United States congressional apportionment
United States congressional apportionment is the process by which seats in the United States House of Representatives are distributed among the 50 states according to the most recent decennial census mandated by the United States Constitution. Each state is apportioned a number of seats which approximately corresponds to its share of the aggregate population of the 50 states. However, every state is constitutionally guaranteed at least one seat.
The number of voting seats in the House of Representatives has been 435 since 1913, capped at that number by the Reapportionment Act of 1929—except for a temporary (1959–1962) increase to 437 when Alaska and Hawaii were admitted into the Union.
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Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons.
The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at least one Representative;…
"Three-fifths of all other persons" refers to the inclusion of 3⁄5 of the slaves in the population base
Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice-President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.
Reapportionments normally occur following each decennial census, though the law that governs the total number of representatives and the method of apportionment to be carried into force at that time are enacted prior to the census.
The decennial apportionment also determines the size of each state's representation in the U.S. Electoral College. Under Article II, Section 1, Clause 2 of the U.S. Constitution, the number of electors of any state equals the size of its total congressional delegation (House and Senate seats).
Federal law requires the Clerk of the House of Representatives to notify each state government no later than January 25 of the year immediately following the census of the number of seats to which it is entitled. If the number of seats has changed, the state determines the boundaries of congressional districts—geographical areas within the state of approximately equal population—in a process called redistricting.
Because the deadline for the House Clerk to report the results does not occur until the following January, and the states need sufficient time to perform the redistricting, the decennial census does not affect the elections that are held during that same year. For example, the electoral college apportionment during 2000 presidential election was still based on the 1990 census results. Likewise, the congressional districts and the electoral college during the 2020 general elections will still be based on the 2010 census.
Number of members
The size of the U.S. House of Representatives refers to total number of congressional districts (or seats) into which the land area of the United States proper has been divided. The number of voting representatives is currently set at 435. There are an additional five delegates to the House of Representatives. They represent the District of Columbia and the territories of American Samoa, Guam, the Northern Mariana Islands, which first elected a representative in 2008, and the U.S. Virgin Islands. Puerto Rico also elects a resident commissioner every four years.
Controversy and history
Since 1789, when the Federal Government began operating under the Constitution, the number of citizens per congressional district has risen from an average of 33,000 in 1790 to over 700,000 as of 2018. Prior to the 20th century, the number of representatives increased every decade as more states joined the union, and the population increased.
The ideal number of members has been a contentious issue since the country's founding. George Washington agreed that the original representation proposed during the Constitutional Convention (one representative for every 40,000) was inadequate and supported an alteration to reduce that number to 30,000. This was the only time that Washington pronounced an opinion on any of the actual issues debated during the entire convention.
In Federalist No. 55, James Madison argued that the size of the House of Representatives has to balance the ability of the body to legislate with the need for legislators to have a relationship close enough to the people to understand their local circumstances, that such representatives' social class be low enough to sympathize with the feelings of the mass of the people, and that their power be diluted enough to limit their abuse of the public trust and interests.
... first, that so small a number of representatives will be an unsafe depositary of the public interests; secondly, that they will not possess a proper knowledge of the local circumstances of their numerous constituents; thirdly, that they will be taken from that class of citizens which will sympathize least with the feelings of the mass of the people, and be most likely to aim at a permanent elevation of the few on the depression of the many; ...
Madison also addressed Anti-Federalist claims that the representation would be inadequate, arguing that the major inadequacies are of minimal inconvenience since these will be cured rather quickly by virtue of decennial reapportionment. He noted, however,
I take for granted here what I shall, in answering the fourth objection, hereinafter show, that the number of representatives will be augmented from time to time in the manner provided by the Constitution. On a contrary supposition, I should admit the objection to have very great weight indeed.
Madison argued against the assumption that more is better:
Sixty or seventy men may be more properly trusted with a given degree of power than six or seven. But it does not follow that six or seven hundred would be proportionally a better depositary. And if we carry on the supposition to six or seven thousand, the whole reasoning ought to be reversed. ... In all very numerous assemblies, of whatever character composed, passion never fails to wrest the scepter from reason.
The Apportionment Act of 1911 (Public Law 62-5) raised the membership of the U.S. House to 433 and provided for an apportionment. It also provided for additional seats upon the admissions of Arizona and New Mexico as states, increasing the number to 435 in 1912.
