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Popular Science Monthly/Volume 65/May 1904/On the Study of Physics

< Popular Science Monthly‎ | Volume 65‎ | May 1904

ON THE STUDY OF PHYSICS.
By Professor FREDERICK E. BEACH,

YALE UNIVERSITY.

THE domain of physics is coextensive with the whole range of phenomena of the material world, but the science of physics, as commonly understood, is restricted to a much smaller field whose boundaries were perhaps first marked by setting off certain groups of phenomena for special study.

In this way there arose five significant branches of physical science: (1) Astronomy, in which are treated the facts and phenomena observed in connection with the heavenly bodies. One peculiarity of astronomical phenomena is worth noting, namely, that they are entirely beyond human control and can not be made the subject of experiment. (2) Chemistry, which treats the relations of different kinds of matter one to another, and those phenomena which accompany material changes, i. e., alteration in the composition of substances. (3) Biology, which deals with vital phenomena; current, as in physiology, or past, as in paleontology. (4) Meteorology, in which are grouped phenomena peculiar to the earth's atmosphere and incapable of repetition at will. (5) All the remaining natural phenomena form the subject of physics, which may be said to treat of mechanics, i. e., the motion and interaction of bodies upon one another, and of those groups of phenomena commonly designated as heat, light, sound and electricity. These conventional divisions of science involve other differences not always clearly apprehended, but important alike to the student and the teacher.

While one may not say that one branch of knowledge is more worthy than another, one set of facts may be more precise or scientific than another, whatever may be the meaning attached to the word. Exactly where to draw the line between science and knowledge has been the subject of some dispute. For the purpose of the present discussion we will adopt as the definition of a science, the precise knowledge of a body of facts accurately verified and erected into a logical system. In this definition, only those branches of learning are intentionally excluded from the rank of science in which the knowledge is either ill defined, or uncertain, or unsystematized, though just what classes of historical investigation or of psychological speculation fail to meet the requirement are of no moment, as we are at present concerned only with branches of physical science. It is obvious that the branches of science thus defined may differ considerably in the precision with which the facts may be stated and in the correlation of the parts of the system, varying all the way from mathematics, which is the science of exactness, to certain branches of natural history, which are no more than catalogues of the forms of life arranged in arbitrary classes and termed science only because they are sciences in the making, the first step always being to arrange the facts in order, even though it be an arbitrary one. Because these differences are often forgotten and sometimes ignored with consequent misapprehension and even serious error, they deserve further review and illustration.

Every science starts with a few postulates suggested by universal experience. These assumptions thus become for that particular science the ultimate things, in terms of which all its conclusions are stated, and they are not, at least as far as that science is concerned, capable of further simplification. The simplest illustration may be seen in the case of mathematics, though many of its followers do not recognize that it is a physical science at all.

Number is a common and easily distinguishable property of all bodies and of all phenomena. By the process of grouping and counting we derive the fundamental laws concerning collections of bodies, such as the associative law, the distributive law, the permutative law, etc. Having secured its fundamental data from the physical world, mathematics withdraws, so to speak, into the realm of thought and, aided by reason alone, weaves its material by successive steps into those remarkable systems which we call arithmetic, algebra, the infinitesimal calculus, the theory of functions, etc. In a sense mathematics is the most fundamental of the sciences, for without it comparisons would lose that element of exactness which alone endows them with the rank of science.

Geomet, though often reckoned as a branch of pure mathematics, is more definitely a physical science than the science of number, since it deals with extension in space, a phenomenon entirely within the domain of physics. But geometry having derived its postulates, or axioms, as they are commonly called in this connection, from experience, like the science of mathematics retires with these data into a world of abstractions; a world deprived of all realities save spatial relations and the laws of thought, and here develops a system self-contained, i. e., requiring no further appeal to the physical world or to human experience.

