**Terminal Ballistics**

Ragnar Benson wishes to determine the correct mixture of ammonium nitrate and nitromethane to achieve complete detonation. A noble endeavor! Accuracy in trench mortars is only useful if the shells go “bang!” wherever they might land. But let us see how Mr. Benson goes about finding this correct mixture. Benson (1992, p. 136) writes:

Despite almost driving our family into poverty by my many costly experiments, I still do not feel I have all the answers pertaining to this process. My experiments indicate that one should use slightly less than one-third nitromethane by volume, but this seems to vary from one gallon of nitromethane to the next and from one bag of ammonium nitrate to the next.

Unfortunately, I know of no formula that states precisely how much nitromethane to use. As a rough starting point, try one part nitromethane to three parts of ammonium nitrate by volume or two parts nitromethane to five parts ammonium nitrate by weight.

In sharp contrast, real science is based on the axiomatic method. Real scientists do not just randomly mix reactants together, stick a blasting cap in the resulting glop and see what – if anything – happens. Calling such guesses “hypothesis” does not make this activity scientific.

Real scientists reason from axioms; in this case, the conservation of mass. Chemical reactions – even very energetic ones like the detonation of high explosives – do not destroy mass. This claim is an axiom; that is, a proposition that is assumed without proof for the sake of studying the consequences that follow from it. An ammonium nitrate/nitromethane explosion just converts a solid and a liquid into three hot gasses; water vapor, carbon dioxide and nitrogen.

The important point here is that we must *assume* that there is exactly the same number of hydrogen, carbon, nitrogen and oxygen atoms in the reactants as there are in these hot gasses. Because water vapor, carbon dioxide and nitrogen are already abundant in the atmosphere, there is really no way that we could catch the products of the explosion, racing away from the blast at 7000 m/s or more, and weigh them. In some endothermic reactions that do not involve gasses, it may be possible to weigh the reactants and products, but even then we must remember that a mechanical scale is limited to only about three significant digits of accuracy. Yet I claim that mass is conserved *exactly*; to 23 significant digits if we could actually count every molecule in a mole of material. Such a bold assertion *must* be regarded as an axiom. There is no way to test it in the great majority of cases and even when it can be tested the accuracy of our measurements is 20 orders of magnitude short of proving our assertion.

So, having established the axiomatic nature of our line of reasoning, let us apply our axioms one-by-one and see what can be deduced from them, without recourse to any experiments or hidden assumptions. (And, as an added bonus, without blowing ourselves up or getting tossed into prison for possessing illegal explosives.)

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

Let x be the number of ammonium nitrate, NH_{4}NO_{3} , molecules and let y be the number of nitromethane, CH_{3}NO_{2}, molecules. From the axiom that there are as many carbon atoms before as after the reaction, we can deduce that there are y carbon dioxide molecules, as shown in equation (1).

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

From the axiom that there are as many nitrogen atoms before as after the reaction, we can deduce that there are (2x+y)/2 nitrogen molecules, as shown in equation (2).

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

From the axiom that there are as many hydrogen atoms before as after the reaction, we can deduce that there are (4x+3y)/2 water molecules, as shown in equation (3).

From the axiom that there are as many oxygen atoms before as after the reaction, we can deduce equation (4).

By multiplying both sides of (4) by two and then subtracting 4x+4y from both sides, we can deduce equation (5), which is all the information we need to deduce equation (6), our result.

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

And so, using only deductive logic based on the axiom that mass is conserved, we have found the formula for an ammonium nitrate/nitromethane reaction.

But we still do not know the relative weights of the two reactants. For this we must introduce a second axiom: The atomic mass of an element is approximately equal to the sum of its protons and neutrons. This is the simple high school model of the atom without isotopes. Specifically:

Hydrogen has one proton and no neutrons.

Carbon has six protons and six neutrons.

Nitrogen has seven protons and seven neutrons.

Oxygen has eight protons and eight neutrons.

