration of solid blocks of elm wood by balls fired in the direction of the fibres. His results are, Powder Mean penetration

(in ounces) (in inches)

2 7

4 15

8 20

He finds "that the penetrations are proportional to the charges" (as he expects) "... except that they are in a lesser ratio at the higher charges"; he does not notice that less resistance to penetration is offered to a ball the nearer it is to the surface of the block (and provided it is several diameters distant from the sides of the block). He comments also that with equal charges, always the velocity increases with the length of the gun barrel. Difficulty arises in con22From pages 269 and 270.

218

firming laws, or otherwise, due to windage, it being observed that up to one "half the powder is lost by unnecessary windage". At this point too, the large sideways deflection of balls is noted - up to 300 or 400 yards in a range of one mile. Tables showing the resistances at speeds from 100 to 2000 f t / s for balls of different weight and diameter are given and it is remarked that, near enough, the resistance, for a ball of given speed, of diameter d, is d2r/4; r is the resistance in lbs. of a "model" ball (approx. 2 in. dia.) at the same speed as that of interest, and given in a table. (How the resistance was determined is not stated on pp. 276/7. ) Alternatively, he derives a general formula,

O.O000257v2-O.OO388v=r

(in lbs. avoirdupois)

provided v is in ft/s; it is not a dimension-less formula. In a problem IX, the"... Space, Time and velocity of a ball Descending through the Atmosphere by its own weight" is discussed through several pages: "terminal or last velocities with which they afterwards descend uniformly" is also discussed - as are many other issues after p. 29. A rule (p.298) that H u t t o n gives is, "when charges bear the same ratio to one another as the weight of the balls, that is when the pieces are said to be like charged, then the velocities will be equal". On p.299 H u t t o n briefly describes how to find the greatest range of a ball given the elevation of the gun piece, for a given speed. An adapted Table is next given deriving from Prof. Robison's article in the Encyclopaedia Brittanica, which gives the maximum range and the associated elevation. These range from 44 ° down to 31 ° . He goes on to say that this elevation is too large for cannon and properly is the task of mortar shells. Of 13 in. mortar shells discharged at 37¼ ° with a speed of approx. 2000 ft/s, H u t t o n remarks that several persons thought that mortars could not achieve this level of speed. The Encyclopaedia Britannica is quoted for rules able to provide range distances for given speed, elevation and ball size. However, H u t t o n believes them to be "much wide of the truth, as depending on very erroneous effects of the resistances". Unreliable rules, he says were given by Mr. Robins and Mr. Simpson.

(ii) The true strength of materials: much about timber H u t t o n introduces a section On the Flexibility, Strength and Rupture of Timber, etc., with a statement that solids (he uses wooden structural elements as examples) may be caused to fracture in simple tension, by crushing, by fracturing transversely in bending or shear, and by twisting. He refers to the following books commenting that, " T h e student who wishes to go into the inquiry with scientific precision, may consult M. Girard's Treatise on the Resistance of Solids, an interesting essay on the Flexibility of Wood, by M. Dupin, in Journal de L'Ecole Polytechnique, tome 10, Tredgold's Principles of Car-

219 pentry, and Mr. Barlow's valuable Essay on the Strength and Stress of Timber23. Having attended many of the experiments recorded in the latter-mentioned work, I can with confidence recommend its principal results as accurate and useful; and shall, therefore, refer to the work itself for the experiments and investigations from which the following formulae and rules are deduced."

Hutton gives a formula for the deflection ~ of an uniform rectangular beam, height h, and depth d, loaded in the middle or elsewhere, by a concentrated load W or a distributed load wl as given by ~=

