Lectures on Ten British Physicists of the Nineteenth Century/Lecture 2

William John Macquorn Rankine was born in Edinburgh, Scotland, on the 5th of July, 1820, He was by descent a Scot of Scots. His father, David Rankine, descended from the Rankines of Carrick, could trace his descent back to Robert the Bruce. Carrick is a hill district of Ayrshire in the southwest of Scotland, famous for its breed of dairy cattle. Before his accession to the Crown of Scotland, Robert the Bruce was Earl of Carrick. In youth Rankine's father was a lieutenant in the regular army, but later in life he became a railroad engineer and eventually Secretary of the Caledonian Railway Company. His mother was Barbara Graham, daughter of a Glasgow banker, and second cousin of Thomas Graham who is celebrated for his investigation of the diffusion of gases and liquids.

Rankine spent his first years in Ayrshire among the Carrick Hills, which he afterwards celebrated in verse, for Rankine, like Maxwell, was an amateur poet:

Come busk ye braw, my bonnie bride,
And hap ye in my guid gray plaid,
And ower the Brig o' Boon we'll ride
Awa' to Carrick Hills, love.

For there's flowery braes in Carrick land,
There's wimplin' burns in Carrick land,
And beauty beams on ilka hand
Amang the Carrick Hills, love.

There dwalt my auld forefathers lang,
Their hearts were leal, their arms were strang,
To thee my heart and arm belang
Amang the Carrick Hills, love.

I'll bear thee to our auld gray tower,
And there we'll busk a blythesome bower,
Where thou shalt bloom, the fairest flower,
Amang the Carrick Hills, love.

In spring we'll watch the lammies play,
In summer ted the new-mown hay,
In harvest we'll sport the lee-lang day
Amang the Carrick Hills, love.

When winter comes wi' frost and snaw,
We'll beet the bleeze, and light the ha',
While dance and song drive care awa'
Amang the Carrick Hills, love.

In these verses we have a description by Rankine of the scenes and pastimes in which he spent his earliest years. Carrick borders on Galloway, and there, ten years later, Clerk-Maxwell grew up in a similar environment. After some preliminary education at home he was sent when eight years of age, to the public school; first to the Academy of the neighboring town of Ayr, afterward to the High School of the City of Glasgow. But his health broke down, and he was restricted for some years to private instruction at his home now in Edinburgh. To his father he was indebted for superior instruction in arithmetic, elementary mathematics, mechanics and physics. When 14 years of age he received from his mother's brother a present, which had a powerful effect on his subsequent career—a copy of Newton's Principia. To his private study of that book and of other books of the like order, he was indebted for his skill in the higher mathematics. While his education proceeded at home, he received instruction in the composition and playing of music, which enabled him in after years to compose the tunes for his own songs.

At the age of 16 he entered the University of Edinburgh. Instead of taking a regular course, he selected chemistry, physics, zoology and botany. Forbes was then the professor of physics; Rankine attended his class twice; the first year he received the gold medal for an essay on "The Undulatory Theory of Light," and the subsequent year an extra prize for one on "Methods of Physical Manipulation." It appears that he did not enter any class of pure mathematics at the University, having already advanced beyond the parts then taught. At this time he was attracted, like many other mathematicians at the beginning of their independent career, by the theory of numbers. In his leisure time he studied extensively the works of Aristotle, Locke, Hume, Stewart, and other philosophers.

Before finishing his studies at the University of Edinburgh, he had gained some practical experience by assisting his father in his work as a railroad engineer. There was then no professor of engineering at the University of Edinburgh; some 20 years later Fleeming Jenkin was appointed, and given that whole province which is now divided at this University into four great departments. Hence, at the age of 18, Rankine was made a pupil of Sir John Macneill, civil engineer, and as a pupil he was employed for four years on various surveys and schemes for river improvements, waterworks, and harbors in Ireland. It was then that he became personally acquainted with the "gorgeous city of Mullingar," which he has described minutely and gracefully in an ode to its praise. He was likewise employed on the construction of the Dublin and Drogheda railway and it was while so engaged that he contrived the method of setting out curves which is known as Rankine's method.

