ordinates, would they not both come to the same conclusions as to the form of the parabola? And suppose that in reality a comet is determined by forces which possess a one-to-one correspondence with these same coordinates, would not the path of the comet possess a one-to-one correspondence to the figures of the parabolas of the two astronomers? The determinedness of all purely formal constructions is truly universal and applies anywhere in the domain of mathematics or pure thought and in any possible real world, also in this our world, i. e., the universe in which we live.

Considering the immanent necessity of the laws of form we can understand that this pre-established harmony has not been made by some supernatural being nor can it have originated gradually by a process of evolution, but it is intrinsically necessary. It is the immanent order which is the condition both of our natural laws and the intelligibility of existence. It is that same intrinsic regularity which can be observed everywhere in nature. This same regularity in the domain of form makes it possible that rational beings originate, that science can be established, that ideals can be proposed and lived up to, that a code of morality and a norm of right conduct can be formulated, and that the universe presents itself as a well-regulated and law-ordained cosmos.

A revision of almost all problems of philosophy from our standpoint will shed new light on their solutions, as will appear when we consider Prof. Hartley B. Alexander's article on "The Definition of Number." When enumerating the different conceptions of the interrelation between logical and mathematical views on the one side and philosophy on the other, he omits to mention the solution offered by the philosophy of form, which alone can be regarded as the philosophy of science.

Mr. Bertrand Russell sees the most essential feature of mathematics in its logical interrelations and goes so far as to claim that mathematics has nothing to do with space. Without objecting to definitions we prefer to regard at least geometry as the purely formal science of extension, which means space, not real space but pure or mathematical space. Mathematics presupposes logic and contains one additional element which is commonly called space, but like all purely formal sciences mathematics produces its objects of investigation by a priori construction. The elements with which we start are products of abstract thought in the realm of pure form. created by thinking away everything that is particular, viz., all concrete objects that consist of matter and energy. Thus we retain

the idea of pure motion and a possibility of establishing pure interrelations.

Pure motion means a change of place without implying energy, and a possibility of pure interrelations is a field of pure motion. We start with these two abstract notions, on the part of the subject an ability to move about, on the part of the object, (i. e., the surrounding world), emptiness; and this emptiness offers a field of possible motion. With these conditions we construct whatever we may be pleased to build up, and observe the result.

In geometry we do something and note what will come of it. For instance, we move and note the trace of our motion. We call it a line. We move again and again, and let the traces of other lines enclose a space; we call the result a figure. Where two lines cross we have a point.

The system under construction may be Euclidean or nonEuclidean according to our start, whether or not we assume we are able to draw straight lines in the Euclidean space. If in our plan of construction we exclude the straight line, we will have to move according to a definite principle in curves of a predetermined constant deviation, in which case our system will be different from the system of Euclid.

If two straight lines cross, the product of our construction is an angle, or rather four angles. The peculiarity of mathematics is to watch and observe the inevitable results of our own constructions, but the main characteristic of our constructions is this, that they are made in a field of anyness, i. e., they apply to any kind of construction made in the same way, not only in emptiness, but in any kind of a world filled with any kind of matter or any kind of energy.

The nature of matter and energy can only be discovered by experience through the senses, but the nature of pure interrelations can be determined by building up constructions in a field of anyness, as they must be under any conditions, which means under all conditions. Therefore the laws of pure form (in other words, the laws of anyness) will be valid for any kind of a world.

Thus we have an explanation why the theorems of pure mathematics are hyperphysical truths, and here we have a specimen of the nature of what theology has called the supernatural. There is only this difference between the old conception of the supernatural and this new conception of it which for the sake of distinction

For an a priori construction of the plane, the straight line and the right angle see the author's Foundations of Mathematics.

we call the "hyperphysical," that the latter is as clear and selfevident as the former is mysterious, hazy, bewildering and mystifying.

The consequence of this conception of mathematics need not be traced here in all details, but we feel assured that in the long run it will solve all the modern problems of philosophy and dispose of the troubles which have been caused by pragmatism, Bergsonianism, by the advocates of the principle of relativity, and also by the logisticians. EDITOR.

LOUIS COUTURAT (1868-1914).

Besides the carnage in battleships and trenches, the great European war carries with it many accidental by-products of disaster not to be overlooked when casting up the grand total of losses the world is suffering. In the early days of last August when the first commotion in the commercial arteries to and from Paris was at its height, a heavy automobile at full speed chanced to run down the carriage in which Louis Couturat was traveling, and his immediate death was the result. Though only forty-six years old he held first rank in France among scientific workers in the philosophy of language, the philosophy of mathematics, and especially in the more modern aspect of logic-for which he agrees with English logicians in preferring the term "logistic," now that this word is but little known in its earlier significations listed in the dictionaries.

