habits of Bacillus mesentericus, which, in its various strains, is responsible for ropy bread, are already well known to bacteriologists, and, empirically at least, to all the better informed among practical bakers. There is no reason to doubt that with the increased knowledge now being acquired any outbreaks of rope will in the future be easily controlled. That the presence in the loaf of cereals other than wheat can be directly harmful is most unlikely. A favorable effect should indeed be seen in a somewhat improved balance in the protein supplied. Maize, it is true, is said to be badly tolerated by certain individuals, though such cases must be rare. It is also stated that the starch of maize is not fully gelatinized when it is cooked in admixture with wheat under conditions suitable for the production of an all-wheat loaf.

These and other points will doubtless receive the attention of the investigating committee. Its most important task, however, will be to decide, by a thorough sifting of the evidence, the more general question as to whether the war bread is, as a matter of fact, producing any ill effects at all upon the public health. The public will be glad to know that the food controller is in possession of the facts.

Meanwhile, since it is of the utmost importance to the nation that a full supply of bread shall be maintained, while the amount of wheat available is not sufficient for the purpose, we are glad to observe that the medical press is urging the profession to see that the privilege of obtaining high-grade wheat flour for cases supposed to have suffered from the war bread is at any rate not abused.-Nature.

SCIENTIFIC BOOKS

The Human Worth of Rigorous Thinking. Essays and Addresses. By CASSIUS J. KEYSER, Ph.D., LL.D., Adrain Professor of Mathematics, Columbia University. The Columbia University Press. 1916. Pp. vi+314. Six of the fifteen chapters of this volume appeared in SCIENCE during recent years,1 while

1 On page 220 it is stated that Chapter XII., on the "Principia Mathematica," had been printed in Vol. XXV. of SCIENCE. It actually had ap

the remaining nine chapters, together with reprints of some of the six which had first appeared in SCIENCE, were published in various other periodicals or by the Columbia University Press. Hence the volume contains nothing new. Its value is due to the convenient form in which these inspiring essays and addresses are here presented. Unfortunately it contains no index and no table of contents besides the chapter or essay headings.

The title of the volume is the same as that of the initial essay, but some of the other essays contained therein could appropriately have appeared under the same heading, while the remaining ones represent somewhat more special developments along the same general line. Hence the title indicates truthfully the subject-matter of the entire collection. The volume might appropriately have appeared also with the following title: Inspiring thoughts relating to the history, bearing and educational value of mathematics with emphasis on the philosophical elements.

The pre-eminent ability of Professor Keyser along the line of presenting the fundamental elements of abstruse subjects in an elegant and popular manner is well known. His style appeals perhaps more strongly to non-mathematicians than to the majority of the mathematicians, who are often so exclusively interested in technical mathematical questions as to be but little concerned with elegance of language and the philosophical question of human worth. Teachers of mathematics should, however, bear in mind that to many of their students technical mathematical questions have little charm, and that some of these students could doubtless be reached by the more subtle but no less real historical and philosophical questions connected with their subjects.

Hence the volume before us can be highly recommended for the prospective teachers of mathematics, as well as for those who are interested in the general cultural values of variscientific subjects. The professional mathematician will, however, also find therein much that is presented from a somewhat new peared in Vol. XXXV., 1912, and Vol. XXXVII.,

ous

1913.

point of view and that throws new light on the philosophical questions which permeate the various mathematical developments. Among the chapters which might appeal especially to such readers we may mention those bearing the following headings: "The axiom of infinity," "Mathematical productivity in the United States," and "Concerning multiple interpretations of postulate systems and the 'existence' of hyperspace."

In Chapter IX. Professor Keyser discusses "Graduate mathematical instruction for graduate students not intending to become mathematicians," arguing that such courses need not presuppose a first course in calculus, but could be based upon the mathematical preparation gained in a year of collegiate study. He would begin such a course" with an exposition of the nature and function of postulate systems and of the great rôle such systems have always played in the science, especially in the illustrious period of Greek mathematics and even more consciously and elaborately in our own time."

The headings of the nine chapters which have not been mentioned in what precedes are as follows: "The human significance of mathematics," ""The humanization of the teaching of mathematics," ""The walls of the world; or concerning the figure and the dimensions of the universe of space," ""Mathematical emancipation; dimensionality and hyperspace," "The universe and beyond; the existence of the hypercosmic," "The permanent basis of a liberal education," ," "The source and function of a university," "Research in American universities," and "Mathematics."

Some of these titles are the subjects of addresses delivered by Professor Keyser before large audiences, and many of those who recall his stimulating language will doubtless welcome the opportunity to secure a collection covering such a wide scope of interests which are common to all, but which should appeal especially to those devoted to the borderland between philosophy and mathematics. One finds here a mixture of the most modern theo

ries and the emotional descriptions of past generations, a charming flow of language il

luminating most recent advances and, above all, an inspiring tableland of thought which is easily accessible to all but which is closely related with fundamental questions of education.