W (or wl) 13 Ehd 3

the value of E, seemingly, is made dependent on the end conditions. It is simply that the modulus ( as we now know it) is modified by introducing a constant for the beam end conditions and the kind of load and its position on the beam (see p. 360). One notes too in this work that the depth of the central axis is defined as the line which separates the stretched from the compressed wood. Formulae for the force of resistance at fracture of wooden structural elements according to how they are loaded and supported, are given. A Table of "constants" is provided; they are in units of lbs and inches, see Fig. 8 (a). Note the variety of woods listed. There is nowhere mention of different strengths for different directions; c denotes the direct cohesion in lb per square inch and s the resistance of a rod in tension in lb. per square inch. u=12/dA, is the ultimate deflection in a beam before fracture but the meaning of s' is unclear; 1 denotes the beam span and A "the last deflection ''24. Many questions (with answers) are set for the reader. Usually, a rule (or two) is given, as for computing the deflection of beams, with Notes for procedure in problems which involve variations of the case to which a rule applies. The foundation of a rule is given in a footnote for one case by starting from formulae stated to have been derived by Euler and Poisson. Under the intriguing title of Promiscuous Exercises, on p. 345, H u t t o n presents a Problem 40 subtitled, To ascertain the Strength of Various Substances. His first paragraph is well worth reproducing: "The proportions that we have given on the strength and stress of materials, however true, according to the principles assumed, are of little or no use in practice, till the comparative strength of different substances is ascertained: and even then they will apply more or less accurately to different substances. Hitherto they have been applied almost exclusively to the resisting force of beams of timber; though probably no materials whatever accord less with the theory than timber of all kinds. In the theory, the resisting body is supposed to be perfectly homogeneous, or composed of parallel fibres, equally distributed round an axis, 2:*See Note 2. 24Without the expenditure of much time and space it is hardly possible to convey quickly the meaning of these terms, i.e. c,s,s', etc.

220 FLEXIBILITY AND STRENGTH

a

Name of the kind of Gray. Spec. Value of o. Valne of~. Wood. Teak . . . . . . . . Poon . . . . . . . . Eng. Oak . . . . Do. Spee. 2 . . .

Canadian Oak Dantzic O a k . . Adriatic Oak.. Ash . . . . . . . . . . Beech . . . . . . . . Elm

........ Pitch P i n e . . . . R e d Pine . . . . N e w Eng. F i r . . Riga Fir . . . . . . Do. Spec. 2 . . .

Mar Forest Fir Do. Spec. 2 . . . Larch . . . . . . . . Do. Spec. 2 . . . Do. Spec. 3 . . . Do. Spcc. 4 . . . Norway Spar..

745

579 969 934 872 756 993 760 696 553 660

657 553 753 738 696 693 531 ;522 556 560 ;377

818 596 598 435 588 724 610 395 6t5 509 588 605

9657802 6759200 3494730 5806200 8595864 4765750 3885700 6580750 5417266

757

596740Q 5314570 13962800 12581400 13478328 2465433 3591133 -1210830 4210830 5832000

588 .. 588 403 411 518 518 518 6,t8

2799347

4900466 7359700

b

I alue OflValue of Value s. 2462 2221 1181 167'2 1766

s'.

c.

[

`2488 2266

14787]

I 1205

9836.I

1736 1803

10855"] 11428:i

1457 I 1477 158.3 1409

738~ 8808 [

2026 1556 1013 1632 1341 1102 1108 1051 1144 1262 653 832 1127 1149 147.t

I

[ I [

{

2124 1586 1042 1666 1368 1116 1131 1081 1168 1310 890 850 1t49 1172 1492

17337 I j 9912 I ] 5767] 10415 I

I100o01 ] 994;~1 10707 I I .... I I 9539] [ 10691 I t .... [ [ .... I [ 7655 I [ 7352] [12180[

iSt. ~]~TALS.

G o l d , cast . S i l v e r , cast . C o p p e r , cast Iron, cast . Iron, b a r . . . Steel, bar

.

.

. . . . .

. . .

Locust t r e e . Jujeb . . . . Beech, Oak . Orange . . . Ahlcr . . . . Ehn . . . . Mulbcny Willow . . . Ash . . . . Plum . . . . I.;Ider . . . Lemon

42,000 34.,000 5(I,000 -/0,003 . 135~000

. . . . . . . . . . .

Pomegranate

.

. . . .

"22,000

Tin~ cast . . : . 5,0t)o L e a d , cast . . . 860 R e g u l u s of Antimony 1 , 0 0 0 Zinc . . . . . '2,600 bismuth . . . . 2,900

~d. WOODS, &c. ]bs. 20~100 Tamarind . . . . 18,500 Fir . . . . . . . 1 7 , 3 0 0 ~,Valnut . . . . 15,500 Pitch pine . . 13,900 Quince . . . . 13,2o0 Cypress . . . . 1o,500 Poplar . . . . . 1'2,500 Cedar . . . . . 12,000 Ivory . . . . . 11,800 Bone . . . . . . 10,000 Horn . . . . . . 9,'750 Whalebone . . 9 ( 2 5 0 Tooth of sea-calf

.

. .

. .

Ibm. 8j750 8,330 8,130 '7,650 6,'750 6,000 5,500 4,~80 16,2'70 5j250 8,750 '7,500 4,075

Other tables and observations on the cohesive strength of metals, &c. are given at the end of this volume. c,~ts. C a cylinder whose diameter is d i n c h e s ,

loaded to ¼ of its absolute strength, w i l l carry permanently as h e r e a n hexed.

