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minutest diameter conceivable as well as for those of the most colossal size we can imagine, and for all intermediate sizes.

Now regard the spheres as equal masses of homogeneous matter endowed with the property of mutual attraction or gravity. They will tend to collect together in a group if free to move relatively, and to remain so. The cubical arrangement would be entirely consistent with complete equilibrium of the attracting forces, but this can not be permanent since it is not a formation of maximum density or concentration. It does not fully satisfy the collecting tendency under the forces of mutual attraction, and the equilibrium of the cubical formation must be unstable. If arranged on the cubical system the group will collapse on the slightest disturbance and the members will seek another arrangement permitting greater concentration. The rhombic dodecahedral grouping affords maximum concentration but it too fails to give complete stability, for it is not perfectly symmetrical and the forces of attraction can not be permanently balanced or in complete equilibrium throughout. Any one group of thirteen of the balls or spheres would be "satisfied" as to concentration and balance or equilibrium by the regular dodecahedral arrangement, but as above set forth this could not possibly obtain as a fixed condition throughout a group of more than thirteen of the spherical bodies. For a group of an indefinite number of the equal spheres greater than thirteen, there is no stable and permanent arangement possible.

We can now in imagination expand the diameter of the balls to any extent limited only by infinity-which means without limitand likewise their size may be reduced to any dimensions short of zero, while their number may be multiplied also without restriction. The above relations are true for the smallest units of matter that can exist as well as for the most gigantic bodies. Furthermore the truth of these principles is not dependent on the complete occupation by matter of each of the individual spherical spaces or volumes considered. These spherical spaces may be only the respective "fields" or space loci of one

or more separate portions of matter in a state of motion respecting neighboring portions in other similar spaces or fields—all having motions of revolution, of vibration, of oscillation, or of pulsation, with limitless combinations and variations as to size and number of the individual portions, their velocity, direction and amplitude of movement, etc.

Every assemblage or group of matter tends to assume the form with an external bounding surface of spherical shape under the mutual attractions of its parts, but however large or small such an assemblage or whatever may be the number of its individual members, its internal structure is governed by the principles above outlined. This indicates an explanation of the paradox involved in the first assumptions or impressions above referred to. The conceit that a perfectly symmetrical grouping of equal spheres with maximum concentration can be made, at first seemingly entirely simple and even axiomatic, turns out to be inconsistent with elementary facts of geometry and therefore impossible.

A direct corollary of this proposition is that a plenum of matter in any form, or any material "continuous medium," is impossible and non-existent. All material substances affected by gravity-which is equivalent to saying all real matter whatever-must be atomic or "granular" in its structure and in its behavior, and this does not depend upon an assumption of "intermolecular repulsion" or of "kinematical energy," nor indeed even upon the theory of energy as a separate entity, nor on any other extraordinary force or attribute. Plain gravitational attraction with the resultant unrelenting stress and struggle for a status which is geometrically unattainable is allsufficient. This may even be made to account for apparent repulsion.

The reason for the conviction and belief that these principles have an intimate and fundamental relation to the universal and eternal unrest of matter and to all physical phenomena of whatever nature will now be apparent and we have at least an interesting and suggestive side light on Boltzmann's demonstration of "the indispensability of atom

istics in natural philosophy," as recently referred to by Professor Bumstead.1

Going back to D'Arcy W. Thompson's book on "Growth and Form," there are found some exceedingly interesting discussions and references pertaining to the various possible divisions of space by plane surfaces that have a direct bearing on this subject. We can assume our equal spheres to be soap bubbles. The shape of a single bubble by itself is determined by the tendency of the enclosing film to contract due to its "tension," or the mutual attraction among its own particles, and the opposition to this contraction tendency presented by the enclosed air. By the same principles that have been explained above it can be shown that in a group of such bubbles the tendency is to assume an arrangement that will give complete symmetry and a minimum total partitioning area, that these conditions can not both obtain as a fixed and simultaneous status for the whole group, and that there can not be a condition of equilibrium and stability throughout such a group. The same will be true of any similar group of compartments or cells enclosing a fluid and with walls or partitions composed of substance that is of a fluid nature. Thompson seems to have fallen into some errors in his discussion, as where he calls the rhombic dodecahedron a "regular solid "2 and where he understands that by means of an assemblage of equal and similar "tetrakiadecahedrons" space may be homogeneously partitioned into similar and similarly situated cells "with an economy of surface in relation to area (volume ?) even greater than in an assemblage of rhombic dodecahedra" (p. 338). The regular" tetrakiadecahedron is a semi-regular polyhedron, a fourteen-sided volume with six equal square faces and eight that are regular and equal hexagons, the sides of these squares and hexagons all being equal. Such a solid may be formed by cutting off the corners of a cube, also by cutting off the corners of a regular octahedron. Space can not be divided into equal volumes of this shape without surplus,

