# A proud hill

Looking at the conical heaps of rubble or sand, one recalls an old legend of the eastern peoples about a proud hill. Once the tsar ordered his soldiers to carry the earth handfuls into a heap, and a proud hill rose. And the king could look at the sea from above, where the ships were running.

This is one of those few legends in which, while seemingly plausible, there is no grain of real truth. It can be proved by geometric calculation that if some ancient despot had taken it into his head to carry out such an undertaking, he would have been discouraged by the meagerness of the result: in front of him would have risen such a miserable pile of land that no fantasy could have blown it into the legendary "proud hill".

Let's make an approximate calculation. How many warriors could an ancient king have? The ancient armies were not as numerous as they are today. Batu came to Rus' with an army of only 3000 soldiers. An army of 100,000 men was already very impressive in numbers. Let us dwell on this number, i.e., let us assume that the hill was made up of 100,000 handfuls. Grab the largest handful of earth and pour it into a glass: you will not fill it with one handful. We will assume that the handful of the ancient warrior was larger than yours and equal to the volume of the glass, i.e. 1/5 liter (cubic decimeter). From here, the volume of the hill is determined:

1/5 x 100000 = 20000 cu. decimeters = 20 cu. meters.

This means that the proud hill was a cone, with a volume of no more than 20 cubic meters. meters. Such a modest volume is already disappointing. But we will continue the calculations to determine the height of the hill. To do this, you need to know what angle the generators of the cone make with its base. In our case, we can take it equal to the angle of repose, i.e. 45 degrees: steeper slopes cannot be allowed, since the earth will crumble (it would be more plausible to take even a gentler slope, for example, one and a half). Stopping at an angle of 45 degrees, we conclude that the height of such a cone is equal to the radius of its base; hence,

20 = pi*x*x*x/3,

where:

x = 2,4 meters.

One must have a rich imagination in order to call an earthen heap 2,4 meters high a "proud hill". Having made a calculation for the case of a one-and-a-half slope, we would have obtained an even more modest result.

Atilla had the most numerous army that the ancient world knew. Historians estimate it at 700000 people. If all these warriors had taken part in raising the hill, a heap would have formed that was higher than we calculated, but not much: since its volume would be 7 times greater than ours, the height would exceed the height of our heap by only 1,9 times : it would be equal to 2,4 x 1,9 = 4,56 meters. It is doubtful that a mound of this size could satisfy Attila's ambition.

From such small elevations, it would, of course, be easy to survey the sea, unless it happened not far from the coast.