which were unknown to the older mathematicians. Inspired by these groups they developed also the theory relating to the groups. which the ancients used implicitly, but it is questionable whether the modern mathematicians would have developed a theory relating to the latter if they had not been inspired by the former. At any rate, they did not take any steps towards such a theory before they had this additional motive. These observations may serve as partial answers to questions raised above relating to the late development of our subject as an autonomous science.

The heading of the present article suggests that some of the developments of our subject can not be properly called easy. In fact, by far the larger part of these developments presuppose a rather extensive technical knowledge and hence they are unsuited for a popular article. Among all the scientists the mathematician works usually at the greatest distance from his postulates, and hence he has the greatest difficulty to exhibit the results of his toil to the public in the hope of securing appreciation, which he craves with the others of his fellowmen. In group theory this distance has become especially long even for a mathematical subject, but this theory does extend also into the experiences of all thoughtful persons. The present article aims merely to direct attention to the richness of the mathematical developments which have contact with these particular experiences and thus to secure an easier approach to some of this richness.

If a group contains a finite number of elements this number is called the order of the group. For instance, the 24 different movements of space which transform a cube as a whole into itself but interchange some of its parts constitute a group of order 24. The most elementary group of a given order g is cyclic; that is, it is composed of the powers of a single one of its elements. For instance, the g numbers which satisfy the condition that the gth power of each of them is equal to unity constitute this group of order g, where g is any natural number. It is evident that the 24 different movements which transform a cube into itself are not powers of a single one of them. Hence they constitute a noncyclic group of order 24. In fact, none of the elements of this group has to be raised to a higher than the fourth power in order to obtain unity, or the identity.

One of the fundamental problems of abstract group theory is the determination of all the possible groups of a given order g. It was noted above that there is one and only one cyclic group of every possible order g. When g is a prime number there is no other group of this order. This is also sometimes the case when g is composite but there is no upper limit to the number of groups

which may have the same composite order. That is, it is always possible to find a number g such that the number of the different abstract groups of order g exceeds any given finite number. In addition to the two groups of order 24 noted above there are 13 others, which were first completely determined in 1896 by the present writer. The lowest order for which the number of groups exceeds the order is 32. There are 51 distinct groups of this order.

The verification of several of these statements would carry us beyond easy group theory. They may serve, however, to exhibit a type of inquiry relating to our subject. Fortunately, some of the most important and far-reaching phases of this subject are also the easiest. For instance, the group of all the transformations of space which leave invariant the distances between every pair of points can easily be comprehended. Those geometric figures which can be transformed into each other under this group may be called equivalent and we may confine our attention to the study of geometric properties which remain invariant under the transformation of this group. We thus obtain a body of knowledge commonly called Euclidean Geometry, but this term is also often used with different meanings. Following the custom introduced by Gauss some still use it to denote all the geometry in which the parallel postulate is assumed.

It is clear that in geometry it is undesirable to endeavor to study every figure as an individual since one could not make much progress in this way. What people have always done in this subject is to confine their attention to invariants under certain infinite groups of transformations. It is true that the ancients did not specify these groups and that we do not usually do this now in a first course in elementary geometry, but for the advanced student, at least, the developments become much clearer if this specification is explicitly made. If we add to the transformations noted above those which do not preserve the size of the figures but do preserve their angles, so that all similar figures are regarded as equivalent, we obtain a larger group, which has been called the principal group of geometry. The body of knowledge relating to the invariants of this group is commonly known as Elementary Geometry. In particular, all circles are equivalent in this geometry and all squares are also equivalent here. Some writers call this geometry Euclidean Geometry. This is done, for instance, on page 61, volume 2, Pascal's Repertorium der höheren Mathematik, 1910.

These observations relating to the groups of geometry may serve also to support the implication noted above that group theory is often a kind of mathematical luxury. One can frequently get along without a knowledge of this subject where a knowledge

thereof would add greatly to the intellectual comfort. The fact that the mathematical world traveled far without making explicit use of this subject which now receives so much emphasis in work closely related to that which they were doing is perhaps best explained by viewing the matter from this standpoint. It need scarcely be added that group theory has also served to point out the way to easier methods of attack and to more powerful means of penetration, but this applies more especially to the more difficult group theory and hence lies outside the domain to which the heading of the present article relates.

