“Is it death?” – thought Andrey Bolkonsky in “War and Peace”. A few moments ago a cannon ball has landed nearby. His adjutant yelled “to the ground”, yet he for a reason unknown to us, hesitated, observing for a few moments the smoking shell on the grass.

Was it fate from the beginning?

The cannon ball was shot by a gun of Napoleon’s army. Based on the position of Andrey’s regiment, it was probably a 12-pounder. For the final outcome it probably does not matter whether the explosive shell of the 12-pounder is to land 12 cm to the left or to the right from the body, hit it directly to the head or to the leg for the radius of explosion is 3-5 meters. Therefore, for the sake of the question we describe the shell as a point in space. Similarly, Bolkonsky’s body can be taken, despite its complex structure seen from above, as a point in space. Accuracy on the scale of 3-5 meters, however, can be of importance.

Imagine the sight of the artillerist as a straight line originating from (roughly) the initial position of the ball. To specify a position of a point, we would need to say how many meters is the point along this line (usually x), how far it deviates to the left (y) or to the right, how many meters it is above or below the altitude of the gun’s barrel (z). Therefore, we need 3 numbers. Taken together, they constitute the coordinate (x,y,z). Let us take the initial position (at t=0 s) of the shell as the origin R0=(0,0,0). That’s the coordinate system, its introduction is attributed to a French mathematician of the 16-17th century Rene Descartes.

We can position Andrey somewhere 1800 m to the East from the French artillery of the 19th century and, for simplicity, on the same ground level. His position, therefore, at t=0, is RB=(1800,0,0), making a sensible assumption that the gunman was looking along the line of fire.

Once the powder in the gun was ignited, the bullet has acquired initial velocity. To define it, we would need to take 2 successive shots separated by a very small time dt. One can be taken at t=0, and another, say, at a moment dt when the ball was x1 meters away from the barrel, at a position r1=(x1, 0, 0).

The velocity is defined as a difference of two positions divided by the time interval between them. Hence it is v=(r1-r0)/dt=(x1,0,0)/dt m/s. In classical mechanics, velocity (v) and coordinate (r) specify the state of the physical object, a cannon ball in this case. In case a system as a whole has many things, the state constitutes coordinates and velocities of all bodies.

Velocity does not represent a position of a point, it is not a coordinate, it is something else. It is a vector. To properly introduce it, we are now to make a detour from jungles of classical mechanics to fields and crops of linear algebra.

In 3D, vectors, just like coordinates, also represented by 3 numbers. These are objects (we can draw it as arrows — objects with length and direction) that we can sum and also multiply by a number. They are usually denoted as lowercase letters with arrows above (or in italic, or with bar, or in boldface). There is no point of summing coordinates of points per se, and the point does not have a direction, however, we can formally put in a correspondence to each point Ri a radius vector ri: a vector emanating from the origin and ending at the desired point.

Another thing, besides summing and multiplying, we can do is to project one vector onto another. For example, to know how fast the bomb falls from above (or rises in the negative direction), we can project its velocity v onto vector (0,0,-1).

Clearly, the projection of a vector v onto itself is a length |v| by definition.

If we know that the bomb is at r(t), we might be interested in learning in how far it is in the direction of Prince Andrey. It will be given by |r(t)|cos(theta), where theta is the angle between r(t) and rB. It can be denoted as (r(t),rB)/|rB| (see Fig.1), where the bracket is an operation called the scalar product, and |r(t)| is to denote the length of the vector rB:

Let us introduce basic vectors: x, y, z, all with the length of 1. The first points towards East, second towards North, third away from the Earth’s core. Any other vector can be represented as a sum of those multiplied by some numbers. A set of vectors with this property called basis.

Note that if we take rt as a hypothenuse and r1 along x, then |rt|r1cos(theta) is rtx r1. Similarly, if r1 is along y, then |rt|r1 cos(theta) is rty d and same for z. Since we can represent a vector rt as a sum of x, y, z, it is clear that the scalar product of two vectors is given by the sum over products of their components:

From here, and the fact that the 2D vector along the hypothenuse (c) can be represented as a sum of two orthogonal sides (a and b), the most apt proof of the Pythagorean theorem follows. Vectors are quite useful.

What about Bolkonsky? With the gun and the ball, it is very simple. Classical mechanics says that the state of a single object composed out of the coordinate and velocity, as well as properties of the media, fully determines its trajectory whenever interactions with other objects are unimportant. It is unlikely that the flight of the cannon ball will be interrupted by a bird, another cannon ball or divine intervention, therefore the only factors substantially affecting trajectory is the initial state of the shell, influence of the media (air), as well as gravity of the Earth.

