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by Dr. Carol Day
Tutor Emerita
Thomas Aquinas College
St. Vincent de Paul Lecture and Concert Series
March 19, 2021

When I first taught Euclid’s Elements, I was puzzled about several features of the “Number Books,” Books 7-9. I was not surprised to find that the students were puzzled too. For the most part, we were used to and comfortable with Euclid’s style and method by the time we got through Book 6, but what was he up to in Book 7? The long list of definitions at the beginning showed that he was launching into arithmetic. But why take up numbers at this point? Why not give us more theorems in plane geometry or perhaps move on to solid geometry? What we got instead were propositions about relatively prime numbers and about numerical ratios. I was puzzled, and I am very sure that I was not the only one in the classroom wondering about what Euclid was up to.

There were other puzzling features of Euclid’s presentation. After the usual enunciation in words, he displays numbers in the setting out as lines. Given the mathematical custom in which I was brought up, I would have liked the setting out to be done using algebraic notation, and I noticed that some of the students wanted that too. This was evident from the way they wrote out their demonstrations. 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? Why in particular does he begin his study of numbers with relatively prime numbers?

Much later on, I had other questions about the arithmetical part of the Elements. Why are these books placed after the treatment of plane geometry and before the treatment of solid geometry? I also wondered whether we should conceive of Book 10, his treatment of irrational magnitudes, as belonging with Books 7-9 rather than as a book standing on its own or just as a necessary preparation for solid geometry. This last question did not occur to me when I first taught the Elements, but it seemed an obvious thing to wonder about when I came back to Euclid many years later.

My plan is to address all these questions, although I can make no promise to settle them all. Since much of my talk will concern the order of the Elements and the number books within it, it would be reasonable to set out the order in which they will be addressed. I will begin with the representation of numbers by lines. This consideration stands apart from the somewhat entangled issues of the order of propositions and of the books themselves. But I hope to show that an understanding of Euclid’s method of representing numbers provides important clues for understanding the rest.

In the second part of this lecture I will ask about the appropriateness of including arithmetical books in a work of geometry, and in the third and fourth parts I will deal with questions about the order of the propositions in Books 7-9 and with the place of the number books in the overall scheme of the Elements.

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. Despite the fact that we disagreed about this, I can understand why she said it. I think she was influenced both by Euclid’s way of depicting numbers and by Descartes’ extension of arithmetical concepts into geometry. In defense of her notion, remember that it has become a common-place to speak of the Cartesian “number line,” as something comprising all the real numbers. To be fair to Mrs. Gustin, I believe that she was trying to give an account that made sense of calling the real numbers ‘numbers’. But if the whole numbers, those with which Euclid was concerned, are also real numbers, the temptation is there to state in a categorical way that numbers are lines. I think the identification of real numbers with lines is a mistake, but my concern here is not with that but with the interpretation of Euclid. Whatever numbers were for Euclid, they were not lines.

Since Euclid defines number as a multitude of units, one need only look at his definition of unit to see what he understands numbers to be. His 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.” [1] Thus we can speak of one line, one sphere, one point, one cow, one instance of blue, one thought, and so on. The unit is something common to them all. He does not make clear what this common thing is, probably because he thought it was enough for the mathematician to see that the unit is the principle of number and that it has some existence apart from its concrete or geometrical manifestations. The determining of its exact nature of its existence belongs to a higher science than mathematics. [2]

In my opinion, then, Euclid’s use of lines to represent numerable things does not imply a thesis about the nature of number. What remains, then, is to explain why lines are suitable and in fact the best way available to him for representing numbers.

At least two methods for depicting numbers were available to Euclid. One was the use of numerals. This method was used by Nicomachus in his Introduction to Arithmetic. In his History of Greek Mathematics, Heath compares Nicomachus’ method to Euclid’s, saying that that the method of representing numbers by lines “has the advantage that, as in algebraical notation, we can work with numbers in general without the necessity of giving them specific values; in Nicomachus numbers are no longer denoted by straight lines, so that when different undetermined numbers have to be distinguished, this has to be done by circumlocution, which makes the propositions cumbrous and hard to follow, and it is necessary after each proposition has been stated, to illustrate it by examples in specific numbers. Further, there are no longer any proofs in the proper sense of the word.” [3]

Consider as an example Book 7, Proposition 1. First, 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.” Following the enunciation comes the “setting out,” as Proclus calls it: “For the less of two unequal numbers AB and CD being continually subtracted from the greater, let the number which is left over never measure the one before it until a unit is left. “ AB and CD refer to a diagram, which 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 usually names it with a single letter. The proof, then, is carried out using these letters as stand-ins for the numbers and their parts.

