Leonhard Euler's letters to a German Princess


Euler's letters to a German Princess were written in French to Princess Friederike Charlotte, a fifteen year old member of the Prussian royal family. When he began writing the letters, Euler was 53 years old and the Director of the Mathematics Section of the Prussian Academy of Sciences in Berlin. Over a period of two years, beginning in 1760, he wrote 234 letters to the Princess. First published in three volumes (the first two in St Petersburg in 1768 and the third in Frankfurt in 1774), they were reprinted in Paris by the Marquis de Condorcet and Sylvestre François Lacroix in 1787, 1788 and 1789. The Paris editions were translated into English by the Scottish clergyman Henry Hunter (1741-1802) and published in 1795. Hunter wrote in the Preface:-
The time, I trust, is at hand, when the Letters of Euler, or some such book, will be daily on the breakfasting table, in the parlour of every female academy in the kingdom; and when a young woman, while learning the useful arts of pastry and plain-work, may likewise be acquainting herself with the phases of the moon, and the flux and reflux of the tides. And I am persuaded she may thrum on the guitar, or touch the keys of the harpsichord, much more agreeably both to herself and others, by studying a little the theory of sound. I have put the means of this in her power; it will be at once her fault, and her folly, if she neglect it.
We give below three examples of these letters. In the first two letters, we note that various different editions of this work contain different measures. We keep Euler's original measures, in which he uses German miles (one German mile equals about 4354\large\frac{3}{5}\normalsize English miles). We also keep Euler's understanding of the solar system. Later editions of the work add in the other planets that had been discovered and minor plants such as Ceres etc. This is confusing, since the letter still has the original 1760 date. We have opted for the closest to Euler's original letter as possible.

1. Of Magnitude, or Extension

The hope of having the honour to communicate in person to your highness my lessons in geometry becoming more and more distant, which is a very sensible mortification to me, I feel myself impelled to supply person instruction by writing, as far as the nature of the subjects will permit.

I begin my attempt by assisting you to form a just idea of Magnitude; producing, as examples, the smallest as well as the greatest extensions of matter actually discoverable in the system of the universe. And, first, it is necessary to fix on some one determinate division of measure, obvious to the senses, and of which we have an exact idea, that of a foot, for instance. The quantity of this, once established, and rendered familiar to the eye, will enable us to form the idea of every other quantity, as to length, great or small; the former, by ascertaining how many feet it contains; and the latter, by ascertaining what part of a foot measures it. For, having the idea of a foot, we have that also of it's half, of it's quarter, of it's twelfth part, denominated an inch, of its hundredth, and of its thousandth part, which is so small as almost to escape sight. But it is to be remarked that there are animals, not of greater extension than this last subdivision of a foot, which, however, are composed of members through which the blood circulates, and which again contain other animals, as diminutive compared to them, as they are compared to us. Hence it may be concluded that animals exist, whose smallness eludes the imagination; and that these again are divisible into parts inconceivably smaller. Thus, for example, though the ten thousandth part of a foot be too small for sight, and, compared to us, ceases to be an object of sense, it nevertheless surpasses in magnitude certain complete animals; and must, to one of those animals, were it endowed with the power of perception, appear extremely great.

Let us now make the transition from these minute quantities, in pursuing which the mind is lost, to those of the greatest magnitude. You have the idea of a mile; the distance from hence to Magdeburg is computed to be 18 [German] miles; a mile contains 24,000 feet, and we employ it in measuring the distance of the different regions of the globe, in order to avoid numbers inconceivably great, in our calculations, which must be the case if we used foot instead of mile. A mile then, containing 24,000 feet, when it is said that Magdeburg is 18 miles from Berlin, the idea is much clearer, than if the distance of these two cities were said to be 432,000 feet: a number so great almost overwhelms the understanding. Again, we have a tolerably just idea of the magnitude of the earth, when we are told that this circumference is about 5,400 miles. And the diameter being a straight line passing through the centre, and terminating in opposite directions, in the surface of the sphere, which is the acknowledged figure of the Earth, for which reason also we give it the name of globe - the diameter of this globe is calculated to be 1720 miles; and this is the measurement which we employ for determining the greatest distances discoverable in the heavens. Of all the heavenly bodies the Moon is nearest to us, being distant only about 30 diameters of the earth, which amount to 51,600 miles, or 1,238,400,000 feet; but the first computation of 30 diameters of the earth is the clearest idea. The Sun is about 400 times farther from us than the moon, and when we say his distance is 9,000 diameters of the earth, we have a much clearer idea than if it were expressed in miles or in feet.

