## Posted tagged ‘Shormann Calculus’

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

July 5, 2018

This is the fifth 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.

### Letter XI. Reflections on Divisibility in infinitum, and on Monads.

In speaking of the divisibility of body, we must carefully distinguish what is in our power, from what is possible in itself. In the first sense, it cannot be denied, that such a division of a body as we are capable of, must be very limited.

By pounding a stone we can easily reduce it to powder; and if it were possible to reckon all the little grains which form that powder, their number would undoubtedly be so great, that it would be matter of surprize, to have divided the stone into so many parts. But these very grains will be almost indivisible with respect to us, as no instrument we could employ would be able to lay hold of them. But it cannot with truth be affirmed that they are indivisible in themselves. You have only to view them with a good microscope, and each will appear itself a considerable stone, on which are distinguishable a great many points and inequalities; which demonstrates the possibility of a farther division, though we are not in a condition to execute it. For wherever we can distinguish several points in any object, it must be divisible into so many parts.

We speak not, therefore, of a division practicable by our strength and skill, but of that which is possible in itself, and which the Divine Omnipotence is able to accomplish.

It is in this sense, accordingly, that philosophers use the word ‘divisibility:’ so that if there were a stone so hard that no force could break it, it might be without hesitation affirmed as divisible, in its own nature, as the most brittle, of the same magnitude. And how many bodies are there on which we cannot lay any hold, and of whose divisibility we can entertain not the smallest doubt? No one doubts that the moon is a divisible body, though he is incapable of detaching the smallest particle from it: and the simple reason for its divisibility, is its being extended.

Wherever we remark extension, we are under the necessity of acknowledging divisibility, so that divisibility is an inseparable property of extension. But experience likewise demonstrates that the division of bodies extends very far. I shall not insist at great length on the instance usually produced of a ducat*: the artisan can beat it out into a leaf so fine, as to cover a very large surface, and the ducat may be divided into as many parts as that surface is capable of being divided. Our own body furnishes an example much more surprizing. Only consider the delicate veins and nerves with which it is filled, and the fluids which circulate through them. The subtility there discoverable far surpasses imagination.

##### *A ducat is a gold coin used in Euler’s day.

The smallest insects, such as are scarcely visible to the naked eye, have all their members, and legs on which they walk with amazing velocity. Hence we see that each limb has its muscles composed of a great number of fibres; that they have veins, and nerves, and a fluid still much more subtile which flows through their whole extent.

On viewing with a good microscope a single drop of water, it has the appearance of a sea; we see thousands of living creatures swimming in it, each of which is necessarily composed of an infinite number of muscular and nervous fibres, whose marvellous structure ought to excite our admiration. And though these creatures may perhaps be the smallest which we are capable of discovering by the help of the microsope, undoubtedly they are not the smallest which the Creator has produced. Animacules probably exist as small relatively to them, as they are relatively to us. And these after all are not yet the smallest, but may be followed by an infinity of new classes, each of which contains creatures incomparably smaller than those of the preceding class.

We ought in this to acknowledge the omnipotence and infinite wisdom of the Creator, as in objects of the greatest magnitude. it appears to me, that the consideration of these minute species, each of which is followed by another inconceivably more minute, ought to make the liveliest impression on our minds, and inspire us with the most sublime ideas of the works of the Almighty, whose power knows no bounds, whether as to great objects or small.

To imagine that after having divided a body into a great number of parts, we arrive, at length, at particles so small as to defy all farther division, is therefore the indication of a very contracted mind. But supposing it is possible to descend to particles so minute as to be, in their own nature, no longer divisible, as in the case of the supposed monads; before coming to this point, we shall have a particle composed of only two monads, and this particle will be of a certain magnitude or extension, otherwise it could not have been divisible into these two monads. Let us farther suppose, that this particle, as it has some extension, may be the thousandth part of an inch, or still smaller if you will; for it is of no importance, what I say of the thousandth part of an inch may be said with equal truth of every smaller part. This thousandth part of an inch, then, is composed of two monads, and consequently two monads together would be the thousandth part of an inch, and two thousand times nothing, a whole inch; the absurdity strikes at first light.

The partisans of the system of monads accordingly shrink from the force of this argument, and are reduced to a terrible nonplus when asked how many monads are requisite to constitute an extension. Two, they apprehend, would appear insufficient, they therefore allow that more must be necessary. But, if two monads cannot constitute extension, as each of the two has none; neither three, nor four, nor any number whatever will produce it; and this complexity subverts the system of monads.

9th May, 1761.

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

May 22, 2017

This is the second 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.

### Letter VIII. Divisibility of Extension in Infinitum

The controversy between modern philosophies and geometricians to which I have alluded, turns on the divisibility of body. This property is undoubtedly founded on extension, and it is only in so far as bodies are extended that they are divisible, and capable of being reduced to parts.