If Euclid had recourse only to determinate numbers, the setting out would have to look something like this: “For the lesser of two numbers, for example 5, being continually subtracted from the greater, for example 93, let the number left never measure the one before it until the unit is left, then 5 and 93 are prime to one another.” The supposed proof would be a calculation. 5 x 18 = 90. So with 5 subtracted from 93 eighteen times, we have 3 left. Now 3 subtracted from 5 leaves 2. and 2 subtracted from 3 leaves 1. Since following the subtracting algorithm leads me to the unit before I find a common measure, I want to assert that there is no other common measure. But how do I prove this? I can go through all the numbers up to 5 to see if they also go evenly into 93, but when I see that none of them do what will I have learned? Only something particular to these two numbers. The only alternative is to suppose some indeterminate common measure other than the unit, give it a name such as G, and work out a proof like Euclid’s, in which thinking of the original numbers as particular examples is pointless, in fact distracting.

It’s tempting to simply write off argument by means of examples as unscientific if not impossible. Nonetheless, we ought to think about how it differs from what Euclid does in proving geometrical theorems. Some concrete representation of the thing to be proved needs to be presented to the imagination, and whatever is in the imagination is singular, not 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 features that are relevant to the argument. For example, in looking at the drawing of a triangle for the purpose of proving Book 1, 5, that the base angles of an isosceles triangle are equal, we need to imagine 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 often 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. I think this has to do with the mode in which they must be defined.

To see why the modes of definition are different in arithmetic and in geometry, consider how the infinite exists in each. In magnitude, we have the infinity of infinite divisibility. As such, the infinite has no relevance to the definition of figures, having to do only with their material aspect, that is, with the continuum in which the figures exist. The formal features of figures arise from their shapes. Geometrical figures are defined by their boundaries, and all their properties flow from the nature of these boundaries. Even in theorems having to do with areas and volumes, where the properties that result from their forms are often harder to know, it is by considering the implications of their boundaries that we learn what we can. This is true even in the use of calculus.

It is quite otherwise with numbers. Numbers are infinite by addition, growing ever greater as we count them. Having no position, they also have no boundaries. One might even say that a given number is a boundary. That is, a number terminates a progression radiating outward from the unit.[4] Whatever is formal in the number comes from the nature of this boundary, which gives rise to the distinctive way in which it is a multitude. It follows that unlike magnitudes, which are defined by way of genus and difference, numbers require a different kind of definition.

Let me explain what I mean by that. Suppose I want to define the number four. It is true but not altogether helpful to say that four is four ones, since it begs the question. [5] Rather, four must be defined at the number that comes next after three, as that number in which a unit is added to three. This may sound like a purely nominal definition, but it is more than that. The nature of any number depends on the nature of the number before it, going all the way back to the unit. All the previous numbers are in it as potency to the next number, which is brought to act by the joining of another unit. The act of joining another unit to three makes what was potentially four to become actually four.

What this amounts to is saying that the additional unit plays the role of form or species in the definition of a number, and the number to which it is added plays the role of matter. St. Thomas makes this point in his Commentary on the Metaphysics, Book 8, Lectio 3: “A number is one per se in as much as its final unit gives to the number unity and species; just as in things composed of matter and form a thing is one through its form and takes on its unity and species.” [6]

Let me say in passing that I do not propose to say how the new unit is added, or exactly what it means to add it. However it comes about in the being of the numbers themselves, what the mathematician sees is that three becomes four when another unit is added to it. As a simple example of how this act of joining the last unit determines a new property, just consider how the new unit changes the number from odd to even. But note how this mode of defining reveals very little about the properties of the number. Though it is obvious that the next number after 4 is odd, we cannot see in any easy way that 5 is the only prime number that is the sum and difference of two primes.[7] 5 = 2 + 3, 5 = 7 - 2.

The discovery and proof of theorems like this would be difficult if not impossible without the analytic methods of modern mathematics. Fortunately, this branch of the science, which pursues the characteristic properties of individual species of number, is not what Euclid was concerned with. This is the branch of arithmetic that is as far from geometry as possible.

If then a specific number is the boundary of an act of accumulating units, so that no other kind of definition can be given by us, how do we translate such a definition into something useful for geometry? I think Euclid had a good answer: we reason about numbers by considering them as measuring and as measured. The ultimate measure of a number is the unit, and its multitude is the distinctive way that the unit measures it. [8]

Measuring is an act of dividing. The geometer, therefore, divides numbers and impose order upon them in order to reveal their properties. Now measure is first known to us in extended things, in things we can sense. 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. The very name of Euclid’s science, Geometry, i.e. earth measurement, refers to this very practical procedure. Although there is something arbitrary in measurement -- one can begin from either end, for example -- there is a comprehensible order of the units from left to right or vice versa. Laying down the unit randomly leads to error and counting the divisions unsystematically leads to confusion. When counting material objects, we tend to imitate this spatial order by systematically ordering the things themselves in space. There is plenty of evidence that in ancient times livestock were counted by associating them one by one with notches in a stick.[9] You may have experienced this in counting pennies by grouping them in groups of five or ten. On the other hand, the units in an abstract number are not laid out alongside each other, nor are they visible in the representation of a number by a numeral. 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! Here we see an advantage of representing numbers by divided lines. By ordering the units in space we give our imagination something to make use of as we go about discovering and proving properties of numbers.

