From the History of Physics

Mikhail Petrovich Avenarius and the Kyiv School of Experimental Physics

A. G. Goldman

In the 1870s, centers of scientific thought were taking shape in Russia, bringing together more or less large groups of physicist-researchers. In 1872, on the initiative of D. I. Mendeleev and F. F. Petrushevsky, the Physical Society was founded at Petersburg University. In the following year, 1873, a physical laboratory was organized at Moscow University, which under the direction of A. G. Stoletov became the center of the Russian school of physicists. From that same year, 1873, works began to be published on the investigation of critical quantities, which brought renown to the Kyiv school of physicists that developed under the leadership of M. P. Avenarius.

Mikhail Petrovich Avenarius developed into a major scholar under the direct influence of N. I. Pirogov, who played a prominent role in the development of Russian scientific and pedagogical thought in the middle of the past century. All progressive Russia listened to the voice of N. I. Pirogov. His ideas corresponded in many respects to the pedagogical ideals of the best part of the society of that time. From 1858, N. I. Pirogov advocated a fundamental renewal of Russian universities, based on a profound connection between teaching and scientific research. Before young scholars N. I. Pirogov set the task of seeking the independent development of our national science and demanded the acquisition of deep specialized knowledge, and above all the mastery of the scientific method. “Show an educated man, on the most limited scale, in some small part of science, only in actual practice, the method and mechanism by which contemporary science arrives at its results—and he will obtain everything else himself, if he truly seeks knowledge” (pp. 571–572)^1.

N. I. Pirogov demanded that future university figures know their business, i.e., science, and, above all, be able “to acquaint their pupils with the mechanism of scientific knowledge” (p. 644)¹. The educational influence of N. I. Pirogov on young scholars—one of them, later rector of Kazan University, Professor N. S. Kovalevsky—defined it in the words: “he taught us to learn.”

MIKHAIL PETROVICH
AVENARIUS

1. THE BEGINNING OF THE SCIENTIFIC ACTIVITY OF M. P. AVENARIUS

Mikhail Petrovich Avenarius was born on September 7, 1835; in 1858 he completed the course of St. Petersburg University with the degree of Candidate of Mathematical Sciences and began working as a teacher of mathematics at the 2nd St. Petersburg Gymnasium. From May 1862 to October 1864, Avenarius, among the “Pirogovites”*) was sent abroad by the Ministry of Public Education for preparation for professorial activity.

*) The “Pirogovites” was the name given to a group of young Russian scholars whose overall educational supervision was entrusted to Nikolai Ivanovich Pirogov.

Avenarius spent most of his period of travel in Berlin, where he attended lectures at the university, took part in Magnus’s physics meetings, and carried out independent scientific work in his laboratory. In the latter half of his stay abroad Avenarius spent one month in Paris, familiarizing himself there with methods of teaching the physical sciences, and then, during the summer semester, worked under Kirchhoff in Heidelberg.

M. P. Avenarius was in many respects dissatisfied with the organization of teaching at the University of Berlin: “... everywhere here, apart from the experiments of Prof. Erman, there is more the professor than the listener,” he writes in his report, and further: “... although I did not neglect the experimental part, so little was allotted for it in the arrangement of things that most of the time remained free.”² He speaks favorably of the physics meetings held once a week at Magnus’s; the surveys on questions of contemporary physics prepared by the participants in these meetings helped one to master scientific questions deeply and comprehensively.²ᵃ His stay in Heidelberg gave Avenarius the opportunity to compare the scientific schools of Kirchhoff and Magnus. He notes as a shortcoming of Magnus’s empirical school that in his exposition “each section represents something independent, closed, contrasting with this the course of Kirchhoff, in which the unity of theoretical thought binds into one whole all parts of physics” [quoted from A. G. Stoletov (p. 420)³].

Of Kirchhoff A. G. Stoletov, who worked in Heidelberg at the same time as Avenarius, wrote: “Physical science in its ideal form, as a combination of theoretical thought with experimental art, appeared to us here for the first time in the person of the creator of spectral analysis” (p. 420)³.

Avenarius began his first scientific investigation in 1862. By this time the law of conservation and transformation of energy had become a recognized principle of physics; in 1860 the first textbook of thermodynamics appeared (Zeuner, Fundamental Features of the Mechanical Theory of Heat). The study of energy transformations was one of the chief problems of experimental physics. The prospects for technical applications of electrical energy prompted the investigation of its various sources; new galvanic cells were being constructed (for example, the Leclanché cell, 1867), new principles were being introduced into the construction of electrostatic machines (the electrophorus machine, 1865), accumulators were being improved (Planté, 1860), and thermobatteries were being built in the hope of obtaining an economically advantageous source of current (Marcus, 1860). The phenomena of thermoelectricity attracted the attention both of engineers and of physicists. The primary task was to master the laws of these phenomena, and Avenarius successfully solved this task.

Despite a number of studies by well-known physicists, the law of dependence of the thermoelectric current on the temperatures of the junctions was not known. In the handbook on galvanism that appeared at that time, the author—G. Wiedemann—having analyzed all the assumptions that had been made concerning the cause of these currents, in conclusion recognizes only the fact that, in order to obtain a thermoelectric current, it is necessary to take two different bodies and to heat or cool the place of their contact.

Avenarius proceeded from the idea that the contact difference of potentials \(e\) existing between two metals depends on the temperature \(t\), and he sought the form of this law as a power series. “I succeeded,” Avenarius writes in the introduction to his doctoral dissertation, “with the aid of thermoelectric investigations, in indicating the existence of three terms of this series; moreover, within the limits of observational error, the presence of terms with higher powers of \(t\) was not detected, and therefore, for determining the complete electrical difference of contact between metals at temperature \(t\), one may use our empirical expression, limiting oneself to three terms”\(^4\).

Thus, Avenarius proceeds from the fact that the contact difference of potentials at the boundary of two metals can be represented, with quite sufficient accuracy, by the expression

\[ e = a + bt + ct^2 . \tag{1} \]

Assuming that the electromotive force \(\mathcal{E}\) of a thermoelectric circuit is due only to the contact differences of potentials at two junctions situated at different temperatures \(t_1\) and \(t_2\), Avenarius obtains the formula

\[ \mathcal{E} = (t_2 - t_1)\,[b + c(t_1 + t_2)] . \tag{2} \]

In it, the absorption and liberation of heat along the entire homogeneous conductor had not yet been taken into account. This phenomenon Avenarius took into account later in the work “On the electromotive force of thermoelectric elements from the point of view of the mechanical theory of heat,” in which he thermodynamically justified formula (2), proceeding from the hypothesis that the specific heat of electricity is proportional to the absolute temperature. In this, the constants \(b\) and \(c\) received a definite physical interpretation. Avenarius concludes this work with the words: “The application of the principles of the mechanical theory of heat to the phenomena of thermoelectricity leads to the same expression for the electromotive force of a thermoelectric element at which experiment had arrived, as an exact empirical expression of the law of dependence of the electromotive force of the element on the temperatures of the junctions”\(^5\).

