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really incites a person to a hard effort, pressure, and discovered certain laws and the gratification which the master- relative to the same, that the specific ing of an important point gives, forms heat of such a gas is independent of its an incentive to further advance which temperature and density, and the proacts more potently than the repulsive duct of the density and the specific heat part of the difficulty. Let us not accept is the same constant for all permanent the valuable principles reached without gases. From the velocity of sound, making at least an honest effort to follow independent of the principles of thermo. the masters who have discovered them, dynamics, the specific heat at constant and thus make the principles in reality volume for a permanent gas can be

But if our comprehension and determined, and these two quantities efforts will not permit us to do this, let when substituted in an equation of the us not cry down the methods by which theoretical value of the mechanical they were attained, and let us study the equivalent of heat, expressed as a funcprinciples themselves thoroughly. tion of the pressure, volume, dilatability,

There is a wonderful harmony in and the two specific heats of a permanent nature—without it all scientific research gas give for this equivalent a theoretical would be of no avail. There are certain value which agrees with that practically great invariable laws, some determined determined by Joule, and recently veriand many more not yet conceived, but in fied by Professor Rowland. I need not our faith in and knowledge of the relia. dwell upon the mathematical exposition bility of those laws exists the only safe- of this, but reference to Stewart (“Eleguard of advance. This opens up that mentary Treatise on Heat,” pages 330 & large realm of reasoning known as “ by 412), Maxwell (“Theory of Heat,” pages analogy,” and enables us to determine 169, 228, & 310), Cotterill (“The Steam thus, and by analysis, laws which are Engine Considered as Heat Manot directly determinable from our prac- chine," page 82) McCulloch ("Mechanical tical experiences or observed phenomena. Theory of Heat," page 92) and Rankine Heat, the observed and usually acknowl- (“Steam Engine," page 321) will serve edged primary physical source of all as a verification of the above summation. energy (though gravitation might on Permit me to call your attention to this some good grounds be selected as our remarkable agreement between theory present ultimatum) beautifully illustrates and practice. The specific heat at conthis harmony when considered as energy, stant volume is not directly determinand thus prescribes laws which are after- able from experiment, so it is developed wards varified in practice, and to which from the velocity of sound, and substipractice conforms and becomes the em- tuting these two independent experibodiment of. This is the strongest plea mental data (the two specific heats) in a which it seems to me can be urged for theoretical equation not involving a the careful study of the mechanical previous determination of the mechanitheory of heat, and we will attempt to cal equivalent of heat, we ascertain such point out some of the work which has equivalent from correct theoretical conbeen done in this respect, and leave it siderations, and it accords with the to your judgment to decide whether a experimental determination.

What a science which has already accomplished beautiful demonstration of the correlaso much is not capable, when many able tion of the different laws and facts of minds are devoted to its study, of an nature do we here perceive exemplified. application and extension which is and But this will appear rather as an illuswill be of incomprehensible value in the tration of the beauty and symmetry of ordinary practice of the engineer, and the science than as a presentation of a which will greatly contribute to the fact of direct utility to the engineer in advance of mechanical engineering. the practice of his profession, and since

The specific heat at constant pressure it is my special object to call attention of any permanent gas can be ascertained to a few of the latter class of facts, I will directly by experiment, not so the spe- have to omit the mention of the interestcific heat at constant volume. Regnault ing relations between the physical prophas made accurate determinations of the erties of bodies, such as between the two specific heat of such gases at constant! elasticities and the two specific heats,

gaseous viscosity and the molecular water of condensation is formed and the
theory of the constitution of bodies, all action continues as indicated above.
of which the science teaches and will at Several methods of decreasing this
once refer to its main application for loss presented themselves:
our own purposes, the work of fluids in 1. The introduction of the compound
engines. It is in this department of our engine which dividing up the range of
profession that thermo-dynamics has temperature between admission and
made the deepest impression in practice, final exhaust, in two or more stages,
and is destined to continue it to a far causes less difference in each cylinder,
greater extent. It has taught us the and therefore decreases the loss by con-
different laws of expansion and work of densation while again the heat abstracted
fluids, and has supplied us with the from the first cylinder at exhaust be-
proper test of the efficiency of different comes available in the second cylinder.
T-T

2. The use of the steam jacket which forms of engines.

the limit of tends to keep the metal of the cylinder T

at uniform temperature and thus prevent efficiency has enabled us to correctly the initial condensation of steam, and appreciate the progress made, and to to supply heat to the steam in the cylindefinitely point out the direction in der to prevent condensation during exwhich the advance in efficiency and per- pansion. fection of heat engines lies. To increase 3. The introduction of a great number the theoretical efficiency, T, the tempera- of strokes per minute so that less time ture of the fluid at its entrance to the will be accorded to the steam to impart cylinder should be raised, and T,, the heat to the metal of the cylinder and to temperature at exhaust lowered as much abstract heat during exhaust. as possible.