Iron .... 135d ~ G o o d r o p e . '22d ~ Oak .... 14d" Fir ..... 9d ~

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8

Fig. 8. Tables of constants and other information. and presenting uniform resistance to rupture. But this is not the case in a beam of timber: for, by tracing the process of vegetation, it is readily seen that the ligneous coats of a tree, formed by its annual growth, are almost concentric; being like so many hollow cylinders thrust into each other, and united by a kind of medullary substance, which offers but little resistance: these hollow cylinders therefore furnish the chief strength and resistance to the force which tends to break them."

It reads as if written by a man who has worked with timber and well done calculations about it! Hutton continues by pointedly noting, "...that we cannot, by legitimate comparison, accurately deduce the strength of a joist, cut from a small tree, by experiments on another which has been sawn from a much larger tree or block. Concentric cylinders are evidently not all of equal strength - those nearest the centre, being the oldest, are also the hardest and strongest; which again is contrary to theory, in which they are supposed uniform throughout. Yet, in some of the most important problems, the results of the theory and well-conducted experiments coincide, even with regard to timber experiments on rectangular beams, afford results deviating but in a very

221 slight degree from the theorem that the strength is proportional to the product of the breadth and the square of the length. Experiments on the strength of different kinds of wood, are by no means so numerous as might be wished: the most useful seem to be those made by Muschenbroek, Buffon, Emerson, Parent, Banks, and Girard. But it will be at all times highly advantageous to make new experiments on the same subject; a labour especially reserved for engineers who possess skill and zeal for the advancement of their profession. It has been found by experiments, that the same kind of wood, and of the same shape and dimensions, will bear or break with very different weights: that one piece is much stronger than another, not only cut out of the same tree, but out of the same rod; and that even, if a piece of any length, planed equally thick throughout, be separated into three or four pieces of an equal length, it will often be found that these pieces require different weights to break them. Emerson observes that wood from the boughs and branches of trees is far weaker than that of the trunk or body; the wood of the large limbs stronger than that of the smaller ones; and the wood in the heart of a sound tree strongest of all; though some authors differ on this point. It is also observed that a piece of timber which has borne a great weight for a short time, has broke with a far less weight, when left upon it for a much longer time. Wood is also weaker when green, and strongest when thoroughly dried, in the course of two or three years, at least. Wood is often very much weakened by knots in it; also when cross-grained, as often happens in sawing, it will be weakened in a greater or less degree, according as the cut runs more or less across the grain. From all which it follows, that a considerable allowance ought to be made for the various strength of wood, when applied to any use where strength and durability are required. Iron is much more uniform in its strength than wood."

His statement about the properties of wood, for the engineer, has not been so greatly improved upon since it was written! Present day engineers have so little knowledge and training with wood that one would be delighted to think they were indeed as well informed as their predecessors. In the few extracted lines below, it is evidently not recognised that the situation is a difficult one to analyse in that it is concerned with either axial buckling imposed on a transverse bending situation or is one of failure requiring a theory of "combined loadings". "Every beam or bar, whether of wood, iron, or stone, is more easily broken by any transverse strain, while it is also suffering any very great compression endways; so much so indeed that we have sometimes seen a rod, or a long slender beam, when used as a prop or shoar (sic) urged home to such a degree, that it has burst asunder with a violent spring. Several experiments have been made on this kind of strain: a piece of white marble, ¼of an inch square, and 3 inches long, bore 38 lbs; but when compressed endways with 300 lbs, it broke with 14½ lbs. The effect is much more observable in timber and more elastic bodies, but is considerable in all."

The researchers named elsewhere from Germany and Holland but especially from France, point up the "route" by which much engineering science was finding its way to England in the early 19th century. The work of the men who were the products of the great new classless French l~coles is well told in Timoshenko's History of Strength of Materials (McGraw Hill, 1953 ). It may be observed that Scotland, historically, has been closer to France in cultural matters t h a n England - certainly its academic development followed