In the opening sentence of this article we alluded to the fact that in the University of Paris there is now a chair entitled "the theory of groups and the calculus of variation." This should not be construed to mean that the developments of these two subjects have as yet much in common. In fact, there are few large mathematical subjects whose developments exhibit as little explicit use of group theory as those of the calculus of variations. Possibly the title noted above indicates that there will soon be a change in this direction. This title also raises the question whether our larger American universities should not have more chairs devoted to special subjects. The creation and occasional renaming of such chairs would tend to direct attention to leading investigators in various fields, an attention which often needs cultivation on the part of administrative officers.

THE HISTORY OF THE CALORIE IN
NUTRITION

By MILDRED R. ZIEGLER

(FROM THE SHEFFIELD LABORATORY OF PHYSIOLOGICAL CHEMISTRY IN YALE UNIVERSITY, NEW HAVEN, CONN.)

THE

HE nomenclature of a science is of vital importance; it is intimately bound up in the subject itself. Special technical words are used to describe the phenomena which are being studied. Lavoisier emphasized the importance of nomenclature in his "Traité Élémentaire de Chimie" (1789) when he stated: "Every branch of physical science must consist of three things; the series of facts which are the objects of the science; the ideas which represented these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact. . .; we can only communicate false or imperfect impressions of these ideas to others, as long as precise terms are lacking."

The calorie as defined in the science of nutrition is a measure of food value. The significance of the word calorie as thus used to-day is not the same as its import when first introduced into the French language. To the student of nutrition the word connotes something quite different from the term as employed by the physicist. It is the amount of a food substance which on combustion in the body will yield energy-heat or work-equivalent to a calorie as understood by the physicists. Even the term as employed by the latter to-day has undergone a change from its original meaning (1845). As first defined the calorie represented the amount of heat required to raise one kilogram of water through one degree centigrade. It is well known that the same term now means the amount of heat required to raise one gram of water through one degree centigrade. The calorie, then, has had-and still does have-two somewhat unlike meanings according as the gram or kilogram of water is the unit of mass, the temperature of which is being changed and the energy required for this change is being considered. Both of these meanings are preserved in the scientific literature of to-day, the larger unit being designated as a large calorie and written Calorie.

It is quite evident that either the small calorie or the large Calorie may be the more convenient unit with which to express heat change depending upon the magnitude of the change being studied. The physicist in expressing the amount of energy supplied in the form of heat required to raise one gram of water from its freezing point to its boiling point obviously is dealing with a relatively small energy change which if expressed in heat units, as discussed above, is in the neighborhood of one hundred calories. In such cases the little calorie is the convenient unit to employ; on the other hand experiments in the field of nutrition have demonstrated that the energy changes are of such magnitude that the Calorie is the more convenient unit for expressing them quantitatively.

The derivation of the word is very interesting; it has developed from the old French word "calorique" which was derived from the Latin term "calor." Lavoisier introduced "calorique," which was defined at that time as an elastic fluid containing the hidden cause of the sensation of heat and to which the phenomena of heat were attributed. This meaning was abandoned with the theory to which it belonged. As far as a review of the literature reveals Bouchardat (1845) was probably the first to define it according to the modern physical conception. In his "Physique élémentaire” we read: "Unité de chaleur-On designe sous le nom d'unité de chaleur ou de calorie celle qui est nécessaire pour élever 1 kilogram d'eau d'un degré du thermomèter centigrade."

The idea of measuring heat in this way had been in general use for some time, but the word calorie had not been introduced earlier.

Pouillet (1832) in his "Physique" defined the heat liberated by combustion of different substances as "Élévation de témperature que 1 gr. de chaque substance en se brûlant avec l'oxigène communiquerait a 1 gr. d'eau." He ascribed the value 6195° to alcohol which checks closely with its caloric value as now known. Although the French had coined the word in the year 1845, it apparently was not in general use for some time; for in 1852 Favre and Silbermann1 in an article on heat wrote: "We shall repeat that the unit which we have adopted is that adopted by all physicists, that is, the quantity of heat necessary to raise 1 gram of water 1 degree and which they call unit of heat or calorie." The Germans evidently took the word from the French, but it is difficult to determine at what time. Gmelin (1817-19) in his

1 Favre and Silbermann: Ann. de chim. et phys., 1852, Ser. 3, xxxiv, 357.