man body in the mucous membrane of the colon and in the liver, and it occasionally, but seldom, infects the ileum. The path of infection is by mouth, the organism being usually ingested in the food and drink. It is not killed by the hydrochloric acid in the stomach, and does not attach itself to the oesophagus or stomach. Infection of the liver is through the portal circulation, and these organisms could easily get into the general circulation if they did not find the liver tissues especially favorable. The latest work seems to indicate that amoebae sometimes do enter the general circulation and infect other tissues, but this is unusual.

In typhoid fever the bacillus typhosus infects only certain tissues, the order of preference being about as follows: The lymphoid tissues of the intestinal mucosa, mesenteric lymphs glands, blood, spleen, liver and gall-bladder. The organisms are found in the circulation shortly after the time symptoms begin, and the isolation of them from the blood constitutes one of the methods of early diagnosis. Several tissues other than the above are occasionally involved, but the rest, constituting the great majority of all the tissues of the body, are not affected at all. This is clearly not due to the character of the blood serum, but must be due to the composition of the tissues themselves.

In diphtheria the Klebs-Löffler bacillus has special preferences of growth in the human body. It affects the pharynx, back of the nose, larynx and trachea. It will much less frequently grow in the bronchi and farther forward in the nose and mouth. The bacilli, of course, must be swallowed in large numbers, but the stomach and intestines are rarely involved. A number of cases of diphtheritic peritonitis have been reported, but they are rare, especially in view of the frequency of this disease. The rest of the tissues of the body are seldom infected.

The micrococcus lanceolatus, which causes lobar pneumonia, also has special preferences. The bases of the lungs are much more frequently involved than the apices. It has been found in pure culture in the pleura, pericardium, endocardium, meninges, joints and abscesses. It is frequently found in the sputum of tuberculosis patients in the absence of pneumonia, and may also be recovered from cultures made from the pharynx and back of the nose. Here also the bacteria are swallowed in large numbers and do not infect the stomach and intestines, but cases of pneumococcus peritonitis are occasionally reported. Other tissues are very rarely affected.

Tuberculosis is one of the best examples of specific infection and immunity that we have and is of universal interest to the residents of the southwest. It infects by preference the tissues of the body in about

the following order: The apices of the lungs, bases of the lungs, pleurae, mediastinal glands, larynx, lymphoid tissues of the intestines and related lymphoid glands, peritoneum, urogenital organs, brain and the bones and serous surfaces connected with them. The tissue that is least often infected, in fact, that is practically never infected, is that of the muscles. The tubercle bacillus gets into the general circulation and goes to all parts of the body, after which time all the tissues have an equal opportunity to be infected, at least, as far as this factor is concerned. The protection against infection is evidently not in the blood serum, but is inherent in the character of the tissues themselves.

The therapeutic application of these principles is very broad, and if only partially successful should permit a great reduction in the mortality from diseases that are either incurable at present, or whose ravages are enormous.

It would not be necessary for a chemical that is given for therapeutic purposes to kill the infecting organism. All that would be needed, in many cases at least, would be to moderately reduce its vitality. In most cases of infection the difference between the ability of the organisms to infect and all the forces of the body to resist must be rather small; otherwise these cases would not only be fatal, but rapidly so, especially in the acute infectious diseases. The more acute and fatal the infectious disease the larger will be this margin of infectivity, as it may be called, while it is proportionately less in the milder and more chronic diseases. In the instance of such a chronic disease as tuberculosis this margin of infectivity must be very small, owing to the long duration of the disease and its slow clinical course.

The composition of the tissues of the various animals used in research and for making biological preparations is not exactly the same as that of the human tissues, but corresponds closely enough for therapeutic purposes, as is proved by the variety and importance of the preparations that are already in use. field from which to choose is almost unlimited.

The

Some animals are not susceptible, and others only slightly so, to the organisms causing many human infections. It is possible that this species immunity may be made use of, in fact, it should be of considerable practical advantage.