In O. D. Khvolson’s physics course this correction is ascribed to the English physicist Tait, who gave it three years later than it had been published by Avenarius.^6

A. G. Stoletov testifies: “And in this correction (the necessity of which was later noticed by Tait) priority belongs to the Russian author” (p. 428).^3

In the autumn of 1864 M. P. Avenarius returned to Russia. He set forth the results of his work in two dissertations. On January 8, 1865, at St. Petersburg University, he defended his master’s dissertation, “On Thermoelectricity,” and on May 10, 1866, there too he defended his doctoral dissertation, “On the Electrical Differences of Metals at Various Temperatures.”

In his thermoelectric investigations Avenarius proceeded from hypotheses constructed at the level of the science of his time.

The phenomena known at that time—the attainment, at a certain temperature difference, of a maximum current, and the phenomena of inversion—were presented by Avenarius as necessary consequences of the fundamental proposition of his theory.

These works of the young scientist received wide recognition, and the formula he found came to be called Avenarius’s law.

Avenarius’s formula entered many Russian and foreign textbooks and monographs. However, some English and American textbooks quite groundlessly call it Tait’s formula (for example, in the textbook of general physics by Hastings and Beach^7).

The priority of Russian science in questions of thermoelectricity, associated with the name of Avenarius, is entirely indisputable and must be reflected in all our literature, as well as in the physics curricula of secondary and higher educational institutions.

From March 1865 M. P. Avenarius was confirmed as docent of Kiev University. From the autumn of that same year he was entrusted with the chair of physics and the direction of the meteorological observatory. From September 1866 he was confirmed in the rank of extraordinary professor, and from November 1867, ordinary professor. His activity at Kiev University lasted 25 years.

During the first ten years M. P. Avenarius had to lecture on the entire course of physics, both experimental and theoretical, as well as lecture on meteorology and direct the meteorological observatory. He himself, with the assistance of one staff member, prepared experiments for the lectures; he also conducted practical classes. The work was carried on without assistants.

M. P. Avenarius quickly won the sympathy of the students.

University and public lectures by Avenarius also aroused great interest. The latter always gathered a large audience, attracted both by the interesting choice of subject matter and by M. P.’s ability to expound difficult scientific questions clearly and intelligibly for listeners with little preparation.^8

In the first years, the teaching of systematic courses covering an entire science absorbed all of Avenarius’s strength. Working conditions improved somewhat when, in 1867, the teaching of the course in theoretical physics passed to N. N. Shiller, thanks to which M. P. Avenarius gained the opportunity to concentrate on experimental work. At first the Physics Cabinet occupied only two small rooms. In 1875 Avenarius succeeded in expanding it and in establishing laboratory work for students. He himself supervised the students’ work; thanks to this he drew still closer to them and attracted the most talented to research work.

The results of Avenarius’s work with young physicists appear as early as 1869, when the first measurement works by Eismont (“Determination of the length of the seconds pendulum”), Zayonchevsky (“Determination of the intensity of the force of terrestrial magnetism in Kiev in absolute units” and “On magnetic inclination in Kiev”), and somewhat later (1873) Kravchenko (“Determination of the elements of terrestrial magnetism in Kiev”^9) were printed. Thus the scientific school of M. P. Avenarius took shape, which, thanks to its research, soon became known. Its members included: V. I. Zayonchevsky, O. E. Straus, K. N. Zhuk, A. I. Nadezhdin, and students G. Kannegisser and D. Dyachevsky*).

2. DEVELOPMENT OF THE SCIENTIFIC SCHOOL OF M. P. AVENARIUS

From 1873, M. P. Avenarius turned to a new field of research: the study of the liquid state and of vapor under changes of temperature and pressure. In this connection he was faced with the task of directly determining critical temperatures.

) O. D. Khvolson (Course of Physics, vol. III, 3rd ed., p. 681) names Pavlevsky among the pupils of M. P. Avenarius; the same is asserted by A. S. Predvoditelev (Essays on the History of Physics in Russia, Uchpedgiz, 1949, p. 216), who calls Pavlevsky Pavlovsky. Pavlevsky in 1878 worked at the Institute of Agriculture and Forestry in Novaya Alexandria (see Zh. M. N. P., part 207, section 4, p. 89, 1880). According to Nadezhdin (p. 83)^13, Pavlevsky’s first note on critical temperatures appeared in 1882 in Chem. Ber.* 1882, No. 4. In 1878 Zayonchevsky was working at the New-Alexandria Institute. It seems likely that the person who guided Pavlevsky when he began work on critical temperatures was precisely Zayonchevsky, and not Avenarius.

The concept of the critical temperature was first introduced into science by D. I. Mendeleev in 1860; namely, he expressed the conviction that for every liquid there exists a limiting temperature above which it remains a vapor or gas, however great the pressure may be. He called this temperature the absolute boiling temperature[^10]; the term “critical temperature” became established later. These considerations of D. I. Mendeleev were confirmed in the works of T. Andrews, which he had been conducting since 1861; in 1869 there appeared a complete description of Andrews’s work on the study of carbon dioxide, which unquestionably demonstrated the existence of the critical temperature. However, there were no exact data concerning the critical temperatures of various substances.

There existed Regnault’s tables of measurements concerning the conditions of evaporation of certain liquids (ether, carbon disulfide). The data contained in them on the dependence of the heat of evaporation on temperature were processed by Zeuner, who represented the course of the so-called “internal heat of evaporation” $\rho$ as a function of temperature by empirical formulas. Regnault’s data, however, did not extend to critical temperatures. Avenarius undertook an experimental determination of the critical temperatures for four of the liquids studied by Regnault. With the care characteristic of Avenarius in all his investigations, a methodology for these determinations was developed and the first direct determinations of critical data were obtained. At the same time Avenarius critically processed Regnault’s data and showed that they lead to values of the critical temperatures agreeing with his experimental data. The work was published in 1873 under the title “On Internal Latent Heat”[^11]; it was the first in a series of investigations of critical quantities to come out of the Kiev laboratory. During the years 1875–1887, numerous critical and other data for various substances were determined there and entered the basic stock of physical constants.