Having finished his term of pupilage, he returned to his father's home in Edinburgh, and commenced the practice of his profession. One of the first projects entrusted to his care was rather singular. In 1842 Queen Victoria visited Scotland for the first time, and resided for several days in the home of her Stuart ancestors Holyrood Palace in Edinburgh. Royal visits to Scotland were not so frequent then as they afterwards became. One manifestation of rejoicing took the form of a large bonfire on the top of Arthur's Seat, a precipitous rock which rises 700 feet above the level of the park surrounding the palace. To Rankine was entrusted the engineering of this bonfire. Applying his knowledge of chemistry he constructed the pile of fuel with radiating air passages underneath.

It was now, when he was 22 years of age, that he published his first scientific pamphlet "An experimental inquiry into the advantage of cylindrical wheels on railways." The course of experiments was suggested by his father, and was carried out by father and son working together. It was followed by a series of papers on subjects suggested by his father's railroad experience; of which one was on the "Fracture of axles." He showed that such fractures arose from gradual deterioration or fatigue, involving the gradual extension inwards of a crack originating at a square-cut shoulder. In this paper the importance of continuity of form and fiber was first shown, and the hypothesis of spontaneous crystallization was disproved. His father was connected with the Caledonian Railway Company, and by that Company young Rankine was professionally employed on various schemes. The work in Ireland had impressed on him the great importance of an abundant supply of pure water to the health of a city. He brought forward a scheme for supplying the city of Edinburgh with water from a lake in the hilly region to the south; a scheme which was thorough and would have solved the problem once and for all. It was defeated by the existing Water Company, with the result that to this day the water supply of the city of Edinburgh is defective.

While engaged in engineering work in Ireland, he had thought much on the mechanical nature of heat, a doctrine which was then engaging the attention of the scientific world. In reading the Principia of Newton, Rankine must have observed how the action of heat was a difficulty in the theory of Dynamics. In France, Carnot had in 1820 given a theory of the heat-engine which assumed that heat was a material substance. Mayer had advanced the theory that heat is a mode of motion. Rankine to explain the pressure and expansion of gaseous substances due to heat, conceived the hypothesis of molecular vortices. He worked out his theory, but owing to the want of experimental data, did not publish immediately. In 1845 Joule brought to a successful result a series of experimental investigations designed to measure the exact mechanical equiva- lent of a given amount of heat. In 1849 William Thomson, professor of physics at Glasgow, gave an account to the Royal Society of Edinburgh of Carnot's theory, and the problem then was, "How must the theory of the heat-engine be modi- fied, supposing that heat is not a substance, but a mode of motion?" Rankine reduced his results to order, and contributed them to the Royal Society of Edinburgh in two papers entitled "On the mechanical action of heat, especially in gases and vapors" and "The centrifugal theory of elasticity as applied to gases and vapors." He was elected a fellow, and read his papers early in 1850. That same year the British Association met in Edinburgh. Rankine was Secretary of Section A, and he had ready an elaborate paper "On the laws of the elasticity of solid bodies," in which the same hypothesis of molecular vortices is the guiding idea.

Rankine was not content to suppose the heat of a body to be the energy of the molecules due to some kind of motion. He supposed, like the other pioneers in thermodynamics, that the invisibly small parts of bodies apparently at rest are in a state of motion, the velocity of which, whether linear or angular, is very high. But he went further; he imagined the motion to be like that of very small vortices each whirling about its own axis; from which it would follow that the elasticity of a gas is due to the centrifugal force of this motion; an increase of angular velocity would mean an increase of centrifugal force. His own statement of the hypothesis is as follows: "The hypothesis of molecular vortices may be defined to be that which assumes that each atom of matter consists of a nucleus or central point enveloped by an elastic atmosphere, which is retained in its position by attractive forces, and that the elasticity due to heat arises from the centrifugal force of those atmospheres, revolving or oscillating about their under or central points." Rankine's molecular vortex is the attracting point of Boscovich surrounded by an elastic atmosphere.

Maxwell wrote in Nature in 1878: "Of the three founders of theoretical thermodynamics (Rankine, Thomson, Clausius) Rankine availed himself to the greatest extent of the scientific use of the imagination. His imagination, however, though amply luxuriant, was strictly scientific. Whatever he imagined about the molecular vortices with their nuclei and atmospheres was so clearly imaged in his mind's eye, that he, as a practical engineer, could see how it would work. However intricate, therefore, the machinery might be which he imagined to exist in the minute parts of bodies, there was no danger of his going on to explain natural phenomena by any mode of action of this machinery which was not consistent with the general laws of mechanism. Hence, though the construction and distribution of his vortices may seem to us as complicated and arbitrary as the Cartesian system, his final deductions are simple, necessary, and consistent with facts. Certain phenomena were to be explained. Rankine set himself to imagine the mechanism by which they might be produced. Being an accomplished engineer, he succeeded in specifying a particular arrangement of mechanism competent to do the work, and also in predicting other properties of the mechanism which were afterwards found to be consistent with observed facts."