We may use tuberculosis to illustrate the therapeutic aspect of this problem. The muscles are not infected in spite of the facts that the tubercle bacilli get into the general circulation and that the resistance of the patient in most chronic pulmonary cases is at a low ebb for a considerable time before death. The patients frequently become exhausted and emaciated to the last degree. The muscles are greatly atrophied,

and would certainly contain tubercles if they did not possess some very special resistance. Presuming that this immunity is due in part to the presence of a definite chemical, several conclusions would then seem warranted. This chemical is not soluble in water, as then it would always be in the circulation in sufficient amounts to prevent and cure the disease. It is destroyed in the process of emaciating, as otherwise, when the muscles atrophy, it would be thrown into the circulation in sufficient amounts to either cure the disease, or by killing the bacteria in large numbers to set free enough endotoxines to kill the patient. Research with the various chemical solvents at our disposal would be indicated.

During the years 1912 to 1915 Dr. R. E. McBride, of Las Cruces, New Mexico, and I did some research work in tuberculosis along the lines indicated above. We inoculated, both subcutaneously and intraperitoneally, guinea pigs with sputum containing tubercle bacilli obtained from human cases. The muscle tissue used was obtained from Angora goats. This animal is readily obtainable in southern New Mexico and was selected on account of the high species immunity to tuberculosis. The muscles of the flank were used, and pieces were placed in a 1 per cent. solution of tricresol, which is strong enough for purposes of sterilization.

In preparing the material for injection a piece was placed in a stone mortar, of course under aseptic conditions, and thoroughly ground up. The connective tissue was discarded. A syringe with a rubber plunger was used, and the mixture readily passed through a 20 gauge needle. The mixture was made fresh each time before using.

At first we injected 1 cc at a time. The injections were made two or three times a week, and were given subcutaneously. Infected animals that did not receive injections of muscle tissue were used as controls. Sometimes the treated animals developed tuberculosis and died in the same period of time that the controls did. Occasionally the onset of tuberculosis was delayed for varying periods, and several times it was prevented. Several instances occurred in which lesions that were apparently tuberculous healed on continued treatment with this preparation.

No local abscesses developed as a result of these injections. Lumps formed which lasted for ten days or two weeks, but these were finally absorbed. There was no general reaction, and at no time was there any evidence of anaphylaxis.

We then decided to make clinical use of the method. The first point to determine was the danger of anaphylaxis in human cases. Dr. McBride injected

subcutaneously into me 3 to 5 cc of this preparation once a week for six weeks. In each instance a small lump formed which was only slightly sore, and which was completely absorbed in from two and one half to three weeks. There was no general reaction.

Dr. McBride then treated three tuberculosis patients who were under his care, of course explaining to them the nature of the treatment, and obtaining their consent in writing. Injections were given twice a week over a period of three months.

I treated a patient in El Paso for five months. A total of forty-eight injections were given, at first every day, and later about every five days. The amount of the mixture injected ranged between one and one half to six cc. The lumps produced were only slightly sore and usually disappeared in three weeks. There was no general reaction.

All these patients were in an advanced stage of the disease, with cavity formation and were not doing satisfactorily.

The results obtained were not conclusive, but were sufficiently suggestive to warrant the statement that improvement resulted from the injections. Other patients under observation, in about the same condition, who did not receive the injections, served as controls.

The resistance, of the tissues is no doubt an important factor in other diseases than those of infectious origin. This is suggested in cases of cancer or epithelial malignancy. Our knowledge of cancer has increased very greatly in the past twenty-five years, especially as the result of modern research methods, but the problem is far from solved. There is one point that may throw light upon one phase of our resistance to this condition, and also possibly upon its treatment. It is well known that epithelial malignancy is common in the esophagus, stomach and large intestine and is rare in the small intestine, especially in the duodenum, and also that it is frequently associated with gastric ulcer and very seldom with duodenal ulcer. The study of the causes of these phenomena may develop some very important facts regarding the nature of malignancy and our resistance to it and is well worthy of research. The epithelial cells of the small intestine must have some defensive factor against malignancy that is not present in the other organs mentioned above, and I suspect that secretin is the protecting chemical.

