Now to say that two quantities are "nearly equal" may be interpreted to mean: either, that the difference between the quantities is nearly zero; or, that the ratio between them is nearly one.

(Here the difference "between" two quantities means the larger minus the smaller. Similarly, the ratio "between" two quantities means the larger divided by the smaller.)

If we adopt the "difference" interpretation, we have:

POSTULATE la. POSTULATE Ib. POSTULATE IC. POSTULATE Id. zero as possible.

The difference between A/a and B/b; or
The difference between a/A and b/B; or
The difference between A/B and a/b; or

The difference between B/A and b/a; should be as near

If, on the other hand, we adopt the "ratio" interpretation, we have: POSTULATE I. The ratio between A/a and B/b (or the ratio between a/A and b/B; or the ratio between A/B and a/b; or the ratio between B/A and b/a; all of which have the same value) should be as near unity as possible.

Since there is no way of choosing, mathematically, between Postulates Ia and Ib or between Postulates Ic and Id, and since these four demands lead to four different results, we shall reject all four of them and adopt Postulate I as the proper interpretation of our Fundamental Principle.

The case of two states is thus disposed of.

For the case of three or more states, one further principle is required, which we state as follows:

POSTULATE II. In a satisfactory apportionment, there should be no pair of states which is capable of being "improved" by a transfer of representatives within that pair-the word "improvement" being understood in the sense implied by the test already adopted for the case of two states, and the rare cases of "no choice" being decided in favor of the larger state.

From these two postulates the following theorem can then be deduced: THEOREM I. For any given values of A, B, C, ... and N, there will always be one and only one satisfactory apportionment in the sense defined by Postulates I and II. No further principles are required.

A working rule for computing this "best" apportionment in any given case is found to be as follows:

Working Rule.-Multiply the population of each state by as many of the numbers

Inf., 1/√1 X 2, 1/√2 × 3, 1/√3 X 4,

as may be necessary, and record each result, together with the name of the state, on a small card. Arrange these cards according to the magnitude of the numbers recorded upon them, from the largest to the smallest, thus forming a priority list for the given states (the cards marked “Inf.” being placed at the head of the list, arranged among themselves in order

of magnitude of the populations of the states). Finally, assign the representatives, from the 1st to the Nth, to the several states in the order in which the names of the states occur in this priority list. (It should be noted that this method satisfies automatically the constitutional requirement that every state shall have at least one representative.)

This method may be called the "method of the geometric mean," since the "multipliers" are the reciprocals of the geometric means of consecutive integers.

The solution of the problem is thus complete.

Alternative Methods.—If we had adopted Postulate Ia or Ib we should have been led, in like manner, to two other methods which may be called the method of the harmonic mean (Ia), and the method of the arithmetic mean (Ib), since the "multipliers" in the working rules are as follows:

It can be shown that method Ia favors the smaller states more than method I does, while method Ib favors the larger states more than method I does. Since there is no mathematical reason for adopting either of the two Postulates Ia and Ib to the exclusion of the other, both should be rejected.

Postulates Ic and Id also determine two distinct methods, which may be called the two methods of similarity ratios. It can be shown that Ic favors the small states even more than Ia does, while Id favors the large states even more than Ib does, so that both should be rejected.

Each of these four methods violates three of the four conditions expressed in our Fundamental Principle, while the method of the geometric mean satisfies all these conditions simultaneously.

The following further methods are suggested by the Theory of Least Squares.

In a theoretically perfect apportionment, A/a would be equal to P/N, and a/A to N/P (where P is the total population). Hence, in place of Postulates I and II, we might consider the following:

POSTULATE IIIa. The sum of the squares of the deviations of the A/a from their true values; or

POSTULATE IIIb. The sum of the squares of the deviations of the a/A from their true values; should be a minimum.

İt can be shown, however, that IIIa favors the smaller states even more than Ic does, while IIIb favors the larger states even more than Id does. In other words, Postulates IIIa and IIIb violate, in opposite directions, all four of the conditions expressed in our Fundamental Principle. Since there is no mathematical reason for adopting either to the exclusion of the other, both should be rejected.

The same remark applies if in Postulates IIIa and IIIb the word "square" is replaced by "absolute value."2

Hence it is clear that if a numerical measure of injustice is desired, both the deviation of A/a and the deviation of a/A should be taken into account together. That is, any formula which reports A/a, say, as too large by a certain amount, should also report a/A as too small by the same amount. The formulas suggested by the simple application of the idea of least squares, as shown in the preceding paragraph, do not have this property. A combination of these formulas suggests, however, the following postulate, in which a denotes the theoretical value of a.

POSTULATE III. In a satisfactory apportionment, the sum T of terms of the form Ae2 where

For purposes of computation, this total error, T, may be replaced by an average error, E= √S/N, where S is the sum of terms of the form αe2

(a — a)2/a.

The method determined by Postulate III is precisely the same as the method of equal proportions based on Postulates I and II.

1 This article contains the substance of two papers presented to the American Mathematical Society, December 28, 1920, and February 26, 1921. Further details, with proofs and examples, will be published either in the Transactions of the American Mathematical Society, or in the Quarterly Publication of the American Statistical Association, or in the American Mathematical Monthly.

For the history of the subject see W. F. Willcox, "The Apportionment of Representatives" (presidential address at the annual meeting of the American Economic Association, December 1915), published in the American Economic Review, Vol. 6, No. 1, Supplement, pp. 1-16, March, 1916. See also 62d Congress, 1st Session, House of Representatives, Report No. 12, pp. 1-108, April 25, 1911, and John H. Humphreys, "Proportional Representation," London, 1911.

