Father. The square-root of any number is that which being multiplied into itself produces the said number. Thus the squareroot of 1 is 1; but of 4 it is 2; of 9 it is 3; and of 16 it is 4, and so on.

Charles. Then if you had at all vessel of water with a cock inserted within a foot of the top, and you wished to draw the liquor off three times faster than it could be done with that, what would you do?

Father. I might take another cock of the same size, and insert it into the barrel at nine feet distance from the surface, and the thing required would be done.

Emma. Is this the reason why the water runs so slowly out of the cistern when it is nearly empty in comparison of what it does when the cistern is just full?

Father. It is: because the more water there is in the cistern, the greater the pressure upon the part where the cock is inserted; and the greater the pressure, the greater the velocity, and consequently the quantity of water that is drawn off in the same time.

In some large barrels there are two holes for cocks, the one about the middle of the cask, and the other at the bottom; now if when the vessel is full you draw the beer or wine from both cocks at once, you will find that the lower one gives out the liquor much faster.

Charles. In what proportion?

Father. As the square-root of 2 is greater than that of 1; that is, while you have a quart from the upper cock, three pints nearly would run from the lower one.

Emma. Are we then to understand that the pressure against the side of a vessel increases in proportion to the square of the depth; but the velocity of a spouting pipe, which depends upon the pressure, increases only as the square-root of the depth?

Father, That is the proper distinction. Charles. Is not the velocity of water, running out of a vessel that empties itself, continually decreasing?

Father. Certainly: because in proportion to the quantity drawn off, the surface de

scends, and consequently the perpendicular depths becomes less and less.

The spaces described by the descending surface, in equal portions of time, are as the odd numbers 1, 3, 5, 7, 9, &c. taken backwards.

Emma. If the height of a vessel filled with any fluid be divided into 25 parts, and in a given space of time, as a minute, the surface descend through nine of those parts, will it in the next minute descend through seven of those parts, and the third minute five, in the fourth three, and in the fifthone?

Father. This is the law, and from it have been invented clepsydras or water-clocks.

Charles. How are they constructed Sir? Father. Take a cylindrical vessel, and having ascertained the time it will require to empty itself, then divide, by lines, the surface into portions which are to one another as the odd numbers 1, 3, 5, 7, &c.

Emma. Suppose the vessel require six hours to empty itself, how must it be divided?

Father. It must first be divided into 36 equal parts; then, beginning from the sur face, take 11 of those parts for the first hour, nine for the second, seven for the 3d, five for the 4th, three for the 5th, and one for the 6th: and you will find that the surface of the water will descend regularly through each of these divisions in an hour.

I believe both of you have seen the locks that are constructed on the river Lea. Charles. Yes and I have wondered why the flood gates were made of such an enor mous thickness.

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Father. But after what you have heard respecting the pressure of fluids, you will see the necessity that there is for the great strength emyloyed.

Charles. I do: for sometimes the height of the water is 20 or 30 times greater on one side of the gates than it is on the other, therefore the pressure will be 400 or even 900 times greater against one side than it is against the other.

Emma. How are the gates opened when such a weight presses against them?

Father. There is scarcely any power by which they could be moved when this weight of water is against them; therefore there are sluices by the side, which being drawn up, the water gets away and passes

into the bason till it becomes level on both sides; then the gates are opened with the greatest ease, because the pressure being equal on both sides, a small force applied will be sufficient to overcome the fraction of the hinges or other trifling obstacles.

Charles. It is this great pressure that sometimes beats down the banks of rivers?

Father. It is: for if the banks of a river or canal do not increase in strength in the proportion of the square of the depth, they cannot stand. Sometimes the water in a river will insinuate itself through the bank near the bottom, and if the weight of the bank be not equal to that of the water, it will assuredly be torn up, perhaps with great violence.

I will make the matter clear by a drawing. Suppose this figure (Plate 11. Fig. 17.)