# “Books VII-X of Euclid’s *Elements*”

By Dr. Carol Day

Tutor Emeritus, Thomas Aquinas College

Tutor Talk (prepared text)

November 28, 2018

When I first taught Euclid’s *Elements*, I was puzzled about several features of the “Number Books,” books VII-IX. Although I had taken a class in Euclidean geometry as a sophomore in High School, we used a textbook, not the original text. I had no notion of anything in classical geometry beyond the properties of basic figures and some of the theory of proportion as applied to figures. The study of numbers was far away in the rear-view mirror. Just like most of our students, I came to the *Elements* with the custom of approaching mathematical theorems and problems using algebra. Euclid’s way of doing things often seemed both strange and inefficient, though I did come to see its beauty and its merit. Right from the beginning, it was fun to master his way of doing math. By the time I got to Book VII, I was used to and comfortable with Euclid. But what was he up to here? The long list of definitions at the start of Book VII showed that he was launching into arithmetic. What was he doing? Why take up numbers when we have been on such a roll doing geometry? I would have expected even more theorems in plane geometry or perhaps the beginning of solid geometry. What we got instead were propositions about relatively prime numbers and about proportions of numbers. I was puzzled. Why was Euclid doing this?

I was also puzzled about something one notices right away. After the usual enunciation of a theorem in words, Euclid displays the numbers in the setting out as lines. I would have liked the setting out to be done using algebraic notation, and I noticed that the students generally wanted that too. I also wondered about the order in which he presented his propositions. Why doesn’t he begin the study of arithmetic from the beginning, as one would expect from his practice in the geometrical books, and then go through theorems about numbers in a systematic way? After studying the entire book of the Elements, I was also puzzled about the order of the number books within the work as a whole. Why are they placed after the treatment of plane geometry and before the treatment of solid geometry? Eventually, I also wondered whether we should conceive of Book X, his treatment of irrational magnitudes, as belonging with Books VII-IX rather than as a book standing on its own as a necessary preliminary for understanding a few things about solid figures. This question did not occur to me the first two times I taught the *Elements*, but it seemed an obvious thing to wonder about when I came back to Euclid many years later.

In this talk, I want to suggests some answers to these questions. I don’t pretend to be a scholar of Greek mathematics or to be able to look back into the past to read Euclid’s mind. Nevertheless, I believe that in thinking about these questions we can discover that they are closely related and that a coherent account can be given that provides a reasonable understanding of them all.

#### Part One: The Representation of Numbers by Lines

*Note: when I say line, I will mean straight line, unless I specify otherwise.*

I remember many times my much beloved colleague and friend, Molly Gustin, would say that numbers are lines. Usually I would strenuously object. Eventually, though, I could see what might lead one to say a thing so outrageous to both kinds of mathematical purists one might find running around campus, that is, to the lovers of the ancients and to the lovers of the moderns. I think Mrs. Gustin was probably influenced both by Euclid’s way of depicting numbers and by the Cartesian extension of arithmetical notions into geometry. In defense of her notion, remember that it has become a common-place to speak of the “number line.” It is only after Dedekind that the modern mathematicians attempted to free the real numbers from the geometrical tether that Descartes used to bind them. I know that Mrs. Gustin was not sympathetic to this program. Despite the fact that we disagreed, I think I can understand why she wanted to identify numbers and lines. But I want to be clear about this: I do not think that by representing numbers by lines Euclid wants to identify them. We can see that he does not by looking at his definition of unit. This definition is broad, to say the least! He says that “An unit is that by virtue of which each of the things that exist is called one.” Thus we can think of one line, one sphere, one cow, one instance of blue, one thought, and so on. The unit is something common to them all! Numbers, then, are simply multitudes composed of these units. From this we see that Euclid is using lines as instances of things which are numerable.