M. Couturat was singularly well informed on many questions, but the particular power and quality of his mind lay in a gift for deductive reasoning combined with the most punctilious intellectual honesty that would never countenance a compromise with the truths of reason. All his work is especially remarkable for the clearness of its representation. His style is never sullied by glittering and bizarre phrases intended to attract attention and admiration, but which often seem to cover a multitude of sins in the way of vague ideas and loose reasoning.

Couturat was first known by his painstaking and illuminating exposition of the mathematical infinite (L'infini mathématique, 1896) in which he discusses the idea of number and analyzes the concepts of continuity and the infinite, refuting practically all of Renouvier's arguments against the latter. His research in this line familiarized him with all the writings of Leibniz, and his next published work was an edition of more than two hundred fragments

from Leibniz's unpublished manuscripts, some of which proved to be of the greatest philosophical interest. This was followed by a scholarly work on Leibnizian logic (La logique de Leibniz, 1901).

It was through their common interest in Leibniz that Couturat became acquainted with the Hon. Bertrand Russell in England, whose Philosophy of Leibniz appeared at this time, and their relation continued to be of the friendliest. Couturat added some notes to Cadenat's French translation of Russell's Principles of Geometry and introduced his Principles of Mathematics to the French public through a series of articles later collected into a book. Readers of The Monist will remember his answer to Poincaré's witty sallies against logistics in the issue of October, 1912. In an introduction to this article, M. Couturat's translator, Mr. Philip E. B. Jourdain, summed up the controversy between these two brilliant Frenchmen.

In the meantime, Couturat had published his Algèbre de la logique. In a small monograph of less than one hundred pages he presents a concise outline of the material contained in the first two volumes of Schröder's prolix three-volumed treatise. He follows Schröder in making the notion of inclusion the fundamental notion in his calculus in preference to the idea of equality, as the English logicians had done and as Schröder also had done in the beginning, though he made the change later under the influence of C. S. Peirce. Besides brevity Couturat's little work possesses the further advantage of clear-cut precision of argument which makes it practically the most easily intelligible presentation of the subject in any language. It is for this reason that the Open Court Publishing Company only last year issued an English edition of it.

Couturat believed thoroughly in the possibilities and desirability of an international artificial language, and he and Professor Ostwald are the two leading scientific men of whom the Esperanto and Ido movements can boast. In the light of M. Couturat's high character, talents and attainments it can only seem trite and trivial to say that the world has suffered an irreparable loss in his death.

CURRENT PERIODICALS.

L. G. R.

The best produced scientific magazine in Great Britain is Science Progress in the Twentieth Century: A Quarterly Journal of Scientific Work and Thought, which is edited by the eminent pathologist Sir Ronald Ross. The first article in the number for April 1915 is "Some Aspects of the Atomic Theory" by Frederick

Soddy. "Either matter must occupy space continuously or it must exist in the form of discrete particles. The historical origin of the atomic theory of matter is to be found in the choice between the two possible answers to these mutually exclusive alternatives......" However, "the true origin of the atomic theory is recognized universally to have been during the first decade of the last century in Dalton's discovery of the simple laws of chemical combination, though, even to the discoverer himself, the laws of gaseous behavior, upon which later the totally distinct but inextricably interwoven molecular theory was to be based, undoubtedly played a part in directing the interpretation he put upon these laws. Henceforth science was to deal no longer with atoms as the end results of a purely mental process of the subdivision of matter, a process which must of necessity have an end if matter does not occupy space continuously, but with atoms of definite mass determinable simply and exactly relatively, that is, the mass of any one kind of atom in terms of that of any other." The article, as we should expect, deals with the modern aspects. Francis Hyndman writes on "The Electrical Properties of Conductors at Very Low Temperatures,” these properties indicating relations between widely different properties of matter. Arthur E. Everest writes on "The Anthocyan Pigments." The term "anthocyan" now denotes a large class of naturally occurring plant pigments, and the present article contains a very valuable account of the advances in this field of research from 1836 up to the present time. Richard Lydekker contributes a summary of "Vertebrate Paleontology in 1914." "The most important part of the year's work is undoubtedly that on the mammallike reptiles and their structural resemblances and relationships." Charles Davison deals with "The Prevision of Earthquakes." "Between foreseeing and foretelling an unexpected event, there would seem to be little if any difference, beyond the fact that the one may be conducted in private while the other implies publication of some kind. But, to the corresponding words 'prevision' and 'prediction,' somewhat different meanings seem to be attributed, prevision being apparently considered as an approximate, and prediction as an accurate, form of forecast." This distinction is assumed in the present paper which contains a very good review of our knowledge on the subject. James Johnstone has an interesting discussion on "Is the Organism a Mechanism?" The concluding sentence of the article must be quoted here: "It may be, of course, that the activities of the organism are capable of reduction to chemical and