In 1921, Congress failed to reapportion the House membership as required by the United States Constitution. This failure to reapportion may have been politically motivated, as the newly elected Republican majority may have feared the effect such a reapportionment would have on their future electoral prospects. A reapportionment in 1921 in the traditional fashion would have increased the size of the House to 483 seats, but many members would have lost their seats due to the population shifts, and the House chamber did not have adequate seats for 483 members. By 1929, no reapportionment had been made since 1911, and there was vast representational inequity, measured by the average district size. By 1929 some states had districts twice as large as others due to population growth and demographic shift.
In 1929 Congress (with Republican control of both houses of Congress and the presidency) passed the Reapportionment Act of 1929 which capped the size of the House at 435 (the then current number) and established a permanent method for apportioning a constant 435 seats. This cap has remained unchanged since then, except for a temporary increase to 437 members upon the 1959 admission of Alaska and Hawaii into the Union.
Three states – Wyoming, Vermont, and North Dakota – have populations smaller than the average for a single district, although none of those states have fewer people than the least populous congressional districts (as of the 2010 census, Rhode Island's two districts). As of May 2016, there is approximately one representative for every 720,000 people in the country.
Clemons v. Department of Commerce
A 2009 lawsuit, Clemons v. Department of Commerce, sought a court order for Congress to increase the size of the House's voting membership and then reapportion the seats in accordance with the population figures of the 2010 Census. The intent of the plaintiff was to rectify the disparity of congressional district population sizes among the states that result from the present method of apportionment. Upon reaching the U.S. Supreme Court in December 2010, the holdings of the lower district and appellate courts were vacated and the case remanded to the U.S. District Court from which the case originated with instructions that the district court dismiss the case for lack of jurisdiction.
The first proposed amendment to the Constitution within the Bill of Rights attempted to set a pattern for growth of the House along with the population, but has not been ratified.
Article the first ... After the first enumeration required by the first article of the Constitution, there shall be one Representative for every thirty thousand, until the number shall amount to one hundred, after which the proportion shall be so regulated by Congress, that there shall be not less than one hundred Representatives, nor less than one Representative for every forty thousand persons, until the number of Representatives shall amount to two hundred; after which the proportion shall be so regulated by Congress, that there shall not be less than two hundred Representatives, nor more than one Representative for every fifty thousand persons.
The proposed Wyoming Rule calls for expanding the House until the standard Representative-to-population ratio equals that of the smallest entitled unit (currently the state of Wyoming). This proposal is primarily designed to address the fact that some House districts are currently nearly twice the size of others; for instance, there are just over 1 million residents in Montana's single district, compared to about 570,000 in Wyoming's. Although a larger House size will generally result in the smallest and largest districts being proportionally closer in size, this is not always the case. Therefore, in some cases, the Wyoming Rule may actually result in an increase in the ratio of the sizes of the largest and smallest districts. For instance, after the 1990 Census and with a House size of 435, the largest district (Montana's at-large district) had 799,065 residents, 76% larger than the smallest district (Wyoming's at-large district with 453,588 residents). The Wyoming Rule would have given a House size of 547 in 1990. Using that size, the largest district (North Dakota's at-large district) would have had 638,800 residents, 92% larger than the smallest districts (Delaware's two districts at approximately 333,084 residents each), which is larger than the 76% figure mentioned above.
Another proposed expansion rule, the Cube Root rule, calls for the membership of the legislature to be based on the cube root (rounded up) of the U.S. population at the last census; this can be split between the House and the Senate, if desired. For example, such a rule would call for 676 members of the legislature based on the 2010 United States Census; this could be 676 members of House, 576 (676 - 100 Senators), or 575 (676 - 100 Senators - 1 Vice President). An additional House member would be added each time the national population exceeds the next cube; in this case, the next House member would be added when the census population reached 308,915,777, and the one after that at 310,288,734.
In 2007, during the 110th Congress, Representative Tom Davis introduced a bill in the House of Representatives that would add two seats to the House, one for Utah and one for the District of Columbia. It was passed by the House, but was tripped up by procedural hurdles in the Senate and withdrawn from consideration. An identical bill was reintroduced during the 111th Congress. In February 2009 the Senate adopted the measure 61-37. In April 2010, however, House leaders decided to shelve the proposal.