You know that the earth performs a revolution round the fun in the space of a year, but that the sun remains fixed. Beside the earth, there are five other similar bodies, named planets, which revolve round the sun; two of them at smaller distances, Mercury and Venus, and three at greater, namely Mars, Jupiter and Saturn. All the other stars which we see, comets excepted, are called fixed; and their distance from us is incomparably greater than that of the sun. Their distances are undoubtedly very unequal, which is the reason that some of these bodies appear greater than others. But the nearest of them is, unquestionably, above 5,000 times more distant than the sun: its distance from us, accordingly, exceeds 45,000,000 diameters of the earth, that is 77,400,000,000 miles, and this again, multiplied by 24,000, will give that prodigious distance in feet. And this, after all, is the distance only of those fixed stars which are the nearest to us; the most remote which we see are perhaps a hundred times farther off. It is probable, at the same time, that these stars taken together, constitute only a very small part of the whole universe, relatively to which these prodigious distances are not greater than a grain of sand compared with the earth. This immensity is the work of the Almighty, who governs the greatest bodies and the smallest.

Berlin, 19 April 1760.

2. Of Velocity

Flattering myself that your Highness may be pleased to accept the continuation of my instructions, a specimen of which I took the liberty of presenting to you in a former letter, I proceed to unfold the idea of velocity, which is a particular species of extension, and susceptible of increase and of diminution. When any substance is transported, that is, when it passes from one place to another, we ascribe to it a velocity. Let two persons, the one on horseback, the other on foot, proceed from Berlin to Magdeburg, we have, in both cases, the idea of a certain velocity; but it will be immediately affirmed, that the velocity of the former exceeds that of the latter.

The question, then, is: Wherein consists the difference which we observe between these several degrees of velocity? The road is the same to him who rides and, to him who walks: but the difference evidently lies in the time which each employs in performing the same course. The velocity of the horseman is the greater of the two, as he employs less time on the road from Berlin to Magdeburg; and the velocity of the other is less, because he employs more time in travelling the same distance. Hence it is clear, that in ,order to form an accurate idea of velocity, we must attend at once to two kinds of quantity: namely to the length of the road, and to the time employed. A body, therefore, which in the same time passes through double the space which another body does, has double its velocity; if, in the same time, it passes through thrice the distance, it is said to have thrice the velocity, and so on. We shall comprehend, then, the velocity of a body, when we are informed of the space through which it passes in a certain quantity of time.

In order to know the velocity of my pace, when I walk to Lytzow, I have observed that I make 120 steps in a minute, and one of my steps is equal to two feet and a half. My velocity then, is such, as to carry me 300 feet in a minute, and a space sixty times greater, or 18,000 feet in an hour, which however does not amount to a mile, for this, being 24,000 feet, would require an hour and 20 minutes. Were I, therefore, to walk from hence to Magdeburg, it would take exactly 24 hours. This conveys an accurate idea of the velocity with which I am able to walk. Now it is easy to comprehend what is meant by a greater or less velocity. For if a courier were to go from hence to Magdeburg in 12 hours, his velocity would be the double of mine; if he went in eight hours, his velocity would be triple. We remark a very great difference in the degrees of velocity. The tortoise furnishes an example of a velocity extremely small. If she advances only one foot in a minute, her velocity is 300 times less than mine, for I advance 300 feet in the same time. We are likewise acquainted with velocities much greater. That of the wind admits of great variation. A moderate wind goes at the rate of 10 feet in a second, or 600 feet in a minute; its velocity therefore is the double of mine. A wind that runs 20 feet in a second, or 1200 in a minute, is rather strong; and a wind which flies at the rate of 50 feet in a second is extremely violent, though its velocity is only ten times greater than mine, and would take two hours and twenty-four minutes to blow from hence to Magdeburg.

The velocity of sound comes next, which moves 1000 feet in a second, and 60,000 in a minute. This velocity therefore, is 200 times greater than that of my pace; and were a cannon to be fired at Magdeburg, if the report could be heard at Berlin, it would arrive there in seven minutes. A cannon-ball moves with nearly the same velocity; but when the piece is loaded to the utmost, the ball is supposed capable of flying 2,000 feet in a second, or 120,000 in a minute. This velocity appears prodigious, though it is only 400 times greater than that of my pace in walking to Lytzow; it is at the same time the greatest velocity known upon earth. But there are in the heavens velocities far greater, though their motion appears to be extremely deliberate. You know that the earth turns round on its axis in 24 hours: every point of it's surface, then, under the equator, moves 5,400 miles in 24 hours, while I am able to get through only 18 miles. Its velocity is accordingly above 300 times greater than mine, and less notwithstanding than the greatest possible velocity of a cannon-ball. The earth performs its revolution round the sun in the space of a year, proceeding at the rate of 128,250 miles in 24 hours. Its velocity, therefore, is 18 times more rapid than that of a cannon ball. The greatest velocity of which we have any knowledge is, undoubtedly, that of light, which moves 2,000,000 of miles every minute, and exceeds the velocity of a cannon ball 4000,000 times.

