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Math for Liberal Studies

The US Senate has 100 members: two for each state

In the US House of Representatives, states are represented based on population

PA has 19 representatives

Delaware has 1

The process by which seats are assigned based on population is called apportionment

The number of seats each state gets is also called that state’s apportionment

Consider a fictional country with four states and a representative legislature with 50 seats

State Population

Angria 80,000

Bretonnia 60,000

Curaguay 40,000

Dennenberg 20,000

Total 200,000

Each state should get a proportion of the seats that is equal to its proportion of the total population

State Population

Angria 80,000

Bretonnia 60,000

Curaguay 40,000

Dennenberg 20,000

Total 200,000

Divide each state’s population by the total population to get the % population

State Population % Pop.

Angria 80,000 40%

Bretonnia 60,000 30%

Curaguay 40,000 20%

Dennenberg 20,000 10%

Total 200,000 100%

Multiply that percentage by the total number of seats (in this case 50) to get each state’s fair share of seats

State Population % Pop. Fair Share

Angria 80,000 40% 20

Bretonnia 60,000 30% 15

Curaguay 40,000 20% 10

Dennenberg 20,000 10% 5

Total 200,000 100% 50

Real world examples rarely work out as nicely as the previous example did

Let’s use more realistic population numbers and see what happens

We will start the problem in the same way

State Population

Angria 83,424

Bretonnia 67,791

Curaguay 45,102

Dennenberg 17,249

Total 213,566

Divide each state’s population by the total population to get the % population

State Population % Pop.

Angria 83,424 39.06%

Bretonnia 67,791 31.74%

Curaguay 45,102 21.12%

Dennenberg 17,249 8.08%

Total 213,566 100%

Multiply that percentage by the total number of seats (in this case 50) to get each state’s fair share of seats


Angria 83,424 39.06% 19.53

Bretonnia 67,791 31.74% 15.87

Curaguay 45,102 21.12% 10.56

Dennenberg 17,249 8.08% 4.04

Total 213,566 100% 50

The problem is that we can’t assign a state 19.53 seats… each apportionment must be a whole number!


Angria 83,424 39.06% 19.53

Bretonnia 67,791 31.74% 15.87

Curaguay 45,102 21.12% 10.56

Dennenberg 17,249 8.08% 4.04

Total 213,566 100% 50

We could try rounding each fair share to the nearest whole number, but we end up with 51 seats, which is more than we have!

State Population % Pop. Fair Share Seats

Angria 83,424 39.06% 19.53 20

Bretonnia 67,791 31.74% 15.87 16

Curaguay 45,102 21.12% 10.56 11

Dennenberg 17,249 8.08% 4.04 4

Total 213,566 100% 50 51

Article 1, Section 2 (1787)…Representatives and direct taxes shall be

apportioned among the several States which may be included within this Union, according to their respective Numbers …

Article 1, Section 2 (continued)…The actual Enumeration shall be made

within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years…

Article 1, Section 2 (continued)…The Number of Representatives shall not

exceed one for every thirty Thousand, but each State shall have at Least one Representative;

State Population

Virginia 630,560

Massachusetts 475,327

Pennsylvania 432,879

North Carolina 353,523

New York 331,589

Maryland 278,514

Connecticut 236,841

South Carolina 206,236

New Jersey 179,570

New Hampshire 141,822

Vermont 85,533

Georgia 70,835

Kentucky 68,705

Rhode Island 68,446

Delaware 55,540

Total 3,615,920

How many Representatives?

State Population

Virginia 630,560




New York 331,589

Maryland 278,514

Connecticut 236,841


New Jersey 179,570


Vermont 85,533

Georgia 70,835

Kentucky 68,705

Rhode Island 68,446

Delaware 55,540

Total 3,615,920

120000,30

920,615,3


State Population

Virginia 630,560




New York 331,589

Maryland 278,514

Connecticut 236,841


New Jersey 179,570


Vermont 85,533

Georgia 70,835

Kentucky 68,705

Rhode Island 68,446

Delaware 55,540

Total 3,615,920

120000,30

920,615,3


How many representatives should each state receive?