Apart from the requirement that each state is to be entitled to at least one representative in the House of Representatives, the number of representatives in each state is in principle to be proportional to its population. No fair apportionment method was devised until recently with five distinct apportionment methods having been used since the adoption of the Constitution, with none of them producing fully proportional apportionment among the states.
The first apportionment was contained in Art. I, § 2, cl. 3 of the Constitution. After the first Census in 1790, Congress passed the Apportionment Act of 1792 and adopted the Jefferson method to apportion U.S. Representatives to the states based on population. The Jefferson method required fractional remainders to be discarded when calculating each state's total number of U.S. Representatives and was used until the 1830 census. The Webster method, proposed in 1832 by Daniel Webster and adopted for the 1840 Census, allocated an additional Representative to states with a fractional remainder greater than 0.5. The Hamilton/Vinton (largest remainder) method was used from 1850 until 1900. The Vinton or Hamilton method was shown to be susceptible to an apportionment paradox. The Apportionment Act of 1911, in addition to setting the number of U.S. Representatives at 435, returned to the Webster method, which was used following the 1910 and 1930 censuses (no reapportionment was done after the 1920 census). The current method, known as the Huntington–Hill method or method of equal proportions, was adopted in 1941 for reapportionment based on the 1940 census and beyond. The revised method was necessary in the context of the cap on the number of Representatives set in the Reapportionment Act of 1929.
The method of equal proportions
The apportionment methodology currently used is the method of equal proportions, so called because it guarantees that no additional transfer of a seat (from one state to another) will reduce the ratio between the numbers of persons per representative in any two states. The method of equal proportions minimizes the percentage differences in the populations of the congressional districts.
In this method, as a first step, each of the 50 states is given its one guaranteed seat in the House of Representatives, leaving 385 seats to assign.
The remaining seats are allocated one at a time, to the state with the highest priority number. Thus, the 51st seat would go to the most populous state (currently California). The priority number is determined by a formula that is mathematically computed to be the ratio of the state population to the geometric mean of the number of seats it currently holds in the assignment process, n (initially 1), and the number of seats it would hold if the seat were assigned to it, n+1.
The formula for determining the priority of a state to be apportioned the next available seat defined by the method of equal proportions is
where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat. An equivalent, recursive definition is
where n is still the number of seats the state has before allocation of the next (in other words, for the mth allocation, n = m-1, where m > 1), and for n = 1, the initial A1 is explicitly defined as
Consider the reapportionment following the 2010 U.S. Census: beginning with all states initially being allocated one seat, the largest value of A1 corresponds to the largest state, California, which is allocated seat 51. After being allocated its 2nd seat, its priority value decreases to its A2 value, which is reordered to a position back in line. The 52nd seat goes to Texas, the 2nd largest state, because its A1 priority value is larger than the An of any other state. However, the 53rd seat goes back to California because its A2 priority value is larger than the An of any other state. The 54th seat goes to New York because its A1 priority value is larger than the An of any other state at this point. This process continues until all remaining seats are assigned. Each time a state is assigned a seat, n is incremented by 1, causing its priority value to be reduced and reordered among the states, whereupon another state normally rises to the top of the list.
The Census 2010 Ranking of Priority Values shows the order in which seats 51–435 were apportioned after the 2010 Census, with additional listings for the next five priorities. Minnesota was allocated the final (435th) seat. North Carolina missed its 14th seat by 15,754 residents as the 436th seat to be allocated; ten years earlier it had gained its 13th seat as the 435th seat to be allocated based on the 2000 census.
Note: The first apportionment was established by the Constitution based on population estimates made by the Philadelphia Convention, and was not based on any census or enumeration.