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1 See SCIENCE for January 18, 1918. 2"Growth and Form," p. 328.

or in other words these volumes can not be stacked together without leaving voids. This may readily be determined by a study of the diedral angles, or practically by constructing a number of the tetrakiadecahedrons and trying it. However, these errors are not material to the present purpose, and it was Thompson's book that first suggested to the writer's mind a still broader generalization of the principles herein referred to.

It may be stated that all processes and phenomena of life are associated directly with some form of fluid substance. This includes not only gaseous matter and liquids but many forms of matter that are not solid in the ordinary sense nor yet liquid, but which have a certain degree of mobility among the constituent particles. The essential primary element of all organisms is the cell. All material substances whatever, whether affected by influences of life or whether only dead matter, are alike governed by the physical and mathematical laws here outlined. Is there not therefore a remarkable and intimate relation between the "simple geometrical principle" above explained and all organic existence and processes;-All life, growth, repair, decay, and dissolution:-Even all mind, intelligence, emotion, and all reasoning and thought. The speculative philosopher might indeed go so far as to add all health, satisfaction, and pleasure; all sickness, distress and pain; all relations and struggles among humans, all endeavors of man, all events of history, everything:-And the psychologist may here note an analogy to the unending strife between good and evil which figures in so many of man's superstitions and religious beliefs, primitive and otherwise.

A further conception of the profound significance of these elementary geometrical relations in connection with all activities and phenomena of the material universe may be formed by imaging a region or space, apart from any known real one, where it is possible for equal spheres to be so grouped that the arrangement will have at the same time maximum concentration and universal symmetry— an imaginary space where the regular dodeca





To accompany paper on "A Geometrical Basis for Physical and Organic Phenomena" by John Millis
Drawn by O. W. Robeson

hedron has each of its diedral angles exactly 120 degrees instead of 116° -33′-54" (or, if preferred, where the full circle is made up of about 349-2/3 real degrees instead of 360), and where equal solids of this form will therefore stack together without voids. The curious and interesting speculations and deductions that follow from imagining a space with more dimensions than real space possesses (as four dimension or n dimension space), or a space having special properties like " ture," etc., are quite well known. All matter in the hypothetical space permitting the special arrangement of equal spheres as above will, if we assume gravity to remain normal, tend to concentration as in real space but this tendency will not be checked or modified or coun


teracted by a departure from equilibrium resulting from an approach or approximation to the rhombic dodecahedral grouping of spherical elements, since the tendency will be to assume the regular dodecahedral grouping throughout. All matter under these conditions must eventually become a stagnant and dead plenum, an amorphous and non granular mass, in which no physical activity or life could possibly have being.

Another curious paradox, not altogether devoid of usefulness, is found in the self contradictory conception, which is at least semi logical, that there could only be a real" continuous ether in a space with the imaginary properties above described! There are as yet unsurmounted (and unsurmountable?) diffi


culties in the way of attempts at such a conception for real space, but even in the hypothetical space a continuous ether would find obstacles to the exercise of its principal function which is "to undulate." Moreover there would be no particular object in undulating— nothing to incite undulation-and so from all angles the ether idea has a rather hard struggle for a real existence.

We deduce properties of the circle by assuming that it is possible to divide the continuously curved circumference into parts so small that each one will be a straight line, and the rigid accuracy of the results so obtained is in no degree vitiated by the fact that the assumption can not possibly be true. Likewise we may deal with physical phenomena as though there were a "medium," an all pervading plenum of substance, itself devoid of gravity and of most other properties of real matter saving only the capacity to undulate. We can deduce, explain and predict with entire success-with consistent results and even astonishing confirmations-notwithstanding our medium or ether may be entirely hypothetical, its assumed properties may be contradictory in themselves, and it may not be possible for such an imaginary medium to have a real objective existence.

To locate and corner the remaining major difficulty in the way of a full comprehension of things of nature will at least contribute to our plans and measures for mobilization of forces and will indicate the main objective and the methods of attack, even if the adversary shall prove forever invincible.

Body B is separated from body A by an intervening distance. It is not possible for an "impulse" of mutual influence which requires time for the passage to be in transit between A and B, to be disconnected from both for the moment-suspended between them in other words-without any intervening medium except space? What a flood of light and clarity would be shed on and through the accumulated mass of physical facts and data, as well as the tangled maze of speculative perplexities, if an affirmative answer to this conundrum could only be given.