The mathematicians, as a class, are perhaps too much inclined to put off the historic, philosophic and didactic questions for later consideration, following the example of the great mathematical encyclopedias which are in course of publication. As a result the majority of them become so engrossed in the technical developments of their subjects as to find little time for the postponed questions of the most fundamental importance-a fate which seemed to threaten the encyclopedias just mentioned. A work in which some of these fundamental questions are handled in an attractive manner is therefore a valuable and timely addition to the mathematical literature.

UNIVERSITY OF ILLINOIS

G. A. MILLER

EQUATIONS AS STATEMENTS ABOUT

THINGS

IN the teaching of elementary physics and mathematics, much trouble is often caused by the fact that students who can readily solve an equation given them are unable to formulate in mathematical terms the data occurring in a practical problem. The purpose of this paper is to report briefly the results of several years' experience with a plan designed to remove as much as possible of this trouble by making the equations show more readily their meanings as shorthand statements of the facts. While there is probably nothing about these ideas that has not been suggested before, such suggestions, when applied at all to teaching, seem to have been rather vague and incomplete, or else applied only to one branch of the subject. In this case the plan to be outlined has been used in a general course of physics and in a course in mechanics, with results much more satisfactory than those obtained by the ordinary method.

To illustrate the difference between the old plan and the new, let us consider a single equation, the falling body law

8=gt2.

This process, simple as it appears to the teacher, is not so simple for the student, as it really involves identifying t as the number of seconds the body has fallen, g as the number of ft./sec. in the gravity acceleration, performing the computation and then interpreting the result as a number of feet. One obvious cause of trouble is the necessity for using certain definite units on each side, with the errors made by the use of the wrong units; and another, perhaps not so obvious, is the fact that the formula itself is not a statement about a real distance of so many feet, a real acceleration of so many ft./sec.2 and a real time of so many seconds, but about pure numbers, mere incomplete "so many s," the most abstract things yet invented by man. Under these conditions is it surprising that a freshman fails to formulate his data into mathematical equations?

On the new plan, the equation is taken as a statement about actual concrete things. In this particular case, the computation would take the form,

The interpretation of the formula is now that s is physically a result of the combination of the gravity acceleration g with the time t, which enters once in producing the final velocity gt, and mean velocity gt and again in combination with this mean velocity to give the distance gt2. The essential feature in the application of this plan is the insertion of each quantity as a quantity, that is, as so

many times another quantity of the same kind,

and not as a mere "so many."

If in computation the boy should happen to forget to square t, he would get

=

1

ft. 8 = X 32 X sec.2 202

1 ft. min.2 25 sec.2

To reduce this to simpler terms he has only to substitute 602 sec2 for min2, exactly as he would perform any other algebraic substitution of equals, and then cancel the sec2 and finish the computation. Or, if he lets

he gets

8 =

= X 22:

X 32 sec.2

X 32 sec.2

=

min.,

min. hr. sec. min. = × 22 3600 sec.2 which is as correct an answer as the other. To reduce units the game is simply to substitute equals for equals and cancel. If this does not give the right kind of an answer, it is a sure indication of an error.

Of course, to play the game fairly, we must abolish formulas with lost units, such as s= 16ť. Examples of these are found most frequently in electricity. The old plan would write such a formula as that for the force on a wire in a magnetic field, as F=IlH with a string of restrictions on units, or FIH with another string. By forgetting the restrictions and using the simpler formula with the most familiar units, the students often achieve remarkable results. On the new plan this would be written F-KIH where

=

dyne

amp. cm. gauss.

=

and all restrictions are removed. It is of of course true that this form of the equation involves more writing than the others; indeed, it may be noted here that the process of treating all equations as physical statements is not necessarily worth while for trained men

doing routine computations, but it is extremely

useful for all sorts of cases where the com

putations are not familiar enough to be clas

sified as routine work. For all such cases it is well worth while to write out the proportionality constant, especially if some one is likely to want l, say, in inches or F in kilograms.

For the sake of such mathematical purists as may not approve of the above on philosophical grounds, a few words should be inserted here on the meaning of the term "multiplication." In elementary arithmetic it means merely repeated addition, but with the introduction of irrational numbers the term is extended by mathematicians to an operation that is not strictly repeated addition. The plan here advocated extends the notion of multiplication still further, to cover a physical combination of concrete quantities. In general the definition of multiplication in each individual case amounts to translating into algebra the ordinary verbal definition of the compound quantity involved (area, velocity, work, etc.). This extension is made practicable by the fact that the operation thus defined obeys the same logical postulates as the corresponding algebraic operation on pure numbers. In other words, the machinery of mathematics can be applied not merely to numbers, but to any group of concepts and

operations satisfying the same postulates. This fact is accepted intuitively by most students; and incidentally the emphasis it puts on the definitions prevents most of the wellknown confusion between acceleration and velocity, power and work, and so on.

