“An Absurdity and Contradiction”
Article the first’s Mysterious Defect
As explained in Section Two of this web pamphlet, the purpose of Article the first of the Bill of Rights was to establish a minimum number of Representatives to complement the maximum number already established by the Constitution. As it turns out, there are two versions of “Article the first”. The first version is the one proposed by the House, which we call the “intended version”. And the second version is the one submitted to the states which, for reasons explained below, we call the “defective version”. Though it is this defective version that was ultimately rejected by the states, historians usually mistake that one for the nondefective version proposed by the House, which is not surprising given that they appear to be identical.
This section explains, mathematically, how both of these versions would work. In so doing, it also shows why the version which was sent to the states could not have been intended by Congress.
Both versions consist of a threetier structure, each tier having its own arithmetical formulation. The intended version will be explained first, followed by an explanation of the defective version.
The intended version of the first Article passed by the House on August 21, 1789, is as follows:
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 less than one Representative for every fifty thousand persons.
Because it is densely worded, the first Article can best be understood by parsing it into its three tiers which, in this analysis, are labeled Tier 1, Tier 2, and Tier 3. As explained herein, the nation’s total population (per the decennial census) determines which of these three formulations is applicable, as follows:
 Tier 1 applies to all population levels less than or equal to three million.
 Tier 2 applies to all population levels greater than three million, but less than or equal to eight million.
 Tier 3 applies to all population levels greater than eight million.
This is summarized in the table below:
Tier  1  2  3  

Applicable population range  Tier 1 ≤ 3,000,000  3,000,000 < Tier 2 ≤ 8,000,000  8,000,000 < Tier 3  
Maximum number of Reps per the Constitution  One for every 30,000. 
Also indicated in the table is the maximum number of Representatives as specified by the Constitution (1:30,000^{1}Per the Constitution: “The number of Representatives shall not exceed one for every thirty Thousand”). This is provided as a point of reference, since the purpose of Article the first was to complement this maximum by establishing a corresponding minimum.
A full explanation of each of these three tiers is provided below.
The Tier 1 Formulation
The Tier 1 formulation, highlighted below, is identical in both versions of 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 less than one Representative for every fifty thousand persons.
Tier 1 stipulates that, if ratified, this amendment would go into effect after the first population census is conducted (which was to commence the following year). Its formulation is simple: “there shall be one Representative for every thirty thousand” until there are 100. Therefore, for any total population below three million (100 × 30,000), the number of Representatives will be exactly equal to that total divided by 30,000, as summarized below.
Tier  1  2  3  

Applicable population range  Tier 1 ≤ 3,000,000  3,000,000 < Tier 2 ≤ 8,000,000  8,000,000 < Tier 3  
Maximum number of Reps per the Constitution  1:30,000.  
Minimum number of Reps per first Article 
1:30,000

Note that the Tier 1 formulation is identical to the Constitution’s maximum House size (of 1:30,000), and so its effect would have been to convert the Constitution’s ratio from a maximum size to an absolute one. They did not appear to anticipate the possibility that such an absolute calculation might result in a fractional remainder to be rounded (e.g. 8.5). However, all indications are that, by the time this proposal was drafted, they already expected that the nation’s total population would exceed three million, thereby mooting the Tier 1 formulation altogether.
The Tier 2 and Tier 3 formulations will avoid any rounding controversy by allowing for a range of values, rather than requiring a specific value.
The Tier 2 Formulation
The Tier 2 formulation, highlighted below, is identical for both versions of 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 less than one Representative for every fifty thousand persons.
Tier 2 goes into effect after the total population exceeds Tier 1’s upper limit of 100 Representatives, “after which” there must be at least 100, or no less than one Representative for every 40,000 persons, whichever is greater. The upper limit of Tier 2 is reached once there are 200 Representatives, which would happen when the population totals 8,000,000 (200 × 40,000). This is summarized below.
Tier  1  2  3  

Applicable population range  Tier 1 ≤ 3,000,000  3,000,000 < Tier 2 ≤ 8,000,000  8,000,000 < Tier 3  
Maximum number of Reps per the Constitution  1:30,000.  
Minimum number of Reps per first Article  1:30,000 
The greater of 1:40,000 or 100.