222 t h e c o n t i n e n t a l r a t h e r t h a n t h e E n g l i s h p a t t e r n . T h i s influence is d o u b t l e s s l y s o m e w h a t r e s p o n s i b l e for t h e r e l a t i v e l y high visibility in t h e 18th a n d 19th c e n t u r i e s of its universities, St. A n d r e w s , Glasgow, E d i n b u r g h a n d A b e r d e e n , a n d t h e subjects t h e y t a u g h t at this time. Recall h o w i m p o r t a n t was t i m b e r ; besides its m a n y o t h e r uses, n a v a l a n d c o m m e r c i a l s h i p s were still b e i n g b u i l t of w o o d w h e n H u t t o n ' s b o o k was writt e n in 1813 - t w o y e a r s b e f o r e t h e b a t t l e of T r a f a l g a r . T h e age of t h e I r o n c l a d was still a g e n e r a t i o n in t h e f u t u r e a n d t h e t r a n s i t i o n f r o m m a c h i n e s of w o o d to o n e s of iron was still t a k i n g place; t h e q u a n t i t y of iron u s e d in t h e e a r l y 1800s was r e l a t i v e l y small. T h e u l t i m a t e s t r e n g t h of s o m e c a s t m e t a l s is given b y H u t t o n , b u t u n f o r t u n a t e l y n o t t h e s t r a i n s at fracture, t h u s "The following list of the absolute strengths of several materials, (see Fig. 8b) is extracted from the collection made by Professor Robison, from the experiments of Muschenbroek and other experimentalists. The specimens are supposed to be prisms or cylinders of one square inch transverse area, which are stretched or drawn lengthways by suspended weights, gradually increased till the bars parted or were torn asunder, by the number of avoirdupois pounds, on a medium of many trials, set opposite each name." O f alloys H u t t o n observes, "...that almost all the metallic mixtures are more tenacious than the metals themselves. The change of tenacity depends on the proportion of the ingredients; and yet the proportion which produces the most tenacious mixture, is different in the different metals." He then remarks, "These are economical mixtures; and afford valuable information to plumbers for augmenting the strength of water-pipes. Also, by having recourse to these Tables...... the engineer can proportion the thickness of his pipes, of whatever metal, to the pressures they are to suffer." R e t u r n i n g to t i m b e r , a list of s t r e n g t h s for a wide v a r i e t y of w o o d s a n d b o n e is given, see Fig. 8 ( b ) . An a p p r e c i a t i o n of t h e c o n s e q u e n c e s o f t h e i n t e r a c t i o n a n d t i m e of loading, ( c r e e p ) is t h e n r e v e a l e d t h u s , "... the numbers express something more than the utmost cohesion; the weights being such as will very soon, perhaps, in a minute or two, tear the rods asunder. It may be said in general, that 2/3 of these weights will sensibly impair the strength, after acting a considerable while, and that one-half is the utmost that can remain permanently suspended at the rods with safety; and it is this last allotment that the engineer should reckon upon in his constructions. There is however considerable difference in this respect: woods of a very straight fibre, such as fir, will be less impaired by any load which is not sufficient to break them immediately." F o r a wide v a r i e t y o f m a t e r i a l s t h e load ( a c t u a l l y p r e s s u r e ) w h i c h m a y be safely s u p p o r t e d is n e x t listed a n d t h e n a " p r a c t i c a l r u l e " due to E m e r s o n for finding t h e load w h i c h c a n be p e r m a n e n t l y s u p p o r t e d b y four m a t e r i a l s , see Fig. 8 ( c ) .

223

Hutton then comes to experiments to determine the transverse strength of bodies. He notes that they are numerous, especially for timber, and that Belidor in his Science des Ingenieurs provided a table of results (though the kind of timber is not mentioned). But as a result of experiments made on small and large bars of oak, clear of knots, Buffon gave the time-deflection results quoted in Fig. 8 (d); the tests were made on simply supported beams, centrally loaded. Hutton goes on to quote from Buffon's surprising results and we read that "oak timber lost much of its strength in ... seasoning or drying"! The following, related observation is remarkable: "Mr. Buffon had found, by many trials, that oak timber lost much of its strength in the course of seasoning or drying; and therefore, to secure uniformity, his trees were all felled in the same season of the year, were squared the day after, and the experiments tried the 3d day. Trying them in this green state gave him an opportunity of observing a very curious phenomenon. When the weights were laid quickly on, nearly sufficient to break the beam, a very sensible smoke was observed to issue from the two ends with a sharp hissing sound; which continued all the time the tree was bending and cracking. This shows the great effects of the compression, and that the beam is strained through its whole length, which is shown also by its bending through the whole length."