State Population Quota

Virginia 630,560 20.93

Massachusetts 475,327 15.78

Pennsylvania 432,879 14.37

North Carolina 353,523 11.73

New York 331,589 11.00

Maryland 278,514 9.24

Connecticut 236,841 7.86

South Carolina 206,236 6.84

New Jersey 179,570 5.96

New Hampshire 141,822 4.71

Vermont 85,533 2.84

Georgia 70,835 2.35

Kentucky 68,705 2.28

Rhode Island 68,446 2.27

Delaware 55,540 1.84

Total 3,615,920 120

How many representatives should each state receive?

A “Fair” Share (or Quota) =

SeatsTotalationTotalPopul

ationStatePopul#

State Population Quota

Virginia 630,560 20.93

Massachusetts 475,327 15.78

Pennsylvania 432,879 14.37

North Carolina 353,523 11.73

New York 331,589 11.00

Maryland 278,514 9.24

Connecticut 236,841 7.86

South Carolina 206,236 6.84

New Jersey 179,570 5.96

New Hampshire 141,822 4.71

Vermont 85,533 2.84

Georgia 70,835 2.35

Kentucky 68,705 2.28

Rhode Island 68,446 2.27

Delaware 55,540 1.84

Total 3,615,920 120

=

Named for Alexander Hamilton, the first U.S. Secretary of the Treasury

Hamilton’s method gives eachstate it’s lower quota (roundeddown shares) at a minimum.

If extra seats exist, they are distributed according to the sizeof each state’s fractionalremainder.

With Hamilton’s method, each state will always receive either their lower quota or their upper quota: this is the quota rule

If a state is apportioned a number of representatives that is either above its upper quota or below its lower quota, then that is a quota rule violation

State Population Quota Round Priority

Virginia 630,560 20.93 20

Massachusetts 475,327 15.78 15

Pennsylvania 432,879 14.37 14

North Carolina 353,523 11.73 11

New York 331,589 11.00 11

Maryland 278,514 9.24 9

Connecticut 236,841 7.86 7

South Carolina 206,236 6.84 6

New Jersey 179,570 5.96 5 1

New Hampshire 141,822 4.71 4

Vermont 85,533 2.84 2

Georgia 70,835 2.35 2

Kentucky 68,705 2.28 2

Rhode Island 68,446 2.27 2

Delaware 55,540 1.84 1

Total 3,615,920 120 111


Virginia 630,560 20.93 20 2




New York 331,589 11.00 11

Maryland 278,514 9.24 9



New Jersey 179,570 5.96 5 1


Vermont 85,533 2.84 2

Georgia 70,835 2.35 2

Kentucky 68,705 2.28 2


Delaware 55,540 1.84 1

Total 3,615,920 120 111


Virginia 630,560 20.93 20 2

Massachusetts 475,327 15.78 15 7


North Carolina 353,523 11.73 11 8

New York 331,589 11.00 11

Maryland 278,514 9.24 9

Connecticut 236,841 7.86 7 3

South Carolina 206,236 6.84 6 4

New Jersey 179,570 5.96 5 1

New Hampshire 141,822 4.71 4 9

Vermont 85,533 2.84 2 6

Georgia 70,835 2.35 2

Kentucky 68,705 2.28 2


Delaware 55,540 1.84 1 5

Total 3,615,920 120 111

State Population Quota Round Priority Seats

Virginia 630,560 20.93 20 2 21

Massachusetts 475,327 15.78 15 7 16

Pennsylvania 432,879 14.37 14 14

North Carolina 353,523 11.73 11 8 12

New York 331,589 11.00 11 11

Maryland 278,514 9.24 9 9

Connecticut 236,841 7.86 7 3 8

South Carolina 206,236 6.84 6 4 7

New Jersey 179,570 5.96 5 1 6

New Hampshire 141,822 4.71 4 9 5

Vermont 85,533 2.84 2 6 3

Georgia 70,835 2.35 2 2

Kentucky 68,705 2.28 2 2

Rhode Island 68,446 2.27 2 2

Delaware 55,540 1.84 1 5 2

Total 3,615,920 120 111 120

State Population Quota Round Priority Seats People/Rep.