It is not hard to see some good reasons for Euclid’s depiction of numbers as lines. If we consider the options available to him, it is clear that other alternatives were either non-existent, unsuitable to the science, or simply awkward. Let’s consider as an example Proposition VII, 1. Here is how Euclid expresses it. “Two unequal numbers being set out, and the lesser being continually subtracted from the greater, if the number which is left never measures the one before it until the unit is left, the original numbers will be prime to one another.” His diagram looks like this:

__________________ ______________ _____

A H E B C F D G

Here AB and CD are numbers being measured, G is supposed as a common measure, and AH is the unit.

It is Euclid’s practice to letter both endpoints of a line, as well as its points of division, assuming that the number it represents needs to be measured. If a number does not need to be measured, he names it with a single letter. The proof, then, is carried out using these letters as stand-ins for the numbers and their parts.

Since algebra had not yet been invented, he could not set out and prove VII, 1 in a fully symbolic way.[1] The only other alternative to a geometrical representation of numbers would be to give a numerical example, something like this. “For the lesser of two numbers, 5, being continually subtracted from the greater 91, let the number left never measure the one before it until the unit is left, then 5 and 91 are prime to one another.” It’s tempting to simply write off this procedure as an argument from example, which is not an appropriate mode of scientific demonstration in any science. This seems true, but we ought to think about how it differs from what he does in the geometrical theorems. For most people, some concrete representation of the thing to be proved needs to be presented to the imagination, and whatever is in the imagination is a singular, not a universal. To prove a theorem about triangles, Euclid must give us a particular triangle, with determinate sides and angles. How is this unlike using a numerical example?

In a geometrical proof, it is not difficult to look at a concrete individual and attend only to the universal features that are relevant to the argument. For example in looking at an isosceles triangle for the purpose of proving I, 5, that the base angles of an isosceles triangle are equal, we need to see that there are two equal and one unequal side and which are the base angles, but we don’t need to attend to the relative lengths of the equal and the unequal sides. It is easy to see that these details do not enter into the argument. We can even see that the proof works if all three sides happen to be equal. The abstraction of the relevant from the irrelevant is easy to do in geometry. But there is something about the way in which concrete numbers exist in our imagination that gets in the way of performing the necessary mental trick.

Here I think we must make a distinction between discovering a theorem and proving it. Clearly, examples are invaluable for discovering theorems, both in geometry and in arithmetic. Someone might see a theorem about numbers from one example, and even get a sense of how to prove it. In very easy theorems anyone of average intelligence could do it. The reason why a (b + c) = ab + ac is not hard to induce from one example, e.g. that 2 (3 + 4) = 2x3 + 2x4. But even in a case like this, a presentation of the proof using an example seems inappropriate.

It seems to me that, for the most part, an examination of a concrete individual is not suitable for proof in arithmetic because of the lack of an inherent order within a number. This lack comes from their character as discrete quantities, as multitudes of units. A number does not have any articulation more complex than its divisibility into units or groups of units. It may or may not be actually divided by some act of counting parts, that is by measurement. Once conceived of as divided into units, these units may be counted in any order we please. There is no order natural to them. In a representation of the number by a numeral, at least in the modern manner using the Hindu-Arabic numerals, the units are nowhere to be seen.[2] This mode of representation makes it hard to see anything mathematically significant in the number, either in itself or in relation to others. There is just not enough to look at. But we do need something to look at, and we often need to see them as divided into parts. This is just what Euclid does when he represents numbers as divided lines!

I even think there is an exception which proves the rule. (By the way, I suppose you know that this phrase refers to a rule holding only for the most part, where the exception illuminates the reason for the rule.) The instances in which a concrete example might be suitable in the theory of numbers are those in which the numbers themselves are grasped using geometrical concepts, as with triangular, square and cubic numbers. Representing these numbers by arrays of dots can indeed serve the imagination well enough. This is a very small territory in the realm of numbers, however. Most theorems about numbers cannot rely on a geometrical crutch. Symbolizing a number such as 7 by a line of dots does not get you anywhere that a line divided into 7 segments won’t get you more suitably. Euclid’s way does not give the false impression that 7 is nothing more than 7 ones side by side, as if it had no character and unity of its own.

