real (or complex, when we deal with multiply ordered series) number system are essential to the theory of functions.

Thus a theorem known by the names of Heine and Borel has been shown by O. Veblen in 1904 to be equivalent to a process which was often used by Weierstrass (it had been used by Bernhard Bolzano in 1817, and by others), when we deal with the aggregate of real numbers, and confine our attention to ordinary mathematics. But the aspect of things is changed when we proceed to the analogous, but far more general, theory of simply ordered aggregates in general. Here, as I showed in a paper published in 1910, the Heine-Borel process has a distinct methodological advantage over the Bolzano-Weierstrass process, in that it avoids the use of a certain axiom. This it does because the Bolzano-Weierstrass process essentially depends upon the successive division (the successive halving, for example) of the interval of the numbercontinuum containing the number-aggregate considered, and there is no known way of defining "division" among non-numerical numbers of any ordered aggregate. This statement requires some explanation. We can, of course, define the phrase "a divided by b❞ to mean anything we like. Thus, if "a" denotes Socrates, "b" Plato, and "divided by" means, say, add together the years (B. C.) of the births of a and b, and divide by some definite number, then "a divided by b" may denote a number which gives the year of birth of, say, Xenophon. But what we want here is a definition which has a meaning when a and b are members of any simply ordered aggregate (as Socrates, Plato, and Xenophon are, if, for example, they are arranged in the order of their times of birth), which meaning reduces to the ordinary arithmetical one when a and bare finite numbers. Now if we are given two transfinite simply ordered aggregates, and a and b are their respective ordinal types (ordinal type is a more general concept than ordinal number3), we can, as Cantor has shown, define a + b and a . b in such a way that, when for a and b we put finite ordinal numbers, a+b and a. b denote the numbers that these notations denote in ordinary arithmetic. And it would be possible to define a-b and a/b; but when M and N are transfinite, these notations would not denote one and only one number, as they do when a and b are finite, but a whole class of them. Thus a a may denote 0 or a itself....,

and so on.

But this would be no use to us in our case. In the

"Quart. Journ. of Math., 1910, p. 218.

Cf. Monist, Jan. 1910, Vol. XX, pp. 96-98.

Bolzano-Weierstrass process, we are given a simply ordered aggregate of real numbers, and begin by halving it. This process of halving determines one and only one point, and it does this, not because the aggregate is simply ordered, but because the aggregate is composed of finite, real numbers. It is because division has been defined as a unique process for the elements of the aggregate. Consider the aggregate of real numbers from 0 to 1, including the ends, whose type Cantor has denoted by 0. When halving this interval we do not look for "the" element which terminates an interval extending from 0 and of type 0/2. If there is such an element, there is an infinity of them. What we do is to determine the one element of the interval whose coordinate is 2; that is to say, the number got by halving the difference (1-0) of the numbers which are the ends of the interval.

In other words, it is because the aggregate of real numbers carries with it a scale of measurement, while a simply ordered aggregate in general does not. Of course, we can give a simply ordered aggregate a scale of measurement which, suo ipso, it does not possess, by correlating it with the aggregate of real numbers or with part of that aggregate. Thus, as Mach has pointed out in his Principien der Wärmelehre, a thermometer-scale primarily only indicates the hotter and the colder of two given states and does not give us any right to speak of a state A as being, say, "twice as hot" as state B, until we have (as we have now, since Thomson's introduction of the absolute scale) a means of correlating the degrees of expansion of a fluid with the real numbers. Not yet have we an absolute scale of hardness of minerals, so that hardnesses form a simply ordered continuum without any scale of measurement. We may remark, quite by the way, that though there seems to be an analogy between a simply ordered aggregate and the series of integers as defined by Richard Dedekind, and though Schröder in the third volume of his Vorlesungen über die Algebra der Logik," which is devoted to the logic of relatives, has pointed out this analogy as a special merit of Dedekind's theory, this analogy on closer inspection seems not to subsist.

The reason why this distinction between the absolute numerical scale and those simply ordered aggregates in which the elements only, have relative positions is so important, will appear presently

'Math. Ann., Vol. XLVI, 1895, p. 510.

* Cf. his Essays on the Theory of Numbers, Chicago, 1901. *Leipsic, 1895.

when we come to consider our treatment of space and time in mathematical physics.