The atomic mass of hydrogen is 1.

The atomic mass of carbon is 12.

The atomic mass of nitrogen is 14.

The atomic mass of oxygen is 16.

Atomic mass is actually the weighted average of the isotopes minus a tiny mass deficit for what is converted into binding energy – the results found in this paper have only three significant digits of accuracy – but the important point is that both this basic high school model of the atom and the more complicated professional model are axiomatic systems.

Nobody has ever seen an atom and nobody ever will. It is smaller than the wavelength of light one would have to reflect off it onto the lens of one’s microscope to observe it. What divides the post-doctoral researcher from the high school chemistry teacher is not that the former has a more powerful microscope for observing atoms – no such microscope exists – but that the former has a more powerful, though also more complicated, axiomatic system.

At this point in the argument, the empiricist will invariably start screeching that he has “refuted” our entire theory by carefully weighing a mole of oxygen and finding that it weighs 15.9994 grams, not 16 as that misguided axiomatist claimed. Well, so it does. But I already noted that the results in this paper were only accurate to three significant digits. If three significant digits of accuracy are sufficient for practical applications, there is nothing wrong with using simple axioms that are known to be approximations.

So, having dispensed with the empiricist’s inevitable criticism that our axioms are not perfect, let us return to our intrepid chemist, Mr. Benson. From our second axiom, we can deduce that the atomic mass of three ammonium nitrate molecules is 3(4 + 28 + 48) = 240 atomic mass units, AMU. Furthermore, we can deduce that the atomic mass of two nitromethane molecules is 2(3 + 12 + 14 + 32) = 122 AMU.

Therefore, using only deductive logic based on two reasonable axioms, we have found that we need 122 parts nitromethane to 240 parts ammonium nitrate, by weight, to achieve complete detonation. This result can be rounded off to one part nitromethane to two parts ammonium nitrate, by weight. And if it fails to detonate? It probably absorbed moisture from the air. Try again. Perfect equations and aesthetic axioms always supersede anecdotal evidence.

Clearly, the axiomatic method is vastly superior to the empirical method of conducting random experiments until something resembling a result appears. But empiricists never learn. Here, Mr. Benson (1992, pp. 139-140) describes a further line of research, conducted with his usual methodology of randomly mixing reactants together to see what happens.

The tip-off to a possible solution came while I was researching World War I’s Messines Ridge sapper attack… Britain’s World War I explosives manufacturers added finely ground aluminum powder to this explosive, called ammonal, to boost its brisance… Having made that discovery, I began to experiment with powdered aluminum. I added it to the ground ammonium nitrate before adding the nitromethane. At a level of about 5 percent (or about 20 grams) mixed thoroughly into 430 grams of NH_{4}NO_{3}, the effect was dramatic.

I leave it as an exercise for the reader to derive equations (7), (8) and (9) employing only deductive logic – no experiments – based on our two axioms. (Note that aluminum has 13 protons and 14 neutrons.) The atomic masses are written below the reactants.

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

_{2}O

_{3}

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

_{2}O

_{3}

_{4}NO

_{3}

_{3}NO

_{2}

_{2}O

_{2}

_{2}

_{2}O

_{3}

It is truly sad that Mr. Benson nearly drove his family into poverty, spending a lifetime and squandering a fortune attempting to accomplish with random experiments what an axiomatist could have achieved in thirty minutes at no cost. My heart breaks for his wife and children! If I were a Liberal, I might advocate a program of socialized explosives so that poor folks could pick up a brick of C4 at the food bank, just as they can now pick up a loaf of bread or a can of soup.

Lest others drive their families into poverty with their many costly experiments, let us be rid once and for all of the idea that doing science means randomly mixing reactants together. In spite of all the highfalutin talk about these random mixtures representing “hypothesis” that are to be tested empirically, accompanied by a barrage of statistical “data” if the experiment is conducted more than once, it is obvious that this is not science. Real scientists employ the axiomatic method.

