by and hence one definition of this term, may be worse than none. Perhaps the most important idea for the student to acquire early in regard to group theory is that there is not only one definition of the mathematical term group but there are several, and there is not only one group theory but there are several such theories.

How then can we know that a certain set of operators constitutes a group? With respect to a finite set there is little trouble, since all the definitions of finite discontinuous groups which are found in good modern literature are practically equivalent. This is, however, not the case as regards the definitions (explicit or implied) of an infinite group. In fact, such a standard work as the Encyklopädie der Mathematischen Wissenschaften contains on page 218 of Volume I a definition which is satisfied by the infinite set of numbers of the form.

when these numbers are combined by multiplication. It is evident that this set of operators does not involve the identity, since 0 is not a natural number, nor does it involve the inverse of any of its operators. If we would let q assume also the value 0, the set would include the identity but not the inverse of any of its other operators.

2

The most comprehensive definition of the technical term group is found on page 243 of tome I, volume 2, of the Encyclopédie des Sciences Mathématiques. It is as follows: "One says of an aggregate of objects that it forms a group in the most general sense of the word, when there is given a rule permitting to deduce from any two A and B of the objects of the aggregate a third object C of this same aggregate." It is evident that this definition of group does not imply the existence of an identity or of the inverse of an operator in the aggregate under consideration. In fact, the aggregate may contain various identities and various inverses of the same operator and still satisfy this definition of a group as results directly from the consideration of algebraic number fields, which are included under the term group in both the German and the French editions of the large mathematical encyclopedia.

One frequently finds in recent mathematical literature criticisms relating to the use of the term group with respect to a

A. Loewy, Archiv der Mathematik und Physik, vol. 9 (1905), p. 105.
'A similar definition appears in tome 3, volume 1, page 334, of this work.

set of operators which are said not to form a group by the authors of these criticisms. For instance, A. Loewy criticizes, in the article to which we referred, the definition of the term group as given in the large German mathematical encyclopedia, stating that it is necessary to add another condition in order to obtain a definition of an infinite group. On the other hand, the conditions imposed in the said encyclopedia definition are already much. more restrictive than those imposed in the definition cited in the preceding paragraph and they include the latter. Hence it is clear that this particular criticism by an eminent authority is based on the assumption that such a definition as the one cited in the preceding paragraph is too general.

We refer to this matter in order to emphasize the fact that what appeared to many as an illegitimate use of the term group due to carelessness or ignorance has been legalized in recent years. The technical term group has thus been given a breadth of meaning comparable to that given to the term function in 1837 by a definition due to Dirichlet. This breadth of meaning is, however, not new but it embodies the fundamental notion which guided the earliest investigators in this field and to which certain explicit restrictions were added by later writers with a view to definite progress in certain directions.

Such restrictions remain desirable and will doubtless be continued, but instead of saying they are imposed on all groups we should say that they are imposed on the particular groups under consideration. Thus there arises the desirability of further classification of groups, and the study of groups in which the inverses of the operators are not present or in which a linear equation may have more than one solution. Up to the present it has been customary to study only those groups in which the inverse of each operator is present and in which a linear equation has only one solution. On the other hand, an equation of higher than the first degree can have more group elements for roots than the degree of this equation. These groups have been divided into four great classes named as follows: Finite, infinite discontinuous, finite continuous and infinite continuous.

The most important property of groups is the existence of subgroups, and hence subgroups should be considered immediately after some of the definitions of groups have been examined and illustrated. In particular, the facts that the order of a subgroup divides the order of the group whenever this order is finite, and that an invariant subgroup corresponds to the identity

of an isomorphic quotient group are of fundamental importance and should be included in the briefest course along this line. The subgroups of the continuous groups of movements of space corresponding to points and to lines respectively, are also of fundamental importance.

An important notion which is intimately connected with group theory but which is not confined to this subject is that of isomorphism. The special isomorphism in which there is a (1,1) correspondence between the elements of two groups is particularly important since it enables us to study once for all the fundamental properties of large categories of groups, and since it gave rise to the notion of abstract groups. Moreover, the possible isomorphisms of a group with itself are of fundamental importance in some of the applications of these groups. Hence the question of isomorphisms might well be considered imme-. diately after that of subgroups, even if a thorough study of this question cannot be taken up until after a considerable body of theorems relating to group properties has been established.

The most important continuous group is composed of the transformations under which the properties of figures studied in elementary geometry are invariant. This group consists of movements, reflections, and similarity transformations and was called by F. Klein the principal or fundamental group. Two of its most important subgroups were noted above. It is interesting to note, in particular, that the dimensions of space are invariant under this group. As an instance of a group, in the most general sense, under whose transformations the dimensions of space are not invariant we may refer to the following:

x' = ax+by

y' akx+bky

which clearly transforms all points of the plane into those of a line passing through the origin, if we regard a and b as parameters and k as a constant.