But before we go any further we must explain a little the nature of the axiom mentioned above. Suppose that at any stage of a process we have to select an element-it does not matter which one-from a class of two or more. This is the case when we extend the Bolzano-Weierstrass process to simply ordered aggregates in general. If we have to do with real numbers, we may choose the definite point 1⁄2 between 0 and 1, the definite point 34 between 2 and 1, the definite point % between 2 and 4, and so on; but with a simply ordered transfinite aggregate in general we do not know of any means of finding a specialized (ausgezeichnetes) element between the ends of the aggregate, and we thus have an arbitrary choice among an infinity of elements between the ends of the aggregate considered. Now a form of the axiom in question is that when there is an infinity of classes of which each contains two or more members, it is possible to carry out for logical purposes-not, of course, carry out in the sense that we "carry out" the counting of a flock of sheep-the series of acts of arbitrary selection when this series is infinite. When the series is finite the axiom becomes a provable proposition; but when the series is infinite the method of proof by enumeration fails us.

This axiom, or rather an equivalent form of it, seems to have been first explicitly published by E. Zermelo in 1904, and it has lately been fully discussed in the part entitled "Selections" of the first volume of Whitehead and Russell's Principia Mathematica.s The chief importance of the axiom lies in its connection with the question as to whether any given aggregate can be well-ordered or not—a question which is outside the range of the subject with which we are at present concerned.

The history of this axiom is sometimes very amusing. Cantor had no doubt about its truth, and avoided using it where he could; Borel felt grave doubts about its validity, and used a form of it without scruple; most mathematicians did not see that it was unproved until Zermelo pointed this out in 1904, and Schoenflies, in the second part (1908) of his Bericht on the theory of aggregates, still failed to recognize this axiomatic character. This sort of

'Cf. pp. 360-366 of my paper "On the Comparison of Aggregates" in the Quart. Journ. of Math. for 1907.

Cambridge, 1910.

'Cf. Monist, Jan., 1910, Vol. XX, p. 116.

thing is partly due to the neglect by mathematicians of logic and ideography.

Of course, we can only practically measure relative positions and motions; but when for scientific purposes we construct a mathematical model of the world, we assume that points of time and space can be represented by real numbers or complexes of real numbers (Cartesian coordinates) or vectors (numbers with two or more units). Now these numbers have, so to speak, an absolute position in their scale: each number has its own individuality. Thus, in the sense in which mathematicians speak of an “arithmetical space," in analytical mechanics we necessarily have to do with an absolute "space."

In analytical mechanics our time and space are necessarily aggregates of numbers because our description of dynamical events is effected by differential equations, and we cannot define a differential quotient, nor indeed division, for aggregates other than those of numbers. We can define many things usually considered to apply only to aggregates of numbers: thus limit, continuity of the independent variable, many of the characteristics-technically known as “closedness," "density in itself," "perfectness" and "compactness"-of aggregates of numbers, the notion of function and even of continuous function can be defined purely ordinally. But beyond thisinto the differential and integral calculus-the purely ordinal theory of functions cannot go. If it could, then it would seem that we should have a means of describing mechanical events in a relative space; for a term in a series which can be fully described (for our purpose) by the giving of its ordinal type has, in general (unless, for example, it is the first or the last term), no property distinguishing it from another. To this circumstance is due the fact we observe when we try to transfer certain theorems of ordinary mathematical analysis to the ordinal theory of functions, namely, that whereas in ordinary analysis-owing to the fact that we have a scale of one measurement with the aggregate of numbers and so can determine uniquely a term (say "half-way") between two given ones certain processes of proof like that of Bolzano and Weierstrass are determined uniquely at every one of an infinity of steps, we require, for the analogous process in the purely ordinal theory, an axiom (known by the name of Zermelo) permitting an infinite series of acts of arbitrary selection.

CAMBRIDGE, ENGLAND.

PHILIP E. B. JOURDAIN.

MAGIC STARS.1

A five-pointed star being the smallest that can be made, the rules will be first applied to this one.

Choosing for its constant, or summation (S) = 48, then:

(5x48)/2 = 120 sum of series.

Divide 120 into two parts, say 80 and 40, although many other divisions will work out equally well. Next find a series of five numbers, the sum of which is one of the above two numbers. Selecting 40, the series 6+7+8+9+10=40 can be used. These numbers must now be written in the central pentagon of the star following the direction of the dotted lines, as shown in Fig. 1.

Find the sum of every pair of these numbers around the circle beginning in this case with 6+9=15 and copy the sums in a separate column (A) as shown below:

Place on each side of 15, numbers not previously used in the central pentagon, which will make the total of the three numbers = 48 or S. 17 and 16 are here selected. Copy the last number of

'We are indebted to Mr. Frederick A. Morton, Newark, N. J., for these plain and simple rules for constructing magic stars of all orders, and to Mr. H. A. Sayles, Schenectady, N. Y., for drawing the diagrams.