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

Posted July 9, 2018 by gensci
Categories: Uncategorized

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.

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

Posted July 5, 2018 by gensci
Categories: Shormann Math

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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 X

Posted July 3, 2018 by gensci
Categories: Shormann Math

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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.

Comparing Khan Academy’s “Procedures” to Shormann Math’s “Story” for Teaching High School Math

Posted June 14, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

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In a 2012 Education Week article, Justin Reich, an internet education specialist, wrote that Khan Academy’s math courses overemphasize procedure. In other words, Khan’s math video lectures explain the steps to solve a problem, but they don’t take time to explain the concept behind the steps. In the words of the author, “writing about mathematics, developing a disposition for mathematical thinking, demonstrating a conceptual understanding of mathematical topics are all more important than procedures.”

But, procedures ARE important, and sometimes, that is what you need. For example, if you are stuck on the side of the road with a flat tire, someone needs to know the steps required to fix the problem in as few steps as possible. You are not interested in the physics behind hydraulic jacks or who invented vulcanized rubber (Charles Goodyear). You have a crisis, and it requires a pragmatic solution. So, maybe you are studying for a big test like the SAT, and it’s in 2 weeks, and suddenly you realize there’s a concept you need to practice some more. This would be a time when Khan Academy might be helpful to you.

But, a good education isn’t really about pragmatism, about skimming through a subject, learning the bare minimum, so you can check it off your transcript or pass a standardized test. And then forget about it.

Shormann Math is different. Certainly, there are times when procedure is emphasized, like our step-by-step video solutions for every homework problem, for example. But there are more things, deeper things, that Shormann Math emphasizes that others, like Khan or Reich or the National Council of Mathematics Teachers either don’t, or won’t.

At the heart of mathematics is a story, an amazing story of the history of humans seeking to discover more about the world around them. And it’s a story that is best understood from a Christian foundation.

Here’s why. To his credit, in Khan Academy’s introductory Algebra 1 lesson, Sal Khan dives into the origins of algebra.  In describing the date of the first algebra book (820 A.D.), Khan makes a well-meaning, respectful attempt to distinguish the “religious” B.C. and A.D. from the “non-religious” common era descriptions of historical dates. Of course, the problem here is that Christianity is the only “religion” considered here. But Christianity is most definitely not a religion, a mere set of rules to keep or procedures to follow.

At its core, Christianity is about a relationship. And that’s a surprisingly hard truth for some to acknowledge. Christianity is about a relationship between God the Father, Son and Holy Spirit, between humans and God, and between humans and humans. Simply put, Christianity is about how this relates to that. But so is mathematics, which is not surprising since its normal for created things to reflect the attributes of their designers. Relationships always tell a story, over time. Which is why history matters in Christianity, and therefore in mathematics, too.

So, with Khan Academy, a student might feel connected to mathematics when they are finished, which is not a bad thing. With Shormann Math though, while our students are connecting to mathematics in measurable ways (see next post on results), we are also connecting them to the story behind mathematics.  And the story is that, throughout history, mathematics is a tool for studying the world around us. So, we give students the mathematical tools to use in their thinking about the world, and give them a chance to practice using them in a variety of situations they might encounter in their everyday lives. But we also tell them who discovered the tools, plus some interesting things about their life stories.

Everyone loves a good story! That the story of mathematics is largely missing from popular, secular math teaching methods like Khan Academy could partly explain why mathematics is often the least-liked, most dreaded subject of all. Overemphasizing procedure may get you an algebra credit, but it won’t necessarily develop the STEM and STEAM skills so often desired.

But what if story is included, so that students get to know the major players influencing modern mathematics? With Shormann Math, we are finding that telling the story of mathematics makes learning math more real, and in the end, easier. If students are inspired by the experiences of historical figures, or by what God’s word says about math, they are more likely to know what to do with the math tools they are becoming fluent with. And if Shormann Math students are more fluent than students using more secular and disconnected methods, then when the opportunity to use math comes along, Shormann Math students will likely be more prepared.

