Commentary on Book VII Definitions¶
Back to Book VII Definitions.
DEFINITION 1.¶
Μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.
Iamblichus (fl. circa 300 A.D.) tells us (Comm. on Nicomachus, ed. Pistelli, p. 11, 5) that the Euclidean definition of an unit or a monad was the definition given by “more recent” writers (οἱ νεώτεροι), and that it lacked the words “even though it be collective” (κἂν συστηματικὸν ᾖ). He also gives (ibid. p. 11) a number of other definitions. (1) According to “some of the Pythagoreans,” “an unit is the boundary between number and parts” (μονάς ἐστιν ἀριθμοῦ καὶ μορίων μεθόριον), “because from it, as from a seed and eternal root, ratios increase reciprocally on either side,” i.e. on one side we have multiple ratios continually increasing and on the other (if the unit be subdivided) submultiple ratios with denominators continually increasing. (2) A somewhat similar definition is that of Thymaridas, an ancient Pythagorean, who defined a monad as “limiting quantity” (περαίνουσα ποσότης), the beginning and the end of a thing being equally an extremity (πέρας). Perhaps the words together with their explanation may best be expressed by “limit of fewness.” Theon of Smyrna (p. 18, 6, ed. Hiller) adds the explanation that the monad is “that which, when the