Thus, in 1878 V. Zajączkowski’s work “Determination of the Elasticity of Vapors of Certain Liquids at High Temperatures” was published[^12]. Zajączkowski measured the elasticity of saturated vapors up to the critical temperature; he determined the critical temperatures and pressures of sulfur ether, sulfurous anhydride, carbon disulfide, benzene, alcohol, acetone, ethyl chloride, ethyl acetate, carbon tetrachloride, and others. The last in this series was Nadezhdin’s work—his dissertation, published in 1886 and containing determinations of the critical temperature, critical pressure, and critical volume of 18 substances[^13].

According to Stoletov, of the entire set of critical temperatures that had been collected in the 2nd edition (1894) of the well-known physi-

... tables of Landolt and Bernstein, about one fourth were obtained in the young Kiev laboratory (p. 429)^3.

Thirty years later the Landolt and Bernstein tables appeared in a fifth edition (1923); many of the former data were excluded from them and replaced by later, more reliable ones, but Zayonchevsky’s data of 1878, as well as Nadezhdin’s data of 1885, were preserved in full; moreover, Zayonchevsky’s data are the earliest of those cited in the tables. Of the 34 determinations of critical volumes indicated in the tables, half belong to Nadezhdin^14.

With the passage of time these determinations not only did not lose their scientific significance, but many of them also found practical application. In connection with the development of refrigerating machines, interest arose in Zayonchevsky’s and Nadezhdin’s measurements of the critical data and specific heat of sulfurous acid. In M. P. Malkov and K. F. Pavlov’s Handbook of Deep Cooling in Technology, published in 1947,^15 the critical temperatures of isobutylene, determined by Nadezhdin, are given as 150.7° and propylene (in the handbook 91.4°, in Nadezhdin 91.6°). In the translated and 1949 edition of the Handbook of the Experimental Physicist (Kay and Laby)^16, the data obtained by Zayonchevsky are given for sulfurous gas, \(t_{\mathrm{cr}} = 155.4^\circ\) and \(p_{\mathrm{cr}} = 78.9\) atm, and for chloroform, \(t_{\mathrm{cr}} = 250^\circ\) and \(p_{\mathrm{cr}} = 54.9\) atm.

These examples show how reliable the experimental determinations of Avenarius’s collaborators were: they withstood verification by experiment over seventy years of scientific development.

The values of the critical quantities for water—a substance of the greatest importance in its applications—aroused particular interest. At the same time, in the case of water, the measurements were connected with overcoming special difficulties. Cagniard de la Tour, the first to study this question, considered his experiments with water unsuccessful; iron vessels were not suitable for the experiment with water, since they were insufficiently hermetic, while glass vessels burst. He believed that the temperature of “complete volatilization” for water is close to the melting point of zinc. In this connection, Musson ascribed to Cagniard the value of this temperature \(t_{\mathrm{cr}} = 362^\circ\)C, and van der Waals—410 and 412°. From Regnault’s data for the latent heat of vaporization, by extrapolating an empirical expression for the latent heat as a function of temperature, they obtained \(t_{\mathrm{cr}} = 873^\circ\); Mendeleev theoretically determined \(t_{\mathrm{cr}} = 543^\circ\), and Clausius—\(328.3^\circ\). Thus there was no definite information whatever about the critical temperature of water.

A method for the experimental determination of the critical temperature of water was found by one of Avenarius’s pupils—O. E. Straus. Working with a mixture of alcohol and ether, he established that a critical state is also observed for a mixture, similar to that which

observed in pure liquids. As a result of a detailed study he established that the critical temperature of a mixture can be calculated with sufficient accuracy by the formula

\[ \vartheta=\frac{\alpha_1\vartheta_1+\alpha_2\vartheta_2}{\alpha_1+\alpha_2}, \]

where \(\vartheta\), \(\vartheta_1\), and \(\vartheta_2\) are the critical temperatures of the mixture and of its two constituents, while \(\alpha_1\) and \(\alpha_2\) are the percentage contents of these parts. Straus’s rule was checked by many investigators, who confirmed it within broad limits, among other things also for air.

Straus published this work in 1880, and already in March 1881 he reported on the determination of the critical temperature of water, based on the use of the established rule. Straus prepared mixtures of water and alcohol in various proportions and measured their critical temperature. The experiment could be carried out with a content in the mixture of up to 50% water. To be sure, some action of the water on the walls of the vessel was observed in repeated determinations with one and the same specimen, but, limiting himself in each determination to three or four experiments, Straus obtained sufficiently reliable data. From the 55 pairs of observations cited with specimens of different water content, O. E. Straus determined the critical temperature of water to be \(370^\circ\text{C} \pm 5^\circ\).

Proceeding from this result, Straus turned to the determination of the critical pressure of water. Van der Waals had attempted to calculate this quantity, indicating once 278 and another time 289 atm., and Clausius, who determined the critical pressure of water as 134 atm.

Straus’s method was based on the law of corresponding states: “At equal reduced temperatures all substances possess the same reduced elasticity of the saturated vapor.” Using the value found for the critical temperature of water, he calculates the reduced temperatures of water corresponding to ether temperatures equal to \(10, 20, \ldots, 80^\circ\), and from tables finds for these temperatures the vapor elasticities of ether and, correspondingly, of water. At equal reduced temperatures the elasticity of water vapor is on average 5.3 times greater than the elasticity of ether vapor. This ratio must also be preserved at the reduced temperature equal to 1, i.e., at the critical temperature. For ether, Zayonchevsky determined the critical pressure as \(36.9\) atm.; consequently, for water it is equal to \(36.9 \cdot 5.3\), i.e. \(p_{\mathrm{cr}} = 195.5\) atm.

By a number of examples Straus showed that this value of the critical pressure, if it is applied to the calculation of the boiling temperatures of various liquids, leads to results very close to observations. The work was published in 1882.^17

In March 1885 A. I. Nadezhdin carried out the first direct determination in science of the critical temperature of water. Since the optical method of establishing the critical state, in which the disappearance of the meniscus is observed, is not applicable to water, Nadezhdin proposed for this purpose a new method, which entered science as the “Nadezhdin method.” Nadezhdin’s instrument, called by him a differential densimeter[^18], consisted of a tube inserted into a frame supplied with a trihedral axis, on which it could swing like the beam of a balance. At first the tube was balanced so that it maintained a horizontal position. Then part of the tube was filled with liquid, and the remaining part with saturated vapor. The tube was then set at an inclination. The critical temperature was determined as the temperature at which the difference between the densities of the liquid and the vapor disappears, which was detected by the return of the tube to the horizontal position. An important advantage of Nadezhdin’s method is its applicability in cases when the substance is strongly colored (such as bromine and iodine), so that it is difficult to see the meniscus, or when the substance acts destructively on glass (such as water).

