The Mathematics of The Second Amendment

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When the Second Amendment to the United States Constitution was ratified 1791, it specified the following:

A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.

In 1791, the Kentucky rifle was the popular weapon of choice among the citizenry. It was named after the region west of the Appalachian Mountains, in the vicinity of the Kentucky River. This was the American frontier of the late 18th century, and the Kentucky Rifle was common among the settlers there. It was also among the rifles bought by the United States government as contract rifles in 1792. Given this, we can assume this is precisely the type of weapon the Framers had in mind when they inked the Second Amendment; it was the weapon to be used by the hypothetical well regulated militia of the era, and both the government and civilian rifles were largely identical.

The Kentucky Rifle in the hands of a skilled user was accurate to a long range compared to other firearms of the day, and a skilled user could get off a whopping two shots per minute. An unskilled user might be able to fire a Kentucky rifle half as fast, if he or she even knew how to reload it at all. These were flintlock, muzzleloaded weapons that had a time consuming reloading process. This procedure is a popular trope in historical films, where the villainous shooter fires a shot that misses and then has to try to reload before the hero rushes in. For purposes of this discussion, we will consider the practical rate of fire to be two shots per minute.

The weapon that has garnered the most notoriety in recent mass shooting events is the Armalite AR-15, a modern assault weapon first produced in 1959. The AR-15 itself is a derivative of previous assault rifles, and was introduced to be a counter to the devastating Russian Kalashnikov rifle of global fame, the AK-47, which was produced in 1947. The AK-47 is arguably the most successful firearm in world history, with over 100 million of the AK-47 and its direct derivates in circulation in the world today, which comprises 20% of all global firearms in existence. The AK-47 is a fully automatic assault rifle (semi-automatic versions also exist) which, when fired in automatic mode, yields a practical rate of fire of 100 rounds per minute. Each of these is a potentially lethal shot, much as each of the Kentucky rifle’s two shots per minute is potentially lethal. In addition, the AK-47 is more accurate, has a greater range, and is far easier to use than a Kentucky rifle. While it may have taken someone hundreds of hours to become lethal with a Kentucky rifle, a relative neophyte can turn the AK-47 into a deadly weapon, as evidenced by child soldiers throughout the world being armed with this firearm.

We see, then, that the rate of fire increased by a factor of 50 between 1791 and 1947. Further developments of assault rifle and personal machine gun technology led to rates of fire of about 120 shots per minute by 1959, a factor of 60 higher than they were in 1791. Thus, in 156 years, the rate of fire increased by 50 times, and in 168 years, the rate of fire increased by 60 times. Data on rifles produced since this time suggest that they are becoming a mature technology, with most new models showing minor, incremental changes, geared more toward decreasing weight, lowering cost, and increasing convenience of use and storage rather than improving rates of fire. But, rifles are not the end of personal firearms, and new technologies for killing are being developed everyday. Given this rate of development in firearms, what can we expect to see in the future in terms of the capability of small arms?

If we call 1791 year 0 and assume that technology in general increases in an exponential fashion, we can estimate the range and rate of fire of personal firearms or their equivalents in the future. We define the base rate of fire as Kentucky rifle units (Kru), or two shots per minute. The AK-47 fires at 50 units and the HK G3 fires at 60 units. Thus, we know that, 156 years after the ratification of the Second Amendment, rifles fired at 50 Kru and, 168 years after the ratification of the Second Amendment, the number increased to 60 Kru. As such, we have two points on two possible exponential curves, (0,1) and (156, 50) for the first, and (0,1) and (168,60). We know the equation for exponential functions is of the form:

y = ab^x.

We have selected units such that y is 1 when x is 0, which means that a is also 1. Computing the value of b for the two different exponential curves — one going through the AK-47 datapoint and the other going through the HK-G3 datapoint — we find it to be 1.02539 in the first case and 1.02467 in the second case. These values are close, suggesting the expected exponential relationship and, for further computation, we average the two figures, giving us 1.02503.

What can we expect for the rates of fire of rifles or analogous firearms in the future? In 2025, some 234 years after the ratification of the Second Amendment, we find that the rate of fire, given by y in the equation y = b^x, comes to 325 Kru, or about 650 shots per minute. In 2050, that number rises to 604 Kru, or about 1,200 shots per minute. In 2100 the number climbs to 2,078 Kru, or about 4160 shots per minute. 500 years after the Second Amendment, in year 2291, the rate of fire of rifles or their equivalents will have become 233,500 Kru, or about 467,000 shots per minute. That is a big number, and, in case the 100 shots per minute that rifles manufactured in the 20th century are able to fire did not already seem large compared to the paltry rate of fire of the Kentucky rifles of the late 18th century, surely 467,000 shots per minute does.

We can perform a similar analysis to discover the ranges of weapons going forward. The Kentucky rifle had an effective range of about 200 yards in the hands of an expert; an untrained shooter would have a much shorter range, if he or she could hope to hit anything at all. Conservative estimates for the effective range of the AK-47 and HK-G3 are 300 m (328 y) and 400 m (437 y), respectively. If we again take the base unit as the Kentucky rifle unit, we have ranges of 1, 1.64, and 2.185. We can use the points (0,1) for the Kentucky rifle, (156, 1.64) for the AK-47, and (168, 2.185) for the HK-G3 to compute exponential curves for range increase just as we did for rate of fire before. This time we find b values of 1.00318 and 1.00466, and we again average the two, to get 1.00392, to calculate values for weapon ranges in the future.