The large number of materials mentioned in this section is remarkable as is Hutton's treatment of the mechanics for design, mostly by consideration of rules of proportion, of physical or material constants and of numbers particular to a specific loading situation 25. With these latter facts in mind we can begin to see how the subject, i.e. The Strength of Materials, came by its name. There can be few mechanical engineers who have not at some time wondered at the discrepancy between the name of this subject and its content. Present day Strength of Materials is about simple stress and strain analysis, not about the strength of materials as such; for this we need to turn to Materials Science. Contemporary Strength of Materials is only about calculations of elastic (mostly) stress distribution and very small deflections in metallic structures. The name has now become a misnomer. In the Age when the subject was being developed, structural materials were mostly of organic origin, inorganic manufactured materials beginning to come into consideration but not possessing a dominating importance. The increasing emphasis, towards the end of the 19th century on iron and steel, and roughly homogeneous materials, made the mathematisation of Strength of Materials easy to establish. Grain, moisture, aelotropy and anisotropy, all natural qualities evolved as part of a plant's response to the forces acting on it, are qualities difficult to work into elementary theories.

2~For an uniform end-loaded cantilever carrying a concentrated load W the deflection is 1/36 of WL3/E(bh3). Our calculations to-day in the usual notation, are done for the purpose of finding the number 1/36; WLa/E (bh 3) is obtainable using simple dimensional analysis.

224

Hutton, Secretary of the Military Society Capt. Alexander Jardine and Capt. Edward Williams, two young artillery officers, saw a need for an "association .. for the improvement" of "... military, mathematical and philosophical information" ... "and a Society to propagate it." This was outlined in a letter from the former to the latter in 1772. It was a few years before Hutton published his Copley Medal paper and the two Captains, ex-R.M.A, cadets, either knew of Hutton's impending research work or maybe even stimulated it. Jardine wrote that, "the mechanical parts of artillery and gunnery such as founding and powder-making etc., are not yet sufficiently established upon principles..." Apparently he gave lists of proposed members, subjects and textbooks for study. He mentioned "the science of metals of ordnance" and of "brass 26 getting worse of late", instancing the improved performance of iron guns in Spain. It was another eighty years before experiments on explosive charges in iron guns were carried out, though in 1816 nearly all brass pieces were abolished. Jardine also noted that "Muller was not properly encouraged" at the R.M.A. and, strangely, "the work he did was not acknowledged, with mechanics and gunnery scarcely if at all taught in the Corps." He urged, "Proper methods ... and building on the basic knowledge of Muller and Desaguliers." The Society proposed was clearly of a professional kind and as such was a kind of forerunner of the I.C.E. and I.Mech.E. It was established in October 1772 at Woolwich, meeting once a month. Coles says that the first paper, length 103 pages, entitled Some Experiments and Observations in Gunnery, was by Lieut. Jardine, Royal Artillery, Gibraltar 1771. Thirty other papers on gunnery and ammunition were listed. However, the Society did not long survive the transferences of its principal enthusiasts and the death of Major General Desaguliers. The Society's papers stayed with Hutton who was its Secretary but were passed to Gregory, to become the centre of a Regimental Library. Presently, they are incorporated into the Library of the Royal Artillery Institution, Woolwich. The belief that there was a need for more military knowledge and theory than for practical military acquintanceship, led to Lieuts. John F. Burgoyne and C.W. Paslay founding a Society for Producing Useful Military Information, along with six other colleagues in the Peninsular War. The Society did not long survive as such though the two principals went on to produce fine work of military importance. Bridge design As a result of a high and rapid flood the River Tyne brought down the Newcastle bridge in 1771. Hutton became interested in this subject and in 1772 he ~ I have d r a w n on Dr. H o w a r d Coles' work for t h i s section [ 1 ].

225 published The Principles of Bridges. It contains "demonstrations of the properties of arches, the thickness of piers (from where derived?), the force of the water against them (how well accepted to-day?), etc., together with practical observations and directions drawn from the whole" [3]. The book was out of print but republished in 1801 when Telford and Douglas's project for erecting an iron bridge over the Thames was made known. Many engineers at this period seemed easily to become consultants with virtually no previous experience of the subject on which they gave their opinions! Conclusion

The paper has endeavoured to sketch the life and professional activities of Charles Hutton. However the writer greatly doubts that he has obtained a balanced view of it. Certainly the critical examination of Hutton's many works is beyond the space available or the time at the author's disposal. However, it should help colleagues peer more deeply and knowingly into the origins of some of their disciplines, hopefully contribute to making lectures on impact topics more interesting, and student audiences more appreciative of their predecessors and their problems. Note 1