For the critical temperature of water Nadezhdin obtained the value \(t_{\mathrm{cr}} = 358^\circ\mathrm{C}\). A. G. Stoletov calls this work the culminating point in the activity of the laboratory.

Several years later these measurements were repeated by Cailletet and Colardeau; for water they obtained \(t_{\mathrm{cr}} = 365^\circ\mathrm{C}\) and \(p_{\mathrm{cr}} = 200.5\) atm.

Later measurements, cited in reference tables, give for water \(t_{\mathrm{cr}} = 374.1^\circ\mathrm{C}\) and \(p_{\mathrm{cr}} = 205\) atm.

As we see, the determination of the critical constants of water carried out by Strauss proved to be correct. Within the limits of the possible error indicated by him, their values agree both with the measurements of Cailletet and Colardeau and with later ones, performed with substantially more advanced experimental technique.

Beginning in the seventies, thanks to work on the critical state, M. P. Avenarius’s laboratory acquired ever greater renown. Its investigations were printed not only in university transactions, but also in the central Russian physics journal (the journal of the Russian Physico-Chemical Society, physical section) and were abstracted in foreign physics journals. Almost every work was subjected to attentive, critical study by researchers in the same field.

The growth of the laboratory was reflected in Avenarius’s scholarly reputation; he became a member of the Russian Physico-Chemical Society, a corresponding member of the Petersburg Academy of Sciences, an honorary member of the Moscow Society of Naturalists, a member of the Kiev Society of Naturalists (and a founding member of the Kiev Physico-Mathematical Society that later separated from it), and a member of the Berlin Physical Society.

By the beginning of the eighties, Avenarius’s laboratory had accumulated extensive scientific experience, obtained numerous new experimental data, and developed a distinctive methodology combining simplicity and precision. The laboratory’s works during this period were published under the unifying title: “From the Physical Laboratory of the University of St. Vladimir”^19.

3. ALEKSANDR IVANOVICH NADEZHDIN

Among the laboratory staff at this time there emerged a young, talented scholar who, through his scientific works, quickly won recognition in the circles of Russian physicists. This was the already mentioned Aleksandr Ivanovich Nadezhdin^20. A. I. Nadezhdin was born in 1858 into the family of a military physician. Having entered the Faculty of Physics and Mathematics of Kiev University in 1877, he early began experimental work and, while a third-year student, received a gold medal together with the N. I. Pirogov Prize for an essay on the competition topic: “On the changes observed in the properties of bodies near the so-called temperature of absolute boiling.”

The range of his interests was very broad: he took part in the Kiev public readings (he lectured on meteorology and devoted much time to preparing experiments); he translated the chapter on Kant in A. Weber’s book History of European Philosophy, which appeared in Kiev in 1882. At the same time he was an amateur member of the Dramatic Society.

On Avenarius’s recommendation, after graduating from the university in 1882, A. I. Nadezhdin was retained as a stipend-holder to prepare for the rank of professor in the department of physics. Simultaneously he was appointed a teacher at the Kiev Women’s Gymnasium.

In the following years he carried out a number of investigations, published in Russian scientific journals and reviewed in foreign publications. Nadezhdin also reported on these works at the Kiev Society of Naturalists, of which he was a very active member. In 1885 he was the first to determine by a direct method the critical temperature of water and of other liquids for which the method of disappearance of the meniscus proved unsuitable. This brilliant work was mentioned above: it was published in the bulletin of the St. Petersburg Academy of Sciences. In the same year he passed the master’s examinations and investigated the elasticity of saturated vapors at high temperatures; this work became part of his master’s dissertation, entitled “Etudes in Comparative Physics,” and was submitted for defense in the spring of 1886.

Already in Nadezhdin’s first student work there appeared the qualities characteristic of him: keen powers of observation and a striving for

to generalization. He investigated the dependence on temperature of the elasticity of saturated vapors of several organic liquids. Determining the critical temperatures of several mixtures, he noted that the critical temperature and the boiling temperature of mixtures of different composition change by one and the same amount: by as much as the boiling temperature rises, by that much the critical temperature also rises[^21]. A. I. Nadezhdin suggested that this law is valid for substances close in composition—polymers, isomers. Therefore he investigates a series of isomers and becomes convinced of the existence of such a regularity[^22]. Somewhat later, in a German chemical journal, a work by Pavlevsky appeared; he expressed the same supposition for homologues and presented the results of measurements on 17 ethers in confirmation of the regularity. Nadezhdin gave further examples of the existence of this relation among homologues[^23],[^23a]. Selecting series or groups with analogous properties (homologues, isomers, derivatives of the same radicals) and establishing particular laws for them, Nadezhdin believed that particular generalizations would serve as steps toward a general synthesis, toward revealing the form of the function that connects the properties of a body with its structure and molecular weight. In the regularity found above he saw an indication that the dependence of boiling temperature on molecular weight and structure has the same form as for the critical temperature.

In the following work—on the heat capacity of liquids[^24]—Nadezhdin establishes a new relation: the ratio of the heat of vaporization of a liquid to the product of the heat capacity and the critical temperature is constant for any liquid, if the comparison is made at temperatures at which the specific volume of each liquid constitutes the same fraction of its critical volume (corresponding volumes). Nadezhdin confirms this proposition with experimental data. He then shows that the ratio of the internal works in vaporization and heating of different liquids, taken at corresponding volumes, is directly proportional to the critical pressure. To interpret these facts Nadezhdin proposes the hypothesis that in a liquid the molecules unite into groups and that the magnitude of the critical pressure is proportional to the number of molecules forming one complex particle of the liquid.

In the introduction to his dissertation Nadezhdin sets forth the initial methodological propositions of his work. “The substrate of all physical phenomena will be matter or substance” (p. 2)[^13], “...the atomic hypothesis has acquired such a degree of probability that it permits the chemical atom to be regarded as a reality” (p. 4)[^13]. Speaking against attacks on the atomic hypothesis, Nadezhdin states that what is beyond doubt is “the existence of that which moves—the smallest particles, atoms” (p. 4)[^13].

For a comprehensive and complete study of physical phenomena, Nadezhdin considers it necessary to investigate the dependence between the physical properties and the composition of bodies. He expects, and regards it as a conclusion from Mendeleev’s law of the periodic system, that there must exist “a close connection between atomic (sometimes particle) weight and the physical and chemical properties of a body” (p. 4)¹³. And he regards it as a program for the future “to find and measure this connection...” (p. 3)¹³.

Taking as the subject of his dissertation the action of heat on solid and liquid bodies, and believing that, in the state of contemporary science, the results of a comparative study of the action of heat on various substances “will for a long time still have a particular character,” Nadezhdin strives for the greatest possible completeness in the presentation and collection of experimental material scattered through an enormous number of memoirs and notes, and in its critical comparison.