A procedure similar to this is indeed frequently employed in the other sciences, where, having selected a few facts derived from experience, we divest them of all associations which are for the moment inconsequent, and proceed to derive by a course of reasoning, new relations which were not obvious in the premises; but there is this noteworthy difference that while in geometry and analysis the appeal to nature is made but once and at the start, in physics new appeals to experience must often be made in the course of the reasoning, and the final relations are accepted only when they are not contradicted by the order of nature. And sometimes, even if the facts do not support the theory, it is still important to observe that the conditions of experiment may not have been so simple as the premises assumed and that the theory may still be true when the proper limitations are introduced. In the minds of certain men who are pleased to call themselves practical, a theory is exploded and to be completely rejected as soon as any discrepancy appears between the observed facts and the reasoned conclusions. To the physicist, however, some sort of theory or rational guide is so important and even necessary that a very imperfect or insufficient theory is preferable to none at all. He is not one of those who delight to hold up to ridicule false and abandoned theories, of which so many examples may be found in the history of physics, for he truly recognizes that though they now seem absurdly wrong, they nevertheless served a useful purpose as a temporary scaffold, without whose aid the more lasting structure might never have been erected. From familiar acquaintance with the imperfections of all experimental data, the physicist grows into the habit of holding his deductions subject to correction in the light of new or more accurate observations. Thus there arises the idea so well expressed by the late Professor Rowland, of degrees of truth or untruth.

There is no such thing as absolute truth and absolute falsehood. The scientific mind should never recognize the perfect truth or the perfect falsehood of any supposed theory or observation. It should carefully weigh the chances of truth and error and grade each in its proper position along the line joining absolute truth and absolute error.

The ordinary crude mind has only two compartments, one for truth and one for error; indeed the contents of the two compartments are sadly mixed in most cases; the ideal scientific mind has an infinite number. Each theory or law is in its proper compartment indicating the probability of its truth. As a new fact arrives the scientist changes it from one compartment to another so as if possible to always keep it in its proper relation to truth and error.

The aim of physical science, according to the earlier writers, was the explanation of the phenomena of nature, i. e., the tracing of occurrences to their causes or the proceeding by logical advance from the cause to the effect. The modern and more acceptable view, due perhaps in large measure to Kirchhoff, is that science aims to state in simple and easily reproducible language the order of the processes of nature. A phenomenon then has received its full explanation when we have presented to the mind a picture or a model in which we may reproduce at will the sequence of events which is observed in nature. All attempts beyond this to satisfy the sense of causation must be futile.

It is a well-known fact that the mind derives a certain pleasure in tracing out similarities in very diverse things. One consequence of this process when systematically carried out was the early recognition of the fact that although the forms of nature were seemingly infinite and exceedingly complex, yet there was discernible throughout something like patterns that had been followed, as though nature were not infinitely varied after all. In anatomy, for example, the similarity in the structural form of fishes, birds and mammals was the subject of attention long before the doctrine of evolution furnished a satisfactory explanation for the resemblance.

Not less striking are the formal resemblances between the laws in widely differing departments of physics, so that, for example, if we have solved a certain problem in the distribution of heat in conductors, the same relation between the symbols furnishes the solution for an important case of electrical currents in conductors, as if the forms of the laws in nature were less complex than the phenomena, the most diversified things having been built up after exactly the same pattern. The recognition of these far-reaching and surprising analogies is found to be most helpful. As Hertz once said, 'it seems as though an independent life and reason of its own dwelt in these mathematical formulas; as if they were wiser than their discoverer, and gave out more than had been put into them.' But one caution is necessary. It must always be remembered that the analogies obtain between the relations and not between the things themselves. Thus we derive valuable mental assistance from the observation that electricity in its relation to potential behaves the same as an incompressible fluid with respect to pressure, but it is a great mistake to think that the thing electricity is like the thing water. Or, to take another illustration, the vibrations which give rise to the sensations of light occur with a rapidity which in an elastic solid would require an enormous rigidity; yet here, just as before, the analogy consists in the relations and not in the things, and those who try to think of an ether at once more rigid than steel and at the same time so tenuous that it produces no perceptible retardation of the planets show that they have missed the point of all analogies which is to furnish a mold in which we can cast our thought concerning the sequence of events.

In like manner, a law in science is now regarded simply as a convenient formula by which we express an observed correlation of properties or a uniformity in the order of nature, and the assumption that a law expresses a compelling and inviolate principle is wholly disclaimed.

The fundamental entities of physics, the ultimates in terms of which it is possible to express all other facts and phenomena of the science, are space, time, energy, matter, electricity and ether. It should, however, be said that it is doubtful whether there is necessity for both electricity and ether. There is a growing tendency at present to explain the properties of matter in terms of electricity. The whole aim of the science is to state the phenomena of nature in terms of these quantities in the most exact language possible. From this point of view, a notable distinction in the kindred science of chemistry is at once apparent in that whereas chemistry introduces no new entity, it subdivides matter into upwards of eighty distinct ultimates called elements, thus enormously enlarging the number of combinations and phenomena with which it must deal.

