Original actions, authorized by Amendment No. 23 to the state Constitution, were filed by Nabors Shaw, of Poinsett county; L. H. Autry and J. Lee Bearden, of Mississippi county; and Girard P. Shofner, of Pulaski county. Each named the Governor, Secretary of State, and Attorney General, as respondents. It was alleged in the petitions that these officers, who constitute a board of reapportionment, erred in the result certified to the Secretary of State January 21, 1941.
In Cause No. 6357 — Shaw v. Adkins, et als. — it is asserted that Poinsett’s population of 37,670 exceeds that of either White, Benton, St. Francis, Hempstead, Miller. *858or Lonoke county, each of which was given two representatives, while but one was assigned to Poinsett.
Cause No. 6358 — -Autry, et als., v. Adkins, et als. — alleges that in assigning two representatives each to Benton, Craighead, Crittenden, Garland, Hempstead, Lonoke, Miller, Phillips, St. Francis, Washington, and White counties, and in assigning but three to Mississippi county, the board acted arbitrarily and in disregard of the county’s population of 80,217. James Terry and Harry W. Haines, and twenty-seven other citizens of Mississippi county, intervened in Cause No. 6258. They adopted the Autry-Bearden complaint and made other allegations.
In Cause No. 6376 — -Shofner v. Adkins, et als. — it is alleged that Pulaski county, with a population of 156,085, is entitled to eight representatives instead of seven as assigned by the board.
Royce Weisenberger intervened in all three cases and seeks to show that under any method of reapportionment Hemjostead county is entitled to two representatives, as now assigned.
The board of apportionment assigned to each of the following counties one representative: Arkansas, Ashley, Baxter, Boone, Bradley, Calhoun, Carroll, Chicot, Clark, Clay, Cleburne, Cleveland, Columbia, Conway, Crawford, Cross, Dallas, Desha, Drew, Faulkner, Franklin, Fulton, Grant, Greene, Hot Spring, Howard, Independence, Izard, Jackson, Johnson, LaFayette, Lawrence, Lee, Lincoln, Little River, Logan, Madison, Marion, Monroe, Montgomery, Nevada, Newton, Ouachita, Perry, Pike, Poinsett, Polk, Pope,. Prairie, Randolph, Saline, Sevier, Scott, Searcy, Sharp, Stone, Yan Burén, Woodruff, and Yell.
To each of the following counties two representatives were given: Benton, Craig’head, Crittenden, Garland, Hempstead, Lonoke, Miller, Phillips, St. Francis, Washington, and White.
To each of the following counties three representatives were assigned: Jefferson, Mississippi, Sebastian, and Union.
*859Pulaski county was given seven representatives.
The board adopted 19,492 as the basis of population for representative.1
Amendment No. 23 required the board to make its first apportionment within ninety days from January 1, 1937, “and. . . thereafter, on or before February 1 immediately following each federal census, the board shall reapportion the state for both representatives and senators.2 . : .”
Section two of the amendment provides that the house of representatives shall consist of 100 members. Each county existing at the time of apportionment shall have at least one representative, “and . . . the remaining numbers shall be equally distributed (as nearly as practicable) among the more populous counties of the state, in accordance with a ratio to be determined by the population of said counties as shown by the federal census next preceding any apportionment hereunder. ”
The board correctly assigned one representative to each county. Of the remaining 25, eleven went to Benton, Craighead, Crittenden, Garland, Hempstead, Lonoke, Miller, Phillips, St. Francis, Washington, and White counties; eight went to Jefferson, Mississippi, Sebastian, and Union counties, and six went to Pulaski, in the proportion heretofore shown.
Was this distribution “in accordance with a ratio to be determined by the population [of the more populous counties] as shown by the federal census”?
If the problem is considered but casually the responding impression is that the “more populous counties” should be grouped in the order of their population, thus permitting familiar mathematical computations to be applied and a definite result ascertained. But-it is not that simple. A somewhat similar provision of *860the federal Constitution3 has been productive of arithmetical and algebraic perplexities since Daniel "Webster’s plan of 1832 proved to be unworkable in-practice because it did not always give the right total.4
According to Professor Huntington (see footnote No. 4), a mathematical study of the problem made in 1921 showed that there are five methods which are workable, and which avoid what the author referred to as the “paradoxes.” The approved concepts are: (1) Method of major fractions. (2) Method of equal proportions. (3) Method of harmonic mean. (4) Method of smallest divisors. (5) Method of greatest divisors.
