evidence that the catastrophes which produce new or temporary stars produce temporary nebulæ, for at a certain stage of their existence the temporary stars have nebular spectra; but in all cases thus far observed as Adams has shown-the nebular spectrum quickly transforms itself into the Wolf-Rayet stellar spectrum. It is not impossible that the planetary nebulæ have in some cases resulted from the more violent catastrophes of distant space and time; that bodies originally stars may have been expanded under the heat of collision or other catastrophe to nebular conditions; but that an ultimate condensation will transform such nebulæ again into the

stellar state, we can not doubt. That such nebulæ as those in the Pleiades (Fig. 13) or as the great Net-Work nebula in Cygnus (Fig. 36) were formed from stars can not be regarded with favor for a moment; but that the many Class B stars existing in those regions should have been formed from nebulous matter, and that others may be forming, implies an evolutionary process that is both natural and easy of comprehension. Transformation from star to nebula is abnormal, is revolution, under the influence of catastrophe. Transformation from nebula to star is normal, is evolution, under the continuous and regular operation of the simple laws of physics.

Those who would suggest that the red stars may be the young stars must start with stars uniformly distributed over the sky, and unassociated with nebulosity; and,

7 This does not include the extremely red (Harvard Class N) stars. We refer to the 457 naked

transforming them through and past the red and yellow stages, must carry them prevailingly into the galactic regions, where, as Class B, or extremely blue stars, they eye red stars of Class M in Pickering's table, page 507, which are prevailingly the more massive members of their spectral class.

preferentially collect in the great nebulous areas, or, in many cases, become enmeshed in details of nebulous structure. We should pause long and consider well before embarking upon a voyage in that direction.

Long centuries of ignorance as to our surroundings gave way, finally, to the enlightening influence of the discovery of the place of the continents upon the earth, and of the place of our planet with respect to the sun. Working at peace and under extreme encouragement, the astronomers of

to-day are learning the place of our star and its planets amongst the other stars. If the Magellanic Clouds, the greater globular star clusters, and the spiral nebulæ prove to be separate and independent systems, we shall bequeath to our successors the mighty problem of finding the place of our great stellar system amongst the host of stellar systems which stretch out through endless space.

LICK OBSERVATORY,

UNIVERSITY OF CALIFORNIA

549

THE FUNCTION OF MATHEMATICS IN
SCIENTIFIC RESEARCH 1

MATHEMATICS embodies some of the earliest scientific developments and hence she was practically unrestricted in regard to the selection of her location in what became later the domain of science. Did she select for herself the most fertile available land or was she misled by superficial attractions in making her choice, while the richest mines were hidden under other land whose surface presented fewer attractions and whose development demanded more 560 complicated machinery? One might naturally expect different answers to this question from the members representing the varied interests of this Science Club.

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It is not our purpose to extol the advantages of location with respect to the mathematical mine. This location was pointed out to you in your youth and the impressions which it has left on your minds are too deep to be modified materially by a few general remarks. Moreover, some of the tunnels of the mathematical mine are used daily by many of you, who gladly speed through them for the purpose of saving time to employ your energies more effectively in the field of your own choice.

Notwithstanding these facts, all will agree that the mathematical mine has been developed extensively, and that its developments have been most helpful and are becoming more useful to various other sciences. As the rivers excavated unknowingly for possible railroad lines through the mountains

1 Read before the Science Club of the University of Wisconsin, April 5, 1917.

long before the construction of such lines was undertaken, so mathematics has been preparing thought roads for sciences long before their development was seriously begun. Hence it does not appear inappropriate for a body of scientists to pause now and then for a few moments to reflect on the methods and ideals which have actuated the mathematical investigator. Such reflections may be inspired by a sense of respect for all that contributes to scientific progress, but they should also prove helpful in the formation of most comprehensive notions in regard to the great problems which confront us as a united band of workers to secure light, to dispel more of the superstitions and to present far-reaching thoughts in the simplest manner.

Among the questions which scientists as a body might be inclined to ask the mathematician is the following: What is the attitude of mind which has contributed most powerfully to mathematical progress Such a profound question would naturally be answered somewhat differently by different men, and your speaker to-night is not so completely ignorant of his own limitations as to suppose that he can furnish a final answer to this question. He hopes, however, that he may not be entirely unsuccessful in making some illuminating remarks on it, and in interesting you by directing your attention to common thoughts which underlie the varied efforts by which we as a body aim to enrich the world.

With this reservation I would be inclined to say that modesty is the attitude of mind which has contributed most powerfully to mathematical progress. The great "Elements" of Euclid, for instance, are candidly based on assumptions or axioms and do not claim to prove everything ab initio. Moreover, this great work does not concern itself directly with such fundamental ques

tions as truth, reality, life, death, etc., but it confines itself to deductions relating to matters which might appear as trivialities when compared with many other problems which then confronted and now confront the human race.

To understand the modesty of Euclid and his geometric predecessors it is necessary to bear in mind the fact that the "Elements" of Euclid were written at a time when other sciences made little or no demand for such results as these "Elements" embodied. Even such a closely related subject as surveying could then make little direct use of these results in view of the theoretic form in which they were presented. The work which is said to have passed through more editions than any other book except the Bible, which considers diametrically opposite questions, must have appeared to many of Euclid's contemporaries as dealing with comparatively trivial matters in a modest way, since it made no attempt to trace its fundamental principles to their sources, but explicitly started with axioms or postulates.

As another evidence of modesty in mathematics I may mention the special symbols for unknowns and the use of equations in these unknowns. The scientific method embodied in equations involving at least one unknown implies the careful study of relations involving something with respect to which we have openly acknowledged our ignorance. Like the axioms or postulates of geometry, it makes no pretention of complete knowledge, but is satisfied with an humble attitude of mind. The basis of mathematical development is thus seen to be characterized by a modesty which has led the investigator to do what he can do thoroughly rather than to try to do that which would naturally interest him more but which lies beyond his power.

Even in its most primitive form, count