Biology, again, while retaining all the postulates of physics and chemistry, introduces a new principle, that of life, whereby the phenomena to be treated become infinitely complex.

If we now recall that it is only the simplest phenomena of nature that can be formulated, it is at once obvious why chemistry and biology are so much more backward in their growth into exact sciences, the former being largely taken up with a description of the properties of different substances, while the latter can do little more than group together different living forms according to some principle of resemblance amid diversity. These differences make it evident that the character of mind best adapted for their investigation may differ somewhat in the different sciences, and also that the effects when taught will be more or less diverse. Different methods of presentation may likewise be found desirable.

In physics three modes of teaching are available, each of which is to be employed in conjunction with the others, each contributing an indispensable, but necessarily different and unequal, portion to the learner. They are (1) recitations upon a text-book, (2) demonstration of phenomena in lectures, (3) work in a physical laboratory. Now while there is nothing in these methods peculiar to the teaching of physics, it is important to observe that not only do the function and service of each method differ considerably in any one science, but that both the function and service of any method are widely different in different sciences. The function of the text-book is chiefly historical, i. e., to record what progress in the development of the science has been effected by our predecessors; but in connection with physics it can not be too strongly emphasized that the science is not a bare record of observations upon natural phenomena. There is nothing more characteristic of the mental attitude of the physicist toward knowledge than the constant desire to answer not alone the question 'how?' but 'how much?' That is to say in other words, we begin to have adequate knowledge of a fact only when we can measure it.

A musician will say without hesitation that one composition is more classical than another. The insufficient and essentially subjective character of such knowledge is at once apparent if we press the question 'when is one composition twice as classical as another?'

Physics is not a bare record of facts, but a highly developed system of quantitative relations between these facts and the order in which they occur. In this respect physics occupies the great middle ground between pure mathematics, in which the physical facts or axioms are few, and the principles or derived logical relations are the whole content,-and chemistry in which the quantitative logical relations are few and the systematic arrangement of the facts forms the body of the science.

The discovery and the elaboration of the more important physical laws are justly reckoned among the grandest achievements of the human intellect. The character of a discovery, the persons to whom it was due, its philosophical importance, or bearing upon other parts of a science, the representation of the quantitative relations by symbols and the development of still other relations by the application of mathematical analysis, some facility in interpreting such short hand quantitative statements of physical principles, in short, the theory of physics, which is certainly the major part of this science, can best be inculcated by the use of a text-book and recitations. Surely any teaching which does not insist upon the philosophical and quantitative relations, however interesting and brilliant the experiments, or however entertaining the facts presented, or however it busies the student with laboratory exercises, does not teach the science of physics.

But granting that the theory of physics is the backbone of the science, there is no necessity of making it bare bone besides. The lectures should clothe it with flesh and blood. Physics is' not an abstract science like mathematics, and the true physicist objects as much to making physics a mathematical gymnasium as he does to its appropriation as a toy for the kindergarten.

The experimental lecture affords the teacher an opportunity to present and explain to the student under the most favorable conditions the comparatively few important phenomena which he has not already met with. By favorable conditions it is meant that these unfamiliar phenomena often require the use of apparatus so delicate and costly that it is not to be trusted in the hands of any but an expert; or that the matter in question may be so overlaid and obscured by contemporaneous phenomena that the learner can recognize and follow it only with the assistance of a guide. Much also can be explained in the lectures as to the apparatus used for the determination of physical constants and the mode of conducting measurements which has no place in the ordinary text-book, and further the language may be less formal and the mode of presentation may embody much of the personal feeling and enthusiasm of the lecturer, both of which are entirely out of place in a text-book of the principles of a science. There will remain, however, a number of phenomena which, on account of their general minuteness, can not be satisfactorily exhibited in the lectures and must be studied by the individual in the laboratory. In any elementary course they are relatively few in number, and even in a fully equipped laboratory are rarely shown to any but the most advanced students.

The value of the lecture demonstrations must again be emphasized in connection with the fact that a number of the most important phenomena, such, for example, as those of electricity, make no direct or at least no unmistakable appeal to our senses, and are apprehended only by some very ordinary occurrence like the movement of two objects in the field of vision. For example, a discovery of such immense importance as that of current induction by Faraday was evidenced to him only by the minute and momentary motion of a needle over a scale. And since throughout the whole range of physics the measurements are rarely direct, so that the interpretation of the reading of any apparatus is of more importance than making the reading, the first introduction to physical apparatus may often better be given on the lecture table than in the laboratory.