In a preliminary discussion of the five methods, Professor Huntington says:
“To meet realistically the actual situation in Congress when an apportionment bill is up for debate, the emphasis is shifted from the process of computation to the test of fairness which the final result should satisfy. The fairness of the final result, not the technical process of achieving this result, is regarded as the important *861thing. For example, suppose an actual apportionment bill proposes to give Alabama nine seats, Arizona one, Arkansas seven, etc., in a house of any given size (say 435). The fundamental question which Congress has to face is this: Does the distribution proposed in the bill put each state as nearly as may be on a par with every other state, or would the bill be ‘improved’ by transferring a seat from such-and-such a state to' such- and-such another state?
“To answer this question, Congress must decide what goal or aim it has in mind when discussing proposed ‘improvements’ in a given bill. It is generally agreed that Congress, consciously or unconsciously, has had two principal aims in view: ’ First, to equalize the ‘congressional districts’ among the several states; and secondly, to equalize the ‘individual shares’ among the several states. What the modern mathematical theory has done is to establish clearly the relations between these two aims and give the possible methods listed above.
“The mathematical facts are as follows: The method of smallest divisors and the method of greatest divisors fail on both these aims; the method of major fractions fails on the first aim; the method of harmonic mean fails on the second aim; the method of equal proportions achieves both aims.
“In view of these facts, the method of equal proportions was approved by two scientific bodies: The Advisory Committee to the Director of the Census, in 1921; and the National Academy of Sciences, in 1929.”
Apportionment methods referred to by Professor Huntington are discussed in “Congressional Apportionment,” by Laurence F. Schmeckebier. The book was copyrighted in 1941 by The Brookings Institution. At page 12 it is said: “While these methods are all mathematically correct, each one starts with a different premise and the results may be different. . . . Two of the modern methods — major fractions and equal proportions —are recognized by statute. The other three methods— harmonic mean, smallest divisors, and greatest divisors *862—have been discussed in committees and in the literature, but have never received statutory recognition. ’ ’5
The problem is to divide 25 representatives among the more populous counties “in accordance with a ratio to be determined by the population of said counties.” It was recognized by those who framed our Constitution that a mathematically exact division would be impossible because there are no fractional representatives; hence, in the Constitution there is authority for making the apportionment “as nearly as practicable.”
First, the ratio must be ascertained. The total population of 1,949,387 divided by 100 shows this factor to be 19,494, minus. The result is termed the natural ratio. But there has been assigned to each county one representative, and 75 times 19,494 gives 1,462,050. This, taken from total population, leaves 487,337. If we divide the remainder by 25 — a number equal to the unassigned representatives — the result is 19,494, the natural ratio.
According to Schmeckebier, no modern method of apportioning representatives uses any ratio in determining the result. A ratio, he says, is often referred to, but it is obtained after the apportionment is made from a so-called priority list. All modern methods — equal proportions, major fractions, harmonic mean, smallest divisors, and greatest divisors — assign the representatives to each state, in the case of congressional action, and to each county, in the case of state procedure, by means of priority lists, which indicate the apportionment to be made from the definite number' — in Congress, ordinarily 435, and in Arkansas definitely 100. As the federal Constitution provides that one representative shall be assigned to each state, no question of priority *863would, exist if the house consisted of 48 members. Each state obtains one representative, regardless of its population. Therefore, the priority list begins with the forty-ninth member of the house, and when completed it shows which states would receive additional members for any size house. As applied to reapportionment in Arkansas, the priority list begins with the member which we may designate as 76. In reality, priorities relate only to 25 members. Under our constitutional provision, the division of these 25 representatives is in accordance with a ratio to be determined by the population of such counties, exclusive of the remaining 75; hence, the priority list, must be compiled and the ratios ascertained.
By the method of equal proportions 6 as applied to Arkansas’ representation in 'Congress, seven congressmen would be retained. By the method of major fractions, Michigan, now having 17 congressmen, would gain one, and Arkansas would be the corresponding loser.
A great deal has been written, by those engaged in the task of testing apportionments, regarding absolute and relative differences. The absolute difference between two numbers is obtained by subtracting the smaller from the larger. The relative difference is the percentage by which the larger exceeds the smaller, and is obtained by dividing* the smaller amount into the difference. An illustration given by Schmeekebier is: If a piece of property costing $500 is sold for $600 and another piece of property costing $1,000 is sold for $1,100, the absolute difference in the profit is the same in each case — $100. But the relative difference in one case is 500 divided into 100, or 20 per cent.; in the other ease it is 1,000 divided into 10Ó, or 10 per cent.
The following table shows population of each of the state’s 75 counties according to the 1930 and the 1940 censuses:
*864Population op Arkansas by Counties
Since priority lists are used in each of the five so-called modern methods of apportionment, it is essential to show the steps by which these lists are prepared. They are obtained by multiplying the population of each county successively by the multipliers applicable to the second representative, the third representative — and so on. It should he pointed out, however, that a different series of multipliers is used for each of the methods.