In his paper on the "Mechanical Action of Heat," Rankine applied the dynamical theory of heat and his hypothesis of molecular vortices, to discuss new relations among the physical properties of bodies, and especially to a relation between the true specific heat of air, the mechanical equivalent of heat, and certain other known constants. He found, using the value for the mechanical equivalent which had just been published by Joule, that the true specific heat of air relative to that of water has the value 0.2378. The best value for that quantity which had been obtained by direct experiment was that of De la Roche and Bérard, 0.2669. Rankine concluded, not that his theory was wrong, but that Joule's result was too small. On further examination of Joule's investigation, just printed in the Philosophical Transactions, he concluded that De la Roche and Bérard's value was too large; and predicted that the true specific heat of air would be found to be 0.2378. Three years later Regnault obtained by direct experiment the value 0.2377.

Soon after this he moved to Glasgow, and founded there the firm of Rankine and Thomson civil engineers. They took up a scheme for supplying the City of Glasgow with water from Loch Katrine. They were not the originators of the scheme, but they were successful in carrying it out. The City of Glasgow solved effectively the problem of an abundant supply of pure water; and in so doing commenced a career which has made it the model municipality of the British Islands. As a resident of Glasgow he became an active member of the Philosophical Society of Glasgow; and to that Society he contributed in 1853 one of his most important memoirs "The general law of the transformation of energy." Two years later he contributed "Outlines of the science of energetics," on the abstract theory of physical phenomena in general, which has now become the logical foundation for any treatise on physics. In it he introduces and defines exactly a number of terms which were then strange or altogether new, but are now familiar concepts in physical science, such as "actual energy" and "potential energy."

To the doctrines of the Conservation and Transformation of Energy, Prof. William Thomson added the doctrine of the dissipation of energy. This doctrine asserts that there exists in nature a tendency to the dissipation or uniform diffusion of mechanical energy originally collected in stored up form; in consequence of which the solar system (and the whole visible universe) tends towards a state of uniformly diffused heat; in which state according to the laws of thermodynamics no further transformation of energy is possible; in other words, nature tends towards a state of universal death. Rankine speculated as to how this dire result may be provided against in nature, and contributed to the meeting of the British Association, held at Belfast in 1852 a paper "On the reconcentration of the mechanical energy of the universe." "My object," he said, "is to point out how it is conceivable that, at some indefinitely distant period, an opposite condition of the world may take place, in which the energy which is now being diffused may be reconcentrated into foci, and stores of chemical power again produced from the inert compounds which are now being continually formed. There must exist between the atmospheres of the heavenly bodies a material medium capable of transmitting light and heat; and it may be regarded as almost certain that this interstellar medium is perfectly transparent and diathermanous; that is to say, that it is incapable of converting heat or light from the radiant into the fixed or conductible form. If this be the case, the interstellar medium must be incapable of acquiring any temperature whatever, and all heat which arrives in the conductible form at the limits of the atmosphere of a star or planet, will there be totally converted, partly into ordinary motion by the expansion of the atmosphere, and partly into the radiant form. The ordinary motion will again be converted into heat, so that radiant heat is the ultimate form to which all physical energy tends; and in this form it is, in the present condition of the world, diffusing itself from the heavenly bodies through the interstellar medium. Let it now be supposed, that, in all directions round the visible world, the interstellar medium has bounds beyond which there is empty space. If this conjecture be true, then on reaching those bounds, the radiant heat of the world will be totally reflected, and will ultimately be reconcentrated into foci. At each of these foci the intensity of heat may be expected to be such, that should a star (being at that period an extinct mass of inert compounds) in the course of its motions arrive at that point of space, it will be vaporized and resolved into its elements; a store of chemical power being thus reproduced at the expense of a corresponding amount of radiant heat. Thus it appears, that although, from what we can see of the known world, its condition seems to tend continually towards the equable diffusion in the form of radiant heat, of all physical energy, the extinction of the stars, and the cessation of all phenomena; yet the world, as now created, may possibly be provided within itself with the means of reconcentrating its physical energies, and renewing its activity and life. For aught we know, these opposite processes may go on together, and some of the luminous objects which we see in distant regions of space may be, not stars, but foci in the interstellar ether."