Because many of the persons mentioned in this account of Hutton were contemporaneous with him and would be well known to readers at that period, I have inserted brief summaries of their biographical dates and works; the information is mostly abstracted from the Dictionary (and the ConciseDictionary) of National Biography (Oxford University Press ), The Encyclopaedia Brittanica (1947), and the second of E.G.R. Taylor's two volumes [4]. This should ease the task of readers in acquainting themselves with the careers of the contemporaries of the Woolwich group of Professors and Hutton's especially. It will also convey an awareness of the variety of activities of the humbler research workers and teachers in the late 18th and early 19th centuries in England. (1) John E. BURGOYNE,1782-1871, was a product of the R.M.A., 1798, who served with great distinction in many military theatres especially in the Corunna campaign in Portugal, 1808-9, as chief of the engineers in the rearguard of the Army. He also served with distinction in the Peninsular War and became the commanding engineer on Wellington's staff. He served at Orleans and Mobile in the U.S.A. in 1814/15. Also, he was principal engineer adviser to Lord Raglan in the Crimean War at the Siege of Sevastopol. He was appointed Inspector General of Fortifications, 1845-68, and a Field Marshal in 1868. (2) Richard DUNTHORNE,1711-1775, is said to have been an astronomer

226

and butler 27 of Pembroke Hall, Cambridge. He published The Practical Astronomy o[ the Moon, 1739, and was scientific assistant to Dr. Roger Long, working on Long's Astronomy, 1770. He made a survey of the fens when superintendent of works for the Bedford Level Crossing Corporation. (3) William E M E R S O N , 1701-1782, of Darlington, is described in the D.N.B. as a mathematician and an unsuccessful private teacher. He was keenly interested in practical mechanics and constructed a spinning wheel for his wife. He was thought of by his contemporaries as an author to be noted for his "judgement and improvement in our subject". He declined to become a Fellow of the Royal Society, [4, p.167], writing that: "The little encouragement there is in this nation for promoting these sciences, is the reason that few people go any length this way. If they have gained so much knowledge as to be able to teach a common school to get a living by, they think it sufficient. Few make further progress, and if they do they get the pleasures and pains of their labour. For when a man is eminent in his discoveries, perhaps he is dignified by the title of F.R.S., but he has to pay a quarterly cess for the honour of it. This is the way that ingenuity is rewarded in England."

Eva Taylor [4 ] writes that he was an eccentric of small private means who devoted himself mainly to turning out mathematical manuals for young students. They were also designed for the self-educating man - "the mechanic" and were held in great esteem. Apparently he was a very "undisciplined" writer, seep. 88 of Ref. [4]. His books included Fluxions, 1749, Cyclomathesis, 1763, The Arithmetic of Infinites, 1767, Dialling, 1770, (also calledgnomonics: this refers to the art and science of constructing sundials for measuring time and for use in surveying mines ), and Principles of Mechanics, 1770. (4) Sir Robert H E A T H , d1770; edited The Ladies'Diary, 1744-53, and helped popularise mathematics in periodicals. After supersession by Thomas Simpson he carried on with rival publications, see Ref. [ 1 ]. (5) George Henry HUTTON, d1827, was the son of Charles Hutton, appointed a Lieutenant-General in the Army in 1821. Apparently he was also an archeologist and was made LL.D. of Aberdeen University. (6) Thomas L E Y B O U R N , 1770-1840, was a teacher of mathematics at the Military College, Sandhurst, 1802-39. He edited the Mathematical Repository during the period 1799-1835 and published A Synopsis of Data for the Construction of Triangles in 1802. (7) John M O N T A G U , 1718-1792, was the 4th Earl of Sandwich. Educated at Eton and Trinity College, Cambridge he was elected F.R.S. in 1740 but had a highly chequered career. He was reappointed First Lord of the Admiralty in 1771 and used the patronage of his office as "an engine for bribery and political 27Originally this was an officer who had charge of the wine for a royal (or high) table and hence was the title of a person of high rank.