Euclid’s method of visually articulating units in numbers by representing them as divided lines serves his science well. The lines may always be made of reasonably short length since any arbitrarily small line can be thought of as the unit. As mentioned above, we are able in this way to grasp the number as a whole containing these parts. Because of the abstractness of the representation, it is not hard to disregard the actual number of divisions in the illustration and to focus on what is essential. In other words, there is no reason to pay attention to the actual count of the divisions, as if one were merely calculating.

Let’s see how this works by looking at proposition VII, 4, which proves that any number is either a part or parts of any number, the less of the greater. The larger number is represented by A. Although it contains the lesser number or parts of it, the actual measuring of it doesn’t need to be seen. On the other hand, the lesser might not measure the greater, but only parts of the lesser will measure it. So, the parts of the lesser have to be shown. The lesser is shown as line BC, divided at E and F, and also another line, D, equal to the number used to measure both BC and A. In the proof, the articulation of the lines into parts helps one to understand the reason for the theorem. The fact that BC is shown as divisible into three parts does not get in the way of understanding the proof, for it is not hard to see that the exact number does not matter to the argument. The proof rests on the nature of measurement, and measure is illustrated in the lines in a way that does not call to mind vividly the particular results of the measurement. Is this not the key to understanding Euclid’s use of lines? Since the truths of arithmetic rest on the idea of measure, we often need to see measurement in action, which requires an order in space and nothing more.

To sum up, then: showing numbers as lines depicts them as discrete quantities relatable to one another either through one measuring the other or through their having some common measure, and in this way Euclid facilitates our grasp of the truths which he wants to prove about them. His technique accomplishes this without suggesting the false notion that a given number has no unity as a distinct species.

#### Part Two: Not Beginning at the Beginning: the Order of Treatment of Arithmetical Theorems

The order we find in Books VII - IX of the Elements is puzzling. A scientific treatment of arithmetic should begin with definitions and postulates and then proceed to prove the simplest properties of numbers first, followed by more complex ones. One would expect that the first theorems would have regard to the first division of number into the odd and the even. Euclid does present such theorems, but not until Book IX, Proposition 21 and following, where they seem to be an afterthought. Only then would one expect a treatment of other kinds of number, such as prime and composite, square and cube, perfect, and so on. And just as the study of proportion in plane figures comes last in his treatment of plane geometry, so one might expect the treatment of proportion in number to come last too. But this is not the case, though it is true that he defers treatment of continued proportions to Book IX. The theorems about the odd and the even come up after propositions about all these other kinds of numbers. In short, the way in which Euclid orders the material in these books does not seem to conform to ordinary scientific procedure. Why not?

Maybe this apparent disorder is a clue, revealing that Euclid does not intend to present the reader with the science of arithmetic as such. After all, these books are embedded in a work of geometry. Could it be that these books really and properly belong to geometry? Could it be that they contain truths about magnitude that involve number? That the proper object of these books is the continuum as perfectly measurable? Perhaps Euclid is concerned not with number in the abstract but with number as a number *of something*. This would give further reason for exhibiting these numbers as straight lines, the simplest of all species of continuous quantity. Clearly, this is not meant to exclude the application of these theorems to any kind of magnitude subject to number. The universality of his definition of unit indicates that the theorems apply to number wherever it is found. An odd number of cats plus an even number of cats is an odd number of cats. Despite the general applicability of the theorems, it seems likely to me that Euclid includes them in the *Elements* because they are things about number that the student of geometry should know and be able to use. If this is correct, the order in which he considers properties of numbers must be understood within the context of the whole book.

In support of this idea, I offer the following observations.