Virginia 630,560 20.93 20 2 21 30,027

Massachusetts 475,327 15.78 15 7 16 29,708

Pennsylvania 432,879 14.37 14 14 30,920

North Carolina 353,523 11.73 11 8 12 29,460

New York 331,589 11.00 11 11 30,144

Maryland 278,514 9.24 9 9 30,946

Connecticut 236,841 7.86 7 3 8 29,605

South Carolina 206,236 6.84 6 4 7 29,462

New Jersey 179,570 5.96 5 1 6 29,928

New Hampshire 141,822 4.71 4 9 5 28,364

Vermont 85,533 2.84 2 6 3 28,511

Georgia 70,835 2.35 2 2 35,418

Kentucky 68,705 2.28 2 2 34,353

Rhode Island 68,446 2.27 2 2 34,223

Delaware 55,540 1.84 1 5 2 27,220

Total 3,615,920 120 111 120

“There is no one proportion or divisor which, applied to the respective numbers of the States will yield the number and allotment of representatives proposed by the Bill.”

When we divide the total population by the number of seats, we get the standard divisor.

Hamilton’s method for apportionment of the House of Representatives results in:

standard divisor = 132,30120

000,615,3

State Population Quota Round Priority Seats People/Rep.

Virginia 630,560 20.93 20 2 21 30,027

Massachusetts 475,327 15.78 15 7 16 29,708

Pennsylvania 432,879 14.37 14 14 30,920

North Carolina 353,523 11.73 11 8 12 29,460

New York 331,589 11.00 11 11 30,144

Maryland 278,514 9.24 9 9 30,946

Connecticut 236,841 7.86 7 3 8 29,605

South Carolina 206,236 6.84 6 4 7 29,462

New Jersey 179,570 5.96 5 1 6 29,928

New Hampshire 141,822 4.71 4 9 5 28,364

Vermont 85,533 2.84 2 6 3 28,511

Georgia 70,835 2.35 2 2 35,418

Kentucky 68,705 2.28 2 2 34,353

Rhode Island 68,446 2.27 2 2 34,223

Delaware 55,540 1.84 1 5 2 27,220

Total 3,615,920 120 111 120

Thomas Jefferson, 1st Secretary of State, 3rd President.

Jefferson’s idea was to modify the standard divisor so that when the shares for each state are rounded down, there are no leftover seats.

State Population Quota Round

Virginia 630,560 20.93 20




New York 331,589 11.00 11

Maryland 278,514 9.24 9



New Jersey 179,570 5.96 5


Vermont 85,533 2.84 2

Georgia 70,835 2.35 2

Kentucky 68,705 2.28 2


Delaware 55,540 1.84 1

Total 3,615,920 120 111

A 120 seat Congress implies standard divisor

Making the divisor smaller will increase each state’s share.

132,30120

000,615,3

State Population Share Round

Virginia 630,560 22.12 22




New York 331,589 11.63 11

Maryland 278,514 9.77 9



New Jersey 179,570 6.3 6


Vermont 85,533 3 3

Georgia 70,835 2.49 2

Kentucky 68,705 2.41 2


Delaware 55,540 1.95 1

Total 3,615,920 120

Modified divisor = 28,500 works!


Virginia 630,560 22.12 22




New York 331,589 11.63 11

Maryland 278,514 9.77 9



New Jersey 179,570 6.3 6


Vermont 85,533 3 3

Georgia 70,835 2.49 2

Kentucky 68,705 2.41 2


Delaware 55,540 1.95 1

Total 3,615,920 120

Modified divisor = 28,500 works!

But it’s unconstitutional


Virginia 630,560 18.31 18




New York 331,589 9.63 9

Maryland 278,514 8.09 8



New Jersey 179,570 5.21 5


Vermont 85,533 2.48 2

Georgia 70,835 2.06 2

Kentucky 68,705 2 1


Delaware 55,540 1.61 1

Total 3,615,920 105 97

A 105 seat Congress implies standard divisor

As before, making the divisor smaller will increase each state’s share.