Changes following the 2010 census
|Gain four||Gain two||Gain one||No change||Lose one||Lose two|
|1. Texas||1. Florida||1. Arizona
4. South Carolina
|(32 states)||1. Illinois
7. New Jersey
|1. New York|
|+12 seats gained total||−12 seats lost total|
List of apportionments
The size of the U.S. House of Representatives has increased and decreased as follows
|Effective date||Size||Change||Legal provision||Reason and/or comments|
|March 4, 1789||59||n/a||Const. Art. I, § 2, cl. 3||Seats apportioned by the Constitution|
|November 21, 1789||64||North Carolina ratified the Constitution with the seats apportioned by the Constitution|
|May 29, 1790||65||Rhode Island ratified the Constitution with the seat apportioned by the Constitution|
|March 4, 1791||67||1 Stat. 191||Vermont admitted|
|June 1, 1792||69||Kentucky admitted|
|March 4, 1793||105||1 Stat. 253 (Apportionment Act of 1792)||Apportionment following the First Census|
|June 1, 1796||106||1 Stat. 491||Tennessee admitted|
|March 1, 1803||107||2 Stat. 175||Ohio admitted.|
|March 4, 1803||142||2 Stat. 128||Apportionment following the Second Census.|
|April 30, 1812||143||2 Stat. 703||Louisiana admitted.|
|March 4, 1813||182||2 Stat. 669||Apportionment following the Third Census.|
|December 11, 1816||183||3 Stat. 290||Indiana admitted.|
|December 10, 1817||184||3 Stat. 349||Mississippi admitted.|
|December 3, 1818||185||3 Stat. 430||Illinois admitted.|
|December 14, 1819||186||3 Stat. 492||Alabama admitted.|
|March 15, 1820||3 Stat. 555||Maine admitted, 7 seats transferred from Massachusetts|
|August 10, 1821||187||3 Stat. 547||Missouri admitted|
|March 4, 1823||213||3 Stat. 651||Apportionment following the Fourth Census|
|March 4, 1833||240||4 Stat. 516||Apportionment following the Fifth Census|
|June 15, 1836||241||5 Stat. 51||Arkansas admitted|
|January 26, 1837||242||5 Stat. 50||Michigan admitted|
|March 4, 1843||223||5 Stat. 491||Apportionment following the Sixth Census, the only time the size of the House was reduced, except for the minor readjustments in 1863 and 1963.|
|March 3, 1845||224||5 Stat. 743||Florida admitted.|
|December 29, 1845||226||5 Stat. 798||Texas annexed and admitted.|
|December 28, 1846||228||5 Stat. 743
9 Stat. 52
|May 29, 1848||230||9 Stat. 58
9 Stat. 235
|March 4, 1849||231||9 Stat. 235||Additional seat apportioned to Wisconsin.|
|September 9, 1850||233||9 Stat. 452||California admitted.|
|March 4, 1853||233||9 Stat. 432||Apportionment following the Seventh Census.|
|234||10 Stat. 25||Additional seat apportioned to California|
|May 11, 1858||236||11 Stat. 166||Minnesota admitted.|
|February 14, 1859||237||11 Stat. 383||Oregon admitted.|
|January 29, 1861||238||12 Stat. 126||Kansas admitted|
|June 2, 1862||239||12 Stat. 411||California apportioned an extra seat|
|March 4, 1863||233||9 Stat. 432||Apportionment following the Eighth Census, in accordance with the 1850 act, which provided for an apportionment of 233 seats|
|241||12 Stat. 353||Supplemental apportionment of 8 seats (1 each for Pennsylvania, Ohio, Kentucky,
Illinois, Iowa, Minnesota, Vermont, and Rhode Island), for an overall increase of 2 seats in the 38th Congress
|June 20, 1863||12 Stat. 633||West Virginia admitted, three seats transferred from Virginia|
|October 31, 1864||242||13 Stat. 32||Nevada admitted|
|March 1, 1867||243||14 Stat. 391||Nebraska admitted|
|March 4, 1873||283||17 Stat. 28||Apportionment following the Ninth Census, replacing the 1850 act|
|292||17 Stat. 192||Supplemental apportionment added one seat each for nine states|
|August 1, 1876||293||13 Stat. 34||Colorado admitted|
|March 4, 1883||325||22 Stat. 5||Apportionment following the Tenth Census.|
|November 2, 1889||328||25 Stat. 679||North and South Dakota admitted, with one and two seats respectively.|
|November 8, 1889||329||25 Stat. 679||Montana admitted.|
|November 11, 1889||330||25 Stat. 679||Washington admitted.|
|July 3, 1890||331||26 Stat. 215||Idaho admitted.|
|July 10, 1890||332||26 Stat. 222||Wyoming admitted.|
|March 4, 1893||356||26 Stat. 735||Apportionment following the Eleventh Census.|
|January 4, 1896||357||28 Stat. 109||Utah admitted.|
|March 4, 1903||386||31 Stat. 733||Apportionment following the Twelfth Census (1900)|
