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

July 17, 2018

This is the seventh 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.

Principle of the satisfying Reason, the strongest Support of the Monadists.

You must be perfectly sensible that one of the two systems, which have undergone such ample discussion, is necessarily true, and the other false, seeing they are contradictory. It is admitted on both sides, that bodies are divisible: the only question is, “Whether this divisibility is limited?”, or “Whether it may always be carried further, without the possibility of ever arriving at indivisible particles?”

The system of monads is established in the former case, since after having divided a body into indivisible particles, these very particles are monads, and there would be reason for saying that all bodies are composed of them, and each of a certain determinate number. Whoever denies the system of monads, must likewise, then, deny that the divisibility of bodies is limited. he is under the necessity of maintaining, that it is always possible to carry this divisibility further, without ever being obliged to stop; and this is the case of divisibility in infinitum, on which system we absolutely deny the existence of ultimate particles: consequently the difficulties resulting from their infinite number fall to the ground themselves. In denying monads, it is impossible to talk any longer of ultimate particles, and still less of the number of them which enters into the composition of each body.

You must have remarked, that what I have hitherto produced in support of the system of monads is destitute of solidity. I now proceed to inform you that its supporters rest their cause chiefly on the great principle of the sufficient reason, which they know how to employ so dexterously, that by means of it they are in a condition to demonstrate whatever suits their purpose, and to demolish whatever makes against them. The blessed discovery made, then, is this, “That nothing can be without a sufficient reason;” and to modern philosophers we stand indebted for it.

In order to give you  an idea of this principle, you have only to consider, that in every thing presented to you, it may always be asked, “Why it is such?” And the answer is what they call the sufficient reason, supporting it really to correspond with the question proposed. Wherever the “why” can take place, the possibility of a satisfactory answer is taken for granted, which shall, of course, contain the sufficient reason of the thing.

This is very far, however, from being a mystery of modern discovery. Men in every age have asked “why;” an incontestable proof of their conviction that every thing must have a satisfying reason of its existence. This principle, that nothing is without a cause,  was very well known to ancient philosophers; but unhappily this cause is for the most part concealed from us. To little purpose do we ask “why:” no one is qualified to assign the reason. It is not a matter of doubt, that every thing has its cause; but a progress thus far hardly deserves the name; and so long as it remains concealed, we have not advanced a single step in real knowledge.

You may perhaps imagine, that modern philosophers, who make such a boast of the principle of a satisfying reason, have actually discovered that of all things, and are in a condition to answer every why that can be proposed to them; which would undoubtedly be their very summit of human knowledge; but, in this respect, they are just as ignorant as their neighbors: their whole merit amounts to no more than pretension to have demonstrated, that wherever it is possible to ask the question “why,” there must be a satisfying answer to it, though concealed from us.

They readily admit, that the ancients had a knowledge of this principle, but a knowledge very obscure; whereas they pretend to have placed it in its clearest light, and to have demonstrated the truth of it: and therefore it is that they know how to turn it most to their account, and that this principle puts them in a condition to prove, that bodies are composed of monads.

Bodies, they say, must have their sufficient reason somewhere, but if they were divisible to infinity, such reason could not take place: and hence they conclude, with an air altogether philosophic, “that, as every thing must have its sufficient reason, it is absolutely necessary that  all bodies should be composed of monads:” which was to be demonstrated. This, I must admit, is a demonstration to be resisted.

It were greatly to be wished that a reasoning so slight could elucidate to us questions of this importance; but I frankly confess, I comprehend nothing of the matter. They talk of the sufficient reason of bodies, by which they mean to reply to a certain “wherefore,” which remains unexplained. But it would be proper, undoubtedly, clearly to understand, and carefully to examine a question, before a reply is attempted; in the present case, the answer is given before the question is formed.

Is it asked, “Why do bodies exist?” It would be ridiculous, in my opinion, to reply, “Because they are composed of monads;” as if they contained the cause of that existence. Monads have not created bodies: and when I ask, “Why such a being exists?” I see no other reason that can be given but this, “Because the Creator has given it existence;” and as to the manner in which creation is performed, philosophers, I think, would do well honestly to acknowledge their ignorance.

But they maintain, that God could not have produced bodies, without having created monads, which were necessary to form the composition of them. This manifestly supposes, that bodies are composed of monads, the point which they meant to prove by this reasoning. And you are abundantly sensible, that it is not fair reasoning to take for granted the truth of a proposition which you are bound to prove by reasoning. It is a sophism known in logic by the name of a petitio principii, or, begging the question.

16th May, 1761.

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

July 9, 2018

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.