The most important of the methods hitherto known are four:

The Vinton method of 1850, long in use in Congress, is known to lead to an “Alabama paradox;" that is, an increase in the total size of the House may cause a decrease in the representation of some state.

The Hill method of alternate ratios, proposed by Dr. J. A. Hill in 1910 but not adopted, comes very near to satisfying the postulates of the present paper, and uses for the first time (though only partially) the idea of the geometric mean. The method is incomplete however, since it can be shown to lead to an Alabama paradox.

The Willcox method of major fractions, devised by Professor W. F. Willcox in 19001910, and now in use in Congress, employs, in effect, a working rule like ours with multipliers: Inf., 2/3, 2/5, 2/7, .; it is essentially the same as the method of the arithmetic mean, and therefore favors the larger states unduly, just as the hitherto unsuspected but equally justifiable method of the harmonic mean favors the smaller states unduly. (It may be noted that the name "major fractions" is somewhat misleading, since the Willcox major fraction is not a major fraction of the true quota, but

a major fraction of an artificial quota, scaled up or down from the true quota to meet the requirements of the computation.)

The d'Hondt method, originated in Belgium and now widely used in European elections, employs "multipliers" 1, 1/2, 1/3, 1/4... and can be shown to favor the larger states to the extent of violating all four of the conditions expressed in our Fundamental Principle.

2 If one should try to minimize the sum of the squares (or the sum of the absolute values) of the deviations of the a's themselves from their true values (with or without "weighting" by the population of the state), the resulting methods would all lead to an Alabama paradox. The same is true of the weighted sum of the absolute values of the deviations of a/A (or of A/a). The same is also true of the absolute values of the logarithms of the ratios between the a'S and their true values.

"This Postulate III was added on April 23, after Professor F. W. Owens had shown (at the meeting of the American Mathematical Society on February 26) that the method of minimizing the sum of terms like A[(a/A) (a/A)] leads to th same result as the Willcox method of major fractions. It may be noted that the method of minimizing the sum of terms like a[(A/a) — (A /α) ]2 leads, not as one might expect, to the method of the harmonic mean, but to the method of the geometric mean.

CURRENT MAPS OF THE LOCATION OF THE MUTANT GENES OF DROSOPHILA MELANOGASTER'

BY CALVIN B. BRIDGES

COLUMBIA UNIVERSITY, N. Y.

Communicated by T. H. Morgan, December 5, 1920

The maps that have been published' showing the distribution of the mutant genes of D. melanogaster can now be much improved because of the discovery of new mutants and the accumulation of crossover data. Figure 1 gives in simplified form the maps that are in use in our laboratory. The distances on the maps are based on the total amount of crossing over between the loci, one unit of distance representing one per cent of crossing over. The map-distances are the same as the observed crossover values or "percentages of exchange" whenever the two loci considered are so close together that no, or only a negligible amount of, double crossing over occurs between them. In the first (X-) chromosome this practical equivalence of map-distance and exchange-value holds for loci not farther apart than about 15 units. In the middle of the second chromosome and of the third chromosome the equivalence holds for only about 10 units. In the end-regions of the second and third chromosomes it holds up to nearly 20 units. For distances somewhat greater than these the mapdistances exceed the observed percentages of exchange by an amount equal to twice the percentage of double crossing over between the given loci. For still greater distances the difference includes also three times the percentage of triples. The number of quadruple crossovers is negligible except perhaps when the whole length of the second chromosome is to

be considered. For each chromosome and for each region within a chromosome the amount of this multiple crossing over is characteristic, and may not be the same in amount for different sections of equal map-distance. Because of this variation, the accurate expression of the relation between map-distance and exchange-values for the more distant loci, requires a table of conversion corrections for each pair of loci considered. Such tables will be published with the more detailed maps of the chromosomes. In general, the correction is relatively slight with distances that are under 20-40, but for longer intervals the correction increases at an accelerated rate. We have not met with percentages of exchange that exceed 50.0, though two of the maps are about a hundred units long.

The map of the third chromosome is the most accurate, since the calculation of the distances between the principal loci is made on the basis of all data up to 1920, and an improved method of weighting and interrelating the data has been followed. Relatively little change in these primary distances the "triangulation" of the map is expected with the further accumulation of data. There is still some uncertainty with regard to the region to the left of spineless, for the different sets of data upon that region may not be comparable because of the possible presence of crossover variations. The data used in the calculations for the primary distances in the first and second chromosomes are homogeneous, and although not including the last four years' work, are still fairly ample in amount. Changes are to be expected when these two maps are recalculated with the improved method and complete data.

The most useful mutants are those that are separable from the wildtype with completeness and ease, that are not inferior to the wild-type in viability and productivity, and that do not interfere with the use in the same experiment of any large class of the other mutant characters. Dominants are more valuable than recessives. Mutants accurately located in the chromosome are more valuable than those whose positions are less well established, though if the other desirable features mentioned above are present in a given new mutant the position will be found rather quickly. A very large factor in the value of the mutant is its position in the chromosome. The end positions are most valuable. Throughout the remainder of the chromosome the most favorable positions are those evenly spaced and just close enough together so that no double crossing over occurs between them. If the interval is too small there is trouble in getting double recessives, and the crossover classes are so small that large totals are required to make differences significant.

The mutants that fulfill all the above requirements most nearly are followed by an asterisk (*) in the maps. Their loci are the primary bases with relation to which the other mutants are located. There are several cases in which two or more excellent mutants affecting quite different