Popular Science Monthly/Volume 83/December 1913/The History of Ohm's Law
|THE HISTORY OF OHM'S LAW|
IN the historical development of any branch of science three steps may generally be traced. First, there is the growth, frequently in disconnected masses, of a body of data. A few of the more readily grasped
facts may find quantitative expression. These formnlæ, whether expressed in words or in mathematical terms, prepare the way for the second stage, in which investigation is directed toward the discovery of connecting links between otherwise isolated observations. This inductive process of framing and testing hypotheses ends with that comprehensive generalization in concise mathematical form which constitutes a "fundamental" law. The third and final stage comprises the deductive application of the newly discovered law to the prediction of new phenomena. The first stage is generally the longest, the second the most contradictory and difficult, while the third is the most fruitful and may perhaps be regarded as the most interesting.
The branch of electrical science to which Ohm's law belongs is now so well advanced into the third stage, into which it may be regarded as having been ushered by Clerk Maxwell, that it is difficult to look back to stages one or two. Yet eighty-five years ago the theory of electricity was in the first stage, and the 3^ears from 1825 to 1860, in some respects, mark the limits of the second. This period includes Ohm's work and that of his immediate successors.
Ohm's work was made possible by the discoveries of Galvani, Volta, Oersted, Ampère, Seebeck and others, whose researches had so broadened the knowledge of phenomena connected with the galvanic circuit as to show the probability of a connecting theory. The formulation of such a connecting theory is the task to which Ohm set himself, approaching the problem from the experimental side. The workers of the early nineteenth century had already in a measure identified and separated the factors at play in the galvanic circuit. Examples of such factors are: the "contact force" at the terminals of the circuit, the "flow of current" along the wire, the "electroscopic force" between any two points of a circuit, the tendency of electricity to escape into the air, and the polarization of the electrodes. There was, however, a lack of definiteness of ideas, as well as of methods of quantitative measurement.
It is no small task to select the necessary from among the incidental factors and to express their relation in concise form. Upon this achievement rests Ohm's chief claim to fame. This is not, however, his only claim to consideration, for besides establishing the law which bears his name, he devised mathematical methods for determining the distribution of electricity in a complex system of conductors, for both steady and variable currents. He did much in clearing up the conception of such terms as electromotive-force, current and resistance. In fact, he did for Volta what Maxwell later did for Faraday. The contributions of Dr. Ohm to the theory of electricity were therefore many sided. They were accomplished because he was a trained mathematician, a skilled experimenter and a keen, logical thinker. A less trained man could not have completed his work; one less honest would have been misled ere the end was reached. Finally, it is interesting to note that recognition of the value of his labors and of the importance of his law came tardily. Like many others, his work was misunderstood and only late in life did appreciation and the ambition of his youth—a university appointment—fall to his lot.
A complete statement of the law discovered by Dr. Ohm involves two independent propositions as follows:
In any given circuit, in a steady state, the current I will he directly proportional to the electromotive-force E, and the constant obtained by dividing the latter by the former is termed the resistance of the circuit.
The resistance of a homogeneous conductor of uniform cross-section is inversely proportional to its area of cross-section and directly proportional to its length. The constant entering into the equation is termed the specific resistance of the conductor, since it is the value of when and are unity. The first proposition states that whatever else the resistance may depend upon, it does not depend upon the current. The second proposition shows how circuits of an elementary type have their resistance affected by a change of dimensions. It would be wrong to infer that part two gives any information as to the effect of changing the material, or the physical state of the conductor; such considerations form the subject of the modern study of conduction, but the results form no part of Ohm's law.
In text-books of physics part I. is generally quoted as Ohm's law, while part II. is discussed under applications of the law. Whether part II. is to be supposed deducible from part I. or is to be regarded as self-evident is not made clear. Of course neither supposition is correct. Part II. must either be taken as the result of experiment, or be justified by some particular hypothesis as to the nature of the electric current. The student of physics taking up the subject of electro-kinetics after that of electro-statics, might, if left to himself, very naturally expect that resistance would vary inversely as the circumference of the conductor rather than as the cross-section. The true circumstances of current distribution are certainly the last which would occur to the student who, for example, conscientiously followed the suggestions in Watson's "Text-book of Physics," where it is asserted that current flow is a fiction, the only real flow being that of the energized field outside the wire. Whether it is or is not worth while in text-books to exercise more care in the elucidation of Ohm's law, in a historical discussion of the subject ambiguity can only be avoided by a precise statement of both propositions.