In conclusion I wish to emphasize the facts that the importance of the resistance of the tissues to infection has never received the attention that it deserves and that this is a promising field for research. ELLIOTT C. PRENTISS

EL PASO, TEXAS

SOME MATHEMATICAL ASPECTS OF

COSMOLOGY

(Continued from page 63)

II. COSMOLOGY

WE come now to a somewhat larger point of view. Cosmogony deals only with the mode of origin of the various celestial objects. But the mode of origin is of no more interest than the mode of dissolution, and both of these are but particular stages in a process of transformation that goes on unceasingly. The study of these transformations in their widest possible aspect is what I understand by the word cosmology. It does not belong to astronomy any more than it does to physics and chemistry, for cosmology is as much concerned with the life history of molecules, atoms and electrons and their inter-relations, as it is with the life history of planets, stars and galaxies. If it were a mature subject, instead of being, as at present, a mere infant, the crystal, the cell and the living organism would play a rôle which we might well call vital. To the cosmologist each of these things is a physical unit which comes into existence, plays its allotted rôle upon the stage of time, and passes out of existence. The mode of its organization is definite, its properties are specific, and its dissolution is liable to be more or less abrupt or catastrophic. Throughout all these transformations we recognize that there is something which persists, and that something we call energy. Energy itself is not defined, but it can be measured and with that measurement we must remain content, for the thing itself escapes us.

I am sure that I could not proceed much further without being assured by some one that I was taking a great deal for granted. It is necessary, therefore, for us to stop and to make some inquiries as to the nature of what we are trying to do. I take it that science aims to extend the boundaries of human experience to the utmost limits, and endeavors to coordinate the experience already acquired for the purpose that it may be available at command and that it may be used as a basis of prediction for the experiences which we anticipate. In doing this, it is merely extending in a purposeful and conscious manner, and intensifying, a process which begins with each individual in the first waking hours of infancy, but which frequently dies out during maturity, or even before maturity is reached. By the time we take up the process consciously we are a long way from the beginning, and it is a very difficult matter to get a correct perspective of our activities. We know that we are on our way, but we do not know, quite, where we are going.

It is in these difficulties that we turn once more to mathematics for aid; and not in vain. The geometers of Alexandria, some two thousand years ago, had trouble over the proofs of their theorems. They could not agree on what constituted a proof, for no two of them would start from the same "obvious" propositions. This situation led Euclid to attempt a unification of geometry; and for this purpose he laid down a system of definitions, axioms and postulates, once for all, to which he could appeal whenever necessary in the course of the argument. Doubtless this system of axioms and postulates covered the points which were of interest and dispute at that time, so that, although the system was by no means complete, it did bring unity and harmony into the science of geometry. The axioms are of the nature of logical statements, while the postulates are statements, supposed to be obvious, about the fundamental concepts of geometry. Evidently the first proof must rest upon propositions which are not proved, and new postulates are necessary whenever new aspects of the subject-matter are considered.

The rapid growth of mathematics during the seventeenth and eighteenth centuries was followed by a second period of sharp criticism early in the nineteenth century associated with names of Gauss, Cauchy, Abel, Riemann and many others. The foundations of arithmetic and geometry were carefully examined for the purpose of determining whether or not the structures built upon them were secure. The result of this scrutiny was that these subjects were removed from the domain of nature altogether. The real number system, for example, is a purely intellectual system. The first steps of its creation were taken unconsciously by rude, perhaps barbaric, people because it was a useful thing. Its completion, through the invention of the irrational numbers, was a definitely conscious operation; but a comprehension of nature of the system was not had until it was derived in a logical manner from a precise set of postulates relating certain undefined elements and undefined terms. There is nothing obvious about the postulates; and other number systems can be had by using other postulates. There is nothing objective about the real number system. It is simply a definite intellectual creation, which is interesting in itself and frequently useful in the many situations in which we find ourselves.