To sum up, it seems to me after several years' experience with this system, that it has the following important advantages: (1) It treats equations as neat shorthand statements about real physical things and emphasizes the esthetic side of mathematics in general; (2) It provides an enlarged principle of dimensions by which equations may be checked during computation; and (3) It removes completely all restrictions on the units to be used and enables the student to concentrate his attention on the facts of nature without the disturbing influence of arbitrary rules.

DAVID L. WEBSTER JEFFERSON PHYSICAL LABORATORY, CAMBRIDGE, Mass.

SPECIAL ARTICLES

ON THE SWELLING AND "SOLUTION" OF PROTEIN IN POLYBASIC ACIDS

AND THEIR SALTS

THERE are available only scattered observations on the absorption of water by proteins in the presence of various polybasic acids and their salts. In order to obtain further experimental data in this field, we undertook a rather detailed study of this problem during the past year. As examples of proteins, dried gelatin discs and powdered fibrin were used. For the polybasic acids we chose phosphoric, citric and carbonic. In connection with the swelling of gelatin, we studied also its "solution." The general results of our experiments may be summed up as follows.

I

The amounts of water absorbed by gelatin from equimolar solutions of monosodium, disodium and trisodium phosphate depend not only upon which of these salts are present, but upon their concentration. Gelatin absorbs but little more water in a solution of monosodium phosphate than it does in pure water.

In low concentrations of disodium phosphate, gelatin swells decidedly more than in pure water, but as these lower concentrations give way to higher ones, the gelatin swells less and less until, when sufficiently high concentrations are attained, the gelatin swells decidedly less than in pure water.

These same general truths may be stated for trisodium phosphate, except that the absolute amounts of water absorbed in solutions of this salt are, at the same molar concentration, decidedly higher than in the case of the disodium salt. Low concentrations of trisodium phosphate bring about much greater swelling than higher ones. With progressive increase in the concentration of the trisodium salt, there is a progressive decrease in the amount of swelling until a concentration is finally reached in which the swelling is decidedly less than in pure water.

Having studied in this fashion the relation of swelling to type of salt and its concentration, we investigated next the amount of water absorbed by gelatin in phosphate mixtures of compositions varying from the extreme of pure phosphoric acid on the one hand through mono-, di- and trisodium phosphate to pure sodium hydroxid on the other. These mixtures were made in different ways. Beginning with pure phosphoric acid, we added successively greater quantities of sodium hydroxid, or beginning with sodium hydroxid, we added successively greater amounts of acid until the theoretical neutralization had been accomplished; or we began with pure acid and replaced this with more and more of the monodi-, or trisodium phosphate until the opposite extreme of a pure alkali was reached; or we began with a definite concentration of any one of the phosphates and added progressively greater amounts of either acid or alkali. The results in all these expriments were practically the same. In 24 to 48 hours the gelatin attained its maximal swelling (practically). When the amount of swelling is plotted on the vertical and the changes in the composition of the solutions from acid through the mixtures of the mono-, di- and trisodium salts to pure alkali on the horizontal, a curve, roughly V

shaped, is obtained. Greatest swelling is observed in the pure acid solution and least in a solution consisting essentially of monosodium phosphate. From this point on, there is a gradual increase in the swelling of the gelatin until the disodium salt is passed, when there occurs a more abrupt rise until the trisodium salt is reached, beyond which the curve rises still more steeply until the sodium hydroxid end of the series is attained.

The swelling of gelatin in monosodium, disodium and trisodium citrate follows the same general laws as its swelling in the corresponding salts of phosphoric acid. Monosodium citrate in all concentrations increases somewhat the swelling of gelatin over the amount of swelling in pure water. The same is true of low concentrations of disodium citrate. But the higher concentrations of this salt depress the swelling to below that attained in pure water. These statements also hold for the trisodium salt. As we succeed in getting more base into the citrate, there appears a distinctly greater tendency to depress the amount of water absorption.

In studying the amounts of water absorbed in citrate mixtures varying between the extreme, on the one hand, of pure citric acid, through mono-, di- and trisodium citrate to pure sodium hydroxid, we observed that the results (when amount of swelling is plotted on the vertical and progressive change in composition of solution on the horizontal) yield a Ushaped curve. Greatest swelling is obtained in the pure acid, the amount of this swelling decreasing progressively as we approach the monosodium salt. From the monosodium to the disodium salt the curve falls more gently, until a minimal point is reached in a mixture of about equal parts of monosodium citrate and disodium citrate. From here on, the curve rises gradually to the trisodium salt, after which it ascends steeply as we pass toward the extreme of the pure alkali.

We have also studied in this fashion the effects of carbonate mixtures. As the sodium bicarbonate in a pure solution of this salt is gradually displaced by a molecularly equivalent amount of sodium carbonate, and this