**Exterior Ballistics**

Historians tell us that the Confederate Army might have won if only they had had more artillery. This weakness is usually attributed to a lack of foundries, of which they had only one, Tredegar Iron Works in Richmond, Virginia. But I tell you, it was not a lack of foundries that cost the South victory; they lost that war for want of the axiomatic method. Yet over a hundred years earlier, in 1745, Leonhard Euler had given the science of ballistics a solid axiomatic foundation. How could this important work have come to be ignored? Because people who hate math were up to their usual tricks 270 years ago, blacklisting mathematicians in the name of pluralism while loudly trumpeting themselves as underdogs struggling against a dogmatic “mainstream” that does not actually exist – then as now, people who hate math *were* the mainstream.

I will discuss these historical events later, but first let us examine the slipshod approach to gunnery that was taken by both sides in the 1861 U.S. Civil War. Civil War Artillery has compiled all that was known about ballistics at the time of the Civil War, obtained from *The Confederate Ordnance Manual* and *The Artillerist Manual*. Basically, artillerists compiled charts empirically, which they then attempted to interpolate on the battlefield.

The following chart gives the range, in yards, on level ground attained by elevating the barrel at empirically tested angles of elevation measured in degrees; the standard field carriage is assumed, though no data is available on exactly how high above the ground this put the muzzle. The “increase” column is the difference in yards from the previous test firing. Note.

Angle ofElevation |
12-Pounder Howitzer1 pound charge, shell |
12-Pounder Napoleon2½ pound charge, shot |
6-Pounder Field Gun1¼ pound charge, shot |

0 degrees | 195 Increase |
325 Increase |
318 Increase |

1 | 539 344 | 620 295 | 674 356 |

2 | 640 101 | 875 255 | 867 193 |

3 | 847 207 | 1200 325 | 1138 271 |

4 | 975 128 | 1320 120 | 1256 118 |

5 | 1072 97 | 1680 360 | 1523 267 |

The first thing that we notice about this chart is that it does not extend beyond a 5° angle of elevation. This is a problem with the empirical method; you cannot fire farther on the battle field than you did at the practice field. But with the axiomatic method, once the parameters (in this case, muzzle speed and ballistic coefficient) have been determined on the practice field, the theory can be extended far beyond any shots fired in practice. These men’s practice range was probably not long enough for them to attempt any higher angles of fire. Also, their chart provided data only for firing on level ground. The axiomatic method is undeterred by slopes because, when solving the equations, one simply looks for the trajectory to cross a slanted line rather than the x–axis. But empiricists are completely stymied by slopes because they cannot possibly test fire their weapon on every uphill or downhill imaginable. Also, my Android application for mortar fire control takes wind into consideration; this too stymies empiricists because they cannot possibly test fire their weapon in every wind condition.

Today, mortars have short barrels and are fired at elevations above 45°; howitzers have medium barrels and are fired at elevations of about 10° to 55°. Assuming uniform air density, the angle of elevation that maximizes range for supersonic projectiles is about 35°, but howitzers are fired at higher angles of elevation to get their shells into the stratosphere where there is almost no air resistance. Guns have long barrels and are fired at elevations below about 10°; the T-90 can elevate its main gun to 14°, but this is for firing uphill, not for attempting long shots.

In 1861, the word “mortar” had the same meaning as it does today except that the shells were gigantic; upwards of 200 pounds. This meant that they were slow enough that a simple parabolic trajectory was not wildly inaccurate and it also meant that it did not matter exactly where they landed because they were timed to explode in the air and shower a wide area with shrapnel or incendiary materials. (What they could do with a 17,000 pound mortar, the Russians can do today with a shoulder-fired Shmel thermobaric weapon.) The shoot-and-scoot tactics that are common today with small man-portable mortars, especially among guerillas, did not exist during the Civil War; for lack of mortar fire control deduced from axioms, they could not possibly expect to hit anything with a man-portable mortar. Back then, mortars were initially used exclusively for seacoast defense and later adapted to sieges, hauled inland on specially reinforced rail cars.