*Observation 1*. A notion that is not defined but is fundamental to all that follows is the notion of measure. Definition 3 says that a number is part of a number, the less of the greater, when it measures the greater. The first notion of measure is of a magnitude laid out along another magnitude so that it goes into it a certain number of times. Although there is something arbitrary in the procedure -- one can begin from either end, for example) there is a comprehensible order of the units from left to right or vice versa. If you try to lay down the unit randomly, you will probably err, and if you count the divisions unsystematically, you will probably get confused. When counting discrete objects, we tend to imitate this spatial order by systematically ordering the things in space. There is plenty of evidence that in ancient times things such as sheep were counted by associating them one by one with notches in a stick. In Middle English these sticks were called tallies or tally sticks. On the other hand, the units in an abstract number are not laid out alongside each other. Where are the units in 7? There is no “where” there! And what happens when we subtract one number from another? When we subtract 3 from 7, we don’t think about which of the units in 7 are being taken away!

That the science of arithmetic treats measure in a unique way is also suggested by the fact that it has special names for the composition and resolution of numbers, that is, addition and subtraction. It is interesting, then, that Euclid begins not with special names for these operations but with measure, the generic term that applies both to measuring things in space and to counting. This suggests to me two further observations. First, that he is associating his study of number with geometry; and second, that he is not interested in calculation, except perhaps in a subordinate role once in a while.

*Objection*: Euclid defines number as a multitude composed of units, not as something measured by a unit. But consider that Euclid does not have us imagine producing the numbers by counting, as a modern text would most likely do. Rather, he starts with a number conceived as a whole -- a multitude -- and then says that it is composed of units. This is verified in the imagination, is it not, by laying the unit (whether conceived as a point, a line, or however) alongside itself so many times until a desired multitude is reached? But unless the multitude is conceived as a whole resulting from this operation, which is in fact just the inverse of measurement, it has no unity but is a mere heap of units. The unity of the number itself is seen to arise from some continuity, or at least contiguity, of the units. Thus we are led in our imagination to the line, and we see once again the appropriateness of Euclid’s representation of number.

*Observation 2*. The beginning of Book VII with numbers prime to one another now seems justifiable. If he were interested in pure numbers, why would he not treat prime numbers and composite numbers before treating of the relatively prime and composite? It is plain that he thinks one should begin with numbers in comparison to each other rather than with them considered in themselves. This makes sense if he is ultimately interested in comparing the measures of magnitudes appearing in geometrical figures. This is in fact what he does in Book XIII, indeed in the very last theorem in the whole work.

*Observation 3*. By defining the even and the odd right after defining part, parts and multiple, and before prime, etc., Euclid seems to acknowledge the primacy of this division of number, but he puts off any theorems about the even and the odd until late in his treatment of number, as I have already remarked. Though I will not take time right now to examine these theorems in detail, I will say that I think they appear where they do because of their connection to the last proposition in the number books, the construction of perfect numbers. This depends on the doubled proportion, which continually produces even times even numbers, as set out in IX, 32. If he wants to tell us about even times even numbers, perhaps he thought it suitable to talk about all the kinds of numbers defined by the even and the odd, and this was a good place to do all that, since it leads to numerical perfection. In this way, the number books are brought to a close with a problem that parallels the theorems in Book IV of plane geometry and in Book XIII of solid geometry.[3] This suggests to me that Books VII-IX should be conceived geometrically.

*Observation 4*: Seeing the number books as ordered to geometry makes sense of the placement of Book X in the *Elements*. If X were included only as needed for later propositions, it would make more sense to place it between Books XI and XII. The most basic theorems of solid geometry do not involve an explicit consideration of the irrational, but the first and foundational theorem of Book X, which we might call Euclid’s Limit Theorem, is the basis for the procedure in XII, 1, the proof that circles are to one another in the ratio of the squares on their diameters. Book XII begins the integration of the theory of irrational magnitudes into solid geometry. So, it is clear that Euclid had other reasons for placing X right after the number books instead of where it would first be needed. This observation brings us to Part Three.

