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

This is the sixth 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.

The partisans of monads are far from submitting to the arguments adduced to establish the divisibility of body to infinity. Without attacking them directly, they allege that divisibility in infinitum is a chimera of geometricians, and that it is involved in contradiction. For, if each body is divisible to infinity, it would contain an infinite number of parts, the smallest bodies as well as the greatest: the number of these particles to which divisibility in infinitum would lead, that is to say, the most minute of which bodies are composed, will then be as great in the smallest body as in the largest, this number being infinite in both; and hence the partisans of monads triumph in their reasoning as invincible. For, if the number of ultimate particles of which two bodies are composed is the same for both, it must follow, say they, that the bodies are perfectly equal to each other.

Now this goes on the supposition, that the ultimate particles are all perfectly equal to each other; for if some were greater than others, it would not be surprizing that one of the two bodies should be much greater than the other. But it is absolutely necessary, say they, that the ultimate particles of all bodies should be equal to each other, as they no longer have any extension, and their magnitude absolutely vanishes, or becomes nothing. They even form a new objection, by alleging that all bodies would be composed of an infinite number of nothings, which is still a greater absurdity.

I readily admit this; but I remark at the same time, that it ill becomes them to raise such an objection, seeing they maintain, that all bodies are composed of a certain number of monads, though, relatively to magnitude, they are absolute nothings; so that by their own confession, several nothings are capable of producing a body. They are right in saying their monads are not nothings, but beings endowed with an excellent quality, on which the nature of the bodies which they compose is founded. Now, the only question here is respecting extension; and as they are under the necessity of admitting that their monads have none, several nothings, according to them, would always be something.

But I shall push this argument against the system of monads no farther; my object being to make a direct reply to the objection founded on the ultimate particles of bodies, raised by the monadists in support of their system, by which they flatter themselves in confidence of a complete victory over the partisans of divisibility in infinitum.

I should be glad to know, in the first place, what they mean by the ultimate particles of bodies. In their system, according to which every body is composed of a certain number of monads, I clearly comprehend that the ultimate particles of a body, are the monads themselves which constitute it; but in the system of divisibility in infinitum, the term ultimate particle is absolutely unintelligible.

They are right  in saying, that these are the particles at which we arrive from the division of bodies, after having continued to infinity. But this is just the same thing with saying, after having finished a division which never comes to an end. For divisibility in infinitum means nothing else but the possibility of always carrying on the division, without ever arriving at the point where it would be necessary to stop. He who maintains divisibility in infinitum, boldly denies, therefore, the existence of the ultimate particles of body; and it is a manifest contradiction, to suppose at once ultimate particles and divisibility in infinitum.

I reply, then, to the partisans of the system of monads, that their objection to the divisibility of body to infinity would be a very solid one, did that system admit of ultimate particles; but being expressly excluded from it, all this reasoning, of course, falls to the ground.

It is false, therefore, that in the system of divisibility in infinitum, bodies are composed of an infinity of particles. However closely connected these two propositions may appear to the partisans of monads, they manifestly contradict each other; for whoever maintains that body is divisible in infinitum, or without end, absolutely denies the existence of ultimate particles, and consequently has no concern in the question. The term can only mean such particles as are no longer divisible, an idea totally inconsistent with the system of divisibility in infinitum. This formidable attack, then, is completely repelled.

12th May, 1761.

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