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

Posted May 29, 2017 by gensci
Categories: Uncategorized

This is the third 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 IX. Whether this Divisibility in Infinitum Takes Place in Existing Bodies?

It is, then, a completely established truth, that extension* is divisible to infinity, and that it is impossible to conceive parts so small as to be unsusceptible of farther division. Philosophers accordingly do not impugn this truth itself, but deny that it takes place in existing bodies. They allege, that extension, the divisibility of which to infinity has been demonstrated, is merely a chimerical object, formed by abstraction; and that simple extension, as considered in geometry, can have no real existence.

*if a body has extension, that means you can measure it (length, mass, etc.).

Here they are in the right; and extension is undoubtedly a general idea, formed in the same manner as that of man, or of tree in general, by abstraction; and as man or tree in general exists not, no more does extension in general exist. You are perfectly sensible, that individual beings alone exist, and that general notions are to be found only in the mind; but it cannot therefore be maintained that these general notions are chimerical; they contain, on the contrary, the foundation of all our knowledge.

Whatever applies to general notion, and all the properties attached to it, of necessity take place in all the individuals comprehended under that general notion. When it is affirmed that the general notion of man contains an understanding and a will, it is undoubtedly meant, that every individual man is endowed with those faculties. And how many properties do these very philosophers boast of having demonstrated as belonging to substance in general, which is surely an idea as abstract as that of extension; and yet they maintain, that all these properties apply to all individual substances, which are all extended. If, in effect, such a substance had not these properties, it would be false that they belonged to substance in general.

If then bodies, which infallibly are extended beings, or endowed with extension, were not divisible to infinity, it would be likewise false, that divisibility in infinitum is a property of extension. Now those philosophers readily admit that this property belongs to extension, but they insist that it cannot take place in extended beings. This is the same thing with affirming, that the understanding and will are indeed attributes of the notion of man in general; but that they can have no place in individual men actually existing.

Hence you will readily draw this conclusion: if divisibility in infinitum is a property of extension in general, it must of necessity likewise belong to all individual extended beings; or if real extended beings are not divisible to infinity, it is false that divisibility in infinitum can be a property of extension in general.

It is impossible to deny that the one or the other of these consequences without subverting the most solid principles of all knowledge; and that philosophers who refuse to admit divisibility in infinitum in real extended beings, ought as little to admit it with respect to extension in general; but as they gran this last, they fall into a glaring contradiction.

You need not to be surprised at this; it is a failing from which the greatest men are not exempt. But what is rather surprising, these philosophers, in order to get rid of their embarrassment, have thought proper to deny that body is extended. They say, that it is only an appearance of extension which is perceived in bodies, but that real extension by no means belongs to them.

You see clearly that this is merely a wretched cavil, by which the principal, and the most evident property of body is denied. It is an extravagance similar to that formerly imputed to the Epicurean philosophers, who maintained that every thing which exists in the universe is material, without even excepting the gods whose existence they admitted. But as they saw that these corporeal gods would be subjected to the greatest difficulties, they invented a subterfuge similar to that of our modern philosophers, alleging, That the gods had not bodies, but as it were bodies, (quasi corpora) and that they had not senses, but senses as it were; and so of all the members. The other philosophical sects of antiquity made themselves abundantly merry with these quasi-corpora and quasi-sensus; and they would have equal reason, in modern times, to laugh at the quasi-extension which our philosophers ascribe to body; this term quasi-extension seems perfectly well to express that appearance of extension, without being so in reality.

Geometricians, if they meant to confound them, have only to say, that the objects whose divisibility in infinitum they have demonstrated, were likewise only as it were extended, and that accordingly all bodies extended as it were, were necessarily divisible in infinitum. But nothing is to be gained with them; they are resolute to maintain the greatest absurdities rather than acknowledge a mistake. You must have remarked, that this is the character of almost all scholars.*

3rd May, 1761

*NOTE: In 2017, beware of the scholar who claims that, in nature, what we see is merely quasi-design, the appearance of design. Making such an absurd statement is evidence of a stubborn refusal to acknowledge the Designer.


