Posted tagged ‘measurable’

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.