and wiry" or "tall and flabby" (p. 381) in placing prospective workers. What we have to know is the limits of physical strength required for specific jobs. Similarly the question of heart efficiency as studied by Lowsley, Crampton and Schneider in this country would be of interest to the industrial physician. The omission of the nine figures illustrating the application of iodine and finger bandages (pp. 397-400) would provide ample space for such a discussion.

The maintenance of high production in any field of activity depends upon the health of the workers. Industrial hygiene need therefore make no appeal to the charity or humanity of industrial managers. It is primarily good business. Efficient, healthy, productive men and women have a social value whether their production is for service or for profits. Under any system of social organization industrial hygiene must therefore play a leading rôle in the future development of the world's industry.

REYNOLD A. SPAETH

SCHOOL OF HYGIENE AND PUBLIC HEALTH,
JOHNS HOPKINS UNIVERSITY

SPECIAL ARTICLES

SOIL ACIDITY THE RESULTANT OF CHEMICAL

PHENOMENA

SALTS of strong acids with strong bases, of strong acids with weak bases, of weak acids with strong bases, of weak acids with weak bases, calcium hydroxide, the lowering of the freezing point, the catalysis of esters and the hydrogen electrode are all in use in one or another of the various methods advocated for the determination of "soil acidity." The results obtained by the different methods show that the condition of a soil at any time can be considered as its progress towards a constantly changing equilibrium according to the principles of Le Chatelier. It is to be remembered that those metallic elements occurring in ordinary soil stand at the top of the electromotive series of elements and that sodium and potassium compounds are all somewhat soluble; whereas, many calcium and magnesium compounds and most iron and aluminum

compounds are very sparingly soluble in

water.

The entirely different results obtained with different salts, and the large variations in soil acidity recently found by Conner when soils were kept at different moisture contents, make it certain that acid soils usually contain many soluble hydrolytic products which are controlled in amount by the quantity of alkaline earths and alkali metals present in the soil.

Carbon dioxide gas has long been known to cause many chemical changes in silicates and phosphates resulting in the increased solubilities of constituents making up these substances. The following results were obtained in recent investigations where soils in culture pots were treated with carbon dioxide. (The details of the different experiments will be published elsewhere.)

1. An "alkaline" sandy soil became acid in reaction in three months treatment with carbon dioxide gas.

2. The acidity of an acid brown silt loam was increased by treating the soil with carbon dioxide gas.

3. Liming this loam decreased its acidity but not as much as the original "lime requirement" determination (Veitch) indicated. One and one half times the total lime requirement did not neutralize the soil.

4. Where the soil was limed, limed and phosphated, and limed and treated with dried blood or sodium nitrate, carbon dioxide gas additions to the soil increased the soil acidity. 5. The specific conductivity of extracts obtained on treating the soils with conductivity water showed that the carbon dioxide gas had changed the constitution of the soil. The specific conductivity of the carbon dioxide treated soils was greater.

6. The acidity of the soils was lowered by extraction with conductivity water and the lowering was greater for those samples which had been subjected to the carbon dioxide treatments. A further evidence that the acidity was due to chemical changes in the soil was that the aluminum and iron in the normal potassium nitrate extracts was effected by the carbon dioxide treatments.

7. The volatile material determination was increased by carbon dioxide treatments, and since this increase could not be accounted for in the determination of total carbon, the carbon dioxide gas must have changed the water of constitution of some of the soil silicates. 8. The composition of the conductivity water extracts from the different soils varied as the fertilizer constituents added would theoretically replace substances known to be present in the soil.

9. The composition of the conductivity extracts from the carbon dioxide treated samples showed that the increased specific conductivity and acidities due to carbon dioxide treatment were associated with substances with low solubility and ionization constants present under conditions where hydrolysis readily took place.

The shifting of the acidity, the chemical changes in the soil and the soil extracts were in accordance with the solubilities of salts of metals high in the electromotive series and their tendencies to hydrolyze. The work leads to the conclusion that soil acidity is the resultant of hydrolytic mass action phenomena and thus the application of the exact amount of lime shown by any method can not be expected to give exact neutrality.

