he called the greatest mathematician in the world, was an independent discoverer of the non-Euclidean geometry, and like Bolyai, he called parallel to a given straight all straights coplanar with it which nowhere met it. But this meaning and use of the word parallel has been superseded by Lobatchevsky's. He has no name for this possible infinity of straights through A coplanar with the straight DC but nowhere cutting it. But that one of them which contains the Bolyai asymptote to the ray DC, Lobatchevsky calls parallel to DC and then says "upon the other side of the perpendicular from A will lie also a line AM, parallel to the prolongation DC' of DC, so that under this assumption we must also make a distinction of sides in parallelism."

This sensed relation then is Lobatchevsky's parallelism, and that it is a sensed relation, a one-sided relation, a one-way relation, a relation which goes toward one side only is stressed and emphasized by his § 24: "The farther parallel lines are prolonged on the side of their parallelism, the more they approach one another." In other words, Lobatchevsky parallels are one-way asymptotes.

So his next theorem, § 25, which tripped Mr. Russell, I might have translated: Two straight lines which are same-way asymptotes to a third are also asymptotes to one another.

So of all straights coplanar with a given horizontal straight which nowhere cross it, two through each given point are parallel to it, one to the right, the other to the left, and all the others I call ultra-parallel.

A parallel to a straight meets it at infinity. An ultra-parallel does not even meet it at infinity. Parallels are straights with a common point at infinity. Parallels are straights which meet on a figurative point. If ultra-parallels determine a point, we must have another name for it. Call it an ideal point. Thus equipped, we are able to answer Mr. Russell's question, Monist, page 621: "How is the professional expert (the man who knows non-Euclidean geometry) better fitted to see more lucidly in dealing with the elements of geometry than any other person of good geometric faculty?"

Just thus, my dear friend: You say, "I will now spread before the reader in detail what seems to me to be good geometrical proof of my proposition. Consider and refer to the following figure." I do, and instantly, at a glance, I see your fallacy, your petitio principii. I am saved the reading of your three pages of pseudoproof. You assume that an angle is determined in size because it is made by the side with the hypotenuse of an isosceles right-angled

triangle. You say, p. 624: "Now the angle MEN being a w-angle equals the angle Bed..."; in other words, you assume that the size of the isosceles right triangle has no effect on the size of its acute angles. This is Wallis's form of the parallel postulate. And so you are guilty of begging the question.

Not only would a smattering of Bolyai have saved you but so would a little excursion into Chapter XV of my Rational Geometry, which starts by saying: "Deducing spherics from a set of assumptions which give no parallels, no similar figures, we get a twodimensional non-Euclidean geometry, yet one whose results are also part of three-dimensional Euclidean."

GEORGE BRUCE HALSTED.

GREELEY, COLORADO.

A REMARK ON F. C. RUSSELL'S THEOREM.

F. C. Russell of Chicago has endeavored in the April number of The Monist to disprove the legitimacy of the non-Euclidean geometry by showing the demonstrability of the parallel postulate. The basis of his considerations has been laid on a simple proposition that the angle-sum of an isosceles right-angled triangle equals two right angles; a proposition the proof of which he does not dare give, saying it would be "spreading an imputation upon the reader," being so simple in nature. But the whole secret of the matter remained concealed under this unknown sort of a proof, and so we are lucky that we had it imparted to us by Russell himself in a subsequent number. In studying it, we have found all that can be desired.

Russell defines his u-angles "as being such angles as the sides of an isosceles right-angled triangle make with the hypotenuse." This definition is of course not in any way objectionable, but when Russell has to consider the u-angles arising from different triangles of unequal sizes, to be always equal, he has unconsciously fallen into a pit of thought, from which he is unable to get out. When we adhere to the Euclidean world, we can well prove the assumption Russell makes, but how can he protest the legitimacy of it, when he is going to show the Euclidean system to be the sole one that can be relied upon? If he wants to be credited by us, he must first prove the assumption he has made; which most probably he cannot do without having recourse to the parallel postulate or some

thing else that may be substituted for it. In a word, Russell has substituted a different axiom in place of the postulate of Euclid. His endeavor and achievement have however left nothing that could make a step towards disarming the pan-geometricians. We stand uninjured on the same ground as before in spite of all the desperate assaults from the strong hand of Russell, who has utterly failed to disground us.

OHARA, KAzusa, Japan.

YOSHIO MIKAMI.

A MATHEMATICAL PARADOX.

The following paradox appears to me to be interesting because it shows how "common sense" breaks down when dealing with a slightly subtle question.

The question to be discussed is: Is the greatest weight that a man can lift the same as the least weight that he cannot lift, or not; and if the weights are different, which is the greater?