No confession of individual faith can be claimed to be of itself a useful contribution to our knowledge of material things, and this is distinctly true so far as concerns the ideas of the undersigned, but as has recently been said by a distinguished physicist it is well sometimes to declare ourselves in this respect "for this naturally has its influence upon all that is said and done and to the end of making the point of view of the writer clearly understood." (Crehore) It will be advantageous for the reader if he can feel that the writer is setting forth ideas with a certain degree of self felt confidence and "intellectual rest." I will therefore add the following, which is stated partly in the first person since it is a sort of individually conceived framework or background on which to pin ideas and new facts; frequently with a satisfactory fit in place, sometimes for the moment quite detached from the general pattern, but as a rule with a constant tendency towards accordance with what may ultimately turn out to be a complete and consistent picture.

There is only one kind of real space, the kind of everyday experience.

There is no material substance that does not have the common attribute of gravity. There is no " force" except gravity, and all physical phenomena are resolvable into this conception.

Gravity is an inherent, essential, and universal attribute of matter. It is and ever will remain inexplainable. What is more (and this is another paradox) if an "explanation" were possible this would actually be a retrograde step in the progress of knowledge, since we would then have at least one remaining mystery on our hands, almost certainly still more troublesome than is that of gravity.

There is no such real thing as a continuous medium or ether. This however in no manner or degree disparages the vast majority of the facts, results and predictions that have been accumulated and accomplished on the "as though there were "assumption regarding an ether, nor is it inconsistent with the confident belief that there will be very many additional real and useful developments and advances in

our knowledge of natural things and phenomena on the same assumption.

It will be recognized that whatever there may be of novelty in the above first principles is found in the combination rather than in any one element.

Finally I concede with the eminent philosophers of long ago that an idea of the real nature of "action at a distance" without any intervening medium is inconceivable to the human mind (my human mind-they no doubt likewise meant theirs) and especially so is the suggestion that an impulse which requires time for transmission from one body to another may have left the one and be on the way to the other-in a state of detachment between-with nothing but empty space along the road. (It is probable that the "velocity of light" as a physical constant is the same as the velocity of transmission of a gravitational impulse or change from one body of matter to another, or at least that there is some very direct relation beween the two.)

Here however is the parting of the ways. I have faith that it will some day be accepted that this inconceivableness is attributable, not to the fact that the suggestion is incompatible with the real workings of nature, but to the limitations in the powers of human comprehension.

If it can be accepted that "philosophy" is only a shorter term for peace of mind arrived at or approximated to after long pondering, then the above may be set down as a sort of personal philosophy of the writer's.


And the path of future progress? We are apt to regard the human intellect of our period as already in a stage of its development which may be called maturity, but this is not at all certain. If something like a curve is plotted to indicate the mental status of man at different periods or ages"-the primitive state, the stone age, the bronze age, the age of iron, etc., its general shape will indicate whether the present is the age of finality in this respect. There was just as much reason for regarding any one of the previous ages as a culmination as there is for assuming that we are now on an ultimate crest of the curve of human

powers of understanding. In fact if we consider the varying rate of change in direction of such a curve, or the rate of its departure from a base line of zero intelligence, there is less ground for thinking our present mental capacity is at a maximum than there was for such a belief at any previous age or period.

Let us therefore "play" that there is an ether, with all its seemingly necessary though improbable attributes, and go ahead with our observations, experiments, studies and researches until the mind of man, now possibly only in the juvenile or youthful stage of its growth, may have so far advanced towards maturity as to be able to put aside this elementary conception and to substitute something more grown up. Meanwhile let us not lose sight of this all-important coordinate part of the program for advancing the development of the human mind in capacity for comprehension so it can assimilate and interpret the facts as they accumulate and keep pace with the general progress. The super intelligence capable of fully comprehending all nature will doubtless always remain a limiting ideal-something to be eternally striven for, to be approached all the while more nearly, but forever unattainable.





As manganese is urgently needed in the war several geologists of the United States Geological Survey, Department of the Interior, have been making systematic examinations of areas that are believed to contain deposits of manganese and manganiferous ores in the United States and the West Indies, in order to appraise our available resources of manganese and to assist in stimulating its production and


Manganese is a metal resembling iron. It is used principally in the manufacture of steel. to which it is added in the form of alloys with iron, such as ferromanganese and spiegeleisen. It is used also in glassmaking, in many chem

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