#### Part Three: How Books VII - X Fit into the Whole Work

What is geometry? One way to define a science is to give its subject genus. The subject genus of geometry is magnitude. Magnitude is divided into that which is in one dimension (the straight line), that which is in two dimensions (the plane) and that which is in three dimensions (the solid). Euclid treats the second species in Books I - IV and VI, and the third species in Books XI - XIII. Book V stands apart as a universal treatment of ratio in magnitude. Does he neglect to treat the first species? Is it not rather reasonable to say that he treats of the first species in Books VII - X? On the face of it, this seems reasonable, in that the illustrations presented to the imagination in these books are of lines. But it is perhaps far-fetched to say that one dimensional magnitude is the subject of these books, given the universal applicability of number. To test the idea, we must see how his treatment of number and the un-countable compares to his treatment of plane and of solid extension.

The name “Geometry” derives from the “measure” and “earth,” and it no doubt originated from the need to define land boundaries. As a theoretician, the geometer is no longer concerned with applications of his science, but it would be wrong, I think, to sever the connection with the idea of measurement. The geometer does not measure as the surveyor does, but he does, as it were, “size up” the ways in which his subject may be divided. Even when he is not comparing magnitudes, which he is not able to do until after Book V, he is distinguishing the various figures from what lies outside them, setting them apart from the infinite and unknowable continuum and making their properties known. In this way, the figures become more and more intelligible to us.

In plane geometry, this is accomplished by delimiting portions of the plane by means of lines, both straight and curved. These lines provide the boundaries which define the various species of figure which the plane geometer considers. For Euclid, these lines are the straight and the circular, each postulated to exist and then used to generate all the other figures which he studies. There are riches of truths to be discovered about such figures, and Euclid gives us many of these, including the most fundamental and important.

In a similar way, the solid geometer takes his part of the subject genus, the three dimensional continuum, and divides it using plane figures and curved surfaces to bring forth the solid figures he wishes to understand. These figures are often very beautiful, and in that they approach more closely to the reality of the world, they seem especially worthy of contemplation. The student of the line, however, has as it were a very limited resource. The only way to divide a one dimensional object is to mark off a point or points, to count the divisions and to look at the ratios of the parts, and perhaps their ratios to other lines that have been set out. That sounds pretty boring, on the face of it! The amazing thing is that, far from being barren of interest, there are such riches to be found in the numbers which may be discerned in the line that we are far from having discovered them all. In addition to the theory of numbers, which is still a very lively field for discovery, there are many truths known and not yet known about irrational quantities, far more than what Euclid himself ever imagined.[4]

So, there is a common notion under which we may understand what Euclid does in treating each kind of continuum. This is the notion of dividing it using beings of the next lower dimensionality,[5] with a view to separating out species which may be explored and understood. Can we say, then, that Books VII-X treat of lines insofar as they are measurable or immeasurable? I think that this is true if rightly understood, though I will suggest a way in which the claim should be qualified to make it more accurate. The great advantage of looking at these books in this way is that it allows us to see the number books not as interlopers but as properly part of Euclid’s subject matter.

Moreover, this idea makes it clear that Book X belongs together with the number books as part of a greater whole. Beyond the fact that lines are presented to our imagination in all four books, there are other reasons for seeing them as belonging together. In Book X, Euclid goes beyond considering incommensurability by defining rational and irrational lines and areas. This seems to me a strong indication that Book X belongs to the same part of the work as the number books. Euclid’s definitions allow us to predicate of a single line its status as rational or irrational, some one line having been set out as the standard. It is reasonable to see this standard line -- which Euclid does not bother to designate -- as the concrete unit. With this line hidden away in the background, one can treat even of these unruly lines as if they were something. While not going so far as Descartes in bringing arithmetic into geometry[6], Euclid doesn’t hesitate to give us a way to name lines, not only as irrational, but even to assign them to distinct species of the irrational, as binomials, apotomes, and the like. Finally, there is a certain parallelism between the ending of Books IX and Book X. Proposition 36 of Book IX shows that an endless sequence of perfect numbers may be generated, while Proposition 115 of Book X shows us that an unending sequence of different irrational lines can be generated. Just as the perfect measures go on forever, so do the imperfect divisions.[7]