In turn with this brief discussion of the use of the text-book and the lecture demonstration the peculiar function of the physical laboratory deserves careful examination. We remark first of all that because experimental examination and dissection are essential to the teaching of biology it does not therefore follow that the laboratory teaching of physics is essential; nor because laboratory manipulation and observation are necessary to the teaching of chemistry does it follow that laboratory manipulation and observation must be necessary and indispensable to the teaching of physics. In such a statement there is not, however, intended the faintest suggestion that work in the physical laboratory is anything but useful and necessary, but rather to emphasize how widely the function of the physical laboratory differs from that of the chemical and biological laboratories. Physics is an exact science (though just what that means can be learned only in a physical laboratory) while biology is not so at all and chemistry is but to a limited degree.

Biology may be said to have formulated a few general laws, such, for example, as the law of biogenesis or the doctrine that life is generated by living beings only; the law of natural selection or the doctrine that the structure and function of any organism are the results of the survival of those members of a class which were best fitted to their surroundings; the law of prepotency which asserts that the probability of any organism approximating to its type increases with the number of its ancestors of that type. None of these laws contain any quantitative elements and consequently are never made the subject of laboratory measurements. As the rest of the science is for the most part a system of more or less rational classification of multitudinous forms of life, or of the relation of the parts of an organism to one another there is obviously nothing in common between the observational deductions made in the biological laboratory and the quantitative measurements of the physical laboratory.

The divergence between the functions of the chemical and the physical laboratory is hardly less marked. While chemistry has irrefragable claims to designation as an exact science, the enumeration of the chief laws discovered up to the present is a simple matter and does not make a very imposing list. They are:

The law of the conservation of matter.
The law of constant proportion.
The law of multiple proportion.
The law of volumes or Avogadro's law.
The law of specific heats or the law of Dulong and Petit.
The law of periodic groups or Mendelejeff's law.
The law of electrolytic dissociation.
The law of isomerism.
The law of organic series.

It is, to say the least, a noteworthy thing that although these laws constitute the true claim of chemistry to be called a science and are moreover essentially quantitative in their character, practically no one ever thinks it necessary to the laboratory study of chemistry that students should carry out measurements looking toward even a rough verification of these laws, nor has the writer heard that the most enthusiastic advocate of the heuristic method has ever cajoled a student into thinking that he (the student) has discovered one of these laws by himself. The real fact which makes the laboratory study of chemistry a totally different one from that of physics is that the student meets even in the elementary stages a multitude of unfamiliar phenomena which can best be comprehended and learned by individual and intimate association with them, while there are altogether but two or three quantitative experiments which are available. In physics, on the other hand, the proportion is quite the reverse. The phenomena are, for the beginner, simple and entirely familiar, especially in mechanics, but the laws or quantitative relations are very numerous. Moreover, the comparisons made by the chemical student are for the most part qualitative in character; that is to say, they involve observation upon such things as the formation of precipitate, the evolution of a gas, a ready solubility or a change in color, and though the result may be more or less the particular amount is of no consequence. It is for this reason easy to arrange a laboratory course whose aim shall be to acquaint the student with such reactions, and we accordingly find him diligently employed in trying to find out what effect sulphuric acid will have on barium chloride or what silver nitrate has done to his fingers. To spend the same amount of time in observing what happens when a brick is allowed to slide down a board or mercury is poured on a glass plate would be nonsense, and the writer of laboratory manuals feels himself driven to a verification, or study, as he may term it, of the quantitative laws. We thus too often find the student emulating Galileo in his discovery of the law of the pendulum. The absurdity of this attitude must be sufficiently obvious from the fact that in practise the student has always to be told what to discover and that it took the greatest men more than one laboratory exercise to find these laws originally.

It is at any rate true that the observation of physical phenomena for the purpose of mere acquaintance forms small part of any laboratory course, except in the more advanced parts of the subject, such, for example, as light and sound, and contrariwise those topics which demand such examination and acquaintance are commonly considered as too difficult of comprehension to be given to a beginner. From these considerations it must be evident that the usefulness of the physical laboratory can not be inferred from the benefits derived from the laboratory teaching of chemistry, but must be judged by a scale of values peculiarly its own. We have called physics an exact science. Now one of the uses of a physical laboratory is to make clear the meaning of that much misunderstood term 'exact.'

