Lead Versus Harmony

I agree. In very long distances, the lead might cover 100 miles while the bladder was traversing 1 mile.

But this phenomenon which you adduce against my proposition is precisely the one which confirms it.

The variation of speed in bodies of different specific gravities is not caused by the difference of specific gravity.

It depends on external circumstances as the resistance of the medium.

If this is removed, all bodies would fall with the same speed.

With bodies which differ widely in weight, their velocities differ more and more as the spaces traversed increase.

This is something which would not occur if the effect depended on differences of specific gravity.

For since these specific gravities remain constant, the ratio between the distances traversed should remain constant.

In reality, this ratio keeps on increasing as the motion continues.

Thus, a very heavy body falling 1 cubit will not anticipate a very light one by so much as the tenth part of this space.

But in a fall of 12 cubits, the heavy body would outstrip the other by 1/3. In a fall of one hundred cubits by 90/100, etc.

A heavy body has an inherent tendency to move with a constantly and uniformly accelerated motion toward the common center of gravity which is the center of our earth.

During equal intervals of time, it receives equal increments of momentum and velocity.

This holds whenever all external and accidental hindrances have been removed.

But of these there is one which we can never remove, namely, the medium which must be penetrated and thrust aside by the falling body.

This quiet, yielding, fluid medium opposes motion through it with a resistance which is proportional to the rapidity with which the medium must give way to the passage of the body; which body is by nature continuously accelerated so that it meets with more and more resistance in the medium and hence a diminution in its rate of gain of speed until finally the speed reaches such a point and the resistance of the medium becomes so great that, balancing each other, they prevent any further acceleration and reduce the motion of the body to one which is uniform and which will thereafter maintain a constant value.

There is, therefore, an increase in the resistance of the medium, not on account of any change in its essential properties, but on account of the change in rapidity with which it must yield and give way laterally to the passage of the falling body which is being constantly accelerated.

Air offers:

• great resistance to the slight momentum of the bladder.
• small resistance to the large weight of the lead

If the medium were entirely removed, the advantage received by the bladder would be so great and that coming to the lead so small that their speeds would be equalized.

My principle is that all falling bodies acquire equal speeds in a medium which, on account of a vacuum or something else, offers no resistance to the speed of the motion, we shall be able accordingly to determine the ratios of the speeds of both similar and dissimilar bodies moving either through one and the same medium or through different space-filling, and therefore resistant, media.

This result we may obtain by observing how much the weight of the medium detracts from the weight of the moving body, which weight is the means employed by the falling body to open a path for itself and to push aside the parts of the medium, something which does not happen in a vacuum where, therefore, no difference [of speed] is to be expected from a difference of specific gravity.

The medium reduces the weight of the body by the weight of the medium displaced.

We may accomplish our purpose by diminishing in just this proportion the speeds of the falling bodies, which in a non-resisting medium we have assumed to be equal.

For example, imagine:

• lead as 10,000 times as heavy as air.
• ebony as only 1,000 times as heavy.

Here we have two substances whose speeds of fall in a medium devoid of resistance are equal: but, when air is the medium, it will subtract from the speed of the lead one part in ten thousand, and from the speed of the ebony one part in one thousand, i. e. ten parts in ten thousand.

While therefore lead and ebony would fall from any given height in the same interval of time, provided the retarding effect of the air were removed, the lead will, in air, lose in speed one part in ten thousand; and the ebony, ten parts in ten thousand.

If the elevation from which the bodies start be divided into ten thousand parts, the lead will reach the ground leaving the ebony behind by as much as ten, or at least nine, of these parts.

Is it not clear then that a leaden ball allowed to fall from a tower two hundred cubits high will outstrip an ebony ball by less than four inches?

Ebony weighs a thousand times as much as air but this inflated bladder only four times as much; therefore air diminishes the inherent and natural speed of ebony by one part in a thousand; while that of the bladder which, if free from hindrance, would be the same, experiences a diminution in air amounting to one part in four.

So that when the ebony ball, falling from the tower, has reached the earth, the bladder will have traversed only 3/4 of this distance.

Lead is 12 times as heavy as water; but ivory is only twice as heavy.

The speeds of these two substances which, when entirely unhindered, are equal will be diminished in water, that of lead by one part in twelve, that of ivory by half.

Accordingly when the lead has fallen through eleven cubits of water the ivory will have fallen through only six. Employing this principle we shall, I believe, find a much closer agreement of experiment with our computation than with that of Aristotle.

In a similar manner we may find the ratio of the speeds of one and the same body in different fluid media, not by comparing the different resistances of the media, but by considering the excess of the specific gravity of the body above those of the media. Thus, for example, tin is one thousand times heavier than air and ten times heavier than water; hence, if we divide its unhindered speed into 1000 parts, air will rob it of one of these parts so that it will fall with a speed of 999, while in water its speed will be 900, seeing that water diminishes its weight by one part in ten while air by only one part in a thousand.

Again take a solid a little heavier than water, such as oak, a ball of which will weigh let us say 1000 drachms; suppose an equal volume of water to weigh 950, and an equal volume of air, 2; then it is clear that if the unhindered speed of the ball is 1000, its speed in air will be 998, but in water only 50, seeing that the water removes 950 of the 1000 parts which the body weighs, leaving only 50.

Such a solid would therefore move almost twenty times as fast in air as in water, since its specific gravity exceeds that of water by one part in twenty. And here we must consider the fact that only those substances which have a specific gravity greater than water can fall through it—substances which must, therefore, be hundreds of times heavier than air; hence when we try to obtain the ratio of the speed in air to that in water, we may, without appreciable error, assume that air does not, to any considerable extent, diminish the free weight [assoluta gravità], and consequently the unhindered speed [assoluta velocità] of such substances. Having thus easily found the excess of the weight of these substances over that of water, we can say that their speed in air is to their speed in water as their free weight [totale gravità] is to the excess of this weight over that of water. For example, a ball of ivory weighs 20 ounces; an equal volume of water weighs 17 ounces; hence the speed of ivory in air bears to its speed in water the approximate ratio of 20:3.