Thus, a very substantial part of the dissertation consists of the analysis and verification of particular generalizations and entire theories on the topic under consideration.

The dissertation consisted of three parts. In the first part Nadezhdin considers the thermal expansion of solids, the relations between the coefficient of expansion, molecular volume, and melting temperature; it is a critical survey of the literature data.

The second part is devoted to the thermal expansion of liquids and to the transition of bodies from the liquid state to the gaseous state. In its first chapter are placed Nadezhdin’s own determinations of the critical state of a number of esters of fatty acids. Nadezhdin’s method is set forth in detail; here, too, is given what has become the classical description of the critical state. At the end of the chapter the results of Nadezhdin’s measurements are brought together. In the second chapter the coefficients of expansion at ordinary temperatures are considered. On the basis of numerous measurements by Nadezhdin, Pawlewski, and others, the question of the dependence between the boiling point and the critical temperature is discussed. Acknowledging that the constancy of the difference between these two temperatures holds for compounds that are metameric and homologous (and only approximately) for boiling temperatures at normal pressure, Nadezhdin relates this dependence to the constancy of the product $a t_k$, where $a$ is the coefficient of expansion, and $t_k$ is the absolute boiling temperature of the liquid. In the third chapter, formulas for the expansion of liquids are compared, and in the fourth, conclusions from equations of state relating to the expansion of liquids are investigated.

In the third part of the dissertation, data on the elasticity of saturated vapors are considered. In the first chapter are given the results of Nadezhdin’s measurements for a number of esters of fatty acids. “These determinations alone of the elasticity of vapors in this work stand, in my

opinion, for a doctorate,” M. P. Avenarius wrote to A. G. Stoletov (March 21, 1886).^25 In the second chapter empirical formulas are compared; in the final chapter the law of corresponding pressures is applied to finding the dependence between the constants of various formulas for the elasticity of vapors.

On March 21, 1886, M. P. Avenarius wrote to A. G. Stoletov: “I have sent you Nadezhdin’s works, hoping that you will be pleased to see such a substantial piece of work, carried out without the assistance of foreign scholars.”^25

A year later, in the posthumous edition of Nadezhdin’s physical investigations, Avenarius, in the preface, assessed this dissertation as follows: “The experimental data offered by him on the critical state of bodies and on the elasticity of vapors of liquids at high temperatures surpass in their significance all those hitherto published, taken together. And since this material is here also processed by him, the present investigation constitutes a substantial work...” (pp. V—VI).^26

A. G. Stoletov, in his works on the critical state of bodies, says of Nadezhdin: “A profound expert on the question that concerns us...”^27

4. THE LAST WORKS OF M. P. AVENARIUS

Beginning in 1887, M. P. Avenarius’s attention was drawn to the question of the thermal expansion of liquids. Many investigators were content with measurements made when temperatures changed within narrow limits, for example up to the boiling point. Avenarius set himself the task of obtaining a complete picture of the changes in the volume of a liquid, and therefore studied the expansion of a liquid under a pressure equal to the critical pressure, thanks to which he could bring the temperature of the liquid almost up to the critical temperature. Thus, Avenarius measured the expansion of ether at temperatures between \(20^\circ,2\) and \(187^\circ,8\).^28

These works were continued by K. N. Zhuk, who in 1881 published measurements of the expansion of ethyl alcohol (between 0 and \(218^\circ,5\)) and of sulfurous anhydride.^29 Under the direction of K. N. Zhuk, the students Kannegisser and Dyachevsky determined the expansion of diethylamine and ethyl chloride.^30 The results of all these measurements proved to be in good agreement with the formula

\[ V = a - b \lg (T_k - T), \]

in which \(V\) denotes the measured volume, \(T_k\) the critical temperature of the liquid, \(T\) the temperature at measurement, and \(a\) and \(b\) constants.

If this formula is differentiated with respect to \(T\), we obtain:

\[ \frac{dV}{dT} = -\frac{c}{T_k - T}, \]

where \(C\) is a constant. Thus the formula takes into account the known fact that the coefficient of expansion of a liquid increases as the critical temperature is approached.

The Italian physicist Grimaldi repeated, very carefully, determinations of the expansion of ether and, comparing the results of his measurements with those calculated by the formulas proposed by various investigators, came to the conclusion that Avenarius’s formula most accurately reproduces the course of the expansion of a liquid (p. 99)\(^{13}\).

After, in 1876, Yablochkov’s candle gave rise to a powerful development of electric lighting, the task arose of ensuring the switching-on, independently of one another, of an arbitrary number of lamps into a single network. This question was called the problem of the “division of electric light.” M. P. Avenarius developed a system for channeling alternating electric current by means of electrolytic capacitors. In 1880 he obtained a privilege for a “method of dividing electric light among an arbitrary number of mutually independent sources or candles”\(^{31}\), tested it on a small scale in Kiev, and then tested and demonstrated it in Paris in the laboratory of the “Société générale d’électricité (procédée Jablochkoff)” during the electrical exhibition of 1881, and delivered a report at a session of the First International Congress of Electricians on October 11, 1881\(^{32}\). Avenarius’s electric capacitors consisted of carbon plates immersed in an aqueous solution of sodium silicate (liquid sodium glass).

M. P. Avenarius describes the action of his capacitors as follows: “When current passes through one of these branches, it goes through the candle, but at the same time polarizes the voltameter, i.e. the latter is charged as if it were a capacitor. When the current changes direction, the voltameter discharges, and this discharge is added to the main current passing through the candle. These discharges of the voltameter, repeated at each change in the direction of the main current, i.e. more than one hundred times per second, will be able to maintain the voltaic arc, provided only that the charge resulting from polarization is sufficiently large”\(^{32}\). Compared with a proper capacitor of corresponding capacitance, the “polarizer,” as Avenarius called his polarization capacitor, has the advantages of compactness and low cost.

The author of a modern course on electric capacitors, V. T. Renné, indicates that the first examples of electrolytic capacitors appeared in 1894—1896\(^{33}\). We believe it inadmissible to overlook the priority of our national science, which, in the person of M. P. Avenarius, as early as 1881 appeared at an international congress with a demonstration of electrolytic capacitors in operation.