In the present discussion let it be remembered that the terms "resistance" and "potential" had not at the time under discussion been applied to electricity. The idea involved in the first term was introduced by Ohm, previous investigators speaking of "conducting power" or "conductivity"; while the term "potential" was brought into the subject later by Green, who borrowed it from Laplace. Ohm does not use it, but speaks of "electroscopic force" and "tension."
Experiments before Ohm.—The list of those who may properly be associated with the historical development of Ohm's law is a long one. Even when one omits those who studied the applicability of the law to electrolytes and gases, and confines it to those who contributed to the subject by the study of solids only, it includes the names of Cavendish 1750-1837, Priestley 1733-1804, Children 1777-1852, W. S. Harris 1792-1867, Davy 1778-1829, Barlow 1776-1862, J. Cumming 17771861, A. Becquerel 1788-1878, Ohm 1789-1854, G. Fechner 1801-1887, Pouillet 1790-1868, Lenz 1804-1865, S. H. Christie 1784-1865, Ritchie 1814-1895, C. M. Despretz 1792-1863, Kirchhoff 1824-1887, J. M. Gangain 1810-1878, Weber 1804-1891, Maxwell 1831-1878, A. Schuster 1851- and G. Chrystal 1851-.
The first person on record to investigate the relation between electromotive-force, current and resistance, afterwards formulated as part I. of Ohm's law, was Henry Cavendish, of England. His work was done prior to 1775, but remained totally unknown to the world until the publication of the Cavendish Researches by Maxwell, in 1879. Besides certain experiments on the relative conductivity of the human body, of iron and copper wire, and of various liquids, he made four series of experiments to determine "what power of the velocity the resistance is proportional to." In these experiments he employed a collection of wide and narrow glass tubes filled with a salt solution. As a source of current he used the discharge from a Leyden jar. The experiment consisted in adjusting the length of the column of liquid in the tube under test until it permitted the passage of a discharge of the same strength as that through a second tube selected as a standard. Under these conditions the resistances of the two tubes were regarded as equal. His method of determining equality of discharge was to place his own body in circuit with the condenser and test-tube, and then to judge by the sensation experienced. This is perhaps the only case on record where the human body has been used as a quantitative instrument in electrical measurements.
As a result of these experiments Cavendish concluded that the "resistance," in his sense of the word, varied as the 1.08, 1,03, 0.976 and 1.00 power of the "velocity" in the respective experiments. Maxwell tells us that by "velocity" Cavendish meant current and by "resistance" the total force opposing the current. This would make the Cavendish resistance equal to the total fall of potential around the circuit and is equivalent to saying that the resistance, in the modern sense of the word, is independent of the current. In his fourth experiment, which was the one most carefully performed, the result is in exact accordance with the modern view, and considering the crudity of his method all four results may be said to check within a reasonable margin of error. The work of Cavendish was on this basis regarded by Maxwell as an experimental proof of Ohm's law, and it was in this light that he left the matter in editing the Cavendish papers. Xo one since then seems to have done anything further than quote Maxwell.
Nevertheless, a closer examination indicates that Maxwell's statement that Cavendish's fourth experiment "is the first experimental proof of what is now known as Ohm's law," must be taken with some reservation. The conclusion one reaches is that Cavendish tacitly assumes part 11. of the law. It is true that in 1775, two years after the above experiments, he states the law for the combination of resistances in parallel and in series though he does not state how he arrived at it, nor does he give any experimental data in proof of his statement. It can therefore hardly be regarded as part of his experimental proof of Ohm's law. With respect to the effect of cross-section on resistance Cavendish's only recorded experiment consists of a comparison of the shock received through nine small tubes with that received through one large tube of equivalent section and the same length. The fact that the two shocks were equal does not settle the relation between resistance and cross-section except for the case of round conductors. Ohm expanded the work to include sections of other shape. It would seem to be clear that Cavendish can not be credited with the establishment of both parts of the law, and strictly speaking it is an error to speak of him as the "discoverer" of Ohm's law. The most significant obstacle in the way of his doing this was, no doubt, the fact that no such thing as a steady current had as yet been discovered.