Representing numbers by lines seems to be an obvious choice, but this was not the only choice Euclid could have made. Another technique was available to him, one which had proved useful to his predecessors. Since the unit is indivisible, it would seem logical to represent it by a point. A number, then, would be represented by a set of points, since the unit is as it were the material from which the number is formed. In Metaphysics XIII Chapter 8, Aristotle describes this approach: “They [that is, some of the Pythagoreans] conducted their inquiry at the same time from the standpoint of mathematics and from that of universal formulae, so that from the former standpoint they treated unity, their first principle, as a point.” [10]

This way of depicting numbers has its uses. When the numbers in question have properties analogous to geometrical properties, this way of depicting them can be helpful for the discovery of theorems. Such for example are square, cubic and triangular numbers. For numbers like these, a visual presentation of their nature is possible by drawing an orderly array of dots. Here is an informal demonstration that summing successive odd numbers produces the sequence of square numbers:

Representing these numbers by arrays of dots can indeed serve the imagination well enough, where they are appropriate. Although the diagram is not a formal proof of the theorem, it is in itself quite convincing. This method covers a very small territory in the realm of numbers, however. Symbolizing a number such as 7 by a line of dots does not have any obvious advantage over using a line divided into 7 segments. Euclid’s way seems to have the disadvantage of not representing perfectly the nature of number as discrete quantity, but it is superior in that it does not give the false impression that 7 is nothing more than 7 ones side by side, as if the number had no character and unity of its own.

Let us now look at a distinctive advantage of visually articulating parts of numbers, whether units or other divisors, by the use of divided lines. 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.

Lets see how this works by looking at proposition 7, 4, which proves that any number is either a part or parts of any number, the less of the greater.

A       D

_____________________________________________ _______

B  E  F C

_________________

Recall that a number is part of a larger number if it measures it without a remainder, but parts if there is a remainder. Thus 3 is part of 6, but it is parts of 7. In the proof, the larger number is represented by A and the lesser by BC. Although it contains the lesser number, the laying out of BC alongside A isn’t necessary. Everything hangs on whether or not A and BC are prime to one another. If BC measures A, it is a part and all is well. If it does not measure it, we need only take the greatest common measure of A and BC, represented by line D. BC is shown as divided into parts BE, EF, EC equal to D, to show that a part existing in BC is also a part of A. That is what it means to say that BC is parts of A; it is made up of numbers which are themselves parts of A.

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 of times D is subtracted 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? Seeing measurement at work requires an order in space and nothing more.

To sum up, then: By representing a number as a divided line, the teacher exhibits its formal character in a sufficiently detailed way, which he could not do by showing it as a numeral. Showing numbers as lines depicts them as quantities relatable to one another either through one measuring the other or through their having some common measure. In this way Euclid facilitates our grasp of the truths which he wants to prove about them.

Part Two: Why does the Elements contain the Number Books?

Before considering the place of the number books in the Elements, we might ask why they are there at all. Euclid does not tell us why he included them, and I have found nothing in Heath’s history or commentaries to shed light on the matter. Since I can’t answer that question, I can at least point out some advantages of including them. The inclusion of the arithmetical books allows Euclid to show that there is an analogy between the subject matter of geometry and the subject matter of arithmetic by proving comparable properties for numbers and magnitudes, each by means of proper principles. . Examples of this abound, but I will illustrate with one. In 7, 13 Euclid proves “If four numbers be proportional, they will also be proportional alternately.” Here is an example: since 2:4 :: 5: 10, so also 2:5 :: 4:10. This proposition is proved from the definitions in Book 7 of part and parts. In 5, 16 Euclid had proven the comparable theorem in geometry: “If four magnitudes are proportional, they will also be proportional alternately.” This proof rests on the definition of same ratio given at the beginning of Book 5. Euclid’s drawing out of the likeness as well as the difference between geometrical and arithmetic theorems is I think the most important result of including the books on arithmetic, at least from a philosophical point of view. I say this because the relation between number and magnitude was a controversial issue for the Greeks and in fact is still a question of great interest.