The heyday of Avenarius’s scientific laboratory belongs to the decade from 1877 to 1886. The talent of the director, in combination with the gift

with young collaborators provided sufficient conditions for this flourishing. But in tsarist Russia there were no necessary conditions for the expansion of scientific research; the newly arisen center of Russian science was not supported, but stifled. M. P. Avenarius had no possibility of keeping his pupils near him and thereby forming a solid nucleus for the development of the scientific field he had created. Zayonchevsky went to work as a docent at the Institute of Agriculture and Forestry in New Alexandria; O. E. Strauss moved in 1881 to Petersburg and soon turned from questions of molecular physics to electrical engineering. K. N. Zhuk almost abandoned scientific work, devoting his time to teaching.

M. P. Avenarius had to bury his beloved and most talented pupil, Nadezhdin.

A. I. Nadezhdin in April 1886 defended his dissertation for the degree of master; in the same April he was assigned a foreign trip for scientific purposes for two years. Great hopes were placed on it, since he intended to rest at least somewhat and restore his strength, which had begun to leave him as a result of excessive work. But it was already too late: on May 6 he left Kiev for Berlin, and from there went for treatment to Franzensbad, where he died in June. His biographer cites the opinion of physicians that work with poisonous liquids may have influenced the rapid course of his kidney disease.

Avenarius wrote with bitterness: “Although Russia cannot complain of a lack of talents among her sons, these talents rarely bear the desired fruit. One misfortune or another halts the fulfillment of scientific tasks that are often very broadly conceived, and one can only regret the loss to science of young powers that had given great promise. With the death of A. I. we have lost very much; despite his young years, he not only gave promise, but had already managed brilliantly to justify the boldest expectations of his mentors and comrades and, at the age of 28, to win for himself an honorable name among European scholars” (p. VI) ^25.

Throughout the entire period of the most intensive activity of Avenarius’ laboratory, working conditions continued to remain difficult.

“If one takes into account that the rooms are 8 feet high, that the windows, correspondingly, are very small, then one comes to the conviction that the premises of our laboratory are wretched to the point of impossibility” (p. 422) ^3. And in these premises, as Avenarius’ pupil E. K. Shpachinsky wrote, “Avenarius spent several hours in succession every day in one of the rooms of his laboratory among lighted gas burners and heated Magnus baths, in an unbearably high temperature, in a dry atmosphere saturated with carbonic acid, all the time on his feet,

patiently watching the readings of the thermometers, pencil in hand, to record changes in volume, etc.” (p. 423)⁸.

Years of such work could not pass without harm to his health, and from the beginning of the eighties M. P. Avenarius began to lose strength; he aged early and, at the age of 55, in 1890, stopped lecturing and was forced to guard himself against physical and mental fatigue.

M. P. Avenarius died on September 4, 1895.

By the end of the eighties, after Nadezhdin’s death, publications under the heading “From the Physical Laboratory of the University of St. Vladimir” ceased.

A. G. Stoletov ends his biography of M. P. Avenarius with the words: “In the annals of Russian physics, Mikhail Petrovich Avenarius will occupy an honorable place both as an investigator and as a teacher. His name must not be forgotten in world science” (p. 432)³.

A. G. Stoletov, who knew him closely, paints an attractive portrait of M. P. Avenarius: “He was a man of gentle and at the same time straightforward character; he never dissembled, spoke and acted always according to his convictions, and his word could be relied upon. He treated science and professorial duties with reverence, as a sacred cause. The testimony of relatives, colleagues, and pupils supplements this personal impression. They unanimously portray the deceased as an excellent family man, warmly loved in his family circle, as a steadfast and highly honorable member of the faculty, as a friend and favorite of the student youth. In official circles, among his comrades, he inspired respect even in people of another camp; he was alien to opportunism and hated formalism. Students valued in him both an engaging lecturer and an untiring worker-mentor, and a reliable defender in a just cause. Always delicate, indulgent without laxity, he knew how to spare youthful self-esteem, knew how to inspire the gifted and encourage the weak; he set a high moral example and, when necessary, did not refuse material assistance” (p. 426)³. “A pure soul,” “a conscientious seeker”—so Stoletov characterizes him in other lines of the same biography.

5. THE KIEV SCHOOL OF EXPERIMENTAL PHYSICS

M. P. Avenarius’s laboratory was one of the first schools of experimental physics in Russia. As is clear from what has been said, the central place in its program was occupied by the study of the properties of liquids over the entire range of their existence and the study of the properties of vapor up to the critical temperature. Throughout the entire nineteenth century, the study of the properties of liquids and vapors was one of the chief problems of physics, to a considerable extent because of the demands that

... placed before science the development of the steam engine. Watt had already determined the latent heat of vaporization (1781) and the elasticity of water vapor up to \(+133^\circ\). In Regnault’s works, published in his “Report on experiments for determining the principal laws and numerical data entering into calculations of steam engines” (vol. I, 1847; vol. II, 1862; vol. III, 1870), various gases and vapors were investigated with the utmost thoroughness. The doctrine of the continuity of the gaseous and liquid states led to the view that critical states are identical or corresponding states of all substances. According to this doctrine, critical data are, as it were, a natural measure of the substance under study, so that pressure, temperature, and volume, expressed in this natural measure (reduced pressure, temperature, and volume), are connected by a single expression for all substances. Therefore the determination of critical quantities was one of the most important tasks of physics in the seventies and eighties of the nineteenth century. As Avenarius wrote: “Using the principles of the mechanical theory of heat and the data on the critical state of bodies, physicists have recently taken up, with particular interest, the question of the changes produced in bodies by heat. But the lack of experimental data, especially concerning the critical state of bodies, has, on the one hand, placed obstacles in the way of further theoretical investigations and, on the other, has made it impossible to decide which of the proposed theories should be given preference” (p. 7)26.

Questions of the critical state were studied in Russia, simultaneously with the work of the Kiev school, from the theoretical side by A. G. Stoletov, P. A. Zilov, and B. B. Golitsyn; experimentally, these questions were investigated by many prominent researchers abroad. Each newly appearing work was received with great attention and subjected to detailed critical study. In this competition among numerous laboratories, an outstanding place belongs to the Kiev laboratory.

Avenarius’s school followed Pirogov’s precept to “value sober knowledge through labor,” and to respect facts. A precisely determined, rigorously verified fact was the starting point in its generalizations. The critical quantities measured in the Kiev laboratory lived for half a century and retained their significance. Numerous measurements carried out during that time in the laboratories of various countries did not displace the earlier measurements made in the Kiev laboratory. Meanwhile, the Kiev laboratory had at its disposal only very modest means. The predominant part of its equipment consisted of instruments for lecture demonstrations.

M. P. Avenarius’s laboratory developed a distinctive, original methodology, which combined simplicity of means with precision of results and therefore deserves attentive study.

To illustrate the skill that Avenarius and his collaborators displayed in their measurements, let us give a few examples.