437,34105

000,615,3


Virginia 630,560 19.11 19




New York 331,589 10.05 10

Maryland 278,514 8.44 8



New Jersey 179,570 5.44 5


Vermont 85,533 2.59 2

Georgia 70,835 2.15 2

Kentucky 68,705 2.08 2


Delaware 55,540 1.68 1

Total 3,615,920 105

Modified divisor = 33,000 works! (and it’s > 30,000)

State Population Jefferson Hamilton

Virginia 630,560 19 18

Massachusetts 475,327 14 14

Pennsylvania 432,879 13 12

North Carolina 353,523 10 10

New York 331,589 10 10

Maryland 278,514 8 9

Connecticut 236,841 7 7

South Carolina 206,236 6 6

New Jersey 179,570 5 5

New Hampshire 141,822 4 4

Vermont 85,533 2 2

Georgia 70,835 2 2

Kentucky 68,705 2 2

Rhode Island 68,446 2 2

Delaware 55,540 1 2

Total 3,615,920 105 105

State Population Jefferson Hamilton

Virginia 630,560 19 18

Massachusetts 475,327 14 14

Pennsylvania 432,879 13 12

North Carolina 353,523 10 10

New York 331,589 10 10

Maryland 278,514 8 9

Connecticut 236,841 7 7

South Carolina 206,236 6 6

New Jersey 179,570 5 5

New Hampshire 141,822 4 4

Vermont 85,533 2 2

Georgia 70,835 2 2

Kentucky 68,705 2 2

Rhode Island 68,446 2 2

Delaware 55,540 1 2

Total 3,615,920 105 105

Jefferson’s hometown = Monticello, VA

Let’s look at a fictional country with four states.

State Population

Biglandia 96,974

Medios 45,902

Regula 44,921

Smallirus 32,764

Totals 220,561

This time the standard divisor is 220,561/50 = 4,411.2


Biglandia 96,974 21.98 21

Medios 45,902 10.41 10

Regula 44,921 10.18 10

Smallirus 32,764 7.43 7

Totals 220,561 50 48

We have leftover seats, so Jefferson’s method tells us to make the divisor smaller


Biglandia 96,974 21.98 21

Medios 45,902 10.41 10

Regula 44,921 10.18 10

Smallirus 32,764 7.43 7

Totals 220,561 50 48

After some trial and error, we find that a divisor of 4,200 works.


Biglandia 96,974 23.09 23

Medios 45,902 10.93 10

Regula 44,921 10.70 10

Smallirus 32,764 7.80 7

Totals 220,561 50

The top chart uses the standard divisor (4,411)

The bottom chart is our Jefferson’s method solution

We have a quota rule violation!


Biglandia 96,974 21.98 21

Medios 45,902 10.41 10

Regula 44,921 10.18 10

Smallirus 32,764 7.43 7

Totals 220,561 50 48

State PopulationModified

ShareRound

Biglandia 96,974 23.09 23

Medios 45,902 10.93 10

Regula 44,921 10.70 10

Smallirus 32,764 7.80 7

Totals 220,561 50

As we just witnessed, Jefferson’s method can violate the quota rule.

This is because lowering the divisor causes the quotient to grow at a faster rate if the numerator is higher.

Thus, this method tends to favor larger states over smaller ones.

John Quincy Adams James Dean Daniel Webster6th President Mathematician 2-time Secretary of State

This method is the same as Jefferson’s method, except you always round the shares up instead of down

This gives you too many seats, so you need to increase the divisor until you get the right number of seats

Let each state’s quota = State Population/Divisor Let n = the rounded down quota. Calculate the harmonic mean of n and n+1

If a state’s quota exceeds the harmonic mean – give the state n+1 representatives. Otherwise, give it n.

Adjust the divisor as necessary to match number of Representatives to the House.

5.0

)1(

n

nn

In this method, we round the shares to the nearest whole number

It’s possible that this gives us the right number of seats, in which case we’re done

Otherwise, we need to increase (if we have too few seats) or decrease (if we have too many seats) the divisor

State Population Rank Adams Dean Webster Jefferson

NY 1,918,578 1 37 38 39 40

PA 1,348,072 2 26 27 27 28

VA 1,023,503 3 20 21 21 21

OH 937,901 4 18 19 19 19

NC 639,747 5 13 13 13 13

TN 625,263 6 12 13 13 13

KY 621,832 7 12 12 12 13

VT 280,657 16 6 6 6 5

NH 269,326 17 6 5 5 5

AL 262,508 18 6 5 5 5

LA 171,904 19 4 4 3 3

IL 157,147 20 4 3 3 3

MO 130,419 21 3 3 3 2

DE 75,432 24 (Last) 2 2 2 1

1842 1852 – 1880s 1880s – 1890s 1900 – 1920s 1929 – 1941 1941 – 2000s

1842

1852 – 1880s (sort of)