Subsequent to these, as yet unknown, experiments of Cavendish, but before the discovery of steady currents, work on the conductivity of different metal wires was undertaken by Van Marum, Priestley, Children and Harris. Using as they did the static discharge as a source of current, their work shows no advance over that of Cavendish either in results or in method. Peter Barlow, of England, was perhaps the first to attempt to use a steady current in the study of resistance. He did this by placing successive wires between the terminals of the same voltaic pile, determining the current strength from the tangent of the angle of deflection of the needle of a "multiplier" (galvanometer). The conclusion that he reached was that the resistance of a conductor is directly proportional to the square-root of the length and inversely proportional to the cross-section. In looking over the data of these experiments one finds discrepancies of 6° to 7° between the observed and calculated deflections based on Ohm's law. This makes it possible to estimate the resistance of the pile, which ought to have been, but was not. included in considering the resistance of the circuit. Such an examination of the data leads to the conclusion that Barlow's failure to reach correct results was due to this neglect of the resistance of his source of current. Had he included this he might have anticipated Ohm, at least to the extent that Cavendish did.
Cumming used the thermoelectric instead of the voltaic pile as a source of current, otherwise his experiments parallel Barlow's, including the same mistakes and reaching the same erroneous conclusions.
Davy: We now come to the first experimenter using steady currents whose results accord with those of Ohm. Sir Humphry Davy about 1820 used a voltaic pile and a divided circuit, one branch of which contained apparatus for the decomposition of water and the other the wire under test.
The experiment consisted in adjusting the length of the wire a-b until its shunting effect was such as to make the potential difference across the water cell just sufficient to cause electrolysis to begin. He found that wires having the same ratio of length to cross-section had the same resistance. This fact, while in accordance with Ohm's law, is a necessary but not a sufficient condition for its establishment. It is, in fact, the same result that Cavendish had reached forty years before.
Becquerel: Antoine Cesar Becquerel, the first of an illustrious line of French physicists, was the discoverer of part II. of Ohm's law. As his rival. Ohm must certainly have been incited by him to greater efforts in his own study of conduction, and in his earliest published papers Dr. Ohm accords to the work of Becquerel both recognition and criticism. Becquerel, like all of the predecessors of Ohm, overlooked the significance of the internal resistance of the source of current, but like Davy his use of a null method eliminated the necessity of taking it into account.
In his experiment Becquerel wound two wires simultaneously on to the frame of a "multiplier" (galvanometer). The terminals of the two coils thus formed being brought out separately could be connected so as either to increase or to oppose each other's effect. In the latter case what is known as a differential galvanometer is formed, the first one on record. Becquerel connected the coils differentially and in parallel. In order now that the coils shall exactly balance each other it is necessary not only that the number of turns of wire be the same on the two coils, but also that their resistance be the same. Since, however, the wires were of different diameter this could only be accomplished by increasing the length, outside of the instrument, of the coil of lower resistance. From one set of experiments Becquerel found, as had Davy before him, that all wires having the same ratio of length to cross-section had the same resistance. But to this experiment he added another showing that the conducting power varied inversely as the length of the conductor. The combination of those two results led for the first time without ambiguity to the conclusion that the conducting power varies directly as the sectional area and inversely as the length of the conductor, thus constituting a complete statement of part II. of Ohm's law. Becquerel also determined by direct experiment that the total current is the same in every part of a series circuit. This fact, so familiar to-day as to seem all but self-evident, was an important one, for without it Ohm's law would be meaningless.
Ohm's Experimental Investigations.—George Simon Ohm was born in Erlangen, Germany on March 16, 1789, After attending the university of his native town he taught in Gottstadt, Neufchatel and Hamburg. In 1818 he became the teacher of mathematics and physics at the gymnasium at Cologne, where he remained for nine years. He was a superior instructor and looked forward with the ambition of securing a university appointment. Then, as now, the best, if not the only, path to preferment lay along the line of scientific research and discovery. To this endeavor Ohm brought three prime qualifications. His father, who was a lock-smith, had trained him as a lad in the use of tools; from his university he gained excellent training in mathematics; in himself he possessed a firm determination to do his best and a strong ambition to succeed. With scant leisure, few books and only the apparatus he himself devised and for the most part built, he had need of patience and perseverance. Difficult as was his progress, he was able in 1825 to publish three papers dealing with the galvanic circuit.
