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ing values of the estimated number of bacteria per c.c.; that is, the odds are even that this number lay between 14.82 and 17.31, about 4.6 to 1 that it lay between 13.72 and 18.72, about 22 to 1 that it lay between 12.69 and 20.22, and nearly 142 to 1 that it lay between 11.74 and 21.86.
This, I believe, answers Dr. Johnstone's question in so far as the small series of counts permit. The problem is typical of many that have not received due consideration by either biologist or statistician; and conclusions departing widely from the truth are continually being reached through failure to apply any criterion of reliability on the one hand, and through an erroneous application of the probable error on the other hand. It is hoped this brief presentation will stimulate discussion. ELLIS L. MICHAEL
SCRIPPS INSTITUTION, LA JOLLA
THE AMERICAN MATHEMATICAL
THE twenty-sixth annual meeting of the society was held at Columbia University on Tuesday and Wednesday, December 30-31, with the usual morning and afternoon sessions on each day. The attendance included 96 members. President Frank Morley occupied the chair, being relieved at the last session by Professor J. L. Coolidge. The following new members were elected: Dr. H. E. Bray, Rice Institute; Professor I. L. Miller, Carthage College; Dr. Helen B. Owens, Cornell University; Professor E. W. Pehrson, University of Utah. Ten applications for membership were received.
At the annual election the following officers and other members of the council were chosen: vicepresidents, C. N. Haskins and R. G. D. Richardson; secretary, F. N. Cole; treasurer, J. H. Tanner; librarian, D. E. Smith; committee on publication, F. N. Cole, Virgil Snyder, and J. W. Young; members of the council to serve until December, 1922, T. H. Hildebrandt, Edward Kasner, W. A. Manning, H. H. Mitchell.
The total membership of the society is now 733, including 80 life members. The total attendance of members at all meetings, including sectional meetings, during the past year was 393; the number of papers read was 187. The number of members attending at least one meeting during the year was
At the annual election 156 votes were cast. The treasurer's report shows a balance of $10,692.23, including the life membership fund of $7,168.87. Sales of the society's publications during the year amounted to $1,811.52. The library now contains 5,690 volumes, excluding some 500 unbound dissertations.
It was decided to proceed with the incorporation of the society under the general law of the state of New York. A committee was appointed to consider plans for the organization and administration of the society after the retirement of the present secretary and librarian from their offices at the close of the present year. A committee was also appointed to consider the formation of an international union of mathematicians. The committee on mathematical requirements presented a report, which was laid over for consideration at the February meeting.
The following resolutions, introduced by Professor R. C. Archibald as chairman of the committee on bibliography, were adopted by the council:
The council regards the preparation and publication, in America, of a dictionary of mathematical terms as not only most desirable but also entirely feasible, provided that financial aid for the preparation of the manuscript can be secured.
Impressed with possibilities for the more extensive development of pure and applied mathematics in America, and with the importance of such development to the nation, the Council records its conviction that there are undertakings whose active consideration would be highly desirable if adequate financial assistance might be regarded as available. Among such undertakings are: 1. The preparation and publication by societies or individuals of surveys, introductory monographs, translations, memoirs, and treatises, in important fields, including the history of mathematics. 2. The organization of research fellowships. 3. The preparation and publication of an encyclopædia of mathematics in English. 4. The preparation and publication of an annual critical survey, in English, of the mathematical literature of the world. 5. The preparation and publication of a biographical and bibliographical dictionary of mathematicians.
The meeting of the society immediately preceded that of the Mathematical Association of America on January 1-2. A very pleasant occasion was the joint dinner of the two organizations on New Year's eve with an attendance of 114 members and friends.
The following papers were read at the annual meeting:
The sum of the face angles of a polyhedron in space of n dimensions: H. F. MACNEISH.
A connected set of points which contains no continuous arc: G. A. PFEIFFER. Fundamental types of groups of relations of an infinite field: C. J. KEYSER.
The theorem of Thomson and Tait and its converse in space of n dimensions: JOSEPH LIPKA. Poncelet polygons in higher space: A. A. BENNETT. Continuous matrices, algebraic correspondences, and closure: A. A. BENNETT.
Concerning points of inflection on a rational plane quartic: L. A. HOWLAND.
Geodesics motion on a surface of negative curvature: H. C. M. MORSE.
The geometry of Hermitian forms: J. L. COOLIDGE. Rotations in space of even dimensions: H. B. PHILLIPS and C. L. E. MOORE.
Note on geometric products: C. L. E. MOORE and H. B. PHILLIPS.
A memoir upon formal invariancy with regard to binary modular transformations: O. E. GLENN. The invariant problem of the relativity transformations of Lorentz appertaining to the mutual attraction of two material points: O. E. GLENN. (Preliminary report.)
The mean of a functional of arbitrary elements: NORBERT WIENER.
Bilinear operations generating all operations rational in the domain : NORBERT WIENER. Fréchet's calcul fonctionnel and analysis situs: NORBERT WIENER.