When Galileo was asked by the perplexed engineers why it was that water would not rise in their pumps to more than thirty feet he is said to have returned their question with another, 'Why does it rise at all?' To which they gave the current explanation, 'because nature abhors a vacuum.' 'Well then I suppose nature's abhorrence must cease at thirty feet' was the philosopher's doubtless knowing but evasive reply. That there is a limit to the elevation of liquids by atmospheric pressure and why is now understood by every educated person, but there are comparatively few who appreciate that exactness, like the schoolmen's horror vacui, ceases after a few significant figures.

The three fundamental magnitudes, time, length and mass, each possess some peculiar property in virtue of which they may be more accurately measured than almost any of the other physical magnitudes. Thus the length of the solar day is said to bear the ratio to the sidereal day of 1.00273791 to 1, an accuracy of one part in a hundred million.

The international kilogram has not been determined beyond 3/1,000 of a milligram, which implies an accuracy of one part in three hundred million.

The international meter has been measured in terms of the wave length of light to about one part in ten million, but such accuracy as that mentioned is attainable only in exceptional instances and enormously exceeds that within reach of ordinary careful work, which rarely extends to one part in ten thousand. In the language of paradox, the physicist is exact because he knows how inexact he is. The phrase 'exact value' is a term with which many well meaning people deceive themselves and others. Every measure is imperfect. Mathematical precision is a fatuous term, except as qualified by the limits within which a statement is true.

In the face of some teaching a denial that physics is an experimental science seems almost to be justified. No law can be proved by one or by a hundred experiments. Suppose, as is sometimes done, that a student is given a bar, a knife edge and a couple of weights, and that he is asked to prove to law of the lever. He balances the bar, determines the weights, measures the lever arms and finds what? That the product of each weight by its corresponding lever arm is constant? By no means. For every time, and with whatever pains he has taken to secure accuracy, the product of the weight by its lever arm will be found different on each side, which proves, if literal interpretation of the figures is demanded, that the law of the lever is false. It is very important to recognize the fact that scientific laws are not proved by perfect corroboration of measurements. The proof of any law is of a negative character. Not even the law of gravitation nor the law of the conservation of energy is proved by any positive demonstration. The probable truth of any proposition is assumed from inability to disprove it. Whence it follows that there is nothing more fundamental to the correct understanding of the science of physics, or indeed of science in general, than the interpretation of measurements according to the theory of probabilities and a rational discussion of the inherent errors.

Now the difficult art of physical measurement can neither be taught nor learned apart from some sort of work in the physical laboratory. In this connection the student should be taught something concerning the different sorts of errors that may arise: (1) Errors of construction or of fluctuations in the measuring instruments. Many otherwise instructed people always start with the assumption that their instruments are correct. A little wholesome yet not unsettling distrust of makers' markings can be taught in a brief examination of scales and thermometers. (2) The limitations of the senses and observational errors may be clearly studied from a series of readings made upon almost any instrument having a moderate degree of sensitiveness. (3) Errors of definition, the personal equation, other constant errors and even out and out blunders demand full illustration and recognition. All these things may be taught from the simplest or from any available apparatus, and the knowledge of them is, in the writer's opinion, of more value to the apprehension of pure science than the exhibition or the so-called verification of any law that may be named.

In this insistence that the chief use of the physical laboratory is instruction in the difficult art of physical measurement, an art difficult on the technical side, because of the patience and manipulative skill required, and difficult on the intellectual side because the comprehension of many measuring instruments is dependent upon advanced theory and considerable analytical power, it has not been forgotten that the laboratory may serve certain other though perhaps minor purposes. Just, for instance, as aid to distinct thinking maybe rendered by the use of mathematical symbols and models, so also it will probably assist the unimaginative student to comprehend the laws under discussion if he can examine under his own manipulation the behavior of the apparatus which embodies those laws. Again there are certain phenomena which should be given the student for personal examination in the laboratory. Thermal phenomena involving the reading of thermometers; the passage of a liquid through the critical state; the study of compound tones; the observation of spectra, diffraction and polarization of light waves are examples of the kind of phenomena which require laboratory instruction. The number, however, of such exercises which appear even in the manuals of a college course is insignificant.