1880s – 1890s 1900 – 1920s 1929 – 2000s

The 1872 apportionment differed from Hamilton’s method in four states

State Population Rank Hamilton Actual

NY 4,382,759 1 34 33

IL 2,539,891 4 20 19

NH 318,300 31 2 3

FL 187,748 33 1 2

Rutherford B. Hayes won the electoral vote 185 to 184 over Samuel J. Tilden

State Population Rank Hamilton Actual Winner

NY 4,382,759 1 34 33 Tilden

IL 2,539,891 4 20 19 Hayes

NH 318,300 31 2 3 Hayes

FL 187,748 33 1 2 Hayes

1842 1852 – 1880s

1880s – 1890s 1900 – 1920s 1929 – 2000s

Hamilton’s method is relatively simple to use, but it can lead to some strange paradoxes:

The Alabama paradox: Increasing the number of total seats causes a state to lose seats

The New States paradox: Introducing a new state causes an existing state to gain seats

The Population paradox: A state that gains population loses a seat to a state that does not

When the population of each state stays the same but the number of seats is increased, intuitively each state’s apportionment should stay the same or go up

That doesn’t always happen

Consider these state populations, with 80 seats available. Watch what happens when we increase to 81 seats

State Population % Pop. Fair ShareLower Quota

Priority Seats

Angria 83,424 39.06% 31.25 31 31

Bretonnia 67,791 31.74% 25.39 25 25

Curaguay 45,102 21.12% 16.89 16 1st 17

Dennenberg 17,249 8.08% 6.46 6 2nd 7

Totals 213,566 100% 80 78 80


Priority Seats

Angria 83,424 39.06% 31.25 31 31

Bretonnia 67,791 31.74% 25.39 25 25

Curaguay 45,102 21.12% 16.89 16 1st 17

Dennenberg 17,249 8.08% 6.46 6 2nd 7

Totals 213,566 100% 80 78 80


Priority Seats

Angria 83,424 39.06% 31.64 31 2nd 32

Bretonnia 67,791 31.74% 25.71 25 1st 26

Curaguay 45,102 21.12% 17.11 17 17

Dennenberg 17,249 8.08% 6.54 6 6

Totals 213,566 100% 81 79 81

After the 1880 Census, it was time to reapportion the House of Representatives.

The chief clerk of the Census Bureau computed apportionments for all numbers of seats from 275 to 350.

Alabama would receive 8 seats if there were 299 total seats, but only 7 seats if 300 were available.

If a new state is added to our country, but the total number of seats remains the same, then we would expect that the apportionment for the existing states should stay the same or go down

This doesn’t always happen

Consider this country with 4 states and 70 seats


Priority Seats

Elkabar 80,424 39.80% 27.86 27 1st 28

Florin 59,902 29.64% 20.75 20 2nd 21

Gondor 48,338 23.92% 16.75 16 3rd 17

Hyrkania 13,405 6.63% 4.64 4 4

Totals 202,069 100% 70 67 70

Now suppose the small state of Ishtar is added to this country


Priority Seats

Elkabar 80,424 39.80% 27.86 27 1st 28

Florin 59,902 29.64% 20.75 20 2nd 21

Gondor 48,338 23.92% 16.75 16 3rd 17

Hyrkania 13,405 6.63% 4.64 4 4

Totals 202,069 100% 70 67 70


Priority Seats

Elkabar 80,424 39.80% 27.86 27 1st 28

Florin 59,902 29.64% 20.75 20 2nd 21

Gondor 48,338 23.92% 16.75 16 3rd 17

Hyrkania 13,405 6.63% 4.64 4 4

Totals 202,069 100% 70 67 70


Priority Seats

Elkabar 80,424 38.72% 27.11 27 27

Florin 59,902 28.84% 20.19 20 20

Gondor 48,338 23.28% 16.29 16 16

Hyrkania 13,405 6.45% 4.52 4 2nd 5

Ishtar 5,611 2.70% 1.89 1 1st 2

Totals 207,680 100% 70 68 70

In 1907, when Oklahoma was admitted to the Union, the total number of seats in Congress did not change

Oklahoma was assigned 5 seats, but this resulted in Maine gaining a seat!