A set of postulates for fields: NORBERT WIENER. On the location of the roots of the jacobian of two binary forms, and of the derivative of a rational function: J. L. WALSH.
On the proof of Cauchy's integral formula by means of Green's formula: J. L. WALSH. On the order of magnitude of the coefficients in trigonometric interpolation: DUNHAM JACKSON. A problem of electrical engineering: P. L. ALGER. Properties of the solutions of certain functional differential equations: W. B. FITE.
Determination of the pairs of ordered real points representing a complex point: W. C. GRAUSTEIN. Sheffer's set of five postulates for Boolean algebras in terms of the operation "rejection" made completely independent: J. S. TAYLOR.
nEr, the magic wand of actuarial theory: C. H. FORSYTH.
A formula for determining the mode of a frequency distribution: C. H. FORSYTH. Asymptotic orbits near the equilateral triangle equilibrium points in the problem of three finite bodies: DANIEL BUCHANAN.
The definition of birational transformations by means of differential equations: C. L. BOUTON. Area-preserving, parallel maps in relation to translation surfaces: W. C. GRAUSTEIN.
Note on linear differential equations of the fourth order whose solutions satisfy a homogeneous quadratic identity: C. N. REYNOLDS, JR.
A practical problem of aerodynamics and thermodynamics: J. E. ROWE.
A property of permutation groups analogous to multiple transitivity: W. B. CARVER and MRS. E. F. KING.
Some pseudo-finiteness theorems in the general theory of modular covariants: OLIVE C. HAZLETT.
Note on the rectifiability of a twisted cubic: Mary F. CURTIS.
The representation of fractions of periods on algebraic curves by means of virtual point sets: TERESA COHEN.
Necessary and sufficient conditions that a linear transformation be completely continuous: C. A. FISCHER.
On the structure of finite continuous groups with a single exceptional infinitesimal transformation: S. D. ZELDIN.
On the location of the roots of the derivative of a polynomial: J. L. WALSH.
Abstracts of the papers will appear in the March issue of the society's Bulletin.
The thirteenth western meeting of the society, being a joint meeting of the Chicago and Southwestern Sections, was held at St. Louis on the same days as the meeting in New York. The next regular meeting of the society will be held in New York on February 28.
THE MESSAGE OF THE BIOLOGIST1 IT is eminently fitting that we biologists, like virile swarm spores, should periodically come together in a holiday spirit of mutual exchange, and after giving and receiving our messages, go back to our life work, reinvigorated and reoriented, to prepare for another brief period of social conjugation.
The messages we send to one another will have little carrying power, and little influence on the receiver, if they are not specific in content, limited in scope, and securely wrapped up in the precise technical terms of our own familiar code.
On the other hand, the biologist would be wholly lacking in social instincts if he failed to recognize that he also has a more comprehensive message for the layman, who is largely dependent on the biologist for his working knowledge of the great domain of nature-life, and by whom the biologist is provided with the necessary means of existence.
This larger message must have a different vehicle. It must first be summarized, digested and metabolized into the vernacular, before it can circulate through the body of social life, reach its terminals, and there accomplish its strengthening and rectifying purpose.
We may well ask ourselves whether we have such a message to give, and if so, what it is, and who, or what, is our authority. And by "we," I now mean all of us, not merely the biologist, but the astronomer, geologist, chemist, physicist and psychologist, for we are what we are to-day because of the underlying community of our methods and purposes, and because, in our concept of evolution, we acknowledge the same mental sovereignty.
This concept, of which we are the trustees,
1 Address of the vice-president and chairman of Section F, Zoology, American Association for the Advancement of Science, St. Louis, January 31,
initiated in man a veritable intellectual mutation, which is now rapidly expressing itself in new phases of social action, and in the emergence, like the parts of a growing embryo, of new types of social architecture. It is our duty to interpret this concept, and to see to it that its real significance is understood, and rightly used in social growth.
The social metamorphosis which historians call the renaissance was largely due to organic improvements in the system of educational circulation and the transmission of mental possessions from man to man. Learning was democratized by translating the bible and the classics into the vernacular, and by this betterment in transmission across time and space, the profits of a dead past were made to flow more freely into a living future, making those profits in some measure the mental heritage of the common people, and their enduring germinal possessions for self-constructive purposes.
In this accelerated social growth, the base line for the orientation of human conduct, and for the measurement of right and wrong, good and evil, was the bible, the classics, and the divine right of civic and religious leadership. The power and stability of these external directive agencies was universally acknowledged, the source of their authority unquestioned, and like radiant beams, their trophic influence was formally expressed in the prevailing architectonics of social procedure.
We are now witnessing, incident to a new birth of social vision, a new social convulsion, much more significant than that of the middle ages, in which science, and especially biological science, unconsciously played, and is still playing, a very important part. For when we recognized a new source of authority in lawful nature-action and in evolution, the old base line for the measurements of human conduct vanished, and many of the old bonds of social allegiance were destroyed; and now we are asked: What shall be the new compulsion to constructive social action, and on what authority can we stay the march of anarchy?