With slight changes of wording the same statements can be made with respect to geometry. The necessary postulates are different from the postulates of the number systems, because their subject-matter is different; but the development of a geometry from a system of postulates has the same abstract char

acter as the development of a number system. Naturally, different geometries result from different systems of postulates, and there is nothing objective about any of them. The same thing is true about dynamics. It is idle to inquire whether the relativistic mechanics is true, or whether the classical mechanics is true. From the postulational point of view they are both true, if they are logically above reproach.

Indeed, having once risen to the level of the postulational method, the construction of intellectual edifices upon new systems of postulates becomes a fine game. Some systems of postulates will be found barren, for apparently nothing can be derived from them. Others are fertile, in the sense that at least a small body of theorems can be derived; while a very few others are extremely fertile, and so useful in their applications that we do not think of them merely as intellectual sports; they become sciences, such as algebra, geometry and mechanics. There is this interesting fact, however; so far as I am aware, no very fertile system has been built upon postulates which were not suggested more or less immediately by our common experiences in life.

On account of its philosophical bearing, I regard the development of the postulational method as the greatest achievement of the mathematicians of the nineteenth century. Not only has it made clear the nature of mathematics, but it has also thrown a flood of light upon the nature of the physical sciences, a fact which is well brought out by E. W. Hobson in his recent book, "The Domain of Natural Science." To him who would gain the widest possible point of view, that is to say, the cosmologist, it is a downright necessity.

There is a fundamental difference, however, between mathematics and the natural sciences. The pure mathematician is interested only in logical systems. He is, therefore, quite free from entanglements with observation and experiment; his postulates can be any consistent set of statements that his fancy dictates. The natural scientist is interested primarily in experience. Logical systems would have no interest to him whatever, if it were not for the extraordinary fact that he finds certain logical systems extremely useful. He is free in the choice of his postulates, therefore, only on those points with regard to which he can have no experience whatever, directly or indirectly. In order that I may speak the same language as the mathematician, I shall understand the word postulate, as used in cosmology, to refer only to statements about matters with respect to which we are and always will be entirely free from experience. Similar statements, which observations or experiments may show to be in harmony or in

conflict with experience, I shall call hypotheses. Hypotheses have the nature of tentative postulates, and are therefore strange things to a mathematician. A mathematical system is closed in the sense that it contains only the assigned postulates and the theorems which are logically derivable from them. A cosmological scheme, which deals with experience, is necessarily an open one. One can not write down all the postulates, once for all, nor the undefined terms, for there is nothing to suggest that we have arrived at the outermost limits of experience, or even that such limits exist.

Notwithstanding the fact that each of us is free in the choice of his postulates, so that no system of postulates merits the claim of exclusiveness, still, on account of our common heredity and experience, it is true that certain postulates are commonly made, and have, therefore, something like a universal appeal to our esthetic sense. Let me write down a few of these postulates which seem to me to belong to a normal system:36

(1) There exists a physical universe, external to myself, with which I have experience.

I am not sure whether or not all the adherents of the modern theory of relativity use this postulate. At times it seems to me they do not. At any rate, there are people who seem perfectly happy with a mathematical formula. As for me, I am not happy unless I can see what lies behind the formula; that is to say, a qualitative understanding of a situation is of even greater importance than a quantitative one. x2, y2 Thus + =1 is an exact relation between the mag16 9 nitudes x and y, whatever they may be. But it makes a great deal of difference whether x and y are to be interpreted as the cartesian coordinates of a point, or as the position and velocity of a particle in simple harmonic motion, or perhaps something else. A mathematical formula is not the goal of cosmology.

(2) The geometry of the physical universe is euclidean.

