 Good afternoon. My name is Jonathan Loss and I'm here today to discuss with you the problem of apportionment or how do we divide items fairly? So I thought we'd start off with a definition of what apportionment is. It's actually the act of dividing items between different groups according to some plan, especially to make a proportionate distribution. So for instance, if I walked into your class today and there were 24 students in it and I had 100 $1 bills, and I wanted to distribute them fairly throughout the class, I would, each of you would get four because 24 of you, four times 24 gives us 96. And then I'd have four $1 bills left over. And what do we do with the leftovers? That's what apportionment deals with. And our country originally dealt with this in its founding. In our Constitution, Article 1, Section 2, we find these words. Representatives and direct taxes shall be apportioned among the several states which may be included with this union according to their respective numbers. 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 in such manner as they shall by law direct. And that's a lot of legal speak and stuff. But the key is that we shall apportion representatives according to their respective numbers. Now, what the Constitution did not say is how we did that. They just said that we should do that. Later on in the same section of the Constitution, we also get this statement that the number of representatives shall not exceed one for every 30,000, but each state shall have at least one representative. So basically you needed 30,000 people to get a representative in Congress, and you should always have at least one. So even the most least populous states deserve at least one representative in the U.S. House of Representatives. So we're going to talk about how Congress originally decided to do this, and we need a few terms to start off with. First of all, we have what's called the standard divisor. And the way you calculate the standard divisor is you take the total population and divide it by the number of things that you want to apportion. So in our case, this would be the population of the United States divided by how many seats we have in the House of Representatives. Originally there were 105 seats to be in the first House of Representatives. Then we get what's called the standard quota. This is something we calculate for each state or each group if we're doing something else. You take the group population and divide by the standard divisor. So we take, and specifically to the case of the United States, we take each individual state and divide by what the standard divisor is to see how many representatives they actually deserve based on the population. Very rarely, if ever, will this be a nice even number. This ends up being a decimal number. So we're going to look at a particular case in history. This is the map of the United States following the 1820 census. So you can see the land boundaries very similar to what we know today, but the shadings and the colorings here are quite different. In fact, everything kind of West of Louisiana, Missouri is some sort of territory and not yet a state. There were 24 states at this time, and we were to apportion 213 members to the House of Representatives following the 1820 census. Now we're going to use the data from the 1820 census, but the first method we're going to talk about was actually proposed by Alexander Hamilton back with the very first census in 1790. He proposed a method that today is called the Hamilton method, and it's probably the easiest method to apply. We're going to start there. So what I want to show you here is these were the 24 states that were with us back following the 1820 census, and then these are their respective populations. And down here at the end, we have the total population of the United States at that point, which was 8,959,313. So the number that I have down here, 42,062.5, is this number I've called the standard divisor. I took the population of the United States, the 8 million, almost 9 million number, and divided by 213, since that's the number of seats that we wanted to apportion. So what this number means specifically at this point in time means that you would need 42,000 people, a little more than that, in order to deserve one member in the House of Representatives. So the states with much greater populations, like Pennsylvania, had over a million people at this time. They would deserve more representatives than, say, Missouri, which only had 62,496. Let's close this. We'll come back to it. Like I said, it was the first method proposed, and Congress actually approved it in 1792. However, if you're aware of the executive branch, the president has the ability to veto items, and the very first use of the presidential veto was to veto Alexander Hamilton's proposal for using his method for apportionment. Now, there's a couple of reasons for why he might have vetoed it. One of them is the less political reason of if we used Hamilton's method, it would have turned out that the state of Delaware would have gotten two members to the House of Representatives, even though they only had 55,000 population, and that's less than 30,000 per member, which if you remember back to the Constitution, they said you needed at least 30,000 per member. The other more interesting and maybe intriguing political view is that the method that Washington later encouraged them to use actually gave his home state more representation. The state of Virginia ended up getting one more representative under the method that was chosen instead of Alexander Hamilton's method. So if you like to play politics and get intrigued with that, you can choose that as your reason. However, even though Hamilton's method was not used at the beginning, it was brought back in 1850 and used. And then we got rid of it again following the 1900 census because we discovered problems with it that I'll tell you about later. So it's had a much storied career. In fact, the apportionment of representatives, the House of Representatives in our country has had quite the rocky history. So let's talk about how to do it. First of all, you determine the standard quota for each state by dividing its population by the standard divisor. So in the Excel spreadsheet that I had up here earlier, the 42,000 number was the standard divisor. We want to divide each state's population by that 42,000 to see how many representatives they each deserve. Then we're going to take each state's standard quota because that's what that number is called after we've divided it, the standard quota and round it down to the nearest whole number. That's really easy to do in mathematics. That means you just drop the decimal and keep whatever the whole number is. We're going to call that the lower quota. Then we're going to add up all the lower quotas to see how many leftovers we have. If we're supposed to apportion 213, if we add up all the lower quotas and it adds up to only say 208, then we've got five