|November 16, 1907||391||34 Stat. 271||Oklahoma admitted|
|January 6, 1912||393||37 Stat. 39, incorporating 36 Stat. 557||New Mexico admitted|
|February 14, 1912||394||37 Stat. 39, incorporating 36 Stat. 557||Arizona admitted|
|March 4, 1913||435||37 Stat. 13 (Apportionment Act of 1911, §§1–2)||Apportionment following the Thirteenth Census (1910)|
|March 4, 1933||435||46 Stat. 26 (Reapportionment Act of 1929)||Apportionment following the Fifteenth Census (1930)|
|January 3, 1943||435||46 Stat. 26 (Reapportionment Act of 1929)
54 Stat. 162
|Apportionment following the Sixteenth Census (1940)|
|January 3, 1953||435||55 Stat. 761||Apportionment following the Seventeenth Census|
|January 3, 1959||436||72 Stat. 345||Alaska admitted|
|August 21, 1959||437||73 Stat. 8, §8||Hawaii admitted|
|January 3, 1963||435||72 Stat. 345
73 Stat. 8
2 U.S.C. § 2a
|Apportionment following the Eighteenth Census|
|January 3, 1973||435||2 U.S.C. § 2a||Apportionment following the Nineteenth Census|
|January 3, 1983||435||2 U.S.C. § 2a||Apportionment following the Twentieth Census|
|January 3, 1993||435||2 U.S.C. § 2a||Apportionment following the Twenty-First Census|
|January 3, 2003||435||2 U.S.C. § 2a||Apportionment following the Twenty-Second Census|
|January 3, 2013||435||2 U.S.C. § 2a||Apportionment following the Twenty-Third Census|
- Delegate counts in italics represent temporary counts assigned by Congress until the next decennial census or by the U.S. Constitution in 1789 until the first U.S. Census.
- Elections held in the year of a census use the apportionment determined by the previous census.
- The 1850 Apportionment bill provided a method to be used in future reapportionments, as well as establishing the then-current 233 as the number of seats to be apportioned after future censuses. Due to census returns being incomplete in California, an additional act provided that California retain the same representation it had when admitted, until a new census could be taken. California would otherwise have lost one seat, and so the total number of seats was increased by one to 234.
- Congress failed to reapportion in 1923, following the Fourteenth Census (1920).
- Pub.L. 77–291 amended section 22 of the Reapportionment Act of 1929 by wholly replacing its text.
- The Reapportionment Act of 1929 stated that the "then existing number of Representatives" would be apportioned after each census, which would have dictated an apportionment of 437 seats, but the Alaska Statehood Act and Hawaii Admission Act explicitly stated that the new seats were temporary increases. Both acts included the phrasing "That such temporary increase in the membership shall not operate to either increase or decrease the permanent membership of the House of Representatives as prescribed in the Act of August 8, 1911 (37 Stat. 13) nor shall such temporary increase affect the basis of apportionment established by the Act of November 15, 1941 (55 Stat. 761; 2 U.S.C. § 2a), for the Eighty-third Congress and each Congress thereafter."
- Kristin D. Burnett (November 1, 2011). "Congressional Apportionment (2010 Census Briefs C2010BR-08)" (PDF). U.S. Department of Commerce, Economics and Statistics Administration. Retrieved February 25, 2015.
- The populations of Washington, D.C. and federal territories are not included in this figure.
- Public Law 62-5 of 1911.
- Rendered moot by the Revenue Act of 1924 and Indian Citizenship Act of 1924.
- 2 U.S.C. § 2c
- Bush signs federalization bill Archived 2009-02-13 at the Wayback Machine, Agnes E. Donato, Saipan Tribune, May 10, 2008.
- Goldberg, Jonah (January 15, 2001). "George Will Called Me An Idiot". National Review. Archived from the original on February 13, 2009. Retrieved April 11, 2018.
- Madison's notes on the Constitutional Convention - Tuesday September 17, 1787
- The Federalist #55
- "Fair Representation, Meeting The Ideal of One Man One vote" - Michel Balinski and H. Peyton Young -- Page 51
- "Archived copy". Archived from the original on February 28, 2009. Retrieved February 15, 2009.CS1 maint: archived copy as title (link)
- "Apportionment of Representatives in Congress". CQ Researcher by CQ Press. ISSN 1942-5635.