In 1853 Rankine was elected a Fellow of the Royal Society of London; and in the following year he sent to that Society one of his important memoirs "The geometric representation of the expansive action of heat."

Glasgow University was in advance of the Edinburgh University in having a chair of civil engineering and mechanics. At the beginning of 1855 the incumbent of the chair was incapacitated by ill health, and Rankine acted as substitute for the remainder of the session. That same year at the age of 35 he was appointed to the chair.

Professor Rankine has been described by an intimate friend, Professor Tait: "His appearance was striking and prepossessing in the extreme, and his courtesy resembled almost that of a gentleman of the old school. His musical tastes had been highly cultivated, and it was always exceedingly pleasant to see him take his seat at the piano to accompany himself as he sang some humorous or grotesquely plaintive song words and music alike being generally of his own composition. His conversation was always interesting, and embraced with equal seeming ease all topics, however various. He had the still rarer qualification of being a good listener also. The evident interest which he took in all that was said to him had a most reassuring effect on the speaker, and he could turn without apparent mental effort from the prattle of young children to the most formidable statement of new results in mathematical or physical science, when his note-book was at once produced, and in a few lines he jotted down the essence of the statement, to be pondered over at leisure, provided it did not at once appear to him how it was to be modified. The questions which he asked on such occasions were always almost startlingly to the point, and showed a rapidity of thought not often met with in minds of such caliber as his, where the mental inertia which enables them to overcome obstacles, often prevents their being quickly set in motion. His kindness, shown in the readiness with which he undertook to read proof sheets for a friend, or even to contribute a portion of a chapter (when the subject was one to which he had paid special attention) was, for a man so constantly at work, absolutely astonishing."

It is customary in the Scottish Universities for a new professor to deliver an inaugural lecture on some subject of general interest connected with his chair; and at that time the discourse was in the Latin language. Professor Rankine chose for his subject "De concordia inter scientiarum machinalium contemplationem et usum"; or the concord in the mechanical sciences between theory and practice; it is printed as a preliminary dissertation in his Manual of Applied Mechanics. In it he traces from ancient down to medieval times the course of the fallacy that there is a double system of natural laws, one theoretical, geometrical, rational, discoverable by contemplation, applicable to celestial ethereal indestructible bodies, and a fit object for the noble and liberal arts; the other system practical, mechanical, empirical, discoverable by experience, applicable to terrestrial gross destructible bodies, and fit only for what were once called the vulgar and sordid arts. And he showed that this fallacy, although no longer formally maintained, still exerted an influence. In reference to this, Professor Greenhill has observed "Although the double system of natural laws mentioned by Rankine is now exploded, we still have a double system of instruction in mechanical textbooks, one theoretical, general, rational; the other practical, empirical, discoverable by experience. It should be the object of modern science to break down the barriers between these two systems, and to treat the subject of mechanics from one point of view."

Appointed to the chair of engineering, Rankine was soon the recipient of many honors. He was made president of the section of engineering, when the British Association met in Glasgow; and the following year, on the occasion of their meeting in Dublin, he received from the University of Dublin the honorary degree of LL.D. The following year he was chosen the first president of the Institution of Engineers in Scotland, an organization of which he had been a principal promoter. Professor Rankine had by this time abundantly proved himself as a pathfinder in the undiscovered regions of science; he was now to prove himself as a roadmaker. His practice as an engineer had made him fully alive to the important difference between the crude results of theoretical reasoning from principles and the reduced formulas adapted to the data obtainable from observation or specification. No sooner was he settled in his chair, than he began the preparation of his celebrated series of engineering manuals. In 1857 appeared Applied Mechanics; in 1859 Steam-engine; in 1861 Civil Engineering; in 1869 Machinery and Mill Work; supplemented in 1866 by Useful Rules and Tables. These manuals have gone through many editions, and there is still a demand for them. Why this phenomenal success? Professor Tait answered, "Rankine was peculiarly happy in discriminating between those branches of engineering knowledge which grow from daily experience, and those which depend on unchangeable scientific principles. In his books he dealt almost exclusively with the latter, which may, and certainly will, be greatly extended, but so far as they have been established can never change. . . . Really original papers and monographs rapidly lose their interest and importance, except as historical landmarks, but Rankine's works will retain their value after this generation has passed away."