The most important finite group is the cyclic group and there is one and only one such abstract group of each order. The cyclic group of order n is perhaps best known in the form of the n nth roots of unity. Among non-cyclic groups of finite order the dihedral groups, or groups of the regular polygons, are perhaps the most important. It must, however, be admitted that it is difficult to decide which are the most important among the hosts of finite and infinite groups intimately related with questions of elementary mathematics. Moreover, groups are social

entities and can be studied better in connection with their social environment than by isolation.

Enough may have been said to show that the suggestion that prospective teachers of elementary and secondary mathematics should know some group theory cannot reasonably be assumed to imply that this subject is less intimately related to their teaching than any of the other eight subjects named above. Nor should it be assumed that the study of this subject normally belongs at the end of the teacher's preparation in pure mathematics. In fact, some group theory should come early among the courses which follow a first course in calculus, and later studies in mathematics should be illuminated by the group concept. The main reason why students often find the study of group theory difficult is that it is usually taken up late in their course and then developed from a very abstract point of view. Groups are intrinsically a part of our natural mathematical endowment and the group concept should be nurtured in our early mathematical development by directing attention to group properties appearing in elementary geometry and in elementary arithmetic. On the contrary, this concept is allowed to famish while other mathematical concepts are nurtured, and then when late in the student's course he turns back to it for light towards the solution of some of his most difficult problems, he naturally finds a weakling whose normal development would demand more time than he feels that he can then give to it. Total starvation is the natural consequence.

BLACK WALNUT WANTED.

The virtue of the black walnut has been its own undoing. Because it does not readily warp or split, it is the best of all woods for airplane propellers and gunstocks, and the Government finds that the demand greatly exceeds the supply. An appeal, therefore, is being made to all owners of walnut, to assist in getting every available log to market. County agents and boy scouts are being pressed into service for locating walnut trees, and everybody interested in winning the war is urged to report the existence of available material. The logs wanted are at least twelve inches in diameter at the small end and eight or more feet long. The Government does not buy this timber but will put owners in touch with manufacturers who are using it. The price at present is about $90 a thousand feet. A tree that will square twelve inches and make a log sixteen feet long is therefore worth nearly $20. We are frequently told that certain products are scarce because of the war, and this is likely to be true of walnuts for some time to come.-[American Botanist.

MR. DOOLEY, 2D, ON THE DISCUSSION METHOD.
(With Apologies to the Original Mr. Dooley.)

BY JEAN BROADHURST,

Teachers College, Columbia University.

Ye' see, Michael, me b'y, since me fayther settled so manny questions av internashunal dispute, it cum to me natshral loike to be inthrusted in the prisint day and its problims. And me posishun av windy washin' at collige is furtherin' me in the way av it, for windy washin' is a quiet job and slow, as is well known. So I hear manny a thing that me puir fayther missed altogither. And the quarest av thim all is the way thim big perfessors is a tryin' to git shut av their jobs, entoirely.

Do ye iver hear me, Michael, a sayin' that the windys don't need washin' or cud wash thimsilves? Be gorrah, a man 'ud be two fules in wan, he wud, to do that-for he'd sure lose his job altogither and the windys wudn't be washed at all, at all.

But this discussin method is a quare one. As I kin make it out, it's a way the perfessors have av lettin' the scholars run on by thimsilves loike a horse widout anny driver at all. No, it's not loike Murphy it is, fur his old white horse stops at ivry house and waits till Murphy gives the baby his bit av pasteurized milk for the day, and comes out wid the empties. That's not loike the discussin way at all, for old Murphy knows where his horse is ivry minute and where he's goin', and what's more, the old horse fetches up at the right places ivry time. But in the discussin way, a horse ud av to av the wings av a bird and legs av a kangrue, for he'd av to jump from one street to anuther and skip whole blocks intoirely; and to kape up wid him Murphy wud av need av an aeroplane wid all the accelirators and patent brakes that av yit to be invinted. Besides, Murphy's horse niver goes backward, me b'y. He's a wise old horse, and he'd know it av he did.

Now, as I kin make it out, in the discussin way the perfessor starts 'em on a little way wid his highbrow stuff, and thin he drops the lines intoirely. And ivry horse pulls a different way, zigzaggin' around as if the royal road to learnin' was the Abbey Inn hill for the steepness av it.

And thin, whin thim as don't discuss, the passingers, so to speak, have been yanked this way and that till they don't know which way they're goin', sumthin' breaks, and the hull thing stops, wid each av thim discussers in a diffrunt place; and the passingers begin wanderin' about aimless-like, some av thim