Given these figures, we find the effective range for a rifle in 2025 to be 2.498 Kru, or about 500 yards, a rifle in 2050 to be 550 yards, a rifle in 2100 to be about 670 yards, and a rifle 500 years after the ratification of the Second Amendment to be about 1400 yards. These figures, again, have been calculated for the minimum effective ranges given for the AK-47 and HK-G3. It does not take much of a science fiction mindset to see how we reach this point when you take into consideration experimental weapons research such as the Metal Storm, rail guns, and computerized targeting scopes that already exist and, while the thought of a personal firearm in 2271 being able to unleash 467,000 shots per minute to a range of 1400 yards is unnerving, it is not the scariest development that the future could hold in personal armaments.

Another direction of analyzing this situation is seeing what things that are now not considered firearms could eventually become portable by a person and, thus, the types of arms someone could bear. We have seen this already; explosives have become smaller over time and now comprise grenades and mines that a person can walk around with. A person can fire grenades from a handheld grenade launcher. When explosives were first introduced, they consisted of bulky devices with fuses that generally required a team of people to set and detonate. Over time, explosives have become far deadlier and more powerful, and now we live in a world with nuclear weapons.

The first nuclear weapon that was detonated in 1945 weighed 110 tons and created an explosion with the force of 10 megatons of TNT. By 1983, thermonuclear warheads that weighed 1.2 tons were capable of creating a 1.2 megatons of TNT equivalent explosion and, by 1991, at the close of the Cold War, a 0.162 ton thermonuclear device could create a 0.2 megaton explosion. In terms of ratios of weight to TNT, the 1945 device was 11 tons per 1 megaton of explosive power, the 1983 device was 1 ton per megaton of explosive power, and the 1993 device was 0.81 tons for a 1 megaton of explosive power. Of note, 0.162 tons is only 324 pounds. This is something that can be transported in an automobile; it does not require a military vehicle for transportation. How long, then, until a thermonuclear device that weighs about the same as a loaded assault rifle (10 pounds) could theoretically be developed to have the explosive power of a megaton of TNT?

We can set 1945 to be year zero, and the basic unit of weight to be 11 tons. We thus have the data points (0,1), (0.0909, 38), and (0.0147, 48) for the devices from 1945, 1983, and 1993. We again solve the same exponential equation to find values of b given by the two different curves which, in this case, comes to 0.938845 and 0.915849. These are farther apart than the values derived from our previous exponential curves, suggesting less accuracy; they average to 0.927342. Using this as our value for b, in what year do we expect to see our first 10 pound, 1 megaton warhead. 10 pounds is 0.005 tons. Expressed in 11 ton units, this is 0.0004545. For our exponential curve to shrink to that value, we find b = 102. Thus, 102 years after 1945, or in 2047, we would expect to see a 10 pound, 1 megaton nuclear device. Of course, nuclear weapons research slowed greatly after the end of the Cold War, so this may never come to pass or, at the very least, it could take a lot longer. But, regardless, technological progress has the potential to deliver a 10 pound device with enough explosive power to level a city and kill hundreds of thousands, or millions, of people in an instant. Did the Framers have that in mind when they wrote the Second Amendment?

I do not want to make any argument about what is right or wrong here. I think the numbers speak for themselves. Are you comfortable with your potentially angry neighbor having a nuclear device? Do you think the only thing that stops a bad guy with a nuclear device is a good guy with a nuclear device? If every human has the potential to kill millions of people in seconds, and if some small fraction of humans are willing to do that, the same way that some small fraction of humans are willing to engage in a shooting spree today, is that a world you want to live in? Would you survive more than a few days? Would anyone? If a mass shooting happens everyday in America, what about in a future where mass shootings are 1,000 times as deadly, or when nuclear weapons a person can carry around are 100,000 times as deadly?

Where do you draw the line? What, exactly, is meant by “common sense” gun regulation? Even the staunchest gun rights activist, who really believes that the right of the people to bear arms “shall not be infringed” at all is unlikely to want everyone to have weapons of mass destruction they can carry around at all times. It is, thus, quite arbitrary where these lines are drawn. It is likely the case that greed control and hate control far outweigh the value of gun control in the great scheme of things, but people are dying in the here and now when it need not be the case, and the math suggests it is only going to get worse. There is, unfortunately, no quick cure for greed and hate to put an end to the killing.

This may seem like a lot of speculation, but we are already living in an era with firearms far more potent than those in 1791, and the disparity between the reality in 1791 and the current day will only continue to grow as time goes on. We can’t stop technology; the human race will have more power to destroy itself everyday. How, then, do we stop that from happening? If it is inevitable that firearms will one day exist that are capable of killing thousands, or millions, or everyone on Earth in seconds, then how do we reconcile this in the face of an amendment penned in 1791 that could allow such firearms to be widespread? Logic dictates that we address this problem sooner rather than later; compassion dictates that we should have addressed it already.

There is a concept known as the Great Filter in the question of whether there is extraterrestrial life to be found out there and, if so, why we haven’t found it. The Great Filter posits that, because we haven’t found any signs of extraterrestrial life, there must be some filter in the development of life that halts the process somewhere between a location’s initial conditions being right for life and a civilization’s eventual expansion into space. It could be that we humans have already passed the Great Filter, and something so improbable happened in our past that it’s a miracle that we’re here today. But, it could be that life up to how it is on present day Earth is not that remarkable, and that the Great Filter looms before us. Ubiquitous, powerful weapons could be such a Great Filter. If that is the case, the Second Amendment, which has already failed countless victims of random acts of violence, will most certainly not serve us very well in the years to come. The mathematics of the Second Amendment dictate that we will eventually reach a population of zero.

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