(3) The time of the physical universe is newtonian. The purpose of postulates 2 and 3 is evident. Previous to the exposition of Einstein's doctrine of relativity they would doubtless have commanded universal assent, but the unusual character and the beauty of Einstein's system, together with the simplicity with which it enables us to anticipate certain very delicate phenomena in the domain of physics and astronomy, have won many adherents to it, so that the classical postulates 2 and 3, at least for the time being, are not universally adopted. Inasmuch

36 See also MacMillan, "Some postulates of cosmology." Scientia, February, 1922.

as the relativists do not concern themselves with a physical basis for the transmission of radiant energy, their scheme being a purely mathematical one, I am not sure that they have any need for postulate 1. As a well-known physicist expresses it, they explain terms of the second order beautifully, but they do not explain terms of order zero at all. There are many of us who prefer the terms of order zero, and are unwilling to sacrifice our intuitions upon the altar of the terms of order two. Let us not forget that success or failure argues nothing for the truth or falsity of either system. The relativists have had great successes at certain points where the classic system has so far failed. That is all. This suggests that great discoveries are waiting for some one among the classicists, and the successes of the relativists should be stimulating.

(4) The physical universe is not bounded in space. Not all people, by any means, think of the universe as unbounded. I think I can safely say that nearly all mathematicians do, and many of the more abstract type of physicists and astronomers; nevertheless, it must be admitted that many scientists prefer to think of it as finite. There is no admitted agreement.

(5) The physical universe is continuous in time. Physical things do not disappear from one position in space, only to reappear at the succeeding instant at some distant position. Discontinuities of this type do not occur. Neither does any body act upon another and remote body instantaneously; which is equivalent to saying that energy is transmitted at a finite velocity. Furthermore, something does not become nothing, and nothing does not become something.

(6) The distribution of matter throughout space is uniform if considered on a large scale, by which I mean, the limit of the mean density of a spherical volume having any center tends towards a definite constant, different from zero, as the radius of the sphere increases indefinitely.

Consider a series of concentric spherical surfaces, the radii of which are proportional to the successive integers 1, 2, 3, and suppose n stars are placed upon the nth surface. We can regard such a system as a universe which is not bounded in space (postulate 4). The total number of stars is infinite, but the mean density of the volume of the nth surface is n+1 proportional to which has the limit zero as n n2 increases. The distribution of matter in such a universe is not uniform. If, however, we place n2 stars upon the nth surface, the mean density of the nth (n + 1) (2n + 1) sphere is proportional to which has n2 the limit 2 as n increases indefinitely. If the stars

were scattered over the surfaces of the spheres at random, so as to avoid peculiar distributions, then we would say that the matter in this universe was uniformly distributed. (Considered on a small scale, matter is never uniformly distributed over any volume; even water is not uniform from this point of view).

If, however, all these stars radiate the same amount of light, and if the law of intensity of radiation is strictly the inverse square law, then the amount of light received at the center of the sphere is the same from each sphere, and since the number of spheres is infinite, the total amount of light received at the center is infinite; if, however, we allow for the occultation of one star by another, the entire sky is only as bright and hot as the disk of the sun. This result follows even for the universe in which n stars only are distributed over the nth sphere, for the amount of light received at the center from the nth sphere 1

is proportional to and, as we know, the series

1

n

Σ is divergent. We shall have occasion to re

n

turn to this point later.

(7) There exist physical units which, for a finite interval of time, preserve their identity and exhibit characteristic properties.

(8) The sequence of physical units is infinite both ways, like the positive and negative powers of a positive number.

The term "physical unit" corresponds largely to the word "object." The smallest physical unit which we recognize at the present time is the positive electron, and the largest one is the galaxy. In ascending scale, we have electrons, atoms, molecules, ordinary masses, stars, star clouds, galaxies. We ourselves and the objects with which our thoughts are normally concerned belong to the class of ordinary & masses, and the variety of the physical units which belong to this class is truly amazing. No two objects are exactly alike, yet resemblances are sufficiently strong to permit classification, and even to suggest the postulates on which the mathematician bases his number system.

Ordinary masses are built out of molecules; mole cules are built out of atoms; atoms out of electrons. Likewise the stars are huge masses of gas; the star clouds are vast aggregations of stars; and the galaxy is an aggregation of star clouds. Each physical unit is built up of units of the next smaller order, and our method of accounting for the properties of objects is to recognize a differentiation in the parts of the object. If there existed a smallest physical unit there would be no differentiation, and hence it would have no properties.