Among measurements of critical quantities, the most difficult are considered to be determinations of the critical specific volume or critical density. How was this quantity determined in the Kiev laboratory? The liquid was placed in a thick-walled glass tube, both ends of which were bent; graduations were marked on one limb and the capacity of all its parts was accurately determined, while on the other an expanded cavity of fairly considerable volume was blown. In the first limb the liquid under investigation was placed; in the second, some auxiliary liquid with a considerable coefficient of expansion; the liquids were separated from one another by a column of mercury. The two limbs were immersed in separate air baths, in which an accurately measured temperature was maintained. By heating the limb containing the auxiliary liquid, its expansion was produced. The liquid exerted pressure on the mercury and, displacing it, reduced the volume occupied by the liquid under investigation and its saturated vapor. By continuing to heat the second limb, it was possible to convert all the vapor of the investigated liquid into liquid. At the moment when the last traces of vapor disappeared, the position of the mercury in the first limb was determined, and thereby the volume of the liquid under investigation. The second limb was then cooled; in this process the liquid contracted, the mercury shifted, the volume for the investigated liquid and its vapor in the first limb increased, and part of the liquid passed into vapor. By suitably cooling the second limb, it was possible to convert all the liquid in the first limb into vapor, and at the moment when the last trace of liquid vanished, to determine the volume of the vapor in the calibrated tube. Such an experiment made it possible to obtain the volume of the body under investigation both in the liquid and in the vapor-like state at a definite temperature.

Repeating the experiment at various temperatures of the investigated liquid, one obtained the dependence of the volume of the liquid and of the vapor on temperature. These curves converge at the point corresponding to the critical state of the body. Although it was not always possible experimentally to reach the critical temperature, by extrapolating the experimental curves one could determine quite accurately the place of their intersection and, consequently, the critical volume^34.

Thus, in order to create the required pressures, Avenarius replaced the absent force pump by the thermal expansion of an auxiliary liquid, i.e., he solved the task set before him by extremely simple technical means, and secured a sufficiently accurate result, though at the cost of a considerable expenditure of time on the measurements.

Nadezhdin considered this procedure difficult and replaced it with another, extremely simple one, based on his observation that the normal critical volume corresponds to the disappearance of the meniscus somewhere in the middle of the tube and to a dense, intense turbidity along the entire length of the tube, accompanying the reverse transition.

Nadezhdin prepared cylindrical calibrated tubes (about 2–3 mm in diameter). He took four such tubes, filled them with liquid, each to a definite fraction of the volume, for example: 0.42; 0.37; 0.34; 0.31, sealed them after boiling, and placed them inside baths for heating. Suppose the observation showed that the critical volume lay between 0.37 and 0.34. Then he filled three more similar tubes to 0.36; 0.35 and 0.33 of the total volume. The new experiment showed that 0.36 and 0.35 were close to the critical value, with 0.36 closer than 0.35 (the boundary disappeared not near the lower edge, but in the middle). Then he cooled the tube with filling 0.36, marked the position of the liquid level with a cutter and, having broken off the tip, removed the liquid and dried the tube; he calibrated it with mercury, so that he obtained the ratio of the vapor volume at the critical temperature to the volume of liquid at the given temperature (pp. 71–72)\(^{13}\).

By this method Nadezhdin determined the critical volumes of 17 liquids.

Success was achieved not thanks to the perfection of complex technical apparatus, but thanks to acute analysis and the resourceful use of observations.

Just as simply, and at the same time reliably, the question of ensuring correct thermometer readings when determining the critical temperature was solved in the Kiev laboratory. The tube with the substance being tested and the thermometer were placed side by side inside an air bath. It was necessary to ensure that, even during gradual heating or cooling, the difference between temperatures at different points inside this bath (box) would be minimal, and also that the thermometer readings would not be distorted by the radiation of the walls of the box. For this purpose the given box with the tube and thermometer was placed in another—similar, but larger—box; this one in a still larger one; then in a still larger fourth one and, if required, in a still larger fifth one. Each box was separated on all sides from the next by an air gap.

A. I. Nadezhdin stated with complete confidence that “our method of determining critical temperatures affords greater guarantees of accuracy—one may vouch for this,” and, criticizing certain publications, pointed out that in the measurements of the Kiev laboratory “the difference between the temperatures of the various parts of the last (inner—A. G.) box, according to repeated observations, does not exceed \(0^\circ,1—0^\circ,2\)”\(^{23a}\).

Thus, the accuracy of the results was achieved by critical control of each element of the experiments being carried out, by a special thoroughness of observations, and by a deep analysis of the observed processes.

In investigating physical facts, the school of M. P. Avenarius studied these facts in their internal development, in their motion: a mastered fact served as a step for the transition to the next, for the expansion of the field under study.

Let us give several examples.

Nadezhdin, having at his disposal commercially available, contaminated amylene, which he had to purify in order to determine the critical temperature, measured the critical temperatures of all the fractions obtained by fractional distillation, with an unknown content of impurities, and established that the critical temperatures change by the same amount as the temperatures of fraction distillation differ, i.e., by as much as the boiling point rises, the critical temperature also rises by the same amount^21.

He then extended this relation to isomers and homologous compounds. Thus, phenomena in various substances were brought closer together.

In a similar way, Straus, from observations of the critical temperatures of two liquids, established a simple formula for determining the critical temperature of a mixture from the critical temperatures of the components, and this made it possible for him, by measuring the critical temperatures of mixtures of water with alcohol, to determine the critical temperature of water^17.

Thus, from determining the critical temperature of a pure substance, a transition was made to determining the critical temperature for a mixture of known composition, and then the reverse transition to determining it for a pure substance for which the critical temperature was inaccessible to direct observation.

The school of Avenarius provided brilliant examples of the use of the theory of corresponding states for mastering phenomena that were difficult of access. It was mentioned above that Straus, having established the critical temperature of water, with the aid of this quantity calculated the corresponding temperatures of water and ether and, from tabulated data on the elasticities of saturated vapors of ether and water, compared the corresponding data on the elasticities of water vapor and ether vapor; from this he established their numerical ratio, and this made it possible, from the critical pressure of ether, to determine the critical pressure of water, still inaccessible to direct determination. Then, from the critical temperature and pressure of water, he calculated the boiling points of various liquids in good agreement with experiment^17.

Nadezhdin, in his work on the heat capacity of liquids, gives numerous examples of similar recalculations.

At the foundation of this entire methodology lay the striving to master the phenomena under study comprehensively and fully.

Avenarius put forward the principle: “it is necessary to consider a phenomenon within the widest possible limits of variability” of the cause that determines it[^35]. His collaborators followed him in this, studying the elasticity of saturated vapors or the expansion of liquids over practically the entire range of existence of these liquids.