Some individuals discover something that changes their life for the better significantly. For many, the event of the discovery is accompanied by the pleasant feeling of excitement.

Often it is something known to many others. People who acquire and systematize new knowledge are, in majority of cases, scientists. However a scientist or not, a human often starts spreading newly acquired knowledge – a behavior trait that probably has an evolutionary origin reinforced by the society. For example, it was forbidden to kill messengers in the Ancient Greece: they had diplomatic immunity and were highly respected. Priests of Ancient Egypt enjoyed a high societal status.

People who ever camped will agree how amazing fire is: you can make food easier to digest and safer to consume, expand the geography and temperature regimes, avoid becoming ill because of the poor weather. It is amazing that an entity using an external object (fire) can expand its range, making one of a multi-continent species without modifying our biology. One of the earliest evidence of usage of fire is in the form of the burned lithics from the Koobi Fora in East Africa and dated 1.5 million years. As far as I know, we did not grow wings or modified our lungs to expand from Africa, but rather resided to advancement of technology and knowledge. When applications of fire were discovered, life of less advanced humans could be dependent on efficiency of communication.

My big discovery, made in the middle school, was a concept of the mathematical proof. At the beginning of the geometry class, I did not understand the purpose of the formal proof. I tend to think that mathematical or logical proofs are unknown to most of the animals, including hunting or gathering primates. It is both the variety of the different external conditions and finiteness of the human brain that led to this type of knowledge storage.

Indeed, what is the point of proving the Pythogerean theorem?

Every pupil equipped with a ruler, pencil and a sheet of paper able to draw can show that a rectangular triangle with two sides of length 3 (a) and 4 (b) has the longest side (c) of length 5. 9 plus 16 is 25, hence a relation c^2=a^2+b^2 between its sides exists. This triangle was known to Ancient Egyptians, who used a rope segmented into 12 pieces to make borders of fields perfectly rectangular.

One can try drawing other rectangular triangles with, say, length of sides 9 and 12 to learn that the largest side has a length of 15. Clearly, a relation c^2=a^2+b^2 exists. What is the point of lengthy and abstract proof residing on some other statements?

The point is that, strictly speaking, we showed that such relation is true for only two cases. It would be sensible to assume that this is case for all such triangles, but we all know that our intuition fails quite often. At this point, it is just an empirical observation that for two particular triangles such relation between the sides holds.

We may ask whether there are more such triangles where the relation holds? We can, for example, make lengthy tables with such triangles, each entry with three numbers (a,b,c):

(3,4,5)

(5,12,13)

(7,24,25)

…

We would then either have such book, or memorize it to be able to build a pyramid or nail windows from vandals.

The mathematical proof, however, will make a relation valid for any triangle: drawn on a paper, on a road with any length of sides. Drawn with a red pen, green pencil or blue chalk. Any kind of triangle! Such proof will contain much more knowledge than a big and possibly expensive book with numbers. It will be a statement applicable for an infinite number of geometrical objects.

This wisdom applicable for any kind of geometrical object of this kind is clearly more than just the empirical observation for a finite number of them, from which we started.

In my opinion, the most apt way to prove such relation comes from the scalar product. Since 3 sides form a triangle, for vectors it should be a+b=c, and hence (a+b,a+b)=(c,c), which for orthogonal a and b implies that a^2+b^2=c^2. It does, however, involve concepts of vectors and scalar products that could be unknown to agriculture workers of the ancient Egypt or Greece.

A simple law for the area of the triangle, however, must have been known for otherwise it was hard to figure out how many crops a given field is supposed to produce.

Consider a triangle drawn on a Figure 1. Triangle (c) can be divided into two, each with area:

The total area of the third is:

Since the area of Sc can be expressed as a sum of Sa and Sb, we complete the second proof of the Pythogerean theorem.

Another way to prove it comes from the way area should scale: the area of a rectangle goes as a square of its diagonal multiplied by the same function of an angle between it and its side and since triangle is a half of rectangle, it should scale quadratically the same way. We can again divide one big triangle into two smaller, and find, just like before, that a^2+b^2=c^2.

We went through three different proofs of the Pythagorean theorem (the first can be also used to prove a cosine formula). I have communicated a discovery I made in a middle school: the infinite value of the mathematical proof as a technology to produce new knowledge.