- "Proportional Representation". Washington, D.C.: Office of the Historian, United States House of Representatives. Retrieved September 21, 2018.
- "Constitutional Amendments Not Ratified". United States House of Representatives. Archived from the original on September 27, 2007. Retrieved September 30, 2007.
- "The "Cube Root Rule": A Push to Make Congress More Representative?". IVN. Independent Voter Network. Retrieved May 31, 2019.
- House of Representatives? Hardly., Alcee Hastings, May 21, 2001.
- Marimow, Ann E.; Pershing, Ben (April 21, 2010). "Congressional leaders shelve D.C. voting rights bill". The Washington Post.
- 3 Annals of Cong. 539 (1792)
- Act of Jan. 14, 1802, 2 Stat. 128
- Act of Dec. 21, 1811, 2 Stat. 669
- Act of Mar. 7, 1822, 3 Stat. 651
- Act of May 22, 1832, 4 Stat. 516
- Act of 25 June 1842, ch 46, 5 Stat. 491
- Act of May 23, 1850, 9 Stat. 432-433
- Act of 1862, 12 Stat. 572
- Act of 1872, 17 Stat. 28
- Act of 1882, 22 Stat. 5
- Act of 1891
- Act of 1901, 31 Stat. 733
- "Congressional Apportionment-Historical Perspective". U.S. Census Bureau. Retrieved October 27, 2013..
- "The History of Apportionment in America". American Mathematical Society. Retrieved February 15, 2009.
- "2 USC §2a". Cornell University Law School, Legal Information Institute. Retrieved May 13, 2008.
- "Computing Apportionment". U.S. Census Bureau. Retrieved February 14, 2009.
- Edward V Huntington (1921). "The Mathematical Theory of the Apportionment of Representatives". Proceedings of the National Academy of Sciences of the United States of America. 7 (4): 123–7. doi:10.1073/pnas.7.4.123. PMC 1084767. PMID 16576591.
- "Congressional Apportionment". NationalAtlas.gov. U.S. Department of the Interior. Archived from the original on October 30, 2008. Retrieved February 14, 2009.
- "PRIORITY VALUES FOR 2010 CENSUS" (PDF). U.S. Bureau of the Census. Retrieved June 7, 2012.
- "Census 2000 Ranking of Priority Values". U.S. Bureau of the Census. February 21, 2001. Retrieved May 13, 2008.
- Congress failed to pass any reapportionment to implement the 1920 United States Census so despite population shift, distribution of seats from 1913 remained in effect until 1933.
- "APPORTIONMENT POPULATION AND NUMBER OF REPRESENTATIVES, BY STATE: 2010 CENSUS" (PDF). US Census. December 21, 2010. Retrieved February 23, 2013.
- The Size of the U. S. House of Representatives and its Constituent State Delegations, thirty-thousand.org.
- See, e.g., section 8 of the Hawaii Admission Act, 73 Stat. 8.
- Balinski, Michael L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven, CT: Yale University Press. ISBN 0-8157-0090-3.
- Foster, Robert (1895). Commentaries on the Constitution of the United States: Historical and Judicial. 1. Boston: The Boston Book Co. pp. 329–446.
- Hamilton, Alexander; Madison, James; Jay, John (1831). The Federalist. Hallowell: Glazier, Masters & Co. ISBN 0-8239-5735-7.
- Edelman, Paul H. (2006). "Getting the Math Right: Why California Has Too Many Seats in the House of Representatives". Vanderbilt Law Review. Nashville: Vanderbilt University. 102 (March): 297.
- Kromkowski, Charles A.; Kromkowski, John A. (1991). "Why 435? A Question of Political Arithmetic" (PDF). Polity. 24 (Fall 1991): 129–145. doi:10.2307/3234988. JSTOR 3234988. Retrieved October 17, 2013.
- Agnew, Robert A. (2008). "Optimal Congressional Apportionment" (PDF). American Mathematical Monthly. Mathematical Association of America. 115 (April): 297–303. JSTOR 27642473.
- Stinebrickner-Kauffman, Taren (2004). "Counting Matters: Prison Inmates, Population Bases, and "One Person, One Vote"". Virginia Journal of Social Policy & the Law. Chicago: Virginia Journal of Social Policy & the Law. 11 (Winter): 229.