In 1859 the volunteer movement spread over Great Britain. In view of possible invasion of the country it was thought that the regular army and the militia ought to be supplemented by bodies of trained citizens; the motto was for defence, not defiance. The movement spread to the University of Glasgow, and Rankine, true to transmitted instincts, gave in his name. He was made captain, and rose to be senior major; but after serving for five years he was obliged to resign on account of the pressure of his professional duties and of the labor involved in the preparation of the manuals. In 1861 he was made president of the Philosophical Society of Glasgow, and from the chair he delivered an address "On the use of mechanical hypothesis in science, especially in the theory of heat." The address shows a clear appreciation of the logical bearing of scientific hypothesis. He had been criticised for holding the hypothesis of molecular vortices. "In order to establish," he said, "that degree of probability which warrants the reception of a hypothesis into science, it is not sufficient that there should be a mere loose and general agreement between its results and those of experiment. Any ingenious and imaginative person can frame such hypotheses by the dozen. The agreement should be mathematically exact to that degree of precision which the uncertainty of experimental data renders possible, and should be tested in particular cases by numerical calculation. The highest degree of probability is attained when a hypothesis leads to the prediction of laws, phenomena and numerical results, which are afterwards verified by experiment, as when the wave-theory of light led to the prediction of the true velocity of light in refracting media, of the circular polarization of light by reflection, and of the previously unknown phenomena of conical and cylindrical refraction; and as when the hypothesis of atoms in chemistry led to the prediction of the exact proportions of the constituents of innumerable compounds. . . . I think I am justified in claiming for the hypothesis of molecular vortices, as a means of advancing the theory of the mechanical action of heat, the merit of having fulfilled the proper purposes of a mechanical hypothesis in physical science, which are to connect the laws of molecular phenomena by analogy with the laws of motion; and to suggest principles such as the second law of thermodynamics and the laws of the elasticity of perfect gases, whose conformity to fact may afterwards be tested by direct experiment. And I make that claim the more confidently that I conceive the hypothesis in question to be in a great measure the development and the reduction to a precise form of ideas concerning the molecular condition which constitutes heat, that have been entertained from a remote period by the leading minds in physical science. . . . I wish it, however, to be clearly understood, that although I attach great value and importance to sound mechanical hypotheses as means of advancing physical science, I firmly hold that they can never attain the certainty of observed facts; and, accordingly, I have labored assiduously to show that the two laws of thermodynamics are demonstrable as facts, independent of any hypothesis; and in treating of the practical application of those laws, I have avoided all reference to hypothesis whatever."

The pressure of a gas is now explained by the impacts and collisions of the molecules. But a sound hypothesis, although displaced, may afterwards turn out to be very valuable. When Crookes started on a search for Newton's corpuscles by constructing a radiometer, he was generally laughed at and his motives explained away by the received hypotheses, but in passing electric discharges through glass tubes exhausted more perfectly than had been done before, he hit on the phenomena of radiant matter, which are now explained by corpuscles much smaller than the atoms.

Rankine was a frequent attendant at the meetings of the British Association, where his social gifts, added to his scientific eminence, made him a conspicuous figure. He was president of the section of engineering, and also of the section of mathematics and physics; and rose to be "King" of the social section known as Red Lions. At the meeting held at Bath in 1864 he produced "The Three-foot Rule," a song about standards of measure, and sang it, to his own accompaniment and in the capacity of a British workman.

When I was bound apprentice, and learned to use my hands,
Folk never talked of measures that came from foreign lands;
Now I'm a British Workman, too old to go to school,
So whether the chisel or file I hold, I'll stick to my three-foot rule.

Some talk of millimeters, and some of kilograms,
And some of deciliters, to measure beer and drams;
But I'm a British Workman, too old to go to school,
So by pounds I'll eat, and by quarts I'll drink, and I'll work by my three-foot rule.

A party of astronomers went measuring of the Earth,
And forty million meters they took to be its girth;
Five hundred million inches, tho', go through from pole to pole;
So let's stick to inches, feet and yards, and the good old three-foot rule.