As time goes on, populations of states change

States that increase population rapidly shouldgain seats over those that do not

However, this doesn’t always happen

Consider this country with 4 states and 100 seats


Priority Seats

Javasu 28,900 18.09% 18.09 18 18

Karjastan 76,200 47.68% 47.68 47 1st 48

Libria 44,200 27.66% 27.66 27 2nd 28

Malbonia 10,500 6.57% 6.57 6 6

Totals 159,800 100% 100 98 100

Suppose that the populations of the states change


Priority Seats

Javasu 28,900 18.09% 18.09 18 18

Karjastan 76,200 47.68% 47.68 47 1st 48

Libria 44,200 27.66% 27.66 27 2nd 28

Malbonia 10,500 6.57% 6.57 6 6

Totals 159,800 100% 100 98 100


Priority Seats

Javasu 28,900 18.09% 18.09 18 18

Karjastan 76,200 47.68% 47.68 47 1st 48

Libria 44,200 27.66% 27.66 27 2nd 28

Malbonia 10,500 6.57% 6.57 6 6

Totals 159,800 100% 100 98 100


Priority Seats

Javasu 28,900 17.97% 17.97 17 2nd 18

Karjastan 76,400 47.51% 47.51 47 47

Libria 45,000 27.99% 27.99 27 1st 28

Malbonia 10,500 6.53% 6.53 6 3rd 7

Totals 160,800 100% 100 97 100

In the 1901 apportionment, Virginia lost a seat to Maine even though Virginia’s population grew at a faster rate!

1842 1852 – 1880s 1880s – 1890s

1900 – 1920s 1929 – 2000s

1842 1852 – 1880s 1880s – 1890s 1900 – 1920s

1929 – 2000s

Determine a priority value (PV) for each state for successive representative numbers until you run out of seats. Let

where n is current number of allotted seats for a given state.

If seats still exist (<436 have been allotted) the state is given the n+1st representative, else it will have n.

)1(

nn

ationStatePopulPVn

Seat # State Population n Priority Value Total Representatives

51 California 37,341,989 1 26,404,773 53

52 Texas 25,268,418 1 17,867,469 36

53 California 37,341,989 2 15,244,803 53

54 New York 19,421,055 1 13,732,759 27

55 Florida 18,900,773 1 13,364,864 27

56 California 37,341,989 3 10,779,703 53

434 California 37,341,989 52 711,308 53

435 Minnesota 5,314,879 7 710,230 8

436 North Carolina 9,565,781 13 709,062 13

437 Missouri 6,011,478 8 708,459 8

438 New York 19,421,055 27 706,336 27

Seat # State Population n Priority Value Reps Need for +1

51 California 37,341,989 1 26,404,773 53 N/A

52 Texas 25,268,418 1 17,867,469 36 N/A

53 California 37,341,989 2 15,244,803 53 N/A

54 New York 19,421,055 1 13,732,759 27 N/A

55 Florida 18,900,773 1 13,364,864 27 N/A

56 California 37,341,989 3 10,779,703 53

434 California 37,341,989 52 711,308 53 N/A

435 Minnesota 5,314,879 7 710,230 8 N/A

436 North Carolina 9,565,781 13 709,062 13 15,800

437 Missouri 6,011,478 8 708,459 8 15,100

438 New York 19,421,055 27 706,336 27 107,100

In 1982 Michel Balinski and H. Peyton Young proved that it is impossible to find an apportionment system that both:

avoids paradoxes

doesn’t violate the quota rule

0

50

100

150

200

250

300

350

400

450

1793 1803 1813 1823 1833 1843 1853 1863 1873 1883 1893 1903 1913 1923 1933 1943 1953 1963 1973 1983 1993 2003 2013

U.S. House of Representatives 1793 - 2013

-

100,000

200,000

300,000

400,000

500,000

600,000

700,000

800,000

1793 1803 1813 1823 1833 1843 1853 1863 1873 1883 1893 1903 1913 1923 1933 1943 1953 1963 1973 1983 1993 2003 2013

Average Number of U.S. Citizens per Representative

Montana

994,416 people

1 representative

Wyoming

568,300 people

1 representative

Rhode Island

1,055,247 people

2 representatives

Within the last 20 years three states (Utah, Montana and Massachusetts) and multiple individuals have filed suits against current apportionment methods and/or census counting methods.

The Supreme Court has upheld the validity of the Huntington-Hill Method and the current apportionment procedure in all cases.

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