The numerical values of all possible weights (both those which the man A can, and those which he cannot, lift at the particular moment considered) form the simply-ordered aggregate of positive real numbers R. Those weights that A can (at this particular time) lift bring about what Dedekind* called a section (Schnitt) in R, and all the members of R fall into the two classes:

a. The class of those numbers r such that A can lift the weight r (then also A can lift any of the weights less than x);

b. The class of those numbers y such that A cannot lift the weight y (then also A cannot lift any of the weights greater than y).

Now, as is well known, there is one, and only one, number which "generates" this section, and this number is either the upper limit of the class (a), or the lower limit of the class (b), but not both.

Thus, our answer to the question about the weights is: Either there is a greatest weight that a man can lift, or there is a least weight that he cannot lift, but not both. The paradox lies in the fact that, to unaided common sense, the existence of a limit seems just as, or even more, plausible in both cases or neither as in one

*Stetigkeit und irrationale Zahlen, Braunschweig, 1872 and 1892 (English translation in Dedekind's Essays on the Theory of Number, Open Court Publishing Co., Chicago, 1901).

only. I cannot see how one is to tell in which case the limit does exist; only that it must in one, and only one, of the two cases.

In my opinion what is paradoxical to the ordinary mind in this is: We have two classes of an infinity of members each (arranged in some order); now ask a person if there is a highest in the first class; if he says "yes" (or "no") he will probably admit by parity of reasoning, that there is (or is not) a lowest in the second.

And yet my case is a translation into picturesque language of an instance well known to modern mathematicians in which the answer must be "yes" in the one case and "no" in the other. PHILIP E. B. JOURDAIN.

BEAMINSTER, DORSET, ENGLAND.

ON THE PROBLEM AND METHOD OF PSYCHOLOGY OF RELIGION.1

In a report before the Congress of Psychology at Geneva Prof. Harald Höffding of the University of Copenhagen undertook to sum up his theory of the psychology of religion with, we must admit, an air of easy and careless assurance. In such a delicate investigation we can not say, "I am right"; much less, "You are wrong." I am not writing at all in this spirit, and I recognize in Professor Höffding too great a degree of culture to assume it in him. However, a fear has taken possession of me and I have not succeeded in freeing myself from it. This is the fear lest Professor Höffding does not take into account so much as they deserve certain difficulties which consciousness raises against the dogmatic presuppositions which form its point of departure,-difficulties which I do not pretend have been solved and much less do I pretend to solve them myself, but whose proper comprehension will always be one step forward.

Professor Höffding's entire conception rests upon the postulate which he lays down as most natural, that the psychology of religion is a part of general psychology. However, religion does not lend itself readily to this classification, but if it did it would be so much like other questions, that if a psychology of religion existed, its first claim would be that general psychology forms a part of the psychology of religion. For it is entirely gratuitous and arbitrary to consider "religion as a particular form and a particular direction 1Translated from the French of Professor Billia by Lydia G. Robinson.

of psychological life." But what constitutes religion in consciousness is precisely the negation of a particular form and a particular direction. Do you wish to eradicate something from your cognizance of a thing? Explain it by another while forgetting the proper and constitutive element it possesses. Had I not a horror of all psychological Baedekers I would say, "Beware of false generalizations." Yes, I learned at an early date, and I still maintain to-day, that nothing is apprehended except in that which is more general and more common. But there is the general and the generic. That with which we have to deal is the general that is called the universal, that does not exclude the essential, the particular, but includes them. This difficulty does not entirely escape Mr. Höffding who endeavors to save himself by analogy. The objection suggests itself that religion must be cognized by its own experience in order to apply psychology to it. He expects this objection and thinks he can overcome it by saying that in default of this experience the psychologist would be able to conceive religious phenomena while seeking to discover what place religion would occupy in his psychical life if it existed for him, in the same way that he might be able to imagine colors beyond red or violet. For this it would be necessary for him to direct his attention to the place which religion occupies in the psychical life of those who are acquainted with it from their own experience. The question is to define the psychological place of religion in the same way that the geometrical locus of a point is defined in mathematics. But there is no psychological locus and least of all for religion. Religion does not occupy a place in the psychical life but it is the whole. It includes and creates all. When it is anything else, it is either no longer religion or it is not yet religion. Religion is quite a different thing to those who are acquainted with it than to those who do not know it. For instance certain rites, sacraments, prayers, ecstacies, clash too much with so-called reason to be comprehended by an indifferent person as they actually exist in a believer who experiences that actual perception upon which alone religion rests. It is like love. Nothing is more absurd than to see an indifferent person judge or laugh at it; as if the external manifestations were the same thing for the one who experiences the passion as for others, and especially as if the principle from which these manifestations arise is in any way comprehensible except to him who has the experience. It is just such a monstrous absurdity as the attempt to impose upon universal culture a history of philosophy conceived and drawn up apart from all