We moderns are accustomed to the idea that there is a universal mathematics, one which is most properly expressed in symbols. There have been various ideas about how universal mathematics stands with respect to arithmetic and geometry. For Viete, Descartes’ predecessor in the invention of algebra, the symbols and the rules for their manipulation are the same for both kinds of mathematics, but each requires a distinct process of interpretation and justification. Another opinion is they are unified by a common subject matter, which might be called quantity as such. A common opinion among the moderns is that mathematics is a branch of logic, so that the symbols themselves seem to be its subject matter.

More importantly for our purposes, though, was the pre-Euclidean opinion that all quantity is the same kind of thing because all quantities are commensurable. This Pythagorean understanding would reduce all mathematics to arithmetic. This view could no longer be held after the shocking discovery that the side of a square and its diagonal are have no common measure. Prior to the scandal of the incommensurable, several of the theorems we find in the first four books Elements had flawed proofs based on a purely numerical theory of proportion. Here is how Heath describes the situation:

After the discovery of this one case of irrationality [i.e. of the square root of two] it would be obvious that proportions hitherto proved by means of the numerical theory of proportion, which was inapplicable to incommensurable magnitudes, were only partially proved. Accordingly, pending the discovery of a theory of proportion applicable to incommensurable as well as commensurable magnitudes, there would be an inducement to substitute, where possible, for proofs employing the theory of proportions other proofs independent of that theory. This substitution is carried rather far in Euclid, Books I - IV.

In other words, all of the demonstrations in the first four books of the Elements are valid quite apart from questions about the divisibility of the continuum. Euclid ingeniously shows that many elementary properties of figures do not an any way rest on the difference between the continuous and the discrete. This sets these books apart from those that follow. Once the universally valid theory of proportion has been established, it is possible to treat the rest of mathematics according to the distinctive principles of the continuous and the discrete.

The treatment of proportion in Book 5 of the Elements makes possible a satisfactory treatment of magnitudes both commensurable and incommensurable. By providing the separate but parallel treatment of proportion in number in Book 7, Euclid shows most clearly that geometry and arithmetic must each be developed from its own proper principles, however many theorems they seem to share in common. This, I propose, is one important reason for including the number books.

A second advantage of including the number books is that it makes the Elements is a more complete elementary treatment of mathematics. When we take a look at the contents of these books, we will see evidence that Euclid desired to aim at completeness, sometimes even at the expense of good order. Taken together with Book 10, the number books give an adequate treatment of the ways in which magnitudes can have ratios to one another. All the potentialities of ratio implicit in Book 5, or rather all those appropriate to beginners in mathematics, are revealed to the student.

A third advantage is that numbers show up from time to time in geometrical theorems.

A most noteworthy example of this is in the very last proposition in the Elements, Book 13, Proposition 18: To set out the sides of the five figures [that is, the five regular solids inscribed in the same sphere] and to compare them to one another. Some of these comparisons involve numerical ratios, as that the square on the diameter of the circle is to the square on the side of the inscribed pyramid as 3:2. We learn other truths along the way involving number, some as simple as 1, 41, which says that the parallelogram having the same base as a triangle and is in the same parallels is double the triangle, or as complex as Book 12, Proposition 10, which shows that any cone is a third part of the cylinder which has the same base and the same height. We see that solid geometry, which is the most complete geometry of the physical world, brings together the discrete and the continuous in a profound way. Here we see the most perfect marriage of geometry and arithmetic.

Part III: The Nature of the Number Books and the Order of their Propositions

If we grant, then, that there are some good reasons for including the number books in the Elements, we may still find them to be unsatisfactory in themselves, as being disorderly. One would think that 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. Arithmetic pursued in isolation from geometry would look quite different from what we find in the number books.

I have already suggested that Euclid treats number in light of the notion of measuring and being measured. This is demonstrated by the way Book 7 begins. Number is defined as a multitude composed of units, and multitude arises from the unit by way of addition. Addition is a kind of measuring, in the sense of meting out, as when we count out 75 cents for a candy bar. This is measuring as composition. But the more common kind of measurement is a process of resolution, in which we begin with numbers or a magnitudes as given and analyze them into equal parts. To account for this kind of measurement, Euclid next defines part. This use of part is not like that in the axiom which states that the whole is greater than the part. Here part means a part which measures a whole. From these basic ideas underlying measurement Euclid goes on to define the two most fundamental divisions of number into species. Even and odd numbers are distinguished by whether or not they can be measured by the dyad, the number two. The notion of measurement is also required for the definitions of prime and composite numbers. Primes are measured by the unit alone, while composite numbers have other measures as well. Thus we see that measure is at the very root of number and its division into species. Let us look more closely now at some of these divisions.