THE two hundred and sixteenth regular meeting of the American Mathematical Society was held at Columbia University, on Saturday, April 23, 1921, extending through the usual morning and afternoon sessions. The attendance included sixtyseven members. Twenty-four new members were elected, and eleven applications for membership in the society were received.

The council voted to accept the invitation received at the February meeeting to hold the next annual meeting of the society at Toronto in connection with the meetings of the American Association for the Advancement of Science.

The following papers were read at this meeting: On the gyroscope: W. F. OSGOOD.

Seven points in space and the eighth associated point: H. S. WHITE.

Most general composition of polynomials: L. E. DICKSON.

Number of real roots by Descartes' rule of signs: L. E. DICKSON.

The Einstein solar field: L. P. EISENHART.

A special kind of ruled surface: J. K. WHITTE

MORE.

On the theorems of Green and Gauss: V. C. POOR.

Pressure distribution around a breech-block: J. E. ROWE.

The mathematical theory of proportional representation. Third paper: E. V. HUNTINGTON. On the apportionment of representatives. Second paper: F. W. OWENS.

On the geometry of motion in a curved space of n dimensions: JOSEPH LIPKA.

Note on an irregular expansion problem: DUNHAM JACKSON.

Hyperspherical goniometry, with applications to the theory of correlation for n variables: JAMES MCMAHON.

On the location of the roots of polynomials: J. L. WALSH.

The kernel of the Stieltjes integral corresponding to a completely continuous transformation: C. A. FISCHER.

On a simple class of deductive systems: E. L. POST.

Topics in the theory of divergent series: W. A. HURWITZ.

A new vector method in integral equations: NORBERT WIENER and F. L. HITCHCOCK.

On a certain type of system of ∞ curves: JESSE DOUGLAS.

Concerning Laguerre's inversion: JESSE DOUG

LAS.

Closed connected point sets which are disconnected by the omission of a finite number of points: J. R. KLINE.

The sum of a series as the solution of a differential equation: I. J. SCHWATT.

Method for the summation of a general case of a deranged series: I. J. SCHWATT.

Higher derivatives of functions of functions: I. J. SCHWATT.

A covariant of three circles: A. B. COBLE.
R. G. D. RICHARDSON,
Secretary

A DECADE OF AMERICAN MATHEMATICS

THE year just closing carries with it into the past another calendar decade, and the fact suggests that I take up with an audience representing the mathematical section of the American Association for the Advancement of Science and the two other mathematical societies meeting with it, a sketch of the progress of our science in this country during the decade. In doing this, I am led to reflect, when I think of the struggle that has marked the period, that though it is difficult to see how a thoughtful and disinterested person can enthuse over international rivalries in territory, dominion, trade advantages or other details of national prestige which are pregnant with dangers of destruction far beyond any possible advantages gained, a desire for national preeminence in scientific attainment is most wholesome and valuable.

I wish I might, therefore, compare the work of America during the decade with that of other countries. But even if this were fair, in view of the handicap the war has imposed on other countries, it would inevitably entail a sitting in judgment on questions of value over a field so broad, with so large a body of workers, that I have hesitated to assume the competency or to appropriate the time requisite to a proper performance of the task.

Instead, I am restricting myself to a review of some aspects of the work of this country alone, seeking to find the directions it has taken, to find some of the respects in which it has been weak, and in which strong, and to draw a few conclusions as to strengthening it in the future.

As to an anlysis of the contributions made, you will agree that since over 1,200 articles 1 Address delivered as retiring vice-president of Section A of the American Association for the Advancement of Science, at Chicago, Dec. 29, 1920.

have been published since 1910, a detailed examination of those articles would be impossible. I have, as a matter of fact, obtained what I believe to be a fairly complete list of these artcles, and made a rough classification of them according to subject matter. Perhaps a quantitative comparison, based on numbers of pages, would have been more informative than one based merely on numbers of titles, but this too would have been open to criticism, and somewhat more difficult to obtain and digest. If you will bear in mind the meaning of the figures given, I have little fear that you will over-estimate their significance, or infer that I have any disposition to propose any quantitative test as the sole measure of the excellence of an individual's scientific output. On the contrary, I should prefer six pages of Fredholm's in the Proceedings of the Royal Academy of Science of Sweden of 1900 to scores of titles and many hundred pages that might be picked out from journals on the other side, or this side, of the water.