4. The use of superheated steam, since The application of this principle to it can emit heat when being admitted to steam engines implies high steam press- a cylinder of lower temperature without ures and great expansion. This was causing water of condensation to form at tried within certain limits, and the bene- entrance or during expansion. ficial results looked for were not realized Neither of the first three remedies to the extent expected. The theory was proves entirely efficacious in practice still recognized as true by those who while all have greatly reduced the loss grasped it, but the practical results had from condensation. An intelligent study not conformed to their expectations. of the theory of heat will show, however, The nature, laws and action of steam that the latter is the only one which were thoroughly investigated, and it offers the possibility of entirely preventbecame apparent that condensation en- ing condensation. Perhaps it will need sued, owing to heat consumed for in- some explanation why the steam jacket ternal work during expansion and to the can never entirely prevent this loss. It detrimental action of the metal of the is owing to the fact that the transfer of cylinder. When the steam enters the heat from the steam in the jacket is not cylinder (of a lower temperature than as rapid as the transfer of the heat from the steam at its initial pressure) it gives the internal sides of the cylinder to the out heat to the metal of the cylinder, steam in cylinder, that the steam on and to do this sets free latent heat, entering the cylinder heats up but a causing water of condensation to be small thickness of metal to its own temformed. When the steam in the cylin- perature, owing to the comparatively der expands it performs internal as well poor power of conduction of iron. And as external work, and becomes partially the steam in the jacket, similarly, heats liquefied; as the steam leaves the cylin- up to its own temperature but a small der, rushing into the condenser, the thickness of the metal of the cylinder water mixed with the steam evaporates, immediately in contact with it; in brief, abstracting additional heat from the the poor heat-conducting power of iron metal of the cylinder. When a fresh does not allow the transfer of heat from volume of steam of initial pressure now the steam in jacket to the steam in enters the cylinder, it comes in contact cylinder to be practically instantaneous. with the metal of lower temperature, There may be some who are opposed

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to the introduction of the bigh speed (as not be discovered that will enable us to they are popularly termed) or rather extend the limits of temperature between “high revolution" engines. But what- which the fluid can expand in a cylinder, ever be their practical defects—and they and thus utilize by far the greater porpo ess some mechanical advantages that tion of the latent power which nature

are inclined to think more than has presented to us in her stores of counter-balance those defects the prin- fuel ! ciple is in the right direction, and al But it may be said: The advances, ready splendid workmanship and refine- pointed out above, in the efficiency of ment of mechanism and machinery engines have not in all cases been institends to confirm the value theory ac- tuted by those who have mastered the cords the system.