I. The first of these, entitled "Vorlaüfige Anzeige des Gesetzes, nach welchem Metalle die Contact-Elektricität leiten," occupies eight pages of Schweigger's Journal für Chemie und Physik, and describes experiments on the "loss of force" (i. e., loss of potential) due to increasing the length of the wire in a simple circuit. In modern language it is the study of the effect on the terminal potential difference Fig. 2. of varying the external resistance of the circuit. The results of these experiments were expressed by Ohm by the following empirical formula,
In this equation V is the loss of terminal potential difference, due to the insertion of an external resistance of length x, a is a constant depending on the length of the connecting wires, and m a coefficient depending (supposedly) upon the electromotive-force of the circuit, the cross-section of the wire and the constant a. The scheme of the experiment is shown in Fig. 2.
In the light of Ohm's later work it is easy to see that this formula is absurd, a conclusion indeed soon reached by Ohm himself. It however marks one of the steps in the discovery of the law, and an examination of the experimental data shows that it represented these data very closely. Ohm was not however to be long misguided by an equation which could be easily checked by increasing the range of the experiment. He saw that, no matter how precise equation (3) might prove as an approximate formula, it could hardly represent a law of nature. He therefore prepared to test an external resistance fifteen hundred feet long. He does not seem to have published the result of this experiment, but he must have seen his mistake, for he promises to develop a new and correct equation. This he did a year later.
II. Two more papers were published by Ohm in the year 1825, as follows: "Über Leitungs-fähigkeit der Metalle für Elektricität," containing a preliminary announcement of his studies on the relative conductivities of different metals. The results were published the following year and are considered under paper IV.
III. "Ueber Electricitätsleiter." This was a discussion of the discrepancy between the results of Barlow and Becquerel, together with the acknowledgment of the inaccuracy of his own formula for the "loss of force" and an intimation of his intention to revise the formula. This closes the work for the year 1825. During these experiments he used for the source of current a cell and measured the potential difference by means of a Coulomb torsion balance. The progress of the work is marked by a clearing of the way, by a grasp of the problem and a development of method rather than by positive achievement.
IV. Two papers appeared during the year 1826: the first of these was by far the most important and was a long one of twenty-nine pages, entitled "Bestimmung des Gesetzes, nach welehem Metalle die Contact-Electricität leiten, nebst einem Entwurfe zu einer Theorie des Voltaischen Apparates und des Schweigger'schen Multiplicators." While not so well known as his book on the mathematical theory of the circuit, puplished a year later, it is in reality his most important work and contains the following results.
1. The data on relative conductivities promised in paper II. Ohm measured the relative conductivities of copper, gold, silver, zinc, brass, iron, platinum, tin and lead. Without going into the details of these experiments it is important to note that Ohm had gained a thorough appreciation of the significance of the specific conductivity of a conductor,
2. Experiments showing that the resistance of two conductors is the same when they have the same ratio of length to cross-section. Ohm's first experiments on cross-section proved the proposition which had already been proven by Davy and Becquerel, namely, that the resistance depends upon the ratio of length to cross-section. The second experiments comprised the comparison of the currents flowing through two conductors of equal length and equal cross-section, one of which was of circular and the other of very flat section. The result showed that with the same electromotive-force the currents were also the same, thus proving the current to be uniformly distributed throughout the section. His third series of experiments gave a verification of the laws of parallel resistances. These results constitute a full experimental proof of part II. of Ohm's law, which, in its entirety, had not been given up to this time.
3. Experiments showing the relation between the magnetic effect of the current, the electromotive-force of the cell, and the length of wire in the circuit. This is the relation which we have denominated part I. of Ohm's law. He first presented it in the rather unfamiliar form of the following equation.
where X is the magnetic effect of the current, a and b are constants depending respectively upon the electromotive-force and the internal resistance of the source of current; x is the length of wire constituting the external resistance under test. In modern language equation (4) may be written,
The experiments by which these results were established constitute the most important of Ohm's work and they well repay careful study. After some preliminary experimentation his apparatus was reduced to the following essential parts: a thermo-couple for generating a steady current and a specially constructed magnetic torsion balance for measuring the current strength. The apparatus and scheme of connections is shown in Fig. 3.