227

jobbery" so that when war with the American colonies took place in 1778, "the navy was inadequate and naval storehouses empty"! (8) C.W. PASLAY, 1770-1861, served in Minorca, Malta, Naples, Sicily, Spain and Holland between 1799 and 1809. He was made F.R.S. in 1816. He introduced courses in military engineering, on sapping, mining, pontooning and exploding gunpowder (electrically) on land and under water. He published many treatises on military engineering. He was made a General in 1860. (9) Richard PEARSON, 1765-1836, was a physician in Birmingham and author of several medical treatises. (10) George SHAW, 1751-1813, some time student of botany and medicine at Oxford and Edinburgh and later lecturer in botany, abandoned the church as a vocation He was a joint Founder of the Linnean Society in 1788, made a F.R.S. in 1789, and was a Keeper of natural history at the British Museum. (11) Robert SIMSON,1687-1768, held a M.A. degree from Glasgow which was awarded in 1711. He was Professor of Mathematics in the University of Glasgow, 1711-61, and had a reputation for his rediscovery of the "Grecian Geometry". He published his Elements of Euclid, 1756, simultaneously in Latin and English, but his Sectionum Conicarum Libri V appeared only in 1835. He attempted a restoration of three lost works, the Porisms of Euclid and two minor works of Apollonius in 1749. (Porisms2S: this is a difficult term to understand but briefly is something directed to finding what is proposed. For Simson a locus is a kind of porism.) Sir Thomas Heath wrote, "The merits of Simson both as interpreter and critic of Euclid are very great". (Introduction to The Elements o/Euclid, 1930, J.M. Dent, Ltd., London. ) The latter should be read for an authoritative and appreciative estimate of Simson's works. Heath republishes Isaac Todhunter's Original Preface in which the latter writes that he reproduces substantially that of Simson. It is this which Sir Thomas Heath uses for Dent's Everyman edition. (12) Matthew STEWART,1717-1785, was a geometrician. He studied at Edinburgh University under Colin Maclaurin and achieved some reputation for his General Theorems, published in 1746. He was a minister of the church 1745-7 but became a professor of mathematics at Edinburgh University and occupied the post 1747-1785. Some of his duties were performed by his son. Dugald, after 1772, but why I do not know. He was elected F.R.S. in 1764, his chief work being Tracts Physical and Mathematical, 1764, which applied geometrical demonstrations to astronomy. (13) Dugald STEWART, 1753-1828, was the son of Matthew (q.v). He was a professor of Moral Philosophy but lectured, instead of his father, in mathematics, after 1772! (14) John TIPPER, d1713, started The Ladies' Diary almanac in 1704 which 2SSee Hutton's 10,000-word article on this topic in his Phil. and Math. Dictionary, 1815.

228

includes a serial collection of mathematical papers. He edited the Diary until his death in 1713. Tipper also founded the Great Britain's Diary in 1710. (15) George TOWNSEND (or TOWNSHEND), 1724-1807. Fourth Viscount and first Marquis Townshend, graduated M.A. from St. John's Cambridge in 1749. He had a tumultuous military career, fighting at Culloden 1746, and with the Duke of Cumberland's army abroad, but retired from it after a difference of opinion in 1750. He took command from General Thomas Wolfe after his death in 1759 in the battle against the French, at the Heights of Abraham, Quebec. He was censured after being accused of ingratitude to him. He was made Lieutenant-General of Ordnance in 1763 but became Lord-lieutenant of Ireland in 1767, where among other things he engaged "in corruption and took to dissipated habits", being re-called to the U.K. in 1772. He was re-appointed Master-General of Ordnance in 1772 and served until 1782, and again from 1783-84. He became a field marshal in 1796. (16) Samuel VINCE, 1749-182t, was originally a Suffolk bricklayer with his father (who was a plasterer) till the age of 12, when the Rev. Mr. Warnes noticed him "sitting reading beside his hod of mortar". He was also at some time an usher (as under-master) at Harleston, Norfolk. After school at St. Paul's, he was admitted as a sizar 2~ at Caius College, Cambridge, at age 21, in 1771. He graduated B.A., Senior Wrangler and First Smith's Prizeman in 1773. He migrated to Sydney but returned, becoming Plumian Professor, 1796-1821. He was ordained and held various benefices. He was elected F.R.S. in 1786 and for his experiments on friction, reported in the Phil. Trans., Vol. 75, 1785, he received the Copley Medal. Vince was described as "an accomplished mathematician and an aimiable man". He was author of texts in particular On Friction and On Resistance of Fluids, as well as works on Astronomy, Optics and Christianity. He illustrated his lectures on the former topics with various machines and "philosophical instruments". ( 17 ) Edward WARING,1734-1798, was Lucasian Professor of Mathematics, Cambridge, during 1760-98, and a F.R.S.; he published mainly on algebra. (See also the footnote on p. 165 of Ref. [ 1 ] ). Note 2

Peter Barlow D.E. Smith, History of Mathematics, Vol. 1, (Dover, 1958) writes of Peter Barlow, on p. 460, 29An undergraduate at Trinity College receiving support from his college but, earlier, one also performing certain menial duties, later taken over by the college servants.