You will recollect that in geometry it is always possible to divide a line, however small, into two equal parts. We are likewise, by that science, instructed in the method of dividing a small line, ai, into any number of equal parts at pleasure, and the construction of this division is there demonstrated beyond the possibility of doubting its accuracy.

You have only to draw a line AI (plate II. fig. 23) parallel to ai of any length, and at any distance you please, and to divide it into as many equal parts AB, BC, CD, DE, etc. as the small line given is to have divisions, say eight. Draw afterwards, through the extremities A, a, and I, i the straight lines AaO, IiO, till they meet in the point O: and from O draw toward the points of division B, C, D, E, etc. the straight lines OB, OC, OD, OE, etc., which shall likewise cut the small line ai into eight equal parts.

This operation may be performed, however small the given line ai, and however great the number of parts into which you propose to divide it. True it is, that in execution we are not permitted to go too far; the lines which we draw always have some breadth, whereby they are at length confounded, as may be seen in the figure near point O; but the question is not what may be possible for us to execute, but what is possible in itself. Now in geometry lines have no breadth*, and consequently can never be confounded. hence it follows that such division is illimitable.

###### *In Shormann Math, a line is defined as a widthless length, which is the same thing Euler is describing. In fact, all normal geometry courses define a line this way. The idea is that we are not concerned with how thick, or wide the line is. When you draw a line though, it has to have some thickness to it in order to be able to see it.

If it is once admitted that a line may be divided into a thousand parts, by dividing each part into two it will be divisible into two thousand parts, and for the same reason into four thousand, and into eight thousand, without ever arriving at parts indivisible. However small a line may be supposed, it is still divisible into halves, and each half again into two, and each of these again in like manner, and so on to infinity.

What I have said of a line is easily applicable to a surface, and, with greater strength of reasoning, to a solid endowed with three dimensions, length, breadth, and thickness. Hence is is affirmed that all extension is divisible to infinity, and this property is denominated divisibility in infinitum.

Whoever is disposed to deny this property of extension, is under the necessity of maintaining, that it is possible to arrive at last at parts so minute as to be unsusceptible of any farther division, because they ceased to have any extension. Nevertheless all these particles taken together must reproduce the whole, by the division of which you acquired them; and as the quantity of each would be a nothing, or cypher (0), a combination of cyphers would produce quantity, which is manifestly absurd. For you know perfectly well, that in arithmetic, two or more cyphers joined never produce any thing.

This opinion that in division of extension, or of any quantity whatever, we may come at last to particles so minute as to be no longer divisible, because they are so small, or because quantity no longer exists, is, therefore, a position absolutely untenable.

In order to render the absurdity of it more sensible, let us suppose a line of an inch long, divided into a thousand parts, and that these parts are so small as to admit of no farther division; each part, then, would no longer have any length, for if it had any, it would be still divisible. Each particle, then, would of consequence be a nothing. But if these thousand particles together constituted the length of an inch, the thousandth part of an inch would, of consequence, be a nothing; which is equally absurd with maintaining, that the half of any quantity whatever is nothing. And if it be absurd to affirm, that the half of any quantity is nothing, it is equally so to affirm, that the half of a half, or that the fourth part of the same quantity, is nothing; and what must be granted as to the fourth, must likewise be granted with respect to the thousandth, and the millionth part. Finally, however far you may have already carried, in imagination, the division of an inch, it is always possible to carry it still farther; and never will you be able to carry on your subdivision so far, as that the last parts shall be absolutely indivisible. These parts will undoubtedly always become smaller, and their magnitude will approach nearer and nearer to 0, but can never reach it.

The geometrician, therefore, is warranted in affirming, that every magnitude is divisible to infinity; and that you cannot proceed so far in your division, as that all farther division shall be impossible. But it is always necessary to distinguish between what is possible in itself, and what we are in a condition to perform. Our execution is indeed extremely limited. After having, for example, divided an inch into a thousand parts, these parts are so small as to escape our senses, and a farther division would to us, no doubt, be impossible.

But you have only to look at this thousandth part of an inch through a good microscope, which magnifies, for example, a thousand times, and each particle will appear as large as an inch to the naked eye; and you will be convinced of the possibility of dividing each of these particles again into a thousand parts: the same reasoning may always be carried forward, without limit and without end.

It is therefore an indubitable truth, that all magnitude is divisible in infinitum, and that this takes place not only with respect to extension, which is the object of geometry, but likewise with respect to every other species of quantity, such as time and number.

28th April, 1761.

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

May 18, 2017

This is the first of eleven Letters of Euler I will rewrite and post on the subject of infinitesmals (the infinitely small), an idea that is fundamental to a good understanding of calculus. I am rewriting them from a 1795 English translation, and will edit some of the awkward character usage (among other things, the first “s” used in any word actually looks like an “f”), but otherwise, for the most part, I will leave it unchanged. Additions and edits will be marked by braces, […].