During the earlier part of his experimental work one thermojuncture was surrounded by a steam-jacket and the other by an icejacket; during the latter part of the work the hot juncture was left at room temperature. The procedure for the experiment was as follows: The steam and ice jackets were first brought to their respective temperatures, the length of the test wire was then adjusted to the required length and lastly the torsion head turned until the magnetic needle was brought back to its zero position. The reading of the torsion head was then recorded and the experiment repeated with a new length of wire.
With the completion of these experiments Dr. Ohm had established both parts of his law, and may be said to have solved the problem to which he had set himself. As in the case of Sir Isaac Newton and the law of gravity, Ohm now found himself in the possession of a key to many doors closed to previous workers, and he proceeded at once to use it, as is shown by his theoretical paper of 1836 and his book of 1827.
V. This paper was published in the Poggendorf Annalen, is ten pages long and is entitled "Versuch einen Theorie der durch galvanische Kräfte hervorgebrachten elektroskopischen Erscheinungen." It is a purely theoretical paper and foreshadows the book which he wrote the following year. In this paper Ohm enunciates his complete law, contrary to the widely accepted statement that the law was first given in the book of 1827 (e. g., Reed and Guthe, "College Physics," 1911). It also contained the correct formula based on this law, for the change of terminal potential difference due to a change in the external resistance. Finally the application of the law to many practical problems is discussed.
VI. The year 1827 furnishes the final paper of the series upon the galvanic circuit, followed by the appearance of the book elaborating the newly discovered relations. This paper appeared in Schweigger's Journal and was entitled "Einige elektrische Versuche." It is a paper of eight pages and contains the results of two experimental investigations confirmatory of the work of the previous year, as follows: (1) A verification of the conclusion as to the uniformity of the distribution of current over the cross-section of the conductor; (2) a verification of the formulæ for the combination of resistances in parallel. This may be said to close Ohm's experimental work in so far as it relates to the establishment of the law under discussion.
Ohm's Theoretical Work.—Turning to the mathematical interpretation which Dr. Ohm gave to the mass of experimental material already considered, we will first examine the paper of 1826 cited above under V. At the beginning of this paper is found the following expression of Ohm's law:
where is the current strength, the specific conductivity of the wire, the cross-section, the electromotive-force of the source of current and the length of the conductor. A second equation brings out for the first time the conception of "reduced length" or resistance. This equation is
where is the length of a hypothetical wire of unit conductivity and unit cross-section and takes the place of the terms in equation (6). Equations (6) and (7) constitute Ohm's formal expression of the law.
In 1827 Dr. Ohm secured leave of absence from the gymnasium in Cologne, and proceeded to Berlin for the purpose of bringing out a book which should contain the theoretical conclusions which he had elaborated from his experiments. This book is entitled "Die Galvanische Kette, mathematisch bearbeitet," "The Mathematical Theory of the Galvanic Circuit." This book is the best known of his works. It contains a comprehensive theory of galvanic electricity, deduced from simple hypotheses and developed mathematically so as to cover a multitude of practical cases. The book may be divided into two parts, of which the first contains an introductory statement of principles on which the theory is based, together with applications to simple problems. Part two involves the use of differential equations and constitutes a more general development of the theory. The absence of a table of contents would seem to indicate haste in getting the book issued. An examination of the text gives the following partial outline of its contents:
In this passage it will be seen that Ohm proposed to follow, in the consideration of the flow of electricity, the lines laid down by Fourier and Laplace for the flow of heat. In fact Ohm does this throughout his book, and in a passage, unfortunately omitted by the English translator, he explains the analogy and acknowledges his debt.
2. After discussing the three principles Ohm applies them to a simple circuit of uniform section and material, in order to obtain a graphical representation of the potential gradient (Gefalle) and its discontinuities.
3. Applications to linear circuits of different material and varying section, with a generalization of the graphical method.