leftover seats. Hamilton's method right here says, what do we do with those leftovers? This was his plan. He says to allocate the leftover seats to the states with the largest decimal remainder in their standard quotas until you've used up all the leftovers. So now we're going to go back to our Excel spreadsheet and see how we can actually do this method in practice if it would have been used in the 1820 census. So we have our standard divisor of 42,000 down here from earlier, and I've put that value in each entry in column E next to the populations of our 24 states. Now, what I'm going to do in the column marked SQ, standing for standard quota, is I'm going to divide each population by the standard divisor. And I'm going to have Excel do that for me. So we will pull up our calculator here, equals. If you want to enter a formula into Excel, you start it with equals. I want to choose that entry, mains population, and divide it by the standard divisor, and then I'm going to hit enter. And it gives me 7.09. The nice thing about Excel is that you can do this once and then if I can get this screen to work, drag it down to fill in all the other, actually I have my assistant here help me, drag it down to fill in all the other blanks. Okay, and so now we have the standard quotas for each of the states. You can see that Maine deserves 7.09 representatives. Massachusetts, for instance, 12.44. New York, one of the larger states, 32.54 representatives. And look, it adds up to a perfect 213. The problem is, I don't think you'll get anyone to volunteer from New York to be the .54 of a person to serve in the House of Representatives. So we've got all these ugly decimal remainders and what do we actually do with them? So remember, Hamilton said, we round everybody down. We totally ignore the decimal portion. So what I've put in this next column, it already had the formula in there, is LQ for lower quota. So 7.09 gets round down to seven. We just ignore the decimal. 5.80 becomes five. Even numbers really large, like 24.95 goes down to 24. And I've had Excel go ahead and add this up for me, we get 200. So according to Hamilton's method now, we are short 13 seats. And we have to decide where those 13 seats go. Those 13 seats, according to Hamilton's method, go to the people, or people, states with the largest decimal remainders. So I'm gonna go and reexamine my standard quota and find 2.99 for Louisiana. That's the .99 is the largest decimal present in that column. So it is going to get the first extra seat and instead of two, it's gonna get three. And then we're gonna have to go through here and find all our largest decimals. In fact, we got 13 of them. So anything in the 90s is gonna be safe. So 9.7, Rhode Island instead of one, they're gonna get two. Oops, backspace, two. There we go. Here we go, Pennsylvania. They're going to get 25. Keep missing that. There we go, 25. Who else do we have? Anybody else in the 90s? No, we're gonna grab all the 80s decimals here. Instead of 13, the state gets 14. And so on until we have them all filled in. Okay, as you can see, we've now filled in 13 extra seats. So New Hampshire was one of the states that got one instead of five. They've now got six. We already did Pennsylvania earlier. Maryland instead of eight, they got nine. It turned out that I believe .53 was the 13th largest decimal. So they were the last state to get an extra C instead of their lower quota. So how we finish this out then is all the other states that we've not already assigned, they get to keep just their lower quota. So Maine, because they did not have one of the 13th largest decimals, they stay with seven. And Massachusetts will stay with 12. And Virginia will stay with 21, and so on until we fill in all those states just with their exact lower quotas. And when we do that, we'll find out that it actually adds up to 213. So I've now gone through and filled in all the states lower quotas that did not deserve one of the extra leftover states. And I've had Excel calculate this value down here as a sum of the 24 entries above it. In fact, if you wanna see the actual formula, Excel has a command called sum. You just type equals sum, open a set of parentheses, and then grab and drag the actual cells that you're interested in. And notice that it adds up to a nice 213. The beauty of computers in Excel is that you can do this a whole lot faster than doing it by hand. Our founding fathers were doing this on parchment, paper, and quill pins. They were also pretty good mathematicians, I guess. So we're going to save this, because we're gonna revisit this later. Well, let's see if it'll let me exit it. There we go, save. And go back and let's do one more example that's actually much smaller and not in historical context. I suppose a state university wants to allocate 100 scholarships to its different schools based on their individual enrollments given as follows. So we've got five schools here, the business department, the engineering school, the education, liberal arts, and fine arts, and these are their associated populations. So the entire enrollment of this school is 45,686, and the university wants to disperse these 100 full scholarships evenly among the different schools based on their enrollment. So under Hamilton's method, we would first find our standard divisor. We take the total population of the school, we divide it by how many scholarships they wanna pass out, and that gives us 456.86. So I have the chart up here again, actually already all filled out, but I have the enrollments copied over again, and then the next row is called our standard quotas. Remember, how you find that is divide the population of the school by our standard divisor. So 27.41 is simply the 12,523 divided by the 456.86, and that's what I've done all the way across that row. I've divided the population by the standard divisor. Hamilton's method next tells us that we round down no matter what the decimal is. So 27.41 becomes 27. All the way over here at 7.03, it becomes seven. If we are to add this row up, you can check this yourself with the calculator. If you add it up, you find that it equals 99, but we're supposed to have 100 scholarships. So who does the university give their extra scholarship to? If they're using Hamilton's method to decide this, they give it to whichever school had the largest decimal portion of its standard quota, which in this case is 0.41, that's the biggest decimal anywhere present in that line. So the business department or business school gets the extra scholarship and they would end up with 28. Each of the other schools would keep their lower quota. And if you add up that bottom row now, you can check that yourself. You'll see that it adds up to 100. So Hamilton's method was not the only method that we could or actually did use in the history of our country. So in our next video, we're gonna find out what other ideas people had and how they addressed the method of apportionment in a way other than Alexander Hamilton.