If this account is correct, if Books VII - X treat of the line, that is, one dimensional extension, as measurable or immeasurable, all the parts of *The Elements* belong to one science, the science of geometry. That X deals with lines as divided in a certain way is clear enough; it belongs properly to geometry, in that the divisions of magnitude which it studies cannot exist outside the continuum. But perhaps I have overstated the case when it comes to Books VII-IX. Here is an objection, courtesy of Mr. Augros: If one says that the number books are about lines as numerable, why are lines never mentioned? And are lines numerable in a way fundamentally different way from the way a plane surface is numerable?

I concede that the number books do not concern themselves with lines in the same was as Book X, and to refer to lines in the enunciations of the theorems would be inappropriate, as this would take away from the universality of their application. It would be better, then, to say that these books deal with number inasmuch as it arises from the division of the line. This is enough, I think, to justify the inclusion of these books under the heading of geometry. And though one must admit that number also arises from the division of the plane and of solids, it is reasonable to say that, for the mathematician, number arises *first* from the division of the line. If the line could not be divided discretely, neither could the plane or the solid.

As with the number books, Book X is not exclusively applicable to lines. Proposition 1, to take an important example, is universally applicable to magnitude, and Euclid applies it without comment in Book XII to circular areas. Nonetheless, most of the 115 propositions of Book X deal explicitly either with lines or with figures having determinate kinds of lines as their sides. The exceptions are 1-8, 11-13, 15 and 16, which are all enunciated about magnitudes. It is clear that the notions of rational and irrational apply at least to quadralinear figures. Although the notions of rational and irrational could be extended to solids, Euclid had nothing to say about this, perhaps because that part of the science was yet to be discovered.

From these considerations, we see that Euclid did not neglect the study of magnitudes in one dimension, but in fact he has an extensive and rich treatment of them, embracing both those with and without numerable parts. He also knew how to apply his knowledge of the rational and irrational lines to figures. We see the fruits of this in the very last proposition of the work, XIII, 18, in which he sets out the ratios of the sides of all the perfect solids to one another.

Finally, let’s consider the placement of these four books in the *Elements*. Recall how the work is organized. There are four books dealing with plane figures as such. Of these, Book II has a unique character, one difficult for the beginner to appreciate, but which gives tools for discovery and construction in the rest of geometry. After this there is a book on ratio and proportion considered in the abstract, followed by an application of these theorems to plane figures. Next come the number books and Book X; finally, there are three books of solid geometry.

Solid geometry is more difficult than the study of two dimensional figures, on the knowledge of which it obviously depends. The need for Euclid to treat the plane before the solid figures is clear. They are most obscure and perhaps the most beautiful, so it is fitting as well as necessary that they come last. But if I am right that Books VII - X deal with divisions of the continuous in one dimension, why does he treat this subject after the plane and before the solids? Why not treat the one-dimensional first?

A few reasons come to mind right away. Pedagogically it makes sense to treat the more knowable (not to mention the more interesting to students) before the less knowable. And whereas the numerable lines do seem more knowable than figures, the same cannot be said of irrational lines. In my experience guiding Freshmen through the *Elements*, I saw that even in the number books, the more complicated theorems often caused the students more problems than the theorems of Books I - IV. It seems to me that the more subtle properties of numbers are less knowable to the students than most of the truths about plane figures. The properties are often well-hidden. Perhaps also the fact that the imagination is less vividly engaged in the number books makes the propositions more difficult to learn and remember.

Although the reasons just given have some force, the more important reason is properly mathematical. Numbers must be drawn out of the line by division. To divide or measure lines, it is necessary to carry out constructions in two dimensions. Euclid bisects a straight line in I, 10 and in VI, 9, he cuts off a prescribed part from a given straight line. In the latter proposition, he shows that any straight line may be measured by any number. He even anticipates irrational divisions of lines in VI, 13, where he constructs a mean proportional between any two lines. These illustrate that, in order to measure a line or to show that it cannot be measured by some other line, it is necessary to carry out constructions in the plane. From this we see that the one dimensional continuum is not only less knowable than the plane figures to us but also in its character as divisible. Thus it is necessary for Euclid to order these books as he does. But is there also a good reason not to defer the treatment of straight lines until after treating solid geometry?