22 April 1760

3. Of the Pleasure derived from fine Music

It is a question as important as curious, whence is it that a fine piece of music excites a sentiment of pleasure? The learned differ on this subject. Some pretend that it is mere caprice, and that the pleasure produced by music is not founded on reason, because what is grateful to one is disgusting to another. Far from deciding the question, this renders it only more complicated. The very point to be determined is, how comes it that the same piece of music produces effects so different, since all admit that nothing happens without reason? Others maintain that the pleasure derived from fine music consists in the perception of the order which pervades it.

This opinion appears at first sight sufficiently well founded, and merits a more attentive examination. Music presents objects of two kinds, in which order is essential. The one relates to the difference of the sharp or flat tones; and you will recollect, that it consists in the number of vibrations performed by each note in the same time. This difference, which is perceptible between the quickness of the vibrations of all sounds, is what is properly called harmony. The effect of a piece of music, of which we feel the relations of the vibrations of all the notes that compose it, is the production of harmony. Thus, two notes which differ an octave excite a perception of 1 to 2; a fifth, of that of 2 to 3; and a greater third, of that of 4 to 5.

We comprehend, then, the order which is found in harmony, when we know all the relations which pervade the notes of which it is composed; and it is the perception of the ear which leads to this knowledge. This perception, more or less delicate, determines why the same harmony is felt by one, and not at all by another, especially when the relations of the notes are expressed by somewhat greater numbers. Music contains, besides harmony, another object equally susceptible of order, namely, the measure, by which we assign to every note a certain duration; and the perception of the measure consists in the knowledge of this duration, and of the relations which result from it.

The drum and tymbal furnish the example of a music in which measure alone takes place, as all the notes are equal among themselves, and then there is no harmony. There is likewise a music consisting wholly in harmony, to the exclusion of measure. This music is the choral, in which all the notes are of the same duration; but perfect music unites harmony and measure. Thus the connoisseur who hears a piece of music, and who comprehends, by the acute perception of his ear, all the proportions on which both the harmony and the measure are founded, has certainly the most perfect knowledge possible of that music: while another, who perceives these proportions only in part, or not at all, understands nothing of the matter, or possesses at most a very slender knowledge of it. But the sentiment of pleasure excited by fine music, must not be confounded with the knowledge of which I have been speaking, though it may be confidently affirmed, that a piece of music cannot produce any, unless the relations of it are perceived. For this knowledge alone is not sufficient to excite the sentiment of pleasure; something more is wanting, which no one hitherto has unfolded.

In order to be convinced that the perception alone of all the proportions of a piece of music is insufficient to produce pleasure, you have only to consider music of a very simple construction, such as goes in octaves alone, in which the perception of proportions is undoubtedly the easiest. Such music would be far from conveying pleasure, though you might have the most perfect knowledge of it. It will be said then that pleasure requires a knowledge not quite so easily attained, a knowledge that occasions some trouble; which must, if I may use the expression, cost us something. But, in my opinion, neither is this a satisfactory solution. A dissonance, the relations of which are expressed by the highest numbers, is caught with more difficulty; a series of dissonances, however, following without choice, and without design, cannot please. The composer must therefore have pursued, in his work, a certain plan, executed in real and perceptible proportions.

Then a connoisseur on hearing such a piece, and comprehending, beside the proportions, the very plan and design which the composer had in view, will feel that satisfaction which constitutes the pleasure procured by exquisite music to an ear accustomed to relish the beauties and delicacies of that enchanting art. It arises, then, from divining in some measure the views and feelings of the composer, whose execution, when fortunate, fills the soul with an agreeable sensation. It is a satisfaction somewhat similar to that which is derived from the sight of a well-acted pantomime, in which you may conjecture, by the gesture and action, the sentiments and dialogue intended to be expressed, and which presents, besides, a well-digested plan. The enigma of the chimney-sweeper, which was so diverting to your highness, furnishes me with another excellent comparison. When you can guess the sense, and discover that it is perfectly expressed in the proposition of the enigma, you feel a very sensible pleasure on making the discovery; but insipid and incongruous enigmas produce none. Such are, if I may be permitted to judge, the true principles on which decisions respecting the excellency of musical compositions are founded.

6 May 1760

Last Updated July 2020