4. Equation for determining the potential at any point, consisting of an algebraic interpretation of the foregoing graphical method.
5. Relation of current strength to the potential gradient. This is Ohm's law expressed in terms of current, electromotive-force and "reduced length" or resistance. It is this statement of the law which is frequently, though wrongly, taken as the earliest expression of the law.
6. Applications of the conceptions of potential gradient and reduced length to special cases.
7. The effect upon the current strength of varying the resistance.
8. Properties of thermo-electric and hydro-electric circuits.
9. The effect upon the electromotive-force and resistance of the number of elements of a battery.
10. Adjustment of the resistance for the best action of a galvanoscope.
11. Divided circuits.
12. The decomposing power of an electric current.
The second part of the book is mathematical in character and need not be outlined here.
Discussion and Summary.—We are now in a position to summarize Dr. Ohm's work in the establishment of the laws of conduction, and to place his experimental work of 1825-26 in proper perspective with respect to his theoretical work of 1827. In doing this it will be necessary first to trace the development of ideas in Ohm's mind, and second to see how the scientific public received the same knowledge.
In tracing this development in the case of Ohm one must infer that the order of appearance of his various published papers marks the stages of growth of his own knowledge. Following this suggestion we find, first, that Dr. Ohm published in 1825 an incorrect empirical formula based upon inadequate experimental data. This paper would seem to show signs of undue haste caused perhaps by a fear that some other worker, Becquerel for example, might anticipate him. If this premature publication can not be placed to his credit his prompt acknowledgment of the error must be. Second, after further experimentation he announced the true law in 1826, in somewhat different form from that in which it is familiar to modern students. In the third place, he framed certain hypotheses from which the law could be deduced. This he does in his book of 1827. In doing this and in elaborating the law and in extending it to a large number of practical cases, Ohm leaves the reader to infer that he starts from the hypotheses and not from experiment. This is a serious mistake on the part of Ohm and it is hard to see just why ho did himself this injustice. He may have assumed that the scientific public was familiar with all of his printed papers—an unsafe assumption at any time—and that direct reference to them was unnecessary. Ohm's book was only made possible by his experimental work and everything of value in it is the direct outcome of the laboratory, yet in the book Ohm writes as if the results reached were deductive and based on the three hypotheses cited above. In this Ohm laid the ground for the misunderstanding of his work by his contemporaries, who did not realize that its basis was experimental and therefore subject only to experimental proof or disproof.
Perhaps Ohm thought that the rather meager foundation of experimental data would be regarded as inadequate for the superstructure, or it may be that he really felt that the experiments had led him to the knowledge of the fundamental causes of the phenomena of conduction, and that his theory was more secure by being logically developed from these supposed fundamental truths. In either event he retarded rather than helped his cause. A third reason that might be assigned would be his desire, supposing him to have it, to be regarded as a deductive rather than as an inductive philosopher. He had, of course, imbibed some of the modern view of the importance of experimentation, else he would not have experimented, but he very likely still retained a good deal of the old Greek notion that by a process of pure reasoning one may reach new truth. In this case experimentation is not so much a source of new knowledge as a new form of thought stimulation. From such a viewpoint the experiments of Ohm had indeed served their purpose so soon as they were completed and he was quite right in ignoring them. Such a view of Ohm's position is strengthened by the fact that he seems to have taken no pains to remove the impression, universal in his day and which persists somewhat even to the present, that his laws were based on theory only and had no experimental origin or support.
So far then as Ohm is concerned we must conclude that however much he may have valued his experimental work for himself, he was well content that the public should consider his laws as being of theoretic and not of experimental origin.
The view which the scientific public early reached as to the value of Ohm's work is well expressed in the following paragraph taken from Cajori's "History of Physics" (pp. 230-1):
In seeking to explain how this extraordinary opinion came to be held the following facts will be of aid:
1. The paper containing Ohm's principal experimental results was published in a German scientific periodical and has never been translated, whereas his theoretical deductions, published in German the very next year, have since gone through two English and one French edition. The theoretical results were therefore far more widely diffused than were the experimental. Indeed it would be easy for a reader of both publications to confuse the priority of two so nearly simultaneous documents.