Considered by scholars as the best mathematician in history, Euler’s influence is everywhere present in modern mathematics. Yet as smart as he was, he still took time to bring difficult concepts down to a level where a non-mathematician might learn some things. And, as you will see, defend the Christianity at the same time.

Although these posts from Letters of Euler are for students in my Shormann Calculus course(available Summer 2018), any curious prince or princess is welcome to read them, too! The idea Euler (and myself) is trying to convey is that any real object can be divided, and divided again. And again, until it is in such small parts (infinitesimals) we can’t see them. Nevertheless, they exist. But how? To understand that, let’s begin with Euler’s description of the properties of any real object, which he refers to as a body.  Enjoy!

### Letter VII. The True Notion of Extension

I have already demonstrated, that the general notion of body necessarily comprehends these three qualities, extension, impenetrability, and inertia*, without which no being can be ranked in the class of bodies. Even the most scrupulous must allow the necessity of these three qualities, in order to constitute a body; but the doubt with some is, Are these three characters sufficient? Perhaps, say they, there may be several other characters, which are equally necessary to the essence of body.

##### *if a body has extension, that means you can measure it (length, mass, etc.); if it has impenetrability, that means you can feel it, which is possible with any solid, liquid or gas; if it has inertia that means it has the physical property of resisting a change in motion.

But I ask: were God to create a being divested of these other unknown characters, and that it possessed only the three above mentioned, would they hesitate to give the name of body to such a being? No, assuredly; for if they had the least doubt on the subject, they could not say with certainty, that the stones in the street are bodies, because they are not sure whether the pretended unknown characters are to be found in them or not.

Some imagine, that gravity is an essential property of all bodies, as all those which we know are heavy; but were God to divest them of gravity, would they therefore cease to be bodies? Let them consider the heavenly bodies, which do not fall downward; as must be the case, if they were heavy as the bodies which we touch, yet they give them the same name. And even on the supposition that all bodies were heavy, it would not follow that gravity is a property essential to them, for a body would still remain a body, though its gravity were to be destroyed by a miracle.

But this reasoning does not apply to the three essential properties mentioned. Were God to annihilate the extension of a body, it would certainly be no longer a body; and a body divested of impenetrability would no longer be a body; it would be a spectre, a phantom: the same holds as to inertia.

You know that extension is the proper object of geometry, which considers bodies only in so far as they are measurable. [Geometry does not consider impenetrability and inertia.] The object of geometry, therefore, is a notion much more general than that of body, as it comprehends not only bodies, but all beings simply extended without impenetrability, if any such there be. Hence it follows, that all the properties deduced in geometry from the notion of extension must likewise take place in bodies, in as much as they are extended; for whatever is applicable to a more general notion, to that of a tree, for example, must likewise be applicable to the notion of an oak, an ash, an elm, etc. And this principle is even the foundation of all the reasonings in virtue of which we always affirm and deny of the species,  and of individuals, every thing that we affirm and deny of the genus.

There are however, philosophers, particularly among our contemporaries, who boldly deny, that the properties applicable to extension, in general, that is, according as we consider them in geometry, take place in bodies really existing. They allege that geometrical extension is an abstract being, from the properties of which it is impossible to draw any conclusion, with respect to real objects: thus, when I have demonstrated that the three angles of a triangle are together equal to two right angles, this is a property belonging only to an abstract triangle, and not at all to one really existing.

But these philosophers are not aware of the perplexing consequences which naturally result from the difference which they establish between objects formed by abstraction, and real objects; and if it were not permitted to conclude from the first to the last, no conclusion, and no reasoning whatever could subsist, as we always conclude from general notions to particular.

Now all general notions are as much abstract beings as geometrical extension; and a tree, in general, or the general notion of trees, is formed only by abstraction, and no more exists out of our mind than geometrical extension does. The notion of man in general is of the same kind, and man in general no where exists: all men who exist are individual beings, and correspond to individual notions. The general idea which comprehends all, is formed only by abstraction.

The fault which these philosophers are ever finding with geometricians, for employing themselves about abstractions merely, is therefor groundless, as all other sciences principally turn on general notions, which are no more real than the objects of geometry. The patient, in general, who the physician has in view, and the idea of whom contains all patients really existing, is only an abstract idea; nay the very merit of each science is so much the greater, as it extends to notions more general, that is to say, more abstract.

I shall endeavor, by next post, to point out the tendency of the censures pronounced by these philosophers upon geometricians; and the reasons why they are unwilling that we should ascribe to real, [measurable] beings, that is, to existing bodies, the properties applicable to [measurement] in general, or to abstracted [measurement]. They are afraid lest their metaphysical principles should suffer in the cause.

25th April, 1761.