A mother humpback whale and calf. That the calf is a scale model of it’s mother is no accident or mere “appearance of design.” It is clear evidence of design, and their Designer.

Comparing Khan Academy’s Mastery Approach to Shormann Math’s Fluency Approach for Teaching High School Math

Posted May 25, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

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This is the first in a series of posts comparing Khan Academy’shormann khan comparison memes online math courses to our new Shormann Math curriculum. Shormann Math is part of DIVE Math and Science.

When Sal Khan started tutoring his cousins in 2003 using digital whiteboard video lectures, my family’s small business was already in its third year of producing similar products. Since then, both Khan Academy and DIVE have continued to offer digital video lectures for learning mathematics and other subjects, leveraging the ever-improving array of digital content delivery methods. Today in 2017, both Khan Academy and DIVE offer self-paced eLearning math courses. So, while there are obvious similarities between the two, there are also some pretty big differences.

Khan’s Store-Bought Layer Cake vs. Shormann Math’s Homemade Pizza

To begin, it is important to understand that Khan Academy teaches a Common Core-based “layer cake” approach to high school math. Shormann Math, on the other hand, teaches an integrated approach pioneered in Europe and Asia (you know, the countries that always beat us on international math tests), and introduced to the United States by the late John Saxon. John Saxon believed in the common-sense idea that results, not methodology are what mattered most(we’ll explore Khan Academy vs. Shormann Math results in a later post).

With the layer-cake approach, high school math is taught in layers, beginning with Algebra 1. Geometry is next, then Algebra 2. Each course is typically covered in one school year.

With the integrated approach, students learn algebra and geometry together. For example, Shormann Math does not have a separate geometry course, because geometry is integrated into Algebra 1 and 2. A geometry credit is included in Shormann Algebra 1 and 2, so students can take 3 years to complete the two courses and be “on track.”

Using a food analogy, both a store-bought layer cake and a homemade pizza can be delicious. With a store-bought layer cake, everybody gets the same thing, which is not always a bad thing, but can be. With a layer cake, if you just like vanilla, you can focus on eating the vanilla and not the chocolate. Or vice-versa.

With a homemade pizza, at least the one my family has made for years, each bite is an integrated medley of cheeses, meats, and vegetables. It is unique, diverse, and anchored in an amazing story that originated on float trip adventures deep in the Alaskan wilderness. Our homemade pizzas are about more than pizza, just like our Shormann Math courses are about more than math.

homemade pizza

Shormann Math, like our family’s legendary pizza, is a savory recipe developed over time with proven ingredients.

From my experience, a layer-cake approach is good for reviewing a particular subject, or specific concepts within a subject. Our CLEP and AP test prep courses employ this method. In other words, the layer-cake approach is good for review, which means Khan Academy math is good for testing your math skills in a particular subject. Khan Academy can help you determine if you have mastered a particular concept for the short-term.

Shormann Math, on the other hand, can help you build fluency in mathematics, which means you develop mastery and you retain it for the long term because you keep using it. In the beginning of every Shormann Math course, we define mathematics as the language of science and a God-given tool for measuring and classifying pattern and shape. I am not sure how Khan Academy defines mathematics, as I couldn’t find a definition anywhere. When you learn a new language, you don’t just learn nouns for a year, then verbs for a year, etc. You learn a little of the basics of everything, then you start putting them into sentences, which you practice. And practice some more. And then you review some more. And practice some more.

Building on fundamentals is how you learn a language, or a sport, or an instrument, or just about anything that you, eventually, hope to excel at. Building fluency takes time. Up next, we’ll explore Khan Academy’s “math procedures” to Shormann Math’s “math story,” followed by real data from real Shormann Math students on how the fluency approach can turn a bad math student into a good one. Stay tuned and thanks for reading!




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

Posted May 22, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

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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.

Plate II, Fig 23, line AI parallel to line ai, each divided into 8 equal segments.

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.