Letters of Euler to a German Princess, Vol. II, Letter XI

This is the fifth of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Click here to read the previous letter.

Letter XI. Reflections on Divisibility in infinitum, and on Monads.

In speaking of the divisibility of body, we must carefully distinguish what is in our power, from what is possible in itself. In the first sense, it cannot be denied, that such a division of a body as we are capable of, must be very limited.

By pounding a stone we can easily reduce it to powder; and if it were possible to reckon all the little grains which form that powder, their number would undoubtedly be so great, that it would be matter of surprize, to have divided the stone into so many parts. But these very grains will be almost indivisible with respect to us, as no instrument we could employ would be able to lay hold of them. But it cannot with truth be affirmed that they are indivisible in themselves. You have only to view them with a good microscope, and each will appear itself a considerable stone, on which are distinguishable a great many points and inequalities; which demonstrates the possibility of a farther division, though we are not in a condition to execute it. For wherever we can distinguish several points in any object, it must be divisible into so many parts.

We speak not, therefore, of a division practicable by our strength and skill, but of that which is possible in itself, and which the Divine Omnipotence is able to accomplish.

It is in this sense, accordingly, that philosophers use the word ‘divisibility:’ so that if there were a stone so hard that no force could break it, it might be without hesitation affirmed as divisible, in its own nature, as the most brittle, of the same magnitude. And how many bodies are there on which we cannot lay any hold, and of whose divisibility we can entertain not the smallest doubt? No one doubts that the moon is a divisible body, though he is incapable of detaching the smallest particle from it: and the simple reason for its divisibility, is its being extended.

Wherever we remark extension, we are under the necessity of acknowledging divisibility, so that divisibility is an inseparable property of extension. But experience likewise demonstrates that the division of bodies extends very far. I shall not insist at great length on the instance usually produced of a ducat*: the artisan can beat it out into a leaf so fine, as to cover a very large surface, and the ducat may be divided into as many parts as that surface is capable of being divided. Our own body furnishes an example much more surprizing. Only consider the delicate veins and nerves with which it is filled, and the fluids which circulate through them. The subtility there discoverable far surpasses imagination.

*A ducat is a gold coin used in Euler’s day.

The smallest insects, such as are scarcely visible to the naked eye, have all their members, and legs on which they walk with amazing velocity. Hence we see that each limb has its muscles composed of a great number of fibres; that they have veins, and nerves, and a fluid still much more subtile which flows through their whole extent.

On viewing with a good microscope a single drop of water, it has the appearance of a sea; we see thousands of living creatures swimming in it, each of which is necessarily composed of an infinite number of muscular and nervous fibres, whose marvellous structure ought to excite our admiration. And though these creatures may perhaps be the smallest which we are capable of discovering by the help of the microsope, undoubtedly they are not the smallest which the Creator has produced. Animacules probably exist as small relatively to them, as they are relatively to us. And these after all are not yet the smallest, but may be followed by an infinity of new classes, each of which contains creatures incomparably smaller than those of the preceding class.

We ought in this to acknowledge the omnipotence and infinite wisdom of the Creator, as in objects of the greatest magnitude. it appears to me, that the consideration of these minute species, each of which is followed by another inconceivably more minute, ought to make the liveliest impression on our minds, and inspire us with the most sublime ideas of the works of the Almighty, whose power knows no bounds, whether as to great objects or small.

To imagine that after having divided a body into a great number of parts, we arrive, at length, at particles so small as to defy all farther division, is therefore the indication of a very contracted mind. But supposing it is possible to descend to particles so minute as to be, in their own nature, no longer divisible, as in the case of the supposed monads; before coming to this point, we shall have a particle composed of only two monads, and this particle will be of a certain magnitude or extension, otherwise it could not have been divisible into these two monads. Let us farther suppose, that this particle, as it has some extension, may be the thousandth part of an inch, or still smaller if you will; for it is of no importance, what I say of the thousandth part of an inch may be said with equal truth of every smaller part. This thousandth part of an inch, then, is composed of two monads, and consequently two monads together would be the thousandth part of an inch, and two thousand times nothing, a whole inch; the absurdity strikes at first light.

The partisans of the system of monads accordingly shrink from the force of this argument, and are reduced to a terrible nonplus when asked how many monads are requisite to constitute an extension. Two, they apprehend, would appear insufficient, they therefore allow that more must be necessary. But, if two monads cannot constitute extension, as each of the two has none; neither three, nor four, nor any number whatever will produce it; and this complexity subverts the system of monads.

9th May, 1761.

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