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. It is certainly the simplest, and the most easily applied to given numbers. Since for every odd there is an even, namely its double, it seems to be the most perfectly symmetrical division of number, and so in some way the most beautiful. The even and odd can themselves be divides into sub-species, and this is what Euclid does next. There are even times even numbers such as 8, even times odd numbers such as 6, and odd times odd numbers such as 9. These subspecies of the odd and even are defined explicitly by measure. For example, an even times even number is one that is measured by an even number an even number of times. Next Euclid gives the other important division of number which comprises them all, namely prime and composite. These two divisions are the only ones given by Euclid which comprehend all numbers. The notions of prime and composite are then extended to relations among numbers. Numbers are prime to one another if they have no common measure and composite to one another if they do. This is not a division of number into species but a description of important properties numbers can have in the category of relation.

It is interesting to note that up to now, Euclid has not defined any of the operations which are taken as fundamental to arithmetic. He assumes both addition and measurement, which is a kind of division, as well-known. Later, subtraction will also be assumed. It is interesting then that he does define multiplication. Perhaps he feels he needs to do this to avoid any confusion with the analogous geometrical operation, the forming of a rectangle. At any rate, rectangles are called to mind in the very next definition, where he defines a plane number as one which has been formed by multiplying one number by another, which in this context are called sides. He similarly defines solid numbers as those produced by the multiplication of three numbers. By defining multiplication in terms proper to arithmetic, he indicates that his use of geometrical terminology in these definitions is only by way of analogy.

The most important examples of geometrical numbers, squares and cubes, are defined next, and then numerical proportion, with a view to defining similar plane and solid numbers. Finally Euclid defines perfect numbers. Recall that these are numbers which are equal to the sum of their measuring parts, including the unit.

It may seem strange that Euclid does not begin with theorems about properties of the odd and even or with the prime and composite numbers. This would be strange if he were interested in pure numbers, in what is now called “Number Theory.” He is concerned instead with numbers in comparison to each other. The principal subject of Book 7 is numerical ratio and proportion. The first proposition gives the criterion by which two numbers are prime to one another, and the second which gives the method for finding the greatest common measure of numbers which not prime to one another. This second proposition is needed for what follows, a series of demonstrations analogous to the propositions about proportional magnitudes in Book 5, up through the sameness of ratio ex aequali. He goes on to present a number of propositions dealing with sameness of ratio that are specific to numbers. Some of these have analogies to geometry. Especially interesting is Proposition 16, which defines cross multiplying. This is comparable to the proposition defining compounding of ratios in Book 6. We see that, right from the beginning, Euclid wants to make us aware of parallels between arithmetic and geometry.

The rest of Book 7 contains more theorems about relatively prime numbers as well as about numbers that are simply prime or composite. Among these are propositions for reducing ratios to their least terms, finding the least common measure, finding the least number measured by two or three numbers, and to find the least number that has given parts. The essence of any numerical ratio is contained most simply in its least terms. This verifies that the subject of this first Book is numerical ratio and proportion and the treatment of them here is the most elementary.

The next two books of arithmetic present many interesting theorems having for the most part some connection with geometry. The principal subject of Book 8 is numbers in continued proportion. These are sequences of numbers in which each one after the first is the geometric mean between the ones prior and posterior to it, for example: 2, 4, 8, 16 ... Starting with Proposition 11, Euclid sets out propositions relating figured numbers (squares, etc.) to continued proportions. The first two of these are particularly important. The first says that between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio that which the side has to the side, and the second says that between two cube numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. More propositions about figured numbers round out the books.

Book 9 at first sight looks very disorderly, as if it were just a catch-all for other interesting theorems. There may be something to that, but I think we can makes some sense of it. It picks up where Book 8 left off, with more propositions about figured numbers, but the point of view is different than in Book 8. In the first seven propositions, he is interested in figured numbers considered in themselves, not in their relation to other numbers. This theme is carried on in the next sequence of propositions, which deal with continued proportions starting from the unit. Such a sequence builds up a series of numbers, each of which is the prior one multiplied by the second in the series. Here is an example: 1, 3, 9, 27... i.e. 1, 3, 3x3, 3x3x3 ... So even here the individual numbers are of interest, as being a sequence of successive squares or cubes, or numbers arising from squares and cubes. More propositions relating to this theme follow, eventually leading to theorems in which prime numbers play a part in these proportions. These propositions culminate in an investigation of the conditions in which it is possible to find a third proportional to two given numbers, i.e. when, given numbers a and b, is there a number x such that a:b::b:x.[11]