The pedagogists who, either with or without any definite knowledge of exact science, are perfectly sure how it ought to be taught assert that the first step in all good teaching is an appeal to the observing powers. "It is a cardinal principle in modern pedagogy that real and adequate knowledge of things can be obtained only in the presence of the things themselves," says one. Assuming that this is as true as the author thought it to be, it is but a half truth, the other half being that the presence merely of the things can not impart any really adequate knowledge. A boy, for instance, might watch the motion of the planets till he was gray without ever learning the first thing about gravitation or the solar system. Facts are but the raw materials of knowledge upon which the reasoning faculties must be exerted in order to extract the hidden principles of nature.

A writer of a well-known series of text-books has adopted as a sort of motto for his pupils, 'Read nature in the language of experiment.' One can not crtiticize an oracular utterance of this sort for the reason that it is not possible to say just what its author had in mind. If it is meant that empirical knowledge derived from the observation of detached facts and not brought into accordance with other facts by means of a hypothesis concerning their relation is sufficient for one to divine the laws of nature, we must certainly dissent. The language of experiment is in general a most difficult one to read, since, as we have been insisting, the measurements are in the great majority of cases indirect and to be interpreted only by tracing through a train of complex relations the consequences of the things observed upon some hypothesis whose truth it is desired to test. Turning now to the more technical side of the science, it is interesting to notice how this indirect character of most physical measurements determines not only the mode of measurements in general, but their precision as well. A measurement, we say, consists in the comparison of any concrete quantity with a definite portion of the same physical magnitude selected as a unit. In a few instances, the comparison is direct, as in the determination of a length by a divided scale, but in the great majority of cases the numerical measure of a quantity is computed by the aid of a relation between other magnitudes which may be more directly, or, at least, more simply, measured. The content of a sphere, for instance, is not determined by successive applications of the unit-cube to the enclosed space, but by first measuring its diameter with calipers and then calculating the volume by the known geometrical relation between the two. Or, to cite another illustration, the direct comparison of a given velocity with the assumed unit of velocity would be a troublesome thing, involving, if they were not very nearly equal in amount, the repeated subdivision of the one or the multiplication of the other. To avoid this, we define the measure of velocity to be the distance traversed per second, and the measurement may then be effected by the simpler process of measuring separately a distance and a time.

In making comparisons, one of the senses must ultimately be appealed to as the judge of the coincidence of two values, but in forming this judgment apparatus is introduced of such sort that the comparison shall contain the least amount of personal bias or subjective impression, thus eliminating as far as possible the psychological element, since the thing desired is a physical equality rather than a psychological one. The former must indeed involve some form of the latter, but equal psychological impressions do not entail equivalent causes. It is a remarkable fact that practically all exact measurements have been reduced to the judgment of the coincidence of two lines by the sense of sight. This universal preference of the eye is probably due not so much to the greater freedom of this sense from illusive deception, as to its unique relation to geometrical space. Various of the other senses are able to distinguish and even to compare degrees or amounts of differences in the sensations peculiar to them, i. e., they are able to estimate a kind of interval or difference in these sensations. The ear in connection with memory, is able to distinguish an interval of time between two successive taps as small as one one hundredth of a second, which is perhaps ten times as well as the eye can do with successive flashes. In another kind of sensation peculiar to hearing, namely pitch, the ear without the aid of beats easily distinguishes sounds whose frequencies are in a smaller ratio than 25/24. Similarly the muscular sense will under proper conditions distinguish an increase in weight of about one per cent., and it is said that experts can distinguish the difference of a fraction of a degree centigrade by the temperature sense alone. Now while the other senses may distinguish two or more kinds of extension, as, for instance, pitch and loudness in the case of hearing, vision is the only sense with quantitative perception in which the extensions are identical in every respect except in their relation to directions, thus giving a field of vision so-called within which individually different marks may be compared. The eye is capable of judging the coincidence of two abutting lines to one minute of arc, which is a more sensitive determination than can be secured from any other sense perception.

The preceding pages may have conveyed the impression that the study of physics is a stern and difficult one. While there was no wish on the writer's part to magnify the difficulties of this most interesting science, it was a definite part of his plan to show that the proper teaching of physics does not consist in the acquisition by the pupil of first-hand knowledge of phenomena; neither does it consist in trying to implant a spirit of inductive reasoning whereby a student is led to divine the great laws of nature as a discoverer; least of all is the obedient following of directions set down in a manual or given by an instructor the study of physics. That alone is true and successful study which cultivates logical power in dealing with phenomena, gives a tenacious hold upon what is known and adds at least something of how the field in present possession of the science was explored and occupied.