Note that Tier 2 supplants Tier 1; it does not augment it. How this works can best be illustrated with real world examples: The first three population censuses (1790, 1800, and 1810). All of them fell into the range between three and eight million.
The very first census, in 1790, tallied a total apportionment population of approximately 3.6 million. Therefore, the first apportionment would skip Tier 1 to be governed only by Tier 2. Based on this census, and pursuant to Tier 2 of the first Article, the minimum size of the House would be 100. This would require that Congress choose any number between 100 and the constitutional maximum of 120^{2}3,615,920 ÷ 30,000 = 120.5 → Round down to 120. (inclusive). In 1792, the second Congress met this requirement by authorizing 105 Representatives, though they had originally proposed 120.
The 1800 census tallied an apportionment population total of 4.9 million, resulting in a minimum House size of 123,^{3}4,889,823 ÷ 40,000 = 122.3 → Round up to 123. with a corresponding maximum of 162.^{4}4,889,823 ÷ 30,000 = 162.9 → Round down to 162. In 1802, the seventh Congress authorized 141 (again meeting the requirement).
And in 1810, the census tallied an apportionment population of 6.6 million, resulting in a minimum House size of 165,^{5}6,584,255 ÷ 40,000 = 164.6 → Round up to 165. with a corresponding maximum of 219.^{6}6,584,255 ÷ 30,000 = 219.5 → Round down to 219. In 1811, the 12^{th} Congress authorized 181 (again meeting the requirement).
Note that all three apportionments complied with Article the first even though it had not been ratified.
The Tier 3 Formulation
This is where the analysis of Article the first becomes interesting: Relative to Tier 3, there is a subtle but profound difference between the intended version passed by the House, and the defective version that was sent to the states.
The Intended Version
The intended version of Tier 3, as proposed by the House, is highlighted below.
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 less than one Representative for every fifty thousand persons.
For any population above Tier 2’s upper limit of 200 Representatives, Tier 3 requires that the minimum number of Representatives be no less than 200, or no less than one Representative for every 50,000 people, whichever is greater, as summarized below.
Tier  1  2  3  

Applicable population range  Tier 1 ≤ 3,000,000  3,000,000 < Tier 2 ≤ 8,000,000  8,000,000 < Tier 3  
Maximum number of Reps per the Constitution  1:30,000.  
Minimum number of Reps per first Article  1:30,000  The greater of 1:40,000 or 100. 
The greater of 1:50,000 or 200.

The effect of Tiers 2 and 3 are illustrated in the chart below. Tier 1 can be disregarded for this analysis.^{7}Tier 1 is simply equal to the Constitution’s formulation for all House sizes less than or equal to three million, which was never applicable for any apportionment.
For any given decennial apportionment, the intended first Article would require Congress to select a number of Representatives in the shaded area (between the minimum and maximum lines) based on the nation’s total population. With respect to Tier 3, how it works can best be illustrated with real world examples: The fourth and fifth population censuses (1820 and 1830).
With a total population of nearly nine million, the 1820 census was the first to be subject to the Tier 3 formulation. In fact, every subsequent apportionment would be subject to this formulation, as there is no upper limit on this final tier. As a result of the 1820 census, the minimum House size allowed was 180.^{8}8,969,878 ÷ 50,000 = 179.4 → Round up to 180. The corresponding maximum size was 289.^{9}8,969,878 ÷ 30,000 = 289.9 → Round down to 289. Therefore, Congress could have chosen any size between 180 and 289 (inclusive). In 1822, the 17^{th} Congress chose 213, in compliance with the intended formulation.
And in 1830, the census tallied an apportionment population of 11.9 million, resulting in a minimum House size of 239.^{10}11,931,000 ÷ 50,000 = 238.6 → Round up to 239. The corresponding maximum size was 397.^{11}11,931,000 ÷ 30,000 = 397.7 → Round down to 397. Therefore, in 1832, the 22^{nd} Congress could have chosen any size between 239 and 397 (inclusive) in order to be in compliance. They chose 240.
Note that in both cases, Congress complied with Article the first even though it had not been ratified.
The Defective Version
Tier 3 of the defective version is highlighted below.
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 less more than one Representative for every fifty thousand persons.
The defective version is absolutely identical to the intended version except that the final instance of the word “less” was incongruously replaced with “more”. It is this alteration that has almost always been overlooked by historians and scholars, perhaps because it breaks the provision’s natural cadence in such a way as to elude perception.^{12}The cadence, or pattern, of the intended proposal, beyond Tier 1, is: “shall be not less than … nor less than” … “not be less that … nor less than”, with four instances of the word “less”. Perhaps the mind continues to perceive this same logical pattern even when the final “less” is replaced by “more”. In any case, even if it is detected by the reader, a literal interpretation of this defective version creates such a patently absurd outcome that it is impossible to imagine that such a result could have been deliberately intended, as indeed it was not.
The effect of this alteration was to obliterate the intended version’s thirdtier formulation by splitting it into two contradictory components, as shown in the table below.
Tier  1  2  3  

