As indicated in the table these varieties were grown under 14 different conditions, covering a period of eight years. In tests one to eight inclusive the varieties were grown individually by the hill method, while in the remainder of the tests the grain was grown in drill row. It will be noted from Table I. that the bearded varieties have produced more tillers per plant in every case.

In Table II. is given the summary of the results with all varieties included in the different tests on tillering. The number of varieties together with their repetition in the 13 different tests amounts to 973 cases. For the sake of convenience the ten highest and the ten lowest tillering varieties in each test are separated into a smooth and a bearded group.

bearded; of the 130 cases of the ten lowest tillering varieties, 106, or 80 per cent., are smooth. The results of these tests indicate that the bearded varieties have a greater capacity for tillering than the smooth.

In the study of environmental factors it was found that the rate of seeding a space per plant has a marked influence on the number of tillers produced per plant. Close seeding resulted in a smaller number of tillers per plant, earlier maturity and a better quality of kernel than wide seeding. The time of seeding determines to a large extent the rate of tillering. Early seeding is accompanied by a larger number of tillers per plant than late seeding. The time of seeding, the number of tillers per plant, the yield per plant and the quality of grain are closely correlated. The competition between plants induced by heavy seeding is more marked among smooth wheats than among the bearded. It appears that heavy seeding has a greater effect in lessening the number of tillers, the length of culm, spike, and yield of grain in smooth wheats than in bearded. The fertility of the soil is a factor that directly affects the rate of tillering. Nitrogen and phosphoric acid seem to stimulate the production of tillers; potash has little or no effect. The relation of tillering to yield is shown by the increase in the yield per spike as the number of tillers per plant increases to 4 or 5, beyond this the yield per spike is more or less uniform. The low tillering plants of a variety produce a smaller yield per spike, and the grain is of poorer quality. These experiments have shown quantitatively the effects on the rate of tillering of such factors as time and rate of seeding, the kinds of fertilizer and the relation of the number of tillers per plant to other characters. The tendency of bearded wheats to tiller more freely than the smooth has not been brought before to the attention of the wheat grower. A close analysis of these results indicates that there is a physiological difference between the two types of wheat which may mean that the bearded sorts are able to make better use of the plant food supplied or are able to extract it from

the soil more easily than the smooth type of grain.

A. E. GRANTHAM DELAWARE AGRICULTURAL EXP. STATION

A MEANS OF TRANSMITTING THE FOWL
NEMATODE, HETERAKIS PAPILLOSA
BLOCH1

A RECENT experiment demonstrated that the fowl nematode, Heterakis papillosa Bloch may be transmitted to chickens by the feeding of a dung earthworm, Helodrilus gieseleri hempeli Smith. The thirteen fowls (three of them controls) used in the experiment were hatched in an incubator, reared in a wormproof field cage, and given food free from animal tissues, while the dung earthworms were taken from a poultry yard in which the fowls were heavily infected with H. papillosa. When these chicks were about five weeks old, they were given dung earthworms every few days until each chick had ingested approximately forty worms. Of ten chicks so fed, four became infected with H. papillosa, the results of these examinations being as follows:

Chick 104, examined sixty-four days after first feeding, nine nematodes in the cæca.

Chick 117, examined one hundred thirtyseven days after first feeding, one nematode in the right cæcum.

Chick 128A, examined twenty-nine days after feeding, two nematodes in the cæca.

Chick 130A, examined twenty-seven days after feeding, two nematodes in the cæca. The six remaining chicks and the three controls were free from nematodes.

As is well known, these small nematodes commonly occur in the cæca of fowls, although

1 Contribution No. 19 from the Zoological Laboratory, Kansas State Agricultural College. Aid of Adams Fund.

2 The identification of this nematode has been verified by Dr. B. H. Ransom, Zoologist, B. A. I., U. S. Dept. Agr., Washington, D. C.

3 The earthworms were identified by Professor Frank Smith, University of Illinois.

The field cage with its floor and eighteen-inch walls of cement is so constructed as to be practically insect-proof also. Examinations of control chickens every few weeks for three years have not yielded a single parasitic worm.

they are not infrequently found in the large intestine. Of three hundred ninety-five chickens taken locally and examined in this laboratory during the last three years, two hundred ninety-three (74.1 per cent.) were infected with H. papillosa. The average infection was 34.4 nematodes, but a single infection of one hundred nematodes is not uncommon, and in one instance a fowl contained three hundred twenty-six of these parasites.

The means by which chickens become infected with H. papillosa is not wholly understood. Evidently, in some cases, a dung earthworm transmits these nematodes, but whether the relation between the two worms is one of parasitism or merely that of an association has not been fully determined. The presence of certain nematodes both free in the nephridia and imbedded in the muscles of earthworms furnishes Dung earthworms are of common occurrence in the local poultry yards, and it might be possible to account for the rather heavy nematode infection of fowls from this source alone. But Leuckart long ago pointed out that H. papillosa may develop directly, according to Railliet and Lucet," who, by feeding to a fowl eggs removed from the uterus of H. papillosa, secured a direct infection of fifteen of these nematodes. The writer, likewise, has obtained direct infections by giving eggs of this nematode to fowls reared under controlled conditions. These data indicate that the relation of the nematode to the earthworm is that of an association, in which case the eggs of the former might be carried on the slimy surface of the earthworm or in its engulfed food. However, the evidence is not such as to preclude the possibility that this earthworm, H. gieseleri hempeli, may, in some way, serve as an intermediate host of H. papillosa, and it is hoped that experiments now under way will reveal the nature of this relation.

a suggestive hypothesis.

MANHATTAN, KANS.