The methodology of scientific work was carefully developed in the pre-revolutionary schools of Russian physicists; under difficult conditions, with the meager funds allotted by the tsarist government for experimental research, only an original, inventive methodology ensured the development of the school and success in competing with the best laboratories of that time in solving difficult physical problems. Questions of research methodology are of enormous importance in our own time as well, especially in connection with the tasks of increasing the productivity of scientific labor. The experience accumulated in the schools of A. G. Stoletov, M. P. Avenarius, and other physicists is highly instructive; it remains to be collected and studied.

Today the conditions for the development of science in our country have changed fundamentally. First-class scientific research institutes have been created, and a vast army of qualified Soviet scientists has been trained.

But even in the prime of its powers, Soviet science does not forget its predecessors—those who, in the difficult pre-revolutionary years, labored honorably under the glorious banner of Russian science.

CITED LITERATURE

1. Works of N. I. Pirogov, vol. I, publ. by the Pirogov Society, Kiev, 1910.

2. Journal of the Ministry of Public Education, part 119, section II, 1863, pp. 60—67.

2a. Journal of the Ministry of Public Education, part 122, section II, 1864, pp. 88—90.

1. A. G. Stoletov, Collected Works, vol. II, Gostekhizdat, 1941.

2. M. Avenarius, On the electrical differences of metals at various temperatures, St. Petersburg, 1866, pp. 2—3.

3. M. Avenarius, On the electromotive force of thermoelectric elements from the standpoint of the mechanical theory of heat, (Kiev) University News, No. 11, Kiev, 1870, pp. 1—5.

4. O. D. Khvolson, Course of Physics, vol. 4, 2nd ed., Petrograd, 1915, pp. 667—668 and 671.

5. Ch. S. Hastings and F. E. Beach, A Textbook of General Physics, Boston, USA, Ginn Co., 1900, p. 445.

6. From a letter of V. Zayonchevsky to A. G. Stoletov of October 4, 1895. (Archive of the A. G. Stoletov Library.)

7. Biographical Dictionary of Professors and Lecturers of the Imperial University of St. Vladimir (1834—1884), ed. Prof. V. S. Ikonnikov, Kiev, 1884, pp. 3—6.

1. D. I. Mendeleev, Sokolov and Engelhardt’s Chemical Journal, 3, 1860, p. 81.

2. M. Avenarius, Ueber innere latente Wärme, Bullet. de Moscou, 47, No. 3, 1873.

3. V. Zayonchkovsky, Determination of the elasticity of vapors of certain liquids at high temperatures, (Kiev) University News, year 18, section III, 1878, No. 4, pp. 21–49 and No. 8, p. 29.

4. Al. Nadezhdin, Studies in Comparative Physics, Kiev, 1886.

5. Landolt-Börnstein, Physikalisch-chemische Tabellen, 5th ed., Berlin, Springer, 1923, vol. 1, table 73, p. 253.

6. M. P. Malkov and K. F. Pavlov, Handbook on Deep Cooling, Gostekhizdat, 1947, table 46, p. 141.

7. D. Kay and T. Laby, Handbook for the Experimental Physicist, Moscow, 1949, p. 81.

8. O. Straus, On the critical temperature and critical pressure of water, ZhRFKhO, physical section 14, 510–517 (1882).

9. A. Nadéjdine, La determination de la temperature critique dans les tubes opaques, Bulletin de l’Académie Imp. d. Sc. de St. Petersbourg, vol. XXX, No. 5, 1886, pp. 327–330.

10. See, for example, ZhRFKhO, physical section 14, p. 536, 15, p. 25, 16, p. 222, etc.

11. Biographical data on A. I. Nadezhdin:
    a) Biographical sketch in the edition: A. I. Nadezhdin, Physical Investigations, Kiev, 1887, pp. VII–XII; signed B.
    b) D. D. Yazykov, Survey of the Life and Works of Deceased Russian Writers, issue 6, Russian Writers Who Died in 1886, St. Petersburg, 1890, pp. 89–90. Also, Bulletin of Experimental Physics and Elementary Mathematics, No. 1, 1886, pp. 16–18; Brockhaus and Efron Encyclopedic Dictionary, vol. XX, 1897, p. 432 et al.

12. K. Zhuk, On the question of the temperature of absolute boiling of liquids (from the manuscript essay of the student A. Nadezhdin), ZhRFKhO, physical section 14, 157–162 (1882).

13. A. Nadezhdin, On the question of the temperature of absolute boiling, ZhRFKhO, physical section 14, 536–541 (1882).

14. A. Nadezhdin, On the question of the critical temperature of isomers and homologous series, ZhRFKhO, physical section 15, 25–30 (1883).

23a. A. Nadezhdin, A few words concerning the article by Mr. Plavsky “Ueber die kritischen Temperaturen einiger Flüssigkeiten,” ZhRFKhO, physical section 16, 74–75 (1884).

1. A. Nadezhdin, On the heat capacity of liquids, ZhRFKhO, physical section 16, 222–237 (1884).

2. From the correspondence of A. G. Stoletov, kept in the library named after him at the Physical Institute of Moscow State University.

3. A. I. Nadezhdin, Physical Investigations, Kiev, 1887, pp. V–VI.

4. A. G. Stoletov, Collected Works, vol. I, 304, Gostekhizdat, 1939.

5. M. Avenarius, Volumveränderung einer Flüssigkeit durch Temperatur und Druck, Bulletin de l’Acad. Imp. des Sc. — de St. Petersbourg, vol. XXIV, 1877, pp. 525–533.

6. K. Zhuk, Volume of a liquid as a function of temperature at constant pressure, ZhRFKhO, vol. 13, pp. 239–241 and pp. 411–413, 1881.

7. K. Zhuk, Volume of a liquid as a function of temperature at constant pressure (abstract from the manuscript essay of the students Kannegiesser and Dyaevsky). Transactions of the Kiev Society ...

naturalists, vol. VII, issue 2, p. LXXXII (1884); ZhRFKhO, physical section 16, 304 (1884).

1. M. Avenarius, Transactions of the Imperial Russian Technical Society, vol. XVII, 1883; Privilege, No. 89, 1880.

2. Congrès International des Électriciens, Paris, 1881; Comptes rendus des travaux, 1882, pp. 373–375.

3. V. T. Renne, Electrical Capacitors, Gosenergoizdat, 1947, p. 8.

4. M. P. Avenarius, “The Critical State of Bodies,” Journal of Elementary Mathematics, vol. 1, No. 5, Kyiv, 1884.

5. M. P. Avenarius, ZhRFKhO, physical section 16, 402–403 (1884).