Applicable population range  Tier 1 ≤ 3,000,000  3,000,000 < Tier 2 ≤ 8,000,000  8,000,000 < Tier 3  
Maximum number of Reps per the Constitution  1:30,000.  
Defective first Article  Maximum 
1:50,000


Minimum  1:30,000  The greater of 1:40,000 or 100. 
200

By substituting “more” for “less”, the 1:50,000 formulation that was intended to be the House’s minimum size would have become its maximum (indicated by the ① in the chart below). For all total population levels above eight million, this would have needlessly contradicted the formulation already established by the Constitution and touted numerous times in the Federalist Papers. In addition, as a consequence of this alteration, the transitional minimum of 200 would continue as the permanent minimum (indicated by ② in the chart below). This results in an absurdity, and then yet another contradiction, as explained with a simple example.
At a total population of eight million, Tier 2 requires that the minimum number of Representatives be 200. However, if the total population increases by just a single person (to 8,000,001), then Tier 3 becomes applicable. At this point, something absurd happens. Because there can not be “more than one Representative for every fifty thousand persons”, the total number of Representatives would be reduced from 200 to 160^{13}8,000,001 ÷ 50,000 = 160.00002 → round down to 160. simply by adding a single person! (This is indicated by the ③ in the chart below.)
In addition, because this provision also requires the minimum number of Representatives to be 200, this defective language would mandate a maximum size (160) which is less than that required minimum! This, of course, is an impossibility (indicated by the ④ in the chart below). This contradiction between the mandatory maximum (of 1:50,000) and the mandatory minimum (of 200) continues until the total population reaches ten million, at which point the two values are equal. Thereafter, for every population size above ten million, the maximum size would exceed 200.
The effect of this preposterous formulation is illustrated in the chart below.
After this proposal was sent to the states, its defective language was evidently detected over time. For example, when the first Article was being considered by the Pennsylvania state legislature in 1791, a member observed that the proposal bore an “absurdity and contradiction”, and proceeded to elaborate with a mathematical proof. It cannot be doubted that this “absurdity”, plus the inconsequentiality of a minimum House size of 200, is what eventually led to the abandonment of the defective proposal.
In order to illustrate the effect of this defective wording, the examples of the 1820 and 1830 censuses will be repeated.
Relative to the 1820 census, this provision would have produced an impossible solution: The maximum size would have been less than the required minimum! Per this census, the total population was approximately nine million. As a result of the specious “maximum” formulation of 1:50,000, the maximum House size would have been 179,^{14}8,969,878 ÷ 50,000 = 179.4 → Round down to 179. which is less than the minimum of 200. As stated above, Congress chose 213, in compliance with the intended version of the first Article.
Because the 1830 population census exceeded ten million, the defective version would have produced a feasible solution (in that the maximum value is greater than the minimum). This census tallied an apportionment population of 11.9 million. Pursuant to the defective version of Article the first, the minimum size would be 200, while the maximum size would be 238.^{15}11,931,000 ÷ 50,000 = 238.6 → Round down to 238. Therefore, under the defective version that was sent to the states, the 22^{nd} Congress could have chosen any House size between 200 and 238, but they didn’t. Instead, as stated above, they chose 240, a number in compliance with the intended version, and in violation of this defective version!
It is worth reiterating that even though neither version of the first Article had been ratified, the first three apportionments complied with Tier 2 of both versions (as they were identical). More importantly, for the next two apportionments, which were subject to the Tier 3 formulation, Congress complied with the intended version, and blatantly disregarded the defective version that had been sent to the states as part of the Bill of Rights.