If I am right to think that book X belongs with Books VII-IX, it becomes clear that it is necessary to put them before the solid geometry. Euclid makes use of propositions from Book X in his treatment of solid figures, and not in trivial ways. Though it is impossible to know the truth of the matter, I wonder whether Euclid would have included Books VII-X in the *Elements* if he hadn’t wanted us to see how the rational and the irrational both play a role in the perfect solids. They are so clearly meant to be the climax of the whole book. That there are such a limited number of these is a very interesting fact. That the plane figures which are their principles involve both simple numerical ratios (as that the diameter of the hexagon is twice its side) and irrational ratios, such as that found in the triangle upon which the regular pentagon is built. Moreover, it is the treatment of the irrational that the argument by approach to a limit and the powerful method of the proof by double reductio are revealed. Using the latter, Euclid is able to establish the important theorems that squares are in the duplicate ratios of their diameters and spheres are in the triplicate ratio of the same. Thus the treatment of the irrational as well as of the rational are made to serve the purposes of solid geometry. It is not Euclid’s way, however, to do the bare minimum. Once he begins his treatment of number, he adds much that goes beyond what we need to see about the possibilities of comparing measures of the divided continuum. In this way Euclid enriches our understanding of number, a knowledge which is certainly valuable for its own sake.

The idea of measure runs throughout all three books, showing that the comparison of numbers is at least as important as properties of the numbers considered in themselves. It is not my intention to consider the internal order (and partial disorder) of the number books. I just want to point out that every indication I can see points to the conclusion that Euclid is not interested in developing the art or science of arithmetic in its own right, but he treats it in the context of geometry for various reasons. Because it enriches geometry by its connections with the perfect solids; because when joined to the treatment of the irrational gives a kind of completion to geometry; and because the previous traditions of connecting numbers with figures (as in square, triangular and cube numbers) suggests that arithmetic and geometry have a kind of affinity, with geometry embracing not only the numerable but also that which cannot be counted.

[1] The algebraic mode of presentation is often advantageous to the student, in that he has less to keep track of in his imagination. In this proposition, not much if anything is gained by using symbols to state the enunciation, but the proof becomes more concise and thus easier to follow when algebraic symbols are used. Once the proposition has been translated into symbols, the

proof looks a lot like a calculation, one carried out according to well known and justifiable rules. But this option was not open to Euclid, nor would it be suitable for our students. For most, if not all of them, algebra is no more than a set of memorized rules which have not yet been subject to critical examination. Besides, the point in studying old texts is not to make them conform to modern methods or standards.

[2] The traditional way in many cultures of representing small numbers by linear strokes -- the Roman numerals are the most familiar to us -- would be at best awkward for proving theorems.

[3] The more advanced treatment of proportion in plane figures must come after the general treatment of ratio and proportion in Book V. In my opinion, deferring Book IV until after Book VI would be inappropriate, as the construction, inscription and circumscription of regular polygons is independent of the treatment of proportion, and so belongs to the elementary part of the science.

[4] In Book X, Euclid treats thoroughly of algebraic irrationals of the second degree, and of a few of degree four. He does not treat of those of the third degree, and he gives no indication of awareness of the transcendental irrationals.

[5] In the case of lines, one uses the point, which has no dimension at all.

[6] “I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness.” *La Geometrie*, p. 4, Dover Publications (1954). Here Descartes understates his achievement, for in bringing about greater clarity he brings about a great power of discovery in geometry.

[7] Little did Euclid know that the irrationals are infinite in a way that exceeds in complexity the infinity of numbers. I am sure he would have like to have known that!

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