mechanical theory of heat, and the We will now briefly refer to the use of advances, therefore, are not in all cases air. Thermo-dynamics demonstrates its the result of the study of the subject. superiority to steam. Its permanently This is partially true, that is while, as gaseous nature enables it to expand far as I can learn, no advances have been without doing internal work, and its secured in the efficiency of heat engines temperature can be raised to a high (not including, of course, reduction of degree without the objection of a high friction and better mechanism for transpressure difficult to control. But prac. forming the rectilinear motion of the tically the few air engines built have as piston) by men who have not been yet not proved a decided success, owing intimately acquainted with, at least, the to the difficulty of obtaining a rapid con- fundamental principles of heat energy, nection of heat to and from the air they have not in all cases mastered the enıployed, and the necessity of a larger whole subject in its highest mathematicylinder than is needed for a steam cal form. But to correctly appreciate engine of the same power. Again, very and define the advance, the theory of high temperatures of the air cause no heat energy has always been called into inconsiderable strains due to irregular play, and if our progress in the efficiency expansion of the metal and a slow oxida- of heat engines has not been as rapid as tion of the metal as well. And so with 'we might have desired, and is not as other forms of fluid engines. Practical rapid at the present time as desirable, it difficulties and obstacles have not always is owing to a lack of knowledge and permitted the rigid teachings of thermo understanding of the subject under disdynamics to be precisely realized, though cussion by some of our more brilliant they have in most cases served as an and experienced minds. There are two exemplification of its truth. But as methods of gaining knowledge. One by long as we have the laws of the mechani- acquiring the laws and results of the cal theory of heat to teach us where the experience of others, the other by acpossibility of or road to progress lies, quiring the laws by our own experience. better steam and better air and better Both constitute the acquisition of pringas and better fluid engines will in time ciples or theory. Both have their uses. be built. Even with the fluids and But it is a loss of time and it is at the materials at present known, the per- expense of many failures and disappointformance of engines are capable of being ments which do not contribute to real doubled, a result by no means too trival advance, if we arrive at the same princito make an honest effort to master the ples by such failures that others have principles which will aid us in securing reached before us by whose experience this end. But who would dare to say in we might have profited. When we have an age when Professor Crooke s fourth acquired the knowledge of the work that state of matter opens up realms of in- others have done, we are prepared to vestigation undreamt of, and when Pro- make further progress, and if the diffifessor Bell's discovery of sound trans- culties and obstacles multiply we will be mitting rays of light potently reminds fairly equipped and fully encouraged to us that we are but at the threshold of meet them. If we then experience failour understanding of the laws of the ures, they will contribute to real advance universe, who would dare to say that instead of demonstrating a fact or law new fluids, materials and conditions will which had already been acknowledged,

and which it was within our power and in my opinion, full justice be done to so province to know. Thus too will in- grand a theme within the limits of a dorsement of, and investment of capital paper of this kind. A complete treatise in, prime movers, advertised as realizing would have to be written to demonstrate fabulous power be avoided, since our its true importance, and some work like knowledge of the laws of thermo-dyna- that of Rankine is the best verification mics will have acquainted us with the of the actual value of the science, and in principles of the conservation of energy, its study does this value become most the impossibility of transgressing cer- potently apparent. tain limits, and will, therefore, indicate But if I have been able to convinceor demonstrate the fallacies of the pro- say, one of you, heretofore uninterested, jected scheme.

of the vitality of the study, I will feel In conclusion, I must say that I am amply compensated, and will offer no aware that full justice has not been done apology for my enthusiasm in the to the theme under discussion, nor can, cause.

THE GEOMETRICAL INTERPRETATION OF IMAGINARY

QUANTITIES. *

Translated from the French of M. Argand by Prof. A. S. Hardy.

Contributed to Van NoSTRAND'S ENGINEERING MAGAZINE.

PREFACE.

with public recognition until after the Tue work now republished † is of that insertion of a note by J. F. Francais, in small number which mark an epoch in the Annales de Gergonne, Vol. IV, 1813, the history of science. In this short 1814, p. 61, in which, at the same time, treatise is found the germ of the true Argand* also published two articles. theory of so-called imaginary quanti- In these articles the subject was so ties. Although generally attributed to exhaustively treated that nothing new the genius of Gauss, this theory was not has since been found to add to them, pointed out by that great geometer until and, unless some older work is distwenty-five years after the publication of covered, Argand must be regarded as Argand's work,ħ and it had been mean- the true founder of the theory of comwhile re-discovered several times in both plex quantities in a plane. France and England. On this point we

• In 1831, Gauss † develcan cite no testimony more convincing oped the same idea, as is well known; than that of a German geometer, whose but, however great his merit, as bringrecent death is deplored by science. ing this idea to the notice of science, it Says Hankel, ş “the first to show how to is none the less impossible to claim for represent the imaginary forms A+Bi by him priority.” points in a plane, and to give rules for From this accurate historical résumé, their geometric addition and multiplica- it is seen that the work of Argand tion, was Argand, who established his remained almost wholly unknown, haytheory in a pamphlet printed in Paris, in ing been distributed but to few persons, 1806, under the title · Essai sur and not put in general circulation. manière de representer les quantités im- Seven years later, Francais, an artillery aginaires dans les constructions géomét- officer at Metz, sent to the Editor of the riques. Yet this paper did not meet Annales the outline of a theory whose

germ he had found in a letter written to * Essay on the Geometrical Interpretation of Imans his brother by Legendre, the latter hav- . preface by M. J. Hoüel, and extracts from the Annales ing obtained it from another author de Gergonne, Paris, Gauthier-Villars, 1874. French, by Prof. A. S. Hardy, Dartmouth College.

whose

he did not give. This + 1st edition, Paris. Duminil-Lesueur, 1806. article came to the aptice si Argand, * Anzeige zur Theoria residuorum biquadraticum Commentalio secunda," 1831 (Gauss Werke, t. II, p. 174).

