Studying His Word and His Works

This is the tenth 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. Yes, this is the tenth letter on infinitesimals, but letter XVI=16 in Volume 2 of Euler’s book.  Click here to read the previous letter.

Continuation.

The system of monads, such as I have been describing it, is a necessary consequence from the principle, that bodies are compounded of simple beings. The moment this principle is admitted, you are obliged to acknowledge the justness of all the other consequences, which result from it so naturally, that it is impossible to reject any one, however absurd and contradictory.

First, these simple beings which must enter into the composition of bodies, being monads which have no extension*, neither can their compounds, that is bodies, have any; and all these extensions become illusion, chimera, it being certain, that parts destitute of extension are incapable of producing a real extension; it can be, at most, an appearance, or a phantom which dazzles by a fallacious idea of extension. In a word, every thing becomes illusion, and upon this is founded the system of pre-established harmony, the difficulties of which I have already pointed out.

*IF A BODY HAS EXTENSION, THAT MEANS YOU CAN MEASURE IT (LENGTH, MASS, ETC.).

It is necessary then to take care that we be not entangled in this labyrinth of absurdities. If you make a single false step over the threshold, you are involved beyond the power of escaping. Everything depends on the first ideas formed of extension; and the manner in which the partisans of the system of monads endeavour to establish it, is extremely seductive.

These philosophers love not to speak of the extension of bodies, because they clearly foresee, that it must become fatal to them in the sequel; but instead of saying, that bodies are extended, they denominate them compound beings, which no one can deny, as extension necessarily supposes divisibility, and consequently a combination of parts which constitute bodies. But they presently make a wrong use of this notion of a compound being. For, say they, a being can be compounded only so far as it is made up of simple beings; and hence they conclude, that every body is compounded of simple beings. As soon as you grant them this conclusion, you are caught, beyond the power of retreating; for you are under the necessity of admitting, that these simple beings, not being compounded, are not extended.

This captious argument is exceedingly seductive. If you permit yourself to be dazzled with it, they have gained their point. Only admit this proposition, bodies are compounded of simple beings, that is, of parts which have no extension, and you are entangled. With all your might, then, resist this assertion: every compound being is made up of simple beings; and though you may not be able directly to prove the fallacy, the absurd consequences which immediately result, would be sufficient to overthrow it.

In effect, they admit that bodies are extended; from this point the partisans of the system of monads set out, to establish the proposition that they are compound beings; and having hence deduced, that bodies are compounded of simple beings, they are obliged to allow, that simple beings are incapable of producing real extension, and consequently, that the extension of bodies is mere illusion.

An argument whose conclusion is a direct contradiction of the premises is singularly strange: this reasoning sets out with advancing that bodies are extended; for, if they were not, how could it be known that they are compound beings, and then comes the conclusion, that they are not so. Never was a fallacious argument, in my opinion, more completely refuted than this has been. The question was, Why are bodies extended? And, after a little turning and winding, it is answered, Because they are not so. Were I to be asked, Why has a triangle three sides? and I should reply, that it is a mere illusion, would such a reply be deemed satisfactory?

It is therefore certain, that this proposition, ‘Every compound being is necessarily made up of simple beings,’ leads to a false conclusion, however well-founded it may appear to the partisans of monads, who even pretend to rank it among the axioms, or first principles of human knowledge. The absurdity in which it immediately issues, is sufficient to overturn it, were there no other reasons for calling it in question.

But as a compound being here means the same thing, as an extended being, it is just as if it were affirmed, ‘ Every extended being is compounded of beings which are not so.’ And this is precisely the question. It is asked, Whether, on dividing a body, you arrive at length at parts unsusceptible of any farther division, for want of extension; or Whether you never arrive at particles such as that the divisibility should be unbounded?

In order to determine this important question, for the sake of argument let it be supposed, that every body is compounded of parts without extension. Certain specious reasonings may easily be employed, drawn from the noted principle of the sufficient reason; and it will be said, that a compound being can have its sufficient reason only in the simple beings which compose it; which might be true, if the compound being were in fact made up of simple beings, the very point in question; and whenever this composition is denied, the sufficient reason becomes totally inapplicable.

But it is dangerous to enter the lists with persons who believe in monads; for, besides that there is nothing to be gained, they loudly exclaim that you are attacking the principle of the sufficient reason, which is the basis of all certainty, even of the existence of God. According to them, whoever refuses to admit monads, and rejects the magnificent fabric, in which every thing is illusion, is an infidel and an atheist. Sure I am that such a frivolous imputation will make not the slightest impression on your mind, but that you will perceive the wild extravagancies into which men are driven, when they embrace the system of monads, a system too absurd to need a refutation in detail. Their foundation being absolutely reduced to a wretched abuse of the principle of the sufficient reason.

26th May, 1761.