2. It was only by virtue of the recognition, tardily but distinctly rendered, on the part of Fechner in Germany, Lenz in Russia, Wheatstone and others in England, that Ohm came out of obscurity. Until this was the case and recognition was given by men of recognized standing, there was little reason why any more attention should be given Dr. Ohm and his meager set of experiments than to a number of equally reliable and equally little-known workers, whose results disagreed with his.
3. To whatever extent the English translation may be supposed to have supplied information as to the degree of interdependence of theory and experiment matters could not have been helped by an inexcusable error of translation of a sentence, the German of which is as follows: "Die Grosse des Überganges zwischen zwei zunachst beisamen elementen habe ich unter übrigens gleichen umständen dem Unterschiede der in beiden Elementem befindlichen elektrischen Kräfte proportional gesetzt." This sentence occurs near the beginning of the book and immediately after an intimation that his hypothesis depends in part on experiment, and a wrong rendering must have conveyed a false impression of the real character of the experiments, and therefore of their value. The rendition of this important sentence is: "The magnitude of the transition between two adjacent particles under otherwise exactly similar circumstances, I have assumed as being proportional to the difference of their temperatures." How the word "temperature" came to be rendered for "elektrischen Kräfte" is difficult to see, and it can not be called an improvement on the original.
4. In his paper of 1826 Ohm did not very fully set forth part II. of his law, the following being his expression: "Cylindrische Leiter von einerlei Art und verschiedenen Durchmcsser haben denselben Leitungswerth, wenn sich ihre Langen wie ihre querschitte verhalten," which may be rendered: "Conductors of the same material have the same resistance if they all have the same ratio of length to cross-section." Now while this condensed statement is equivalent to the more elaborate statement contained in the book of 1827, this fact might easily be overlooked by a casual reader. It is also to be remembered that in the earlier stages of the discussion of Ohm's law part I. received general acceptance, while part II. was by no means universally agreed upon.
5. Lastly the fact that he wrote his book from a theoretical and not an experimental point of view invited the judgment passed upon it that his conclusions were "a web of naked fancies" without "the semblance of support from even the most superficial observation of facts."
From a modern point of view it may well be questioned whether the two propositions constituting Ohm's law could ever have been arrived at by any other than an experimental route. Weight is given to this conclusion by the following: (1) Our present knowledge of certain deviations from Ohm's law are accounted for only by the present corpuscular theory of electricity. Now Ohm, so far as he developed his ideas theoretically, did so on the basis of heat flow and the theory of heat was not corpuscular. While such ideas may not be opposed to the corpuscular conception, we can not expect an inadequate conception at the basis of a theory to lead, by a process of deduction, to correct predictions. The same partial conception may, however, prove of great value in an inductive process which is checked at every step by experiment. (2) In the formulation of a theory so essentially simple as is Ohm's law, one must look for a background of clear ideas, and we can admit of but one source of data for this purpose—namely, experiment. The absence of clear ideas of such terms as current flow, resistance and electromotive-force, at the time of, and their presence after Ohm's work is direct evidence of an experimental source of information. Thus the mathematical theory of electrostatics was based on Coulomb's law experimentally established, and a similar experimental basis was necessary for Ohm's law.
The discussion of the origin of Ohm's law may then be summarized as follows: Dr. Ohm carried along his experimental or inductive work simultaneously with the theoretical or deductive work; first the one then the other was to the front, until finally in 1826 he was able, from his experimental data, to announce the true law. In 1827 he ill-advisedly advanced his hypotheses as the origin of his theory without making it sufficiently clear that they were based on experiment. As an example of deductive reasoning the law means little, while as an example of inductive reasoning the law marks an important stage in the progress of science, and in its simple, generalized form is found to be identical with our present wider knowledge, so long as we avoid such phenomena as the skin effect, the Hall effect and conduction through gases.
Ohm's law thus experimentally discovered has stood the most exacting tests that modern methods have made possible. It has served as a firm basis for the progress of electrical science and marks the transition from the somewhat haphazard disconnected researches of those who preceded, to the well-directed, consistently developed labors of those who followed him. In a less degree and in a more limited field Ohm did for a branch of electrical science what at an earlier date Newton did for the great field of mechanics.
- A correct translation is given on a former page.