JAMES E. ACKERT

Railliet, A., et Lucet, A., "Observations et expériences sur quelques helminths du genre Heterakis Dujardin,'' Bull. Soc. Zool. France, Par., 17: 117-120, 1892.

forum for the discussion of the problems of collegiate instruction in mathematics.

As retiring president of the association, I know of no more fitting topic than that I have chosen. It vitally concerns us; it is bound up with the functions of this association; and the times in which we live seem to point forcibly toward its consideration. I shall attempt to outline to you my own views on the true significance of mathematics, and to sketch what I for one would be glad to see this association promote.

In speaking of the significance of mathematics, I understand that we mean not at all the baser material advantage to the individual student, not at all a narrow utilitarianism, but rather a comprehensive grasp of the usefulness of mathematics to society as a whole, to science, to engineering, to the nation. Any narrower view would be unworthy of us; any narrower demand by educators means a degraded view of the purposes of education in a democ

racy.

Especially under the stress of war, public attention may be secured for the real claim of mathematics as a public necessity, not only to be employed by a few specialists, but also to influence and to determine the conduct and the efficiency of thousands.

Thus a knowledge of trigonometry and of the trigonometric theorems of geometry is a prime requisite for the successful and efficient conduct of our armies, not only by a few engineers who are to make maps and to train artillery, but also for all officers to whom the lives of men are entrusted. Any one of these officers, cut off with his force, without a superior engineer at hand, may lose his position and the lives. of his men if he is ignorant of the significance of these propositions. Ignorance at such a crisis would be next to treason; it would be incompetence.

Do we, in trigonometry, so bring out the significance of the fundamental ideas on right triangles that the officer who faces. such a test will sense the possibility of finding a range, or estimating a distance, without help and without instruments or tables? Frankly I do not believe that we have been doing this, even in such a practical subject as trigonometry. We have been too often content, and too often solely seeking, even here, the knowledge of intricate formalisms, of formulas and rules and theorems, of operations done mechanically. Too often we have omitted, even here, to give insight into the rather obvious significance of these rules and formulas.

On the whole, however, trigonometry is the one subject in which some small measure of insight has usually been secured.

If I now turn to other topics of our curriculum, may I not name scores of equally vital topic, usually studied by our students, in which insight is rarely gained? Let me mention some such instances:

In algebra, as taught in colleges, among the topics always considered are fractional exponents, logarithms and arithmetic and geometric progression. To many, fractional exponents remain a pure formalism, learned by rote and unappreciated, connected neither with the other topics just mentioned nor with any realities of life. That fractional exponents occur in expressions for air-resistance (as in airplanes), in water resistance (as in measuring streamflow), in electricity (as in induction), would surprise most students who pass our courses. That these exponents are determinable and are determined by logarithms would surprise students and some teachers, even if the essential equivalence of exponents and logarithms is adequately emphasized. The idea of a compound interest law, namely, that one quantity may proceed in arithmetic progression as another

related quantity proceeds in geometric progression, is ordinarily not brought out, nor is the fact that this same situation leads to a logarithmic law.

The omission of these and similar vital connections, both of mathematics to the exterior world and of one topic in mathematics to another, is directly responsible for the failure of algebra to reach the hearts of our students, and for the failure of the students to gain real insight into the significance of the subjects they so dully learn.

I shall not dwell long on any one topic, for I desire to emphasize the existence of significance for life and society in the entire range of mathematical courses, and I desire to call your attention to the failure shall I not say our failure?-to bring to light that significance.

Let me turn to analytic geometry for another instance of our traditional blindness, if it be that our sin, if it is not blindness. Here, as before, applications abound. Most of the results of scientific experiment to-day are known and are recorded not by algebraic formulas of traditional form, but solely by curves traced in our tradi-, tional style, showing graphically the functional relations between two or more interdependent variables. Laws of physics, of chemistry, of every quantitative science, expressed by such means abound. The effort of science may well be said to be to deduce from such graphical functions the corresponding laws in algebraic or formalistic form.

Yet to most students of analytic geometry, precisely the reverse view seems to be our aim. The significance of analytic geometry as a piece of scientific machinery is totally lost, and the subject sinks to the level of dubious value in the minds of our students and of half-informed educators. In the present emergency, popular conviction of the real significance of analytic geometry for society is being attained, and

may be fostered, through the occurrence of just such graphical laws in the dynamics of airplanes, in artillery performance (ballistics) and in wireless telegraphy. Here as in general in science, most of our information on functions is now in graphical form, and the desire to express the function in equation form illustrates the fundamental demand of science, and the fundamental significance of analytic geometry.

That the calculus is regarded as dry and uninteresting by many students, and that its value is occasionally doubted, is the strongest proof possible that its significance is not grasped. Here the connection with realities is so easy and so abundant that it is actually a skillful feat to conceal the fact. Yet it is done. I know personally of courses in the calculus (and so may you) in which the pressure to obtain and to enforce memory of formal algebraic rules has resulted in absolute neglect of the idea that a derivative represents a rate of change! I know students whose whole conception of integration is the formalistic solution of integrals of set expression by devices whose complexity you well know. That an integral is indeed the limit of a summation, and that results of science may be reached through such summation is often nearly ignored and not at all appreciated. That the ideas of the calculus should fall so low as to consist mainly in formal differentiations and integrations of set expressions must indeed astound any one to whom the wonderful significance of the subject is at all known. Moreover, it must convince any liberally minded educator who takes our own courses as a true representation of mathematical values that even the calculus is of no importance for real life or for society.

I might proceed to other courses-differential equations as given by Forsyth, the theory of equations as by Burnside and Panton or as by even the most recent