On the Square Root of Time

Was Mr Jules Regnault the source of inspiration of Mr Gann?
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Jules Regnault [1863]
Law on the Square Root of Time

After much thought, we realized that it is not possible to find a relation between stock market gains and losses. It is ... with respect to time ... that we found a relation ...
In decreasing the time periods to 5 days, 3 days, 2 days ... the mean deviations decrease steadily.

Consequently, the deviations are smaller for shorter time intervals and larger for longer time intervals.

Finally, if one tries to find how these different deviations are related to the different times in which they occur, one notices that as the period decreases by half, the deviation decreases not by half but, roughly, in the proportion 1:1.41; for a period which is three times shorter, the deviation decreases in the proportion 1:1.73, for a time period which is four times shorter, the ratio is 1:2.

There exists therefore a mathematical law which regulates the variations and the mean deviation of stockmark et prices, and this law, which seems never to have been noticed, is given here for the first time:

THE PRICE DEVIATION IS DIRECTLY PROPORTIONAL TO THE SQUARE ROOT OF TIME.

Hence the investor who wants to sell after the deviation doubles, that is with a difference twice as large between the buy and sell price must wait four times longer, if he wants to sell with triple deviations, [he must wait] nine times longer, and so forth. One multiplies the time by the square of the deviations.

One who leaves only one day between [his buying and] selling, would sell with a deviation which is smaller by one half than one who sells every four days, three times smaller than one who sells every nine days, etc..., dividing the deviations by the square root of time.

Quite a large number of transactions is required, however, in order to make these ratios clearly apparent, and they become strictly correct when the number of transactions is exceedingly great.

Let us understand the reason for this remarkable law:

The security varies but is always looking for its real price or an absolute price, which one can represent as the center of a circle whose radius represents the deviation, which may be anywhere on the surface. Time is equal to the surface and the points on the circumference represent extreme deviations. As it varies, the security moves either away from or closer to the center, and the basic notions of geometry teach us that the radii or deviations are proportional to the square root of the area, that is of time.

Why is it that the reciprocal law holds when dealing with either gravity or the oscillations of a pendulum, where [in one case] the space traveled or [in the second case] the deviation of the oscillations is proportional to the square of time? It is only because these falling bodies go from the circumference to the center, whereas the stock price in its greatest deviations, is pushed away from the center towards the circumference.

How astonishing and admirable are the ways of Providence, what thoughts come to our mind when observing the marvelous order which presides over the most minute details of the most hidden events! What! The changes in stock market prices are subject to fixed mathematical laws!

Events produced by the passing fancy of men, the most unpredictable shocks of the political world, of clever financial schemes, the outcome of a vast number of unrelated events, all this combines and randomness becomes a word without meaning! And now worldly princes, learn and be humble, you who in your pride, dream to hold in your hands the destiny of nations, kings of finance who have at your disposal the wealth and credit of governments, you are but frail and docile instruments in the hands of the One who brings all causes and effects together in harmony and who, as the Bible says, has measured, weighed and parceled out everything in perfect order.

Man bustles but God leads.

The price of the “Rente”, while fluctuating capriciously, remains influenced in final instance by constant causes. The most important one, clearly defined and whose existence is without doubt, is the interest rate. This cause, so feeble in appearance, finally dominates all others. The accidental causes [will] have totally disappeared and, however powerful their effects, however strange and irregular they appear, they always end up after a while cancelling almost completely, revealing the influence of constant and regular causes, however weak the effect [of these regular causes] is.

The causes for a drop [in price] are fewer than those for a rise [in price] but, while they are few in number, they make this up by their strength; so that by multiplying number by strength one would obtain a constant value.

The price variations obey two distinct laws. The first is that the deviations are proportional to the square root of time ... The second is that the value [of the stock] whatever its deviation, is constantly attracted towards its average price as the square of its distance [to that price].

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Source:
Jules Regnault (1863): Calcul des chances et philosophie de la Bourse
. Mallet-Bachelier et Castel, Paris, 219 pages.

[above quotations translated from pages 49-52, 154, 161, 187]
There is one copy of Regnault's book at the British Library