& Vorlesungen uber die complexen Zahlen und ihre func * Vol. IV, p. 183, and Vol. V, p. 197. tionem. (Leipzig, 1867, p. 82).

+ Works, Vol. II, p. 174,

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who immediately wrote Gergonne a note long enveloped negative and imaginary in which he made himself known as the quantities, as well as upon the general author of the work cited in Legendre's theory of functions, by affording a definletter, and in which he gave quite a ite geometrical interpretation. Others, complete summary of his pamphlet of as yet of less importance, but perhaps 1806. This double publication gave rise destined in the future to render great to a discussion in the Annales, in which services, have resulted in the creation of francais, Gergonne and Servois took new methods in analytical geometry, part, closing with a remarkable article, among which may be cited those of in which Argand explained more satis- Möbius, Bellavitis, Hamilton and Grassfactorily certain points in his theory, man. Unable to avoid the constant especially his demonstration of the presence of negative and imaginary fundamental proposition of the theory quantities in the results of analysis, or of algebraic equations, the simplest yet to surrender the important advantages given, which subsequently Cauchy only following the use of their corresponding reproduced, in a purely analytic, but less symbols, mathematicians had for a long striking, form. These various articles, time been content to employ them withthe natural sequel to Argand's pamphlet, out fully accounting for their true published in a Recueil now very rare, nature, regarding them as signs of operare collected in an appendix to this ations which in themselves had no meanvolume. Notwithstanding their appear- ing, yet which, under certain rules, led ance in a scientific journal so well surely and directly, though in an obscure known, the views of Argand were wholly and mysterious manner, to results which unnoticed, as appears from the fact that other quantities would not have yielded, twenty-two years after the publication of except indeed by long and difficult prothe essay they were re-stated both by cesses, involving the discussion of an Warren, in England, and Mourey, in indefinite number of particular cases. France, apparently without any knowl. It is at last seen, however, that the edge on their part of their earlier expo- impossibility of negative quantities is, in sition. Nor did they themselves succeed general, only apparent, and results from in attracting the attention of geometers, a generalization of the idea of quantity although the researches of Mourey were without any modification of the corregiven in the Lecons d'Algèbre by Leféb- sponding analytical operations.

An ure de Fourcy, and two articles, sup- analogous case is found in the very eleplementary to his first work, had been ments of arithmetic, which, however, has published by Warren in the Philosophi- given rise to no difficulty. The operacal Transactions. Only after Gauss had tion of division cannot be exactly perspoken, were these views taken up in formed if we are restricted to whole Germany. They soon became familiar numbers. But if unity be divided into to English geometers, and were the equal fractions, the division is always starting point of Hamilton's theory of possible, and the result becomes a Quaternions, while, in Italy, Bellavitis complex expression, consisting of two made them the basis of his Méthode des numbers, one indicating multiplication, Equipollences.* In France, Argand's the other division. Hence arises a new theory was worked over, without material class of quantities, fractions, subject to addition, till its adoption by Cauchy, operations to which are applied the same who expounded it in his Exercices names given to the operations on inted'Analyse et de Physique mathémati- gers, which they include as particular que,f with a complete historical notice cases. But the definitions of multiplirendering Argand full justice.

cation and division have been therefore In the work of this modest savant of carefully modified, to render them appliGeneva is to be found the origin of cable to the new quantities. By promany subsequent researches, some of ceeding in an analogous manner in addiwhich have thrown unexpected light tion and subtraction, the meaning of a both upon the mystery which has so negative quantity has been definitely

* Esposition de l ethode des Equipollences Guisto fixed. So long as the problem is re-
Bellaritis. Traduit de l'Italien par c. A. Laisant, Paris. stricted to the simple determination of
Gauthier-Villars, 1874.
† Vol. IV, p. 167.

magnitude, the subtraction a-6 is imVol. XXIV.-No. 1-2.

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