In 1861, the difference between guns and howitzers was miniscule. The ballistics for both guns and howitzers was known only in the same narrow 0° to 5° range. The only practical difference was that guns were of smaller caliber and, with the higher velocity provided by slightly longer barrels, they could fire solid shot at masonry walls, while the black-powder-filled shells of howitzers bounced off before exploding for lack of contact detonation. (Contact detonation only works for elongated projectiles that hit pointy end first; spherical shells can strike the target on any side.) Because Civil War artillerists lacked sound gunnery deduced from axioms, they knew nothing about angles of elevations between 5° and 45° and simply never fired their weapons that high. Had they made use of Euler’s century-old work, they might have had a real howitzer, in the modern sense of the word. Note.

This low angle of fire is also a major contributor to why the chart is so wrong, as will be discussed below. When firing at high angles near the angle that maximizes range (about 40° for subsonic projectiles), small variations in muzzle speed and undulations in the Earth are of no consequence. But, when firing at low angles, the shell just skims over the surface; tiny variations in muzzle speed and small undulations in the surface can change the point of impact by a hundred yards or more. For that matter, a spherical shell skips when it strikes such a glancing blow and, if the ground is hard, it may not leave a mark but will just bounce joyfully along until it finally comes to rest hundreds of yards downrange. This effect is well known to artillerists and is why so much effort is put into the maneuver of guns so they can enfilade trenches, while high angle fire can be directed from anywhere within range of the trench.

Not only does the chart not extend beyond a 5° angle of elevation, the data that we do have is all wrong. Consider the 12-pounder howitzer. How could raising the barrel from 1° to 2° add 101 yards to the range, but raising the barrel from 2° to 3° add 207 yards to the range? And further increases in elevation again result in about 100 yards per degree? The 12-pounder Napoleon chart indicates that raising the barrel from 2° to 3° adds 325 yards to the range, but raising the barrel from 3° to 4° adds only 120 yards to the range, but then raising the barrel from 4° to 5° adds 360 yards to the range. The gun data is just as erratic. The 6-pounder field gun chart indicates that raising the barrel from 2° to 3° adds 271 yards to the range, but raising the barrel from 3° to 4° adds only 118 yards to the range, but then raising the barrel from 4° to 5° adds 267 yards to the range.

This makes no sense: Even the most casual analysis indicates that angle/range charts should be negative monotonic; that is, the greatest increase in range is achieved by raising the barrel from 0° to 1°, almost no increase is achieved by raising the barrel from 39° to 40°, and intermediate increases in range smoothly decline. To have the increase sometimes great and sometimes small is clear evidence that the data is worthless.

The men conducting these experiments should have noticed that their data did not make sense and tried again with more care. This is another problem with the empirical method; interpolation only works if you fire in increments of one or – better yet – half degrees. But that requires constantly adjusting the angle of elevation and there are sure to be mistakes made with so many adjustments. The data compiled by *Civil War Artillery* shows evidence of mistakes having been made in the firing of every one of these weapons.

The axiomatic method requires firing at only one angle of elevation to determine the ballistic coefficient. Thus, the test barrel can be set in concrete so it does not budge even one minute of angle between shots. This one angle can be chosen to be 40° to minimize the effect of variations in muzzle speed and undulations in the Earth. There is no reason to have the shells skimming inches over the surface and then skipping when they hit, which makes it unclear where the first touch down was. The empiricists are doing these tests all wrong!

The axiomatic method also requires that the muzzle speed of the weapon with each powder charge be known precisely. *The Confederate Ordnance Manual* and *The Artillerist Manual* are conspicuously silent on muzzle speed; where it is known today it was measured by modern hobbyists firing replicas past electronic chronographs and is reported only as a gee-whiz fact, not for actual fire control purposes. But Benjamin Robins invented the ballistic pendulum in 1742. How could this simple device have been forgotten? It was not forgotten but *suppressed* by the pluralists, who insist that all of science consists of compiling empirical charts and all of mathematics consists of linear interpolation of empirical charts.

Given the available budget for practice ammunition, many shots can be fired at this one angle of elevation (40°) so, if one shot is a “flyer” because something went wrong with the firing of it (e.g. the carriage got wedged against a rock and bounced under recoil), that data can be discarded. When you fire only one shot at each of many different angles of elevation, flyers go unnoticed and throw a monkey wrench into the whole procedure. Also, sometimes there are mistakes in the loading of the charge or variations in the dampness of the black powder; the axiomatic method allows designers to notice these intermittent problems when firing repeatedly into a ballistic pendulum, while they would go unnoticed when firing at the practice range.

But to assume that one can hit targets at any range based on having zeroed the weapon at only one angle of elevation requires axioms that are always true. What are these axioms? Leonard Euler proposed three:

- Constant atmospheric density from the ground to the apogee.
- Drag is proportional everywhere to the square of the speed.
- Gravity is everywhere pointed downwards; e.g. the Earth is flat.

And, yes, I am well aware that that high-pitched shrieking noise in the background is all the empiricists informing us that exactly none of these three axioms are always true. Admittedly, they are not… But trench mortars, which a Civil War vintage cannon aimed upwards basically is, do not go high enough for the air to become noticeably thinner. They do not fire at high enough speeds – 240 m/s is the limit – for drag to be proportional to higher powers of speed than two. And they do not fire far enough for it to matter that the Earth is a sphere.

The beauty of the axiomatic method is that the axioms can always be modified later to deal with more complicated situations. For instance, I developed an Android application for mortar fire control based on three similar but slightly modified axioms. The atmosphere becomes progressively thinner as altitude increases, as described by the axiomatic system proposed by Lewis Fry Richardson, which I take into account. And drag is proportional to the cube and then to the fifth power of speed at higher speeds, which I also take into account, as well as the effects of decelerating through the sound barrier (343 m/s). But I am sticking with gravity being everywhere pointed downwards, in spite of the sneering “flat Earther” comments that it generates from those who insist on interpolating from empirically generated tables as Civil War artillerists did. Note.

Modern howitzers can fire on targets over the horizon and do take the curvature of the Earth into consideration, as well as many other things, like humidity, that have a negligible effect on mortar gunnery. (Incidentally, almost everybody, including FM 23-10, has it backwards; the more humid the air, the thinner it is. Clouds float, don’t they? H_{2}O has an atomic mass of 18 while N_{2} has an atomic mass of 28.) But the theory employed by the modern artillerist is essentially that of Euler. If he could be resurrected and given the opportunity to talk to them, he would immediately recognize everything they are doing as being based on his 1745 annotated translation of Benjamin Robin’s 1742 book, *New Principles of Gunnery.*

If Civil War era artillerists could be resurrected, they would be astonished to see modern howitzers being fired at angles of elevation well above 5° and hitting point targets over the horizon. And they would be absolutely appalled to learn that the theory which defines modern gunnery pre-existed them by 100 years and had been blacklisted by the self-described pluralists who insist that all of science consists of interpolating empirically generated tables.

It is really frustrating for mathematicians, who know what they are doing, to find that politics has placed them under the thumb of self-aggrandizing tyrants who, incapable of adding two fractions if they have different denominators, yet feel themselves qualified to abruptly blacklist anybody who so much as utters the forbidden phrase, “axiomatic method.” And it is particularly galling that these tyrants cannot string two sentences together without mentioning pluralism and proudly describing themselves as employing the “scientific method” – a method no real scientist uses – while sneering at those whom they have blacklisted as “flat Earthers.”

**REFERENCES**

Benson, Ragnar. 1992. * Big Book of Homemade Weapons.* Boulder, CO: Paladin Press