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major fraction, and that fact constitutes their only novel feature. If I am rightly informed, no series of tables has been prepared heretofore possessing both of these characteristics.
The results of this method are simple, but the method by which they have been reached is somewhat difficult to explain. If a ratio of 240,000 persons to each Representative be assumed arbitrarily as a starting point, that number divided into the constitutional population of each State and one Representative assigned for each whole number and each major fraction in the series of quotients a total of 383 Representatives is reached. If the ratio be then diminished by 10 to 239,990, no difference in the apportionment will result, but the decimal in each quotient will be slightly increased. The amount of this increase in each decimal will depend upon the population of the State in question, that of the largest States increasing most rapidly. If the ratio be further reduced to 239,980, 239,970, etc., the decimals continue to increase with each change of ratio, but with varying rapidity.
rapidity. It is a simple problem to compute in which State the decimal will first pass 0.500 and become a major fraction and at just what ratio the change will occur (see Table 2). In the present case the State whose decimal first reaches 0.500 is Illinois, and the corresponding ratio is 239,940 (see Table 3). Accordingly, Illinois is entitled to the next Representative, who might be called No. 384. The ratio of 239,940 has been called the boundary ratio. The next smaller boundary ratio is 239,615, at which point the decimal for Mississippi rises to 0.500, and that State therefore is entitled to Representative No. 385.
Any ratio between these two numbers, 239,940 and 239,615, will yield a table apportioning 384 Representatives; but the nearer the ratio is put to 239,940 the weaker becomes the admitted claim of Illinois to an additional Representative. A ratio of 239,935 gives a quotient for Illinois of 23.5005. So the nearer the ratio is put to 239,615 the stronger becomes the rejected claim of Mississippi to a Representative. The ratio of 239,617 divided into the constitutional population of Mississippi gives a quotient of 7.49994. In order to exclude such difficult cases the ratios for division have been fixed midway between the two boundary ratios; but it must be clearly understood that a table computed from any ratio whatever between the two boundary ratios and applying rigidly the fundamental principle of major fractions would yield precisely the same apportionment.
The method followed in finding these boundary ratios for the whole series of apportionable Representatives from 49 to 440 will be clear, I hope, to anyone who will examine Tables 2 and 3. If it is not, Í shall be glad to submit a further statement or to explain
it fully when I appear before the committee. Yours, respectfully,
TABLE 1.-Apportionment of each number of Representatives between 390 and 440,
inclusive, by method of major fractions.
Table 1.-Apportionment of each number of Representatives between 890 and 140,
inclusive, by method of major fractions—Continued.
TABLE 1.-Apportionment of each number of Representatives between 890 and 440,
inclusive, by method of major fractions Continued.