229 "... the British mathematician of this period who showed the greatest genius, or at any rate the greatest perseverance was Peter Barlow, 1776-1862. Born of humble parents, he became one of the leading writers of England on the theory of numbers ~°, professor of mathematics at the Woolwich Military Academy and a Fellow of the Royal Society, (1823). His contributions to the magnetic theory, the strength of materials and optics were also noteworthy, and his mathematical dictionary 3' is still a valuable source of information. His tables of factors, reciprocals, powers, roots, hyperbolic logarithms and primes, should also be mentioned as the best of the earlier publications of the kind, being still looked upon as standards,~. ''

The dictionary (unpaginated) in one volume contains about 1000 pages. In the Introduction to it, Barlow writes that "Stone's mathematical dictionary is one small octavo volume whilst Dr. H u t t o n ' s dictionary is in two volumes quarto, the first is too small to convey much essential information on such an extensive subject ... the latter though an excellent and comprehensive performance, cannot of course contain many of the more recent improvements besides it being necessarily limited to a comparatively small number of purchasers, its size and price being such as not to meet the views or suit the conveniences of a number of persons ..." Barlow's book as he himself says, contains nothing on the "exploded science of Astrology ... Fortification and Military Affairs..." Barlow is writing of course of the first edition of Hutton's Dictionary, i.e. 1795 and 1796. The much extended edition of 1815 stands, in the writer's opinion, as a great achievement; H u t t o n was then 78 years of age. It is in two volumes, totalling 1500 pages, quarto. A great pity of both these dictionaries is that the origin of the information contained in either is not stated. There is no doubt that though Barlow's work is a significant one, H u t t o n ' s is much greater. Their dates of publication incidentally, 1814 and 1815, make for thoughts of mutual competition! The articles on Porisms may, in both dictionaries, stand as an example of the erudition of both writers; one wonders if the subject has been anywhere better discoursed upon in subsequent years. The remarks of present day writer, J.F. Bell in his Mechanics of Solids, Vol. I (Springer Verlag, 1984), show him to have been led to a different assessment of Barlow's work; he is very severe (footnote to p. 17), writing of Barlow that "his mathematical limitations are a matter of public record". This is unproven and his censure seems to be based on a careful examination of only one of Barlow's works, namely, A Treatise on the Strength of Timber, Cast Iron ..., J. Weale, London, 1833. Bell's remarks pertain to his perceived performance of Barlow as an experimentist in respect of this one book and not as an early 19th century student of mathematics, magnetism and optics. A fair assessment of ~°An Elementary Investigation of the Theory of Numbers, London, 1811. :~'A New Mathematical and Philosophical Dictionary, London, 1814. :~2NewMathematical Tables, London, 1814. De Morgan published an edition in 1856.

230 Barlow for m o s t readers will a t t e n d at least on a reading of the e n t r y a b o u t him in the Dictionary of National Biography, Vol. I, p. 1142, Oxford University Press.

Acknowledgments I wish to acknowledge t h e help a n d assistance of Mr. A.J. Clark of the Royal Society of L o n d o n ' s library in supplying me with copies of p a p e r s of y e s t e r y e a r a n d in seeking answers to some of t h e questions I a s k e d of him. I also e x t e n d m y a p p r e c i a t i o n to P r o f e s s o r A.G. A t k i n s for initially sending to me e x t r a c t s f r o m H u t t o n ' s Course of Mathematics, Vol. III. I t h a n k P r o f e s s o r F.W. T r a v i s for reading t h e p a p e r a n d helping it to publication. Lastly, I express m y deep a p p r e c i a t i o n of t h e l e n g t h y efforts required of m y wife, in t y p i n g a n d in various o t h e r ways bringing m y m a n u s c r i p t to a publishable form.

References 1 W. Johnson, The Woolwich Professors of Mathematics, 1743-1900, J. Mech. Working Technol., 18 (1989) 145-194. 2 W. Johnson, The Magnus Effect: Early investigations and a question of priority, Int. J. Mech. Sci., 28 (1986) 859-872. 3 O. Gregory, The Imperial Magazine, March, 1823. 4 E.G.R. Taylor, The Mathematical Practitioners of Hannoverian England, 1714-1840, Cambridge University Press, 1966. 5 H. Douglas, Naval Gunnery, 1855, Conway Maritime Press, 1982. 6 D. Stimson, Scientists and Amateurs: A History of the Royal Society, Sigma Books Ltd., 1949. 7 W. Johnson, Benjamin Robins's New Principles of Gunnery, Int. J. Impact Eng., 4 (1986) 205-220. 8 B. Brodie and F.M. Brodie, From Cross-bow To H-Bomb, Indiana University Press, 1973.