The limitations on the statistical field before us must first be stated. It includes no historical, biographical, or philosophical contributions, and only such in applied mathematics as were contributed by men primarily mathematicians, or appeared in journals devoted entirely to mathematics. It does not, moreover, contain articles contributed to journals of primarily didactic emphasis. Otherwise, it is intended to be complete, and contains contributions to a considerable number of foreign periodicals.

I wish to consider first the distribution of effort amongst various sub-fields of mathematics, and then to comment on some other aspects of interest presented by the data collected. In the matter of classification, in addition to certain customary headings, I have endeavored to separate out a few other classes of subjects of interest for the purposes in hand: first, certain topics whose present vitality and interest among mathematicians generally have been pointed out by Bliss, Van Vleck and others on occasions similar to this, and secondly some topics characteristically American in that Americans have taken a significant or preponderant part in their develop

It will be noticed that algebra and analysis constitute about two thirds of the whole, though this is not surprising in view of their large variety of phases and methods. Their share, however, is larger than in most countries, doubtless because of the prevailing tendencies in the countries to which our mathematicians went for training during the closing decades of the last century. I can not help feeling that a more even balance would be desirable, because of the considerable suggestive help of the more intuitive branches of mathematics. Particularly does it seem regrettable that mathematical physics has not received more attention from mathematicians. It is true that some work has escaped a place in the data of the present study because it has not found its way into mathematical periodicals. For instance, a former member of the ordnance department has told me that he has in his possession over a hundred copies, mostly unpublished blue-prints, of articles on ballistics. But in view of the reputed practical temperament of the American people, in view of the racial traditions we might naturally have inherited from Great Britain, in view of its service to mathematics through its great suggestiveness of interesting problems, and in view of the service of mathematics, through mathematical physics, to physics and engineering, it does seem clear that a greater cultivation of this field in this country is most desirable. In fact, it might almost be considered as characteristic of the decade that this desideratum has been repeatedly and forcefully pointed out.

One reason for the situation which exists is to be found in our tendency to early and

over specialization. Our physics departments are apt to load their students with their own courses, with emphasis on the experimental side, often content to have their graduates equipped with the calculus and a formal course in differential equations; while, on the other hand, little physics is usually required of students concentrating in mathematics. This is in part due to lack of mutual confidence, and in part to the student's own haste to receive his degree. Instruction in mathematical physics should be given by mathematical physicists. But until we have produced a more adequate supply of these, mathematician and physicist must cooperate. We can at least offer courses in those parts of mathematics which are of fundamental importance to physics, and in which details of rigor are replaced by cautions, in case of real danger, and in which a sympathetic attitude toward a desire to find out how nature works replaces a disdain for everything aside from the mathematical game, the instructor bearing in mind that the physicist has always the appeal to experiment with which to check his logic. On the other hand, it is probable that lecture courses in physics would be more frequented by students of mathematics if an attempt were consistently and constantly made to draw a clear line between mathematical consequences of previously established results and fresh appeals to experiment or new physical hypotheses. The more this distinction can be made, and the more the physical assumptions can be simplified and gathered into groups at the beginning of course or topic, the more will the course be likely to appeal to the student with mathematical predisposition.

Returning to our table for a glance at the distribution of effort we find the place occupied by algebra even higher than we should expect. This is largely due to the work of two men, Dickson, in the theory of numbers, of groups, and in allied subjects of algebra, and Miller, in the theory of groups. Other investigators whose work has enriched this field include Blichfeldt, Carmichael, Vandiver, Bell and Lehmer, in the theory of numbers; Glenn, Carmichael, Coble, Curtiss, Bennett, Metzler,