And you, as biologists and American men of science, can not shirk the grave responsibilities of social leadership now thrust upon
you, for it requires little gift of prophecy to forsee that America is destined quickly to become the world's chief center of biological learning, as she is to-day the center of the broadest sympathy with human life and
Perhaps it may clarify our vision if we first ask, not what biology is, but what science, as a whole, does, and what she tries to do. It will little help us to enumerate all the sciences, or be told there is pure" science and applied science; science experimental, and descriptive. Behind and beyond all these varied aspects of science there must be common motives, and common purposes in the scientists, if we are rightly to include them as intelligent beings in the same class.
Let us therefore precipitate and remove these adjective purities and impurities, and you will then agree with me, I believe, that there still remain in science several overlapping functions and purposes. First to explore and to chronicle. To that end, she aims to discover what things are contained in nature, where they are, what they do, what the order is, step by step, of their coming in, their growing up, their going out. And then to memorize, to conserve her mental possessions, to register, in convenient and enduring symbols the result of her explorations, for future usage. Second, to compare and explain. To that end, she aims to discover why things are as they are, in what respects they differ, in what they agree, how one thing influences another, constructively, or destructively, and to distinguish the right ways of doing things from wrong ways. Her third function is to do things rightly. In that respect, she is artistic, architectural. To that end, by conforming her ways of doing things to nature's ways, she aims to create, and to conserve, and to use her records and her knowledge of right and wrong profitably.
Thus three qualifying motives pervade science: the acquisitive, the ethical and the moral. She seeks knowledge through experience, wisdom through understanding, and profit through obedience. One purpose is self-constructive, or egotistic, the other, self
giving, or altruistic. Both are cooperative functions; in action, continuous; in rightness, cumulative; in effect, creative.
The renaissance of to-day has its chief creative impulse in the consciousness of evolution. This revelation of modern science, which we all acknowledge as our guiding star, has come to mean world-growth, or the progressive organization and architectural upbuilding of nature. Nature is now the source of our authority, and creative nature-action, as expressed in nature-growth, is the standard of all our values. Science is therefore compelled to express all her measurements in positive and negative constructive terms, which ultimately must be oriented in reference to this gradient base line of natureprogress, called evolution.
In this nature-growth, we fail to discover any gain or loss, either in basic constructive matter, or in energy. But gain there must be, if evolution is a reality. That gain is, in reality, a moral and ethical gain, or a gain in that creative action and constructive rightness which we call organization and directive discipline. There are no better positive and negative terms to express those gains, both relatively and absolutely, than the familiar terms, right and wrong, good and evil.
On this point, therefore, there need be no equivocation in our message. The profit in evolution is in better constructive action. By the conservation of these profits, nature augments her capital in constructive right
But how is this profit made and conserved? That is the really vital question. Until it is answered there can be no underlying intellectual stability in human life, individually, or socially; no basic unity of purpose in human conduct. Here our vision is not so clear. Many of us believe that on this point we have no comprehensive message to give. The most familiar attempts to explain how evolution takes place are restricted to special aspects of evolution, and are often epitomized in personal names, such as Darwinism, Lamarkism, Weismannism, Mendelism. Among
us there are naturalists, morphologists, physiologists, and psychologists; breeders, experimentalists, and bio-chemists. And surrounding us on all sides are the physicists, chemists, geologists, and astronomers, with whom we must reckon, for their domains and their subject matter overlap ours in countless ways.
But unfortunately between all these workers there is little common understanding and much petty criticism.
Are we building out of aimless contributions to science a new Babel's tower of disjointed, slippery words, with nothing to hold them to constructive lines, and preserve the unity of purpose in our social architecture?
Perhaps the most comprehensive terms, although they have little meaning outside the organic world, are natural selection," the struggle for existence," and the "survival of the fittest." But granting their validity within the organic world, they have no definite moral significance. They convey no implication as to how man, or anything else, must act in order to exist, to say nothing of surviving. What is the fittest? Why is it fit? Why does it survive? If right combinations happen primarily by chance, why, or how, do they come to happen regularly? How can "right accidents" become cumulative, or lawful, or determinate, unless there is a saving, or more enduring, directive element in that something we call rightness?
When the layman makes his holiday call on his biological menagerie and points his umbrella at us, hoping to receive through that safety-first device a brush discharge of information, we fail to come across with illuminating answers to these very pertinent questions. But to conceal our low potential, and preserve our self-respect, we all resort to certain unintelligible sounds, or warning signals, according to the particular pen in which we have been bred and exercised, and which are guaranteed to scare away, or charm into inaction, the most intrepid questioner. One mumbles something about "environment" and “ecology," and crawls back into the bushes. Another wheezes something about "enzymes" and "vitality" and goes on with his experimenting. Another climbs to the top of his