Did Congress deliberately alter the first Article?
Some may imagine that twothirds of the members of both chambers of Congress changed their minds at the last minute and, without any debate, deliberately created the version that was sent to the states. However, there are several significant problems with that theory. First, had that been the case, it is unimaginable that not one of them would have detected the resulting defect in its formulation. In addition to being very well educated, many of these gentlemen had years of professional experience that would have further honed their mathematical abilities. Nor would twothirds of both chambers have ever agreed to knowingly send such a defective proposal to the state legislatures, if only because of the embarrassment that would result.
Second, had it truly been Congress’s intent to simply replace the proportional minimum (of 1:50,000) with a fixed minimum of 200, then they would have simply deleted the final ten words as follows:
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 less than one Representative for every fifty thousand persons.
This deletion would have established the fixed minimum size without contradicting the maximum size already established by the Constitution. In fact, had that been their goal, there would be absolutely no reason for not making this alteration. Because this modification would have avoided the irresolvable math problem, it would have produced feasible solutions at every population size. Moreover, it seems possible that such a proposal could have been ratified by the states even though it did not ensure the level of representation that many had believed necessary.
Alternatively, could Congress have deliberately made this alteration, without realizing the resulting defect, in order to reduce the maximum size of the House from 1:30,000 to 1:50:000? Such an outcome would have been a complete non sequitur to everything that preceded it: Never in the debates that led up to the creation of the first Article was there any broad support for overriding the maximum size already established by the Constitution and repeatedly promised in the Federalist Papers.
And finally, had this version of the first Article truly been intentional, then Congress’s fifth apportionment (based on the 1830 census) would have complied with it, rather than conspicuously violating it (in order to comply with the intended version instead). Across the ages, the 22^{nd} congress is telling us what was truly intended.
In any case, the important point to understand is that it was the defective version of the first Article that ultimately failed to be ratified, even though it was twice only one state short of the required number to be ratified (thanks largely to the fact that its defect was initially undetected). Because so many states had demanded such an amendment, had the intended (nondefective) version been sent to the states, it would have undoubtedly been ratified.
©ThirtyThousand.org [Article published 03/21/22; updated 05/09/22.]
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 1Per the Constitution: “The number of Representatives shall not exceed one for every thirty Thousand”
 23,615,920 ÷ 30,000 = 120.5 → Round down to 120.
 34,889,823 ÷ 40,000 = 122.3 → Round up to 123.
 44,889,823 ÷ 30,000 = 162.9 → Round down to 162.
 56,584,255 ÷ 40,000 = 164.6 → Round up to 165.
 66,584,255 ÷ 30,000 = 219.5 → Round down to 219.
 7Tier 1 is simply equal to the Constitution’s formulation for all House sizes less than or equal to three million, which was never applicable for any apportionment.
 88,969,878 ÷ 50,000 = 179.4 → Round up to 180.
 98,969,878 ÷ 30,000 = 289.9 → Round down to 289.
 1011,931,000 ÷ 50,000 = 238.6 → Round up to 239.
 1111,931,000 ÷ 30,000 = 397.7 → Round down to 397.
 12The cadence, or pattern, of the intended proposal, beyond Tier 1, is: “shall be not less than … nor less than” … “not be less that … nor less than”, with four instances of the word “less”. Perhaps the mind continues to perceive this same logical pattern even when the final “less” is replaced by “more”.
 138,000,001 ÷ 50,000 = 160.00002 → round down to 160.
 148,969,878 ÷ 50,000 = 179.4 → Round down to 179.
 1511,931,000 ÷ 50,000 = 238.6 → Round down to 238.