Major fractions and equal proportions, having been recognized as the two methods generally approved, the three other processes will not be analyzed, although tables showing results under each are included in this opinion.
*865 Major Fractions. — The priority list is obtained by dividing the population of each county successively by the arithmetic mean between succeeding representatives.7 The priority list obtained by the process shown in the seventh footnote (major fractions) is made use of in this way: Assign the seventy-sixth representative (first disposed of in the additional group of 25) to the county having the highest number in the priority list, the second to the next highest number in the priority list, and so on until the full 25 have -been disposed of. Pulaski county, having the highest number in the priorit}7 list,8 would receive the first additional representative, and Miller county would be recipient of the twenty-fifth representative. The order of assignment is shown by the figures inclosed within parentheses, following priority numbers. (See table identified as footnote 8.) For instance, Pulaski county, instead of having its seven representatives assigned successively, receives the first, second, fourth, seventh, twelfth, and twenty-second of the 25 to be assigned, and other counties, as shown by the figures inclosed in parentheses in tabulations appear as footnotes 8, 10, 11, 12, and 13, receive additional representatives in the numerical order shown.
*866 Equal Proportions. — The priority list is obtained by dividing the population of each county by the geometric mean of successive numbers of representatives.9 The *867priority list obtained by the process shown in the ninth footnote (equal proportions) is made use of in the manner explained as pertaining to major fractions. The priority list and other data for use in the equal proportions method are shown as the tenth footnote.10
Other Methods.- — Footnotes 11,12, and 13 are priority lists and other data for harmonic mean method, smallest divisors method, and greatest divisors method.11, 12, 13
Proof of product obtained under the two methods discussed in detail — major fractions and equal proportions- — may be shown in the following manner, the result in each case being the same:
Lonoke county has a population of 2.9,802, and has two house members. Theoretically each represents a population of 14,901. Mississippi county has a population of 80,217 and has three members. Each, therefore, represents a population of 26,739. In theory-each Mississippi county member represents 11,838 more constituents than does each of Lonoke’s members, the differential being 79.4 per cent. If one member should be taken from Lonoke county and given to Mississippi county, each of Mississippi’s four members would represent 20,054 persons and Lonoke’s one remaining member would represent 29,802, an absolute difference of 9,748, or 48.6 per cent, against Lonoke. The disparity difference against Mississippi county at present is much larger than the disparity against Lonoke county if the latter is reduced to one-member.
Union county’s population is 50,461. It has three members, each of whom, theoretically, represents a con*868stituency of 16,820. Poinsett’s population is 37,670. The county has one house member, and this member, in theory, represents 20,850 more in population than do each of Union county’s members, the differential being 124.0 per cent. If one member should be taken from Union and assigned to Poinsett, the two Poinsett members will each represent 18,835 in population. Each of the remaining Union county members will represent a *869population of 25,231 — an absolute difference of 6,396. The percentage against Union county in its relation to Poinsett would then be 34.0, as compared with a former percentage of 124.0 against Poinsett.
By either of the five methods Lonoke and Union counties each lose a representative, and by either Poinsett gains.
*870By the method of smallest divisors Pulaski county would lose one member, while by the method of greatest divisors it would g'ain two. According to either of the other three methods, Pulaski’s representation would be unchanged.
By four of the five methods Mississippi county gains a member, while under the smallest divisor method its representation remains static.
*872By the method of smallest divisors Ouachita and Greene counties would each gain a member, but under either of the other four methods there would be no changes.
By the method of greatest divisors Hempstead and Miller counties would each lose a member, while under either of the other four systems these counties would not be affected.
These results are presented in the following table. It shows assignment of the 25 additional representatives by counties according to the apportionments of 1937 and 1941. Also, there is shown results obtained by the five apportionment methods discussed in this opinion:
Results Obtained by Different Methods
It is not our purpose to assert that all possible methods of computation have been tested, or to say that in certain circumstances relating to population gains or losses a more equitable apportionment could not be made. We have given earnest consideration to discussions by authorities who have been classed as experts. That the methods of major fractions and equal proportions give uniform results when applied to Arkansas is *873significant. The fact that the National Academy of Sciences has given its indorsement to the method of equal proportions in preference to other systems is highly persuasive. This method, therefore, is adopted by us, with the result that Lonoke and Union counties each lose a representative, and Mississippi and Poinsett each gain a member.
The board of apportionment’s findings in respect of house membership, while remarkably accurate in the main, must be revised to the extent indicated. It is so ordered.