Reply to the Objections of the Monadists to Divisibility in infinitum.

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

July 3, 2018

This is the fourth 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, which I posted over a year ago. Sorry, I’m a little slow on posting these!

Letter X. Of Monads.

When we talk, in company, on philosophical subjects, the conversation usually turns on such articles as have excited violent disputes among philosophers.

The divisibility of the body is one of them, respecting which the sentiments of the learned are greatly divided. Some maintain, that this divisibility goes on to infinity, without the possibility of ever arriving at particles so small as to be susceptible of no farther division. But others insist, that this division extends only to a certain point, and that you may come at length to particles so minute, that, having no magnitude, they are no longer divisible. These ultimate particles, which enter into the composition of bodies, they denominate simple beings, and monads.

There was a time when the dispute respecting monads employed such general attention, and was conducted with so much warmth, that it forced its way into company of every description, that of the guard-room not excepted. There was scarce a lady at court who did not take a decided part in favour of monads or against them. In a word all conversation was engrossed by monads, no other subject could find admission.

The Royal Academy of Berlin took up the controversy, and being accustomed annually to propose a question for discussion, and to bestow a gold medal of the value of fifty ducats on the person who in the judgment of the Academy has given the most ingenious solution, the question respecting monads was selected for the year 1748. A great variety of essays on the subject were accordingly produced. The president Mr. de Maupertuis named a committee to examine them, under the direction of the late Count Dohna, great chamberlain to the queen; who, being an impartial judge, examined with all imaginable attention, the arguments adduced both for and against the existence of monads. Upon the whole, it was found that those which went to the establishment of their existence were so feeble, and so chimerical, that they tended to the subversion of all the principles of human knowledge. The question was, therefore, determined in favour of the opposite opinion, and the prize adjudged to Mr. Justi, whose piece was deemed the most complete refutation of the monadists.

You may easily imagine how violently this decision of the Academy must irritate the partisans of monads, at the head of whom stood the celebrated Mr. Wolff. His followers, who were then much more numerous, and more formidable than at present, exclaimed in high terms against the partiality and injustice of the Academy; and their chief had well nigh proceeded to launch the thunder of a philosophical anathema against it. I do not now recollect to whom we are indebted for the care of averting this disaster.

As this controversy has made a great deal of noise, you will not be displeased, undoubtedly, if I dwell  a little upon it. The whole is reduced to this simple question, “Is the body divisible to infinity?” or in other words, “Has the divisibility of bodies any bound, or has it not?” I have already remarked as to this, that extension, geometrically considered, is on all hands allowed to be divisible infinitum; because, however small a magnitude may be, it is possible to conceive the half of it, and again the half of that half, and so on to infinity.

This notion of extension is very abstract, as are those of all genera, such as that of man, of horse, of tree, etc., as far as they are not applied to an individual and determinate being. Again, it is the most certain principle of all our knowledge, that whatever can be truly affirmed of the genus, must be true of all the individuals comprehended under it. If therefore all bodies are extended, all the properties belonging to extension must belong to each body in particular. Now all bodies are extended; and extension is divisible to infinity; therefore every body must be so likewise. This is a syllogism of the best form; and as the first proposition is indubitable, all that remains, is to be assured that the second is true, that is, whether it be true or not, that bodies are extended.

The partisans of monads, in maintaining their opinion, are obliged to affirm, that bodies are not extended, but have only an appearance of extension. They imagine that by this they have subverted the argument adduced in support of the divisibility in infinitum. But if body is not extended, I should be glad to know, from whence we derived, the idea of extension; for, if body is not extended, nothing in the world is, as spirits are still less so. Our idea of extension, therefore, would be altogether imaginary and chimerical.

Geometry would accordingly be a speculation entirely useless and illusory, and never could admit of any application to things really existing. In effect, if no one thing is extended, to what purpose investigate the properties of extension? But as geometry is, beyond contradiction, one of the most useful of sciences, its object cannot possibly be a mere chimera.

There is a necessity, then, of admitting, that the object of geometry is at least the same apparent extension which those philosophers allow to body; but, this very object is divisible to infinity: therefore existing beings, endowed with this apparent extension, must necessarily be extended.

Finally, let those philosophers turn themselves which way soever they will in support of their monads, or those ultimate and minute particles, divested of all magnitude, of which, according to them, all bodies are composed, they still plunge into difficulties, out of which they cannot extricate themselves. They are right in saying, that it is a proof of dulness to be incapable of relishing their sublime doctrine; it may however be remarked, that here the greatest stupidity is the most successful.

5th May, 1761.

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