The great Egyptian pyramid's a thousand yards about;
And when the masons finished it, they raised a joyful shout;
The chap that planned that building, I'm bound he was no fool,
And now 'tis proved beyond a doubt he used a three-foot rule.

Here's a health to every learned man, that goes by common sense,
And would not plague the workman by any vain pretence;
But as for those philanthropists who'd send us back to school,
Oh! bless their eyes, if ever they tries to put down the three-foot rule.

This song indicates the great inconvenience and expense which would for a short time follow the change to the metric system; but it says nothing of the enormous inconvenience and expense which must always accompany the continued use of that muddle of units which prevails in Great Britain, and to a lesser degree in the United States. The want of system in the units obscures and clouds the whole subject of arithmetic; the school boy's time is spent on artificial reductions instead of the real relations existing between quantities. Consider the great convenience of the American decimal system of coinage, compared with the pounds, shillings, pence and farthings still inflicted on commerce in the old country. It was learned men of the three-foot rule type who prevented the decimal reform of the coinage advocated by De Morgan.

"Too old to go to school" is a sentiment worthy of the Chinese, and its prevalence in Great Britain for generations is a cause which at the present moment threatens her industrial supremacy. The argument drawn from the length of the polar axis of the Earth, is said to be due to Sir John Herschel. At one time it was possible to choose the yard and the pound, but that time has been allowed to slip away. The system of electric units, universally adopted, calls for a change to the meter and the kilogram. Had Rankine received any part of his education abroad, he would probably have opposed this insular idea; his colleague, Sir William Thomson was so educated, and has all his life been an enthusiastic advocate of the metric system.

When one sails up the river Clyde towards Glasgow, he sees on either bank a long succession of shipbuilding yards. Glasgow was in Rankine's time famous for its naval architects and shipbuilders, and they were Rankine's special friends. Hence, he was led into a number of investigations which are of importance in navigation. One of his papers is on the exact form of waves near the surface of deep water, and another investigates the lines of motion of water flowing past a ship. M. Napier, a naval architect, asked him to estimate the horsepower necessary to propel at a given rate a vessel which he was about to construct; and supplied him in confidence with the results of a great number of experiments on the horsepower required to propel steamships of various sizes and figures at various speeds. Rankine deduced a general formula, which he communicated to Napier directly and to the world at large in the form of an anagram: 20A. 4B. 6C. 9D. 33E. 8F. 4G. 16H. 10I. 5L. 3M. 15N. 14O. 4P. 3Q. 14R. 13S. 25T. 4U. 2V. 2W. 1X. 4Y.

The meaning of this anagram was afterwards explained as follows: "The resistance of a sharp-ended ship exceeds the resistance of a current of water of the same velocity in a channel of the same length and mean girth, by a quantity proportional to the square of the greatest breadth, divided by the square of the length of the bow and stern." Rankine and his naval friends prepared an elaborate Treatise on Shipbuilding which was published in 1866.

Rankine's only brother had died while yet young, and it seems that in later life his father and mother lived with him. In 1870 his father died, and in the following year his mother. Rankine never married; when he composed the song about a bridal tour to the Carrick Hills, his eyesight had failed so that he could not read. He had undertaken to write the memoir of John Elder, a shipbuilder, and this he was able to finish in 1872. Mrs. Elder endowed his chair so that it is now called the John Elder Chair of Engineering; it was however too late to benefit Rankine. A substitute had to be appointed to take charge of his classwork; and at the close of the year he died suddenly; not of any special disease, but as the result of overwork. His death occurred on the 24th of December, 1872, in the 53d year of his age.

When I first came to this country and attended a meeting of the American Association for the Advancement of Science, I was eagerly sought out by a professor of the Stevens Institute who was a great admirer of Rankine and desired to learn about his personality. I had to say that I had never met Rankine but that he could learn something of the man from the Collection of his Songs and Fables. The fables are founded on the curious signs which distinguish inns in England, such as the "Swan with two necks," the "Cat and Fiddle," etc.; they are illustrated by a lady who was a cousin of Maxwell and who also depicted scenes in Maxwell's country life. From my conversation with this professor I learned how widely the engineering manuals of Rankine were used in the United States, that his thermodynamic researches were well known, and that his name was everywhere held in high honor.

  1. This Lecture was delivered on March 18, 1902.—Editors.