Knorr - Greek number theory

W’KNORR

PROBLEMS IN THE INTERPRETATION OFGREEK

NUMBERTHEORY

EUCLID AND THE ‘FUNDAMENTAL THEOREM OF ARITHMETIC’

IT w o u LD appear that the historian of mathematics has a special advantage among historians of thought, in that the object of his study has a universality and an independence from contingent cultural considerations which other fields do not possess to a comparable degree. For instance, 2 + 3 = 5 independent of who makes the assertion and under what circumstances.’ But this advantage brings with it certain problems: (1) What can we mean in claiming that some early mathematical achievement is an equivalent of some later achievement expressed in different (e.g. more familiar modern) terms? (2) How appropriate is it to employ later standards of logic and rigor in the assessment of earlier mathematical works? I should like to take up these two issues in the context of a specific case, Euclid’s version of the ‘fundamental theorem of arithmetic’, which has been the subject of some recent and especially unsophisticated analyses.

Two authors, M. D. Hendy ( 1975) and A. A. Mullin (1965) have examined the Euclidean theorem on the assumption that ‘to judge for ourselves the depth of Euclid’s propositions we need to translate his geometrical ideas into the modern language of arithmetic.‘2 They indicate certain defects in Euclid’s treatment, some of which are real, but others merely alleged, as I shall show. They correctly observe that the Euclidean theorem (Elements IX, 14) is not technically equivalent

lIf, however, we distinguish the fact that 2 + 3 = 5 from the statement “2 + 3 = 5”, we may recognize that the latter is culture-dependent. The statement would have different meanings for, say, Plato and Peano-one considering it the assertion of an absolute and irreducible truth, the other viewing it as a theorem. A more detailed examination of this distinction is not necessary in the present context, but will be alluded to again briefly in our concluding remarks.

2Hendy, 190. The same assumption is implicit in Mullin and in the authorities which both Hendy and Mullin rely upon: Hardy and Wright and Buchner.

Stud. Hist. Phil. Sci. 7 (1976), No. 3. Printed in Great Britain

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354 Studies in History and Philosophy of Science

to the complete ‘fundamental theorem of arithmetic’ (FTA), but at most to a restricted case of it. But they make three additional claims which I maintain are invalid: (1) Euclid’s proof holds only for numbers factorable into three distinct primes, rather than into any finite number of primes; (2) this limitation is rooted in the geometric intuition underlying the Euclidean arithmetic; and (3) ‘Euclid having gone so far could not prove the fundamental theorem itself’,3-that is, technical considerations are such that ‘his method cannot be readily adapted’ to the general case. In opposing each of these three claims, I shall show how they result from the authors’ inappropriate use of criteria based on modern number theory.

Let us state the two theorems in question: (1) FTA: The factorization of any natural number into prime factors is unique, save for order. (2) Elements IX, 14: If a number be the least that is measured by prime numbers, it will not be measured by any other prime except those originally measuring it.4

The authors (Hendy and Mullin) correctly assert that the two theorems are not technically equivalent. For Euclid’s theorem does not cover the case of numbers which possess a square factor. The mistaken claim that the theorems are equivalent is explicit in T. L. Heath’s comments on Euclid [1921, I, 401; 1926, II, 4031 and has insinuated itself into the standard secondary literature on the history of mathematics.’ The authors thus perform a useful service in resisting this error. Unfortunately, they propagate several other erroneous or misleading claims.

(1) Hendy severely circumscribes the generality of Euclid’s proof:

To demonstrate this general proposition [IX, 141 Euclid proves only the explicit

case where a number A is the product of three primes B, C and D. Heath

generously, but not rigourously, generalises this to an unspecified number of factors

again.

The precise force of Hendy’s objection is not clear. Does he mean (a) that Euclid himself thought his proof valid only in the case of products of three primes? or (b) that by the standards of modern logic the validity of the proofs as given must be so restricted? We may show that

3Hendy, 191. Italics arc mine. 4Tbis and other quotations of Euclid follow Heath’s translation. As the proof will be

discussed later, a sketch of it will be useful. Let N be the least common multiple of primes A, B,

C. If a prime p divides N, we may set N = p x M, where M is less than N. If p equals none of the primes A, B, C, then since each is a measure of N, it must measure M (by VII, 30). ‘Ibis makes M a common multiple of A, B, C. But N was constructed as the Eeast common multiple of these numbers. Ibis contiadiction establishes that p must equal at least one of the terms measuring N.

sBut one may note that many accounts avoid Heath’s error. Cf. Bourbaki, 111; Bell, 49, 22Of; Taisbak, 109; and Davenport, 19.

Problems in the Interpretation of Greek Number Theory 355

(a) is just false, while (b) imposes an inappropriate conditicn upon the historical interpretation of Euclidean arithmetic.

Without doubt, Greek arithmetic suffered the handicap of an under-developed algebraic notation. Euclid could operate rhetorically with unspecified integers (or, for that matter, lines, plane and solid figures) via a literal symbolism; but to examine the case of an indefinite number of such unspecified numbers presented difficulty. Euclid’s solution here was to consider a definite number of terms, where this would not affect the pattern of the argument. For instance,

IX, 20: Prime numbers are more than any assigned multitude of prime numbers.

Let A, B, C be the assigned numbers . . . .

Does Euclid imagine he has only established that the multitude of primes is greater than three? Or, again: VII, 33: Given as many numbers as we please, to find the least of those which have

the same ratio with them. Let A, B, C be the given numbers, as many as we please . . .

In this case it is evident that Euclid assumes an explicit number of terms only for reasons of illustration. He certainly does not recognize this assumption as a restriction on the general applicability of the theorem or its proof.

This same stylistic tactic appears throughout the arithmetic books.6 In the determination of the greatest common measure of given integers, Euclid develops the procedure first for two integers [VII, 21 then for three [VII, 31 . This suffices to establish the general recursive nature of the procedure; and, in fact, the ancient commentator Hero already made explicit the ability to carry through the process for any number of given integers [Heath, 1926, II, 302fl. The procedure for finding the least common multiple is also developed first for two, then for three given integers [VII, 34, 361. Implicit is the indefinite extendability of the process; and it is precisely this process which underlies the notion of product used in the factorization theorem [IX, 141.

The ‘perfect-number theorem’ [IX, 361 illustrates the same tactic in an unusual way. Here, Euclid proves that a specified construction gives rise to perfect numbers, i.e. those equal to the sum of all of their own proper divisors. In this construction, one sets out the geometric progression 1, 2, 22, 23, . . . , 2”, where n is such that the sum of the terms of the progression is a prime. For instance, 1 + 2 = 3, or 1 + 2 + 4 = 7. When this prime is multiplied by the final term in the progression, a perfect number results (e.g. 3 x 2 = 6 and 7 x 4 = 28 are perfect). But in the proof, Euclid hypothesizes ‘as many numbers as we please, A, B,

C, D, beginning from a unit . . . in double proportion, until the sum of

60ther instances are VII, 12, 14; VIII, l-4,6-10; IX, 8-13, 17, 21-23, 32, 35, 36.


all becomes prime’. Taking him literally, we are examining the terms 1, 2, 4, 8, whose sum is 15; this is not a prime, so the hypothesis of the proof does not in fact lead to the construction of a perfect number. It is thus clear through this example, as through those mentioned above, that Euclid does not view as restrictive any such assumption of a particular number of terms; the assumption is made for the convenience of the exposition of a property or process applying to ageneral class of an indefinite number of terms.

One may note that this use of explicit values for an intended indefinite index appears elsewhere in the Elements. For instance, the Euclidean division algorithm for integers [VII, l-31 and for magnitudes [X, 2-41 requires an unspecified number of subtractive steps; but Euclid assumes in VII, 1 that the process terminates after three steps; a similar assumption is made in X, l-4. The Euclidean conception of proportion [V, Defn. 51 requires the introduction of arbitrary integral multiples of given ma<gnitudes. But in practice a definite multiple is taken, as in VI, 1 where Euclid assumes the doubles

of Cgiven lines. Hence, if we are to adopt the critical position expressed in (b) above, we must sacrifice the major portion of the Elements as not rigorously general.7

What, then, is our judgment of Euclidean method to be? Shall we reject Euclid, concluding with Hendy and Mullin that ‘it is reasonable to assume that formal induction either did not occur to [the Greeks], or else was considered logically unacceptable’? I believe not; we do better to inspect Euclid’s methods and the dialectical commentary on these by his contemporaries. On the matter of formal induction, one may see that Euclid establishes results obtaining for all the natural numbers by an argument very similar to our complete-inductive proofs. For instance, as indicated before, his treatments of the procedures for finding the greatest common measure and the least common multiple fall into two parts: first Euclid verifies the procedure for two integers; then he shows how the case for three integers can be reduced to the case for two. In like fashion, a modern complete-inductive proof of a proposition P ranging over the natural numbers would first establish the

validity of P (1) and then show that P (n) implies P (n+l) where n is an unspecified integer. Euclid’s use of the analogous recursive pattern thus behes the claim that he lacked or distrusted the principle of induction.

‘Modern treatments of number theory are inspired by the Disquisitiones Arithmeticae (1801) of Gauss. It is interesting that Gauss expresses general theorems in a fashion close to Euclid’s. For instance, in section 15 he proves that ‘if none of the numbers a, 6, c, d, etc. can be divided by a prime p neither can their product abed etc’. Shall we then judge that Gauss’ proofs are defective, just as Hendy and Mullin allege Euclid’s to be? Or are we to believe that the mere insertion of ‘etc.’ is sufficient to rehabilitate the Euclidean proofs?


Rather, it was a technique of general quantification which eluded him. Where the modern treatment can introduce the dummy-variable n and validate the generality of the argument by implicit appeal to an axiom or theory of algebraic representation, Euclid proves his theorem for n = 3, leaving it understood that the proof applies for any other value of n.8 Now, the precise logical justification of universal proof via general variables has been attempted in a variety of ways and at least one logician recognizes the matter as problematic.’ In view of this, we should be permitted to ask whether a defence of Euclid’s approach can be given.

Euclid could present two arguments. First, the primary objective of the Elements is pedagogical. In the context of an elementary introduction to geometry and number theory it can be quite effective to substitute specific values for general terms in the investigation of a proposition, especially when such substitution does not affect the generality of the concepts or procedures under discussion. This manoeuvre is common enough in modern teaching as not to require further remark.

Second, when one proves a proposition obtaining for a set of entities which have certain defining properties in common by effecting the proof for just one member of the set, one is employing this member as a representative instance. The proof is generalizable to the extent that one has employed only those properties of the representative which it has in common with all the others in the set. Precisely this justification of mathematical generality is given by Proclus. Commenting on Elements I, 1, the construction of an equilateral triangle on a given line, he writes thus:

[geometers] pass therefore to the universal conclusion in order that we may not

suppose that the result is confined to the particular instance. This procedure is

justified, since for the demonstration they use the objects set out in the diagram

not as these particular figures, but as figures resembling others of the same sort.’ ’

In so stating this principle of representation, Proclus follows the traditional Aristotelian view. In Aristotle’s logic, most notably the

sThis suggests a reason for Euclid’s frequent choice of 3 as the particular instance examined. When a proposition or procedure involves recursive steps, the case of 2 will not suffice for showing the recursion, whereas the cases of 4 onward will introduce patent redundancies into the argument. Reduction of the case of 3 to that of 2 is both necessary and sufficient for illustrating the recursion.

9Russell writes: ‘The notion of the variable is one of the most difficult with which logic has

to deal’ [1937, 5-61 and disclaims the goal of providing a ‘satisfactory theory’ of it in the principles of Mathematics; nevertheless, he devotes chapter VIII (esp. section 93) to the discussion of the variable and general propositions. W. van 0. Quine introduces a ‘rule of universal generalization’ for the purposes of his own theory of quantification [1959, 159-1671. 1 o~roclus (Friedlein), 207; Morrow, 162.


Posterior Analytics, problems in the execution of formal proofs are taken up. The aspects of generality in proofs are of specific interest. Aristotle insists that we can have scientific knowledge only of universals, not of particulars-that is, of attributes of things, not of the particular things themselves. We can know scientifically that all pairs are even, for instance, even though we cannot know (in another sense) all pairs [Post. Anal. I, 11. But attributes do not exist apart from things. How, then, can one demonstrate a proposition like ‘A is true of B’ in general?-e.g. that ‘having angles which sum to two right angles’ is true of ‘being a triangle’. Aristotle indicates this is done by assuming a random instance of the class in question, here a triangle chosen at random (ibid., I, 4). Care must be taken to choose the instance as representative of the widest possible class. For instance, in demonstrating the property of the sum of the angles, one should choose a triangle at random, not an isosceles triangle at random; otherwise, the proof holds for the subclass only (ibid., I, 5). ” Aristotle’s remarks thus provide guidelines for the setting out of general propositions and proofs. Euclid’s use of particular instances in the demonstration of general theorems is seen to conform to these Aristotelian rules.”

Precisely such a principle of representation answers the objections raised by Hendy against the generality of Euclid’s proofs. Of course, we cannot retrieve the particular thoughts of Euclid on this matter; but a historical analysis of the nature of Euclid’s procedure does not have such a specific objective anyway. The Elements (both those of Euclid and those, no longer extant, of his Academic predecessors), by virtue of their project of organizing mathematics systematically and their presumption of offering complete formal proofs of mathematical truths, became the focus of inquiries on the nature of proof itself. Logical problems related to the Elements were debated both before and after Euclid’s time, and some answers were proposed by scholars in the circles of Plato and Aristotle; moreover, Euclid’s presentation meets the test of many of these answers. These facts must certainly interest the historian of logic and mathematics.

11A remark on this theorem by Geminus (quoted by Eutocius) has been interpreted to indicate that the proof was originally divided into the cases of equilateral, isosceles and scalene triangles. But Heath argues this refers not to the original proof historically, but to pedagogical treatments familiar in antiquity [ 1926, I, 317-320; 1949,431.

‘2Related to these remarks on universal proof is another point by Aristotle: that the geometer does not err in assuming a line is straight or one-foot-long, even though the drawn representative has not these properties; for the actual straightness or the actual length of the particular line examined is not an element in his syllogism [Prior Analytics, 49, b33-7; 50 al-4; Metaphysics, 1089 a21-5; cf. Heath, (1949) 26f, 219f]. E. Beth discusses these Aristotelian passages in the course of the presentation of his own tableau for general deduction via specific instantiation [ 1964, 190-1941.


But to go further than this-to press the issue of rigor in accordance with more stringent modern canons-is in my view less fruitful. For instance, the objection that a representation-principle in the form given above is not adequate for the purposes of number theory is a legitimate one. For if we assume a recursive procedure terminates after three

steps, say, how can we avoid appealing to the specific nature of three in the demonstration? But this objection holds equally against Aristotle and the fourth-century logicians as against Euclid. What I wish to emphasize is that Euclid’s procedure in this regard was well-considered and co-ordinate with the dialectical critique of mathematical methods in antiquity. By imposing strict modern canons in our assessment of his treatments one would become guilty, thus, of two major oversights: ignorance of the state of foundations studies in antiquity: and ignorance of the pedagogical context of the Elements.

(2) I have suggested that the reason why Euclid chooses three as the representative instance in his arithmetic proofs follows from the recursive nature of his arithmetic processes. It has nothing to do with some specious restriction to three factors as being the maximum ‘that [he] clearly conceived with [his] geometrically oriented notation’.’ 3 There is a widely-circulating impression that Euclid’s number theory is essentially geometric in its underlying intuition. I wish now to show the untenability of this view.

First, let us consider the features which have led to the adoption of such a view. (a) Euclid represents integers by line segments in the propositions in Books VII, VIII and IX. (b) Euclid introduces certain classifications of numbers, such as ‘plane’, ‘solid’, ‘square’ and ‘cube’ [VII, Defn. 16-191 which have an obvious geometric motivation.

The linear representations in (a) are firmly established in the Elements. They have their utility for helping us keep in mind the order of the terms introduced in successive steps of the arguments. Line segments are especially well-suited for displaying the additive relations between integers. But other than this, Euclid does nothing with the geometric mode, and it is this fact which we should find surprising. For example, the theorem on the commutativity of multiplication [VII, 161 ought to be absolutely trivial. To show A x B = B x A for given integers A and B, could we not figure each product as a rectangle? The equality would then follow from Euclid’s axiom of superposition and the invariance of area under translation. But no; Euclid provides an arithmetic demonstration based on numerical proportions and an asymmetrical definition of multiplication [VII, Defn. 151 involving

13Hendy [1975, 1901 owes this view to Mullin [1965, 2181, who lifts it from Hardy and Wright [ 1938, 1821, who in turn cite S. Buchner.


repeated addition. Geometric properties and propositions are not utilized here, nor anywhere else in the arithmetic books.

As for (b), these quasi-geometric classifications of numbers had their origins in the concrete Pythagorean arithmetic of the fifth century. But already by Plato’s time, mathematicians viewed them as geometric merely by analogy [cf Theaetetus 148A]. Euclid’s definitions are completely arithmetic. A ‘square number’ is the product of two equal factors, a ‘cube’ the product of three equal factors. The ‘product’ [i.e. of multiplication] is in turn defined as the repeated addition of one integer according to the multitude of units in another [VII, Defn. 151. Significantly, when such plane and solid integers are discussed, as in VII, 16-18 or VIII, 1 l-27, the Euclidean manuscripts retain the linear representation. But what is the geometric sense of visualizing a square or cubic integer as a line segment?

A glance at VII, 27 or VIII, l-10 and IX, 8-13 shows that Euclid’s concept of the power of an integer [i.e. the equivalent to our An] was not restricted to second- or third-powers, and hence a fortiori not bound by alleged geometric limitations. Euclid obtains the higher powers by means of indefinitely extended progressions in ‘continued proportion’. For instance, let us generate a progression for a given integer A according to the following recursive rule: 1, A, A x A, A x (A x A), A x

(A x (A x A)), etc. Th en what we call An would be for Euclid the (n+l)st term of this progression. For reasons given earlier, Euclid might find tricky the examination of the general term; but he would certainly have no problem in discussing, say, the seventh powers, if he cared to.

If my argument is accepted, one might wonder why Euclid adopted the linear representation at all. I believe the answer to this lies in the theory of incommensurable and irrational magnitudes developed in Book X. That theory is founded upon an arithmetic distinction: magnitudes are commensurable if and only if they have the ratio of integers [X, 5-61. As Euclid can construct incommensurables only by geometric means, the entities of Book X are of necessity geometric: line segments and rectangles. But as proportions involving integers appear throughout, it becomes convenient to operate with integers as if they were commensurable line segments. I expand upon this view elsewhere [Knorr, 1975, 241-3, 26Of, 3091.

In brief, the arithmetic books VII-IX do not rely upon geometric properties or intuitions. In less than one-fourth of the theorems is even the linear representation of integers contained in the text of the propositions. i4 The adoption of this limited degree of geometrization

14The text of a demonstration points necessarily to an implied linear representation in the

following way only: ‘an integer AB is the sum of integers AC and CB’. Such dictions appear in


is accountable with reference not to the Euclidean theory of number in itself, but to such external considerations as its applications to the theory of incommensurables.

(3) We may now turn to the relation of Euclid’s IX, 14 to the familiar modern theorems on the uniqueness of factorization. As already said, Hendy and Mullin and others have correctly observed that Euclid’s theorem is at most a restricted case, equivalent to the FTA for integers free of square factors. It might be objected that even this is not accurate, since Euclid does not explicitly introduce factorization in the theorem, asserting instead a condition on the divisibility of a constructed composite number by a single prime. But a careful look will show that the transition to a statement on uniqueness of factorization for such square-free integers is immediate from Euclid’s result. Euclid has shown that any prime which divides the least common multiple of given primes must equal one of those primes. That is, if we express the least common multiple of primes a, b, c, . . . as A, and if we discover an alternative expression for A as the least common

multiple of primes a’, b’, c’, . . ., by Euclid’s theorem each of the primes a’, b’, c’, . . . appears among the terms a, b, c, . . ., and by the same token, each of the a, b, c, . . . appears among the a’, b’, c’, . . . Hence, the two sets are identical, save for the order of the terms and possible redundancies. The equality of these sets is a precise technical equivalent of the assertion of the uniqueness of factorization into primes of a square-free integer.

We must now consider the relation of these results to the complete FTA: does Euclid’s omission of the complete form indicate his inability

to provide a proof or even to express the complete theorem, as Hendy and Mullin charge? I argue not. Here, three points may be made:

First, although textbooks today commonly assign to the FTA the status of a fundamental theorem, it should be recognized that such is hardly necessary. If one chooses to begin from the definition of prime numbers and the uniqueness of factorization into primes, developing from this the properties of relatively prime integers, then the FTA is indeed central. r5 But Euclid organizes his number theory in just the reverse order: he starts from the division algorithm for finding the greatest common measure of integers; this gives him an operative definition of relatively prime integers (i.e. those whose greatest common measure is 1). From the examination of relative-primality, he

VII, l-2, 4-11, 15, 20, 28; IX, 15, 20-27, 35. Otherwise, Euclid refers to integers via single-literal symbols: ‘let the integers A, B, C, and D be given,’ for instance. There is nothing intrinsically geometric in the latter mode of reference.

1 SFor such an approach, see Gauss [section II: ‘Congruences of the First Degree’] ; Hardy and Wright [ 19881; and Davenport [ 1952, chapter I.$51.


eventually obtains results on primes, in particular, the important theorem VII, 30: if a prime measures a product of two integers, then it must measure at least one of those integers. l6 In this theory, the FTA loses much of its significance. Far from being ‘fundamental’, it is slipped in well toward the end of Euclid’s arithmetic theory. Placed in the middle of Book IX, it obviously has no importance for the verifications of the prior theorems; as it makes use of no theorems other than VII, 30 and 36, it cannot be viewed as the culmination of any major portion of the theory; nor is IX, 14 applied in any subsequent theorem. Its inclusion by Euclid was probably prompted by the familiarity of the unique factorization into primes within arithmetic practice and the suitability of demonstrating such a result in a theory of numbers.

Second, we must be highly sceptical of any claim that Euclid could

not prove the complete FTA. As shown above, the theorem was not particularly important for Euclid’s theory, so that the omission of the general FTA had no intrinsic significance. Moreover, when one perceives how readily the result is proven in modern school-text introductions to number theory-for which the only requisite is the equivalent of VII, 30, as for Euclid in IX, 14-the hypothesis of essential limitations on the Euclidean approach becomes even more unconvincing.”

Third, to counter Hendy and Mullin’s view of Euclidean incompetence, I propose to offer a proof accessible within the ancient number theory. But did Euclid even &-row the general theorem? We must come to terms with this question first. It is easy to accept a priori

that Euclid must have known the more general result; for it emerges quite readily from the practical arithmetic experience of finding the prime factors of numbers. But we can find within the Elements a more explicit indication of this. The theorem on perfect numbers [IX, 361 requires the designation of all the divisors of the number Yp, where p is a prime. Euclid sets these divisors out as the numbers 1, 2, 2* : 23, . . . , 2”,p, 3% 2*p, * * *, 2”-‘p. In doing so, Euclid must be aware of the relation of the divisors of a number to a factorization of it;

t61n the proof of this theorem, Euclid does not require the full condition of primality. For

he deduces from the hypothesis that the prime p measures the product A x B, butp does not

measure (or equal) A, that p is relatively prime to A and, consequently, measures (or equals) B.

Thus, if we consider the structure of Euclid’s proof of theorem VII, 30, we can perceive that

this Theorem is actually the corollary to a more general assertion: that if a number (not

necessarily prime) measures a product A x B and is relatively prime to A, it must measure (or

equal) B. This more general form will be used later, in our construction of a Euclidean proof of the FTA.

t 7Cf. Davenport’s alternative proof of the FTA [ 1952, 291. But he employs implicitly a

non-Euclidean assumption, the order-independence of n-long products; cf. note 23 below.


moreover, when n is 2 or greater, the factored number possesses a square factor, so that we meet a case not covered in IX, 14. Now, in the course of the proof of IX, 36, Euclid demonstrates that the number 2”~ has no other divisors than those designated. It is thus clear that Euclid possessed a result illustrative of the principle of the uniqueness of factorization into primes, viewed it as a result on factorization, and realized that it required a proof. ‘a We may be satisfied that Euclid also knew the more general result treated in the FTA and its requirement of proof. On this base, we may inquire whether it was within the power of his methods.to produce such a proof.

The construction of a Euclidean proof of the FTA, while presenting little difficulty, will utilize a few corollaries to Euclidean theorems, as well as some properties of the powers of numbers. The class of numbers we term powers are distinguished by the neo-Pythagoreans, who call them a special class of composite numbers, ‘those measured by a single prime only’ [cf. Nicomachus, I, 1 l-131. Diophantus provides a nomenclature and a notation for powers up to the sixth order [Atithmetica, I, preface] and Archimedes denominates very large powers of 10 in the Sand-Reckoner. As said above, Euclid introduces in VII, 27 and in Books VIII and IX a construction of powers paralleling our recursive definition An =A x An- ’ . He sets out a finite geometric progression, where the first term is unity, and examines the properties of the final term. What we denote An is thus for Euclid the (n+l)st term in such a progression. Among the results given by him are these:

(i) Any term measures the final term according to the units in a term within the progression [IX, 1 l] ; that is, An = Ar x Afldr.

(ii) Any prime which measures the final term measures the second also [IX, 121 ; that is, if a prime p measures An, it measures A also.

(iii) The final term in a progression whose second term is prime is measured by all the numbers in the progression and by no other numbers [IX, 131; that is, for a prime p, the set 1, p, p2 . . . , p”- ’

exhausts the proper divisors of p” .

(iv) An additional result, not given by Euclid, follows at once from (iii): if two numbers are powers of different primes, they are relatively prime to each other.lg

We state, further, two theorems which complement IX, 14: (v) If Nis the least common multiple of a set of numbers A, B, C, . . .,

where each term in the set is relatively prime to each other term in the

’ 8 Euclid’s proof in IX, 36 is not generalizable in this form to the general case of uniqueness of factorization into primes, however.

‘9F’roof: if pn and qm are not relatively prime, there must exist some prime r which is a common measure; by IX, 12 (ii) r must measure both p and q; hence, r = p = q. This contradicts the choice of p and q as different primes. The special case where n = m is proved in VII, 27.


set, then it is impossible that any divisor of N be relatively prime to all terms in the set. 2o

(vi) If a prime measures a number N, as in (v), then it measures exactly one of the terms A, B, C, . . .

The FTA will develop as a corollary to the following theorem: let N be the least common multiple of a set of terms A, B, C, . . . , taken as powers of different primes; let p be a prime divisor of N; let p” be a divisor of N, but p n+l not be a divisor of N; then p” equals exactly one of the original terms measuring N.

The proof may be given as follows: Since A, B, C, . . . are powers of different primes, by (iv) they are relatively prime, each to all the others. The condition of (v) and (vi) is thus satisfied; since p measures N, it measures exactly one of the terms of the set, say A. By (ii) A is a power of p. We claim A = p”. For if they are unequal, one must measure the other, by (i). (1) Suppose p” measures A; we set A = p" x q, where by (iii) p measures q. As A measures N, it follows that p n+l or even some higher power of p measures N. This contradicts our hypothesis that

P n ’ ’ is not a divisor of N. Then (2) we suppose A measures p”. If p” = A x q, p is a measure of q, as before. Set N = p” x M. Now, B measures N and by (iv) is relatively prime to p”; hence, B measures M.21 The same follows for C and all the other powers (other than A) originally measuring N. Hence, A x M is a common multiple of A, B, C, . . . But if A measures p”, then A x M measures p” x M = N. This contradicts the construction of N as the least common multiple of A, B, C ) . . . 22 Thus, p” = A.

The FTA follows as a corollary. Let N be expressed as the least common multiple of A, B, C, . . . , powers of different primes; let N be expressed as the least common multiple also of A', B', C', . . . , as an alternative set of powers of different primes. By the previous theorem, each term A', B', C', . . . equals exactly one of the terms A, B, C, . . . ;

conversely, each term A, B, C, . . . exactly equals one of the terms A', B', C', . . . . Thus, the two sets are identical, save for order.

The proof thus constructed establishes that the Euclidean number theory is perfectly adequate for verifying the complete form of the FTA. Although the proof given may appear intricate, even unwieldy, especially in comparison with familiar modern textbook treatments of the theorem, this cannot be supposed to indicate any inherent shortcomings in Euclid’s approach. In fact, modern elementary treatments commonly assume without proof certain steps which do require

z”A proof may be modelled directly on that of IX, 14, where the appeal to VII, 30 is altered

to appeal to VII, 24. Cf. note 4. 2 1 This step may be justified via a corollary to VII, 30; cf. note 16. 22The steps in this argument should be compared with Euclid’s proof of IX, 14; cf. note 4.


proof; permitted such assumptions, the above argument could be considerably abridged. 23 But it is clear that there can be no question of Euclid’s inability either to express the complete factorization theorem or to provide an acceptable proof.

This discussion addresses the second of the two issues with which we began: one cannot hope efficiently to understand or fairly to assess a mathematical system, like Euclid’s number theory, in strict accordance with the concepts, organization and logical standards of the corresponding modern systems. By imposing modern standards of logic, authors like Hendy and Mullin presume to detect flaws-here, the alleged lack of generality of Euclid’s proofs-and thus fail to see that Euclid’s procedure was in accord with the studies of proof-technique of his time. In accounting for the alleged flaw, they misconstrue the geometric intuition underlying the Greek arithmetic. Finally, in drawing from this the conclusion of Euclid’s incompetence to prove the more general factorization theorem, they miss entirely the force of those Euclidean theorems and concepts, through which the construction of such a proof can become straightforward.

This being so, shouldn’t Euclid’s omission of the theorem be problematic? Not necessarily. As we have seen, the proof of an equivalent of the FTA would require a detailed set of extensions, albeit direct ones, of given Euclidean theorems. Such detail might well be deemed unsuitable for an introductory work such as the Elements.

Moreover, the results at issue are sufficiently straightforward as not to overreach the abilities of a competent student of the Elements. Finally, as we have also noted before, the general theorem would not be ‘fundamental’ within the organization chosen by Euclid. Thus, its omission does not detract in any way from the development and utility of Euclid’s theory.

231n particular, elementary treatments usually accept that the value of the product of n-many integers is independent of the order of multiplication A proof might require not only the commutativity and associativity of multiplication, but also a systematic inquiry into the permutations of finite sets. Such is the approach adopted by B. L. van der Waerden [1949, 11-18, 53-621. This result is possible in a Euclidean form: if N is the least common multiple of A, B, C, . . . , each term being relatively prime to all the others, and if one of those terms (say, D) be chosen; then setting N = D x M, we can prove that M is the least common multiple of the other terms originally measuring N. The proof follows immediately from the construction of the least common multiple in IX, 36. By means of this result, we may provide an alternative proof for the FTA. We choose those two terms in the two factorizations of N which by (v) are not relatively prime, say A and A’; these are powers of the same prime p. If they are unequal, one measures the other: we set A = A’ x q, where p is a measure of Q also. If now we set N = A x M = A’ x M’, M will be the least common multiple of B, C, . . . and M’ will be the least common multiple of 8, C’, . . . We have also that M’ : M = A : A’= q : 1. By VII, 20 it follows that q measures M’. But 4 is relatively prime to all the terms B’, C’, . . . of which M’ is the least common measure; this is impossible by (v). Thus A = A’, and so on for each other term in the expression for N.


This brings us to the other general issue we raised: how can the historian avoid the risk of distortion in speaking of an ancient result as the equivalent of a modern result? In a very real sense, he cannot. Consider first that the Euclidean theorems and the modern factorization theorems presume certain concepts: the ‘common multiple’ and ‘least common multiple’ of specified integers, on the one hand; the ‘n-long product’, on the other. Are these concepts the same? Detectable in Euclid’s treatment is an element of sophistication which introductory modern treatments lack. Euclid specifies a recursive construction for the extended product [VII, 361; he proves the commutativity of binary products [VII, 161 -this is not taken for granted; while the commutativity and associativity of the extended products are not proved, he does not apply these properties either. But implicit is a realization that such results as the order-independence of the procedure for common multiples require proofs. By contrast, modern elementary treatments of the FTA typically assume such results without proof (see notes 17 and 23). Complete treatments are deferred to the more formal introductions to ‘higher arithmetic’. These latter develop arithmetic in the context of more general algebraic structures, in particular, algebraic fields and integral ideals [Bell, 220, 223-51. Now, Euclid’s assumption of the order-independence of products could go wrong only in the instance of non-commutative systems; but his number theory is of course not such a system, and he does not recognize the ability to generalize beyond the natural numbers. His formal handling of the concepts in number theory thus distances him both from modern elementary accounts and modern advanced accounts.

If the respective notions of product are not strict equivalents, then a fort&i the results which require them cannot be equivalents either. We must then seek another sense of ‘equivalence’, if any such exists, which might relate the Euclidean and the modern theorems. As indicated earlier, the role played by each theorem within its respective theory of numbers will not provide us one. In modern number theory, following Gauss, the FTA is indeed ‘fundamental’; by contrast, it is the algorithm for finding the greatest common measure and the theorems on relative primality which are fundamental for Euclid. One sees that considerations based on the theory and the order and use of concepts within different systems compel us to distinguish the ancient works from the modern, rather than to formulate equivalences between them.

If we had at our disposal a concept of mathematical fact, an absolute about which theorems are made and proved, as a Platonist conception might offer, then the problem of equivalence could easily be answered.


But mathematical statements do not have any such absoluteness or independence from cultural considerations. As modern views emphasize the conventional character of mathematics, we must here give up the ability to appeal to the absolute. For if a mathematical statement refers to elements, operations and identities whose meaning and validity are accepted by postulate alone, how can we speak of the equivalence of two statements founded upon different postulates?

I believe the historian can retrieve a serviceable notion of equivalence from the practice of mathematics. For instance, Euclid’s IX, 14 contains a partial answer to the practical question: what are the divisors of a number which is constructed as the product of specified numbers? The FTA may also be viewed as addressing this question. I maintain that to the extent the two theorems imply justifications for the same procedures in mathematical practice, to that extent they are legitimately termed equivalent. Such a rule can establish an equivalence between theorems and also between concepts, since concepts regularly have a component rooted in practice. Now, our discussion of Euclid’s theorems and the FTA has never claimed an equivalence between their proofs. But it sometimes happens that we do wish to speak of an equivaIence of this kind. We might term two proofs equivalent if they order equivalent concepts and equivalent theorems according to the same pattern.

The historian of mathematics constantly speaks of equivalents, and this is inevitable if study of the history of mathematics is to have any relevance to what mathematicians do today. On the other hand, the philosophical description of historical equivaIence must surely be considered problematic; I should be greatly amazed if the sohrtion proposed above survives intact a close philosophical scrutiny. But it is in the clarification of such important presuppositions of historical work, I believe, that the philosopher of mathematics can abet the labors of the historian. Moreover, subjecting ancient work to the test of modern logic can point out the differences in philosophical outlook entailed by the ancient procedure. But to go beyond this-to interpret deviations as logical errors, to misconstrue concepts, or to invent essential limitations on the capacities of early mathematical systems-as the authors here criticized have attempted, serves no purpose. Their effort typifies a perversion of the logician’s role in the study of the history of mathematics.

Bibliography

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vol. 2, pp. 189-191, 1975. Knon, W. R, The Evolution of the Euclidean Elements, Dordrecht, Netherlands, 1975. Mullin, A. A. ‘Mathematico-philosophical remarks on new theorems analogous to the

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the text of G. Friedlein, Leipzig, 18671, Princeton, New Jersey, 1970. Q.uine, W. van 0.. Methods of Logic, revised edn, New York, 1959. &.sei, B., Principles of Mathemu&, London, 1903 [2nd edn, 19371. Taisbak, C. M., Division and Logos, Odense, Denmark, 1971. van der Waerden, B. L, Modern Algebra, New York, 1949.

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