What happens next, leading up to the end of the number books, is puzzling. Prop. 20 shows that there is no end to the generation of prime numbers. This complicated and very important proposition could have been proved in Book 7. The only rationale I can see for including it here is that Book 9 has been more closely focused on the properties of given numbers taken in themselves than were the first two books. Proposition 20 is followed some others (21-30) which are elementary and even trivial, having to do with odd and even numbers. It is hard to see how these are not out of place, and I am inclined to doubt that Euclid put them there. But if I had to speculate: One possible way to relate these propositions to the ones that come before in Book 9 is to note that most of them relate to the unending production of numbers from other numbers. All the kinds of numbers he has dealt with so far have been showed to be indefinitely many. This is shown explicitly for primes and implicitly for the rest. The continued proportions produce endless series of squares, cubes, and similar plane and solid numbers. Similarly but more obviously, addition produces as many even and odd numbers as we want. Multiplication gives us even times even, even times odd, and odd times odd numbers. Propositions 32 -34 deal explicitly with these kinds of composite numbers.

This brings us to the final propositions, 35 and 36. Proposition 36, which requires 35, gives us a way to make perfect numbers.[12] Recall that perfect numbers are those whose factors when added together make up the number. The number 6 is the first of these, since 6 = 1 + 2 + 3. Is it interesting that Euclid ends the number books with this difficult construction of perfect numbers, just as he ends the work as a whole with the construction of the prefect solids. The number of perfect solids is finite. It is still not known whether or not there are infinitely many perfect numbers.

To sum up my account of the order within the number books:

1. First we have the fundamentals of how numbers relate to one another as prime or composite and how they relate to one another in ratio and proportion.

3. Next we have continued proportions and how they relate to figured numbers. The view shifts from division, that is, measurement, to multiplication.

4. Next we have truths about the figured numbers themselves, their simple production by multiplication and their special production in continued proportions beginning from the unit.

5. Next we have the production of numbers of various other kinds, culminating in perfect numbers.

Part Four: The Ordering of Books in The Elements

To what extent may the order of the Elements be attributed to Euclid? Heath gives this answer: Euclid’s own works “show no signs of any claim to be original; in the Elements, for instance, although it is clear that he made great changes, altering the arrangement of whole Books, redistributing propositions between them, and inventing new proofs where the new order made the earlier proofs inapplicable, it is safe to say that he made no more alterations than his own acumen and the latest special investigations...showed to be imperative in order to make the exposition of the whole subject more scientific than the earlier efforts of writers of elements.” [13] If Heath is right, we must take seriously not only the inclusion of the number books in the Elements, but also their place in the work as a whole. We ought to be able to find a rationale according to which Euclid’s order makes sense. As we look into this, it is important to keep in mind that the order appropriate to arithmetic considered within the framework of geometry is not the same as the best order for teaching arithmetic as an art of calculation.

We must begin by considering how Euclid structures his treatment of geometry. The subject of geometry is magnitude. Magnitude is divided into extension in one dimension (straight lines), in two dimensions (plane figures) and in three dimensions (solid figures). This division gives structure to Euclid’s work, but not in a simple way. It is not possible to begin with lines and work up to solids or to begin with solids and work down to lines. Euclid does of course begin with lines in one sense. Straight lines and circles are postulated, but they appear in Book 1 only as boundaries or as auxiliary lines within figures. This is not the same thing as a study the one dimensional as such. He does makes such a study but it is difficult and has to be deferred. Solid geometry cannot be studied before plane geometry, since the properties of solids can’t be known without knowing about their boundaries. Solids are the figures most difficult to understand and perhaps also the most beautiful, so it is fitting as well as necessary that they come last in the work. Plane geometry must therefore be treated first.

Plane geometry is developed at length in Books 1-4, and this section ends with the inscribing and circumscribing of regular polygons in and about circles. In Book 4, we join together the most perfect plane figures in constructions that are beautiful both to the mind and to the eye. This book brings to completion a distinct part of Euclid’s subject, but the treatment of plane figures is far from being completed. Book 6 deals with the important subject of ratio and proportion in two dimensional figures. This study demands knowledge of ratio and proportion in a more universal way, so the development of plane geometry is necessarily interrupted by Book 5. Book 6 brings to completion the elements of plane geometry.

Book 10 is the only book which deals expressly with lines. It is the science of the one dimensional carried on as far as Euclid’s method allows. I have already said that Euclid could not begin with the science of the one dimensional, and now I must explain why not. Extension, whether in one, two or three directions, serves as the material component of geometry. A science first of all considers formal properties of its subject. Since the material causes of a thing are as such unintelligible, they must be made intelligible through form. This is true at every level. Wood, the material of furniture, is known through the formal properties of wood. Elementary particles are known through their forms, and the obscurity we find when we try to understand them comes from the elusiveness of their forms. Prime matter is known only insofar as it is demanded by generation and corruption. In geometry too the forms, which are the shapes, must be understood before we can know the areas and volumes contained by them.

Now consider how lines can be known. First we can see that there is a difference between the straight and the curved, and that while there is only one way of being straight, there are infinitely many ways of being curved. Some kinds of curvature are intelligible and some are not, and the geometer is interested in those that are. Euclid deals with curved lines only as boundaries of figures, but Apollonius and other ancient geometers begin to study curved lines as such. But is there a possibility of studying lines as material, that is, in the way in which we study area and volume? Yes. Euclid begins this study in Book 10 and is able to carry it quite far. That the subject of Book 10 is straight lines in particular and not magnitude in general, as in Book 5, is made clear by the words he uses in enunciating the theorems. He even goes so far as to posit a common standard by which all such lines can be classified as either rational or irrational. He comes ever so close to Descartes’ idea of introducing a unit into geometry.

It remains to be seen why the study of lines is not a good place to begin. Every species of magnitude is divided by the species below it. In plane geometry, a portions of the plane is delimited by lines, straight or curved, producing triangles, circles and so forth. In solid geometry, the three dimensional continuum is delimited by plane figures and curved surfaces to bring forth cubes, pyramids, spheres etc. 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 division of the line that we are far from having discovered them all.[14]

Although the line is the simplest geometrical object, there are good reasons not to treat it first. Pedagogically it makes sense to treat the more knowable before the less knowable. And whereas commensurable, that is numerable, lines do seem more knowable than figures, the same cannot be said of incommensurable lines. Books 5 and 10 make the greatest demands on the student’s powers of abstraction. The fact that incommensurables were not known to the earliest geometers points to their obscurity. Perhaps also the fact that the imagination is less vividly engaged when studying lines makes the propositions about them more difficult to learn and remember.

Although the reasons just given have some force, the more essential reason is mathematical. To divide or measure lines, it is necessary to carry out constructions in two dimensions. Euclid bisects a straight line in Book 1, 10 and in Book 6, 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 Book 6, 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. Finally, it is worth mentioning that Book 10 includes some applications of proportion to squares and thus implicitly to other rectilinear figures. This is a hint that the distinction between rational and irrational is applicable to all kinds of extension. For all these reasons, it is necessary for Euclid to put Book 10 after Book 6. But are there also good reasons not to defer the treatment of straight lines until after solid geometry, to the very end of the book?

That we should not put Book 10 last can be seen in at least two ways. First, the treatment of incommensurables underlies the argument by approach to a limit and the powerful method of proof by double reductio. Using the latter, Euclid is able to establish the important theorems that circles 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. Second, the perfect solids are clearly meant to be the climax of the whole book of the Elements. That there are only five of these is a source of great wonder. The plane figures which bound these figures 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. This could now be known without Book 10. For these reasons alone, it is right for Book 10 to come before Book 11, the first book on solids.

To sum up the order I have justified so far: plane geometry in Books 1-4, proportion in magnitude generally and then in plane figures in Books 5-6, rational and irrational lines in Book 10, solid geometry in Books 11-13. All that is missing from this picture are the number books. Where do they fit in?

First, we can see the number books as belonging to the study of ratio and proportion, since this is implied in the understanding of number through measurement. Since arithmetic is easier than geometry and prior to it, one might wonder why Euclid does not put these books right before Book 5, so that ratio and proportion are treated first in number and then in magnitude. A simple answer might be that this would unnecessarily interrupt his treatment of plane figures. Another possibility, which would avoid this problem, would have been to begin the entire work with the books on number. After all, the theorems in this part are independent of the geometrical theorems, and starting with arithmetic would have some advantages. Numbers are better known and more accessible to the student than magnitudes. Also, separating off the treatment of multitude from that of magnitude would reinforce the idea that arithmetic is not simply reducible to geometry but is a science in its own right.

Despite the fact that arithmetic is an independent science, the number books do not stand alone in the Elements as pure arithmetic. I think they must be seen as integral to Euclid’s approach to geometry. I have spoken at length about how Euclid considers number by means of the idea of measurement. This is not the only possible way to conceive them. For instance, Richard Dedekind writes: “I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding.”[15] Starting with addition rather than with measurement could form the basis of a work on arithmetic and no doubt is the best way to go for treating the art of calculation. Euclid is not interested in calculation, and Dedekind’s way of developing a science of arithmetic departs from the path of the ancients. Euclid seems also not to have been very much interested in the kind of arithmetic one finds in Nicomachus and other ancient writers on the subject. For Euclid, the path to a scientific treatment of arithmetic passes through geometry.

Seeing the number books in the context of geometry allows us to understand the placement of the number books. The reason for placing them between 6 and 10 cannot be necessity, since they are independent of the rest. If necessity is not the reason, he must have seen it as appropriate to place them where he does, right before Book 10.

This idea is supported by the character of the first few propositions in Book 10. A careful look reveals some parallels between how Books 7 and 10 begin. Proposition 10, 1[16] has no equivalent in the number books, but Proposition 2 parallels Book 7, Proposition 1. Here is 7, 1: “Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the numbers will be prime to one another.” Compare this to 10, 2: “If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.” This proposition reveals the distinctive character of magnitude, as opposed to multitude. But Euclid goes on to make explicit how multitude can exist in magnitude. Propositions 3 and 4 exactly parallel 7 2 and 3 by finding common measures of two and three commensurable magnitudes respectively. Propositions 5 - 8 nail down the difference between commensurable magnitudes and numbers. He shows that commensurable magnitudes have the ratio of a number to a number and conversely, and that incommensurable magnitudes do not have the ratio of a number to a number, and conversely.

From these considerations, we may infer that Euclid wanted us to think about number and magnitude in contrast and comparison to one another. The number books should not be thought of as standing on their own as an independent treatment of arithmetic randomly inserted into a book of geometry. By looking first at number as manifest in lines, and then at lines where number fails to cover all its possibilities, Euclid manifests the necessity to go beyond the flawed geometry of his predecessors to a true and more complete science of magnitude and figure.

Here, then, is my account of the order of the Books of the Elements. The first four books contain the elementary propositions about figures that can be known without having to invoke a theory of proportion. Of these, Book 1, which deals with rectilinear figures is the foundation of all the rest. This book ends with the important and beautiful Pythagorean theorem and its converse. Book 2 gives analytical tools for the books to follow. Book 3 deals with properties of circles, the simplest figure after the straight line and next in the order of learning after the basic rectilinear figures. It presupposes book 1. The last three propositions of the book reveal the power of theorems in Book 2 to reveal complicated properties of simple figures. These only hint at what the geometer can figure out with the second book in his tool-kit. Book 4 beautifully brings together the regular polygons, of which the square and the equilateral triangle are the most perfect, with the circle, by inscribing and circumscribing. Book 5 teaches the fundamental theorems about ratio and proportion in magnitude, and this doctrine is applied in Book 6 to reveal many important and beautiful theorems about plane figures. Books 7-9 treat of numbers, while Book 10 deals comprehensively (as much as was possible for Euclid) with lines both rational, which correspond to numbers, and irrational, which do not. Book 11 presents the most fundamental theorems about constructions in three dimensions and with solids formed by planes. Book 12 deals with solids based upon the circle, and with ratios of volume found in various kinds of solids. Finally, Book 13 constructs the five regular solids within a sphere and considers the ratios of their sides to one another. Some of these ratios can be expressed in numbers and some cannot. Euclid shows that the regular solids draw together the rational and the irrational, number and that which cannot be numbered. In this way he brings the elementary study of geometry to a beautiful and fitting conclusion.

[1] Elements, Book VII, definition 1.

[2] See Aristotle, Metaphysics Book XIII, for an extensive discussion of various ideas about the nature of the unit, of number and of other mathematical objects.

[3] Sir Thomas Heath, A History of Greek Mathematics, Vol. 1, p.98,  Dover Publications (New York, 1981).

[4] To describe this in a neo-Platonic way, one might say that numbers are emanations from the One.

[5] Russell’s definition of two as the set of all twos seems to me to have the same problem.

[6] St. Thomas Aquinas,  Commentary on Metaphysics Books 7-12, 1725 (Augustine Institute, Green Bay, Wisconsin, 2020), my translation: “Est enim per se unum numerus, inquantum ultima unitas dat numero speciem et unitatem,; sicut etiam in rebus compositus ex materia et forma, per formam est aliquid unum, et unitatem et speciem sortitur.”

[7] Joe Roberts, Lure of the Integers, The Mathematical Association of America (1992), p. 52.

[8] Other measures are of course possible.  For example, two, what the Greeks call the dyad, measures all even numbers.

[9] In Middle English these sticks were called tallies or tally sticks.

[10] 1084b 24-26.  In this passage, Aristotle is criticizing their confusion of mathematical conceptions with metaphysical ones.

[11] A proposition purporting to show how to know whether 3 numbers have a fourth proportional exists in the Greek, but the text is corrupt.  See the note in Densmore’s edition, p. 26.

[12] Without high speed computers, it is not possible to find many of these, since we are required to sum up the continued proportion starting from one and based on 2 until the sum becomes prime.

[13] A History of Greek Mathematics, Vol. 2, p.357.

[14] In Book 10, 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.

[15] “Continuity and Irrational Numbers,” in Essays on the Theory of Numbers, Dover Publications (New York, 1963), p. 4.

[16] “Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this procedure be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.”  The porism extends the theorem to the subtraction of halves.

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