Pythagoras would have been a good rifleman, because he understood his angles. The Pythagorean Theorem states that, in a right-angled triangle, the square of the hypotenuse (“c”) is equal to the sum of the squares of the two other sides (“a” and “b”), that is, a2 + b2 = c2. The Pythagorean Theorem comes into play when shooting at an angle, i.e., uphill, or downhill, especially at distance. Here’s an interesting phenomenon: When shooting at a distant target, whether shooting uphill or downhill, absent appropriate aiming-point correction (using either an external optic or iron sights), a bullet will tend to hit high. To new shooters, this seems odd. How can it be that, whether shooting uphill or down, the bullet will tend to hit high? Pythagoras has part of the answer. Newton has the other half.
Try this demonstration: Hold one of your arms straight-out in front of you, level (i.e., parallel) to the ground, and pointing straight ahead, with fingers extended. Now, hold your other arm out, but pointed upward, at a 45-degree angle, as if you were aiming a rifle at an uphill target, also, with fingers extended. Now, think of an imaginary plumb line (a string, weighted on one end), descending, from the fingers of your upward-pointed hand, and the finger tips of your other hand (which, of course, is pointed straight ahead). The angle between the plumb line and your level arm would be exactly 90 degree; a right angle. Your upward-pointed arm is the hypotenuse of the triangle formed by your two outstretched arms and the plumb line. Thanks to Pythagoras, we know that the hypotenuse of a right-angled triangle is longer than either of the other two sides of the triangle.
Now, give your arms a rest and imagine yourself holding and aiming a rifle at an upward angle. The untrained shooter, who also happens to intuit a bit of geometry, thinks, “Hmm, if the line of sight between my rifle and my target is the hypotenuse of a triangle, and the hypotenuse is longer than the other sides of the triangle, my bullet will have to travel farther to get to the target versus a bullet shot at a level target, so, I have to aim higher, to account for the longer distance.” However, it turns out that bullets don’t think like Pythagoras. Bullets don’t think at all. Once a bullet exits the muzzle, it just reacts to external factors. Key among those factors, especially in the vertical direction, is the force of gravity, which works at a right angle to the earth’s surface. In other words, all objects, including bullets, fall straight down. A bullet shot at an angle, regardless of whether that angle is upward or downward, falls straight toward the earth’s surface. Thus, at any given point in its flight, a bullet is pulled directly toward the ground, along an imaginary line that is always level–or parallel–to the ground. A bullet does not “care” whether it is aimed at an uphill or downhill target (as if flying along the hypotenuse of triangle). A bullet “cares” about gravity. The problem, here, is that inexperienced shooters who don’t think about trigonometry and the physics of gravity let their eyes deceive them. When shooting uphill, these shooters think they must “aim high” to account for the effect of gravity on a bullet traveling along the hypotenuse of an imaginary triangle, but gravity only effects the bullet along an imaginary line that lies parallel to earth. Yet, from Pythagoras, we know, in a triangle formed by the line of sight to a target located either uphill or downhill from the shooter, a line parallel to the ground, and a third line connecting the other two lines, the line parallel to the ground is shorter than the line of sight to the target.
Again, why does a bullet aimed either uphill or downhill tend to hit “high?” Because the inexperienced shooter is thinking about the distance to the target along the line of sight, whereas the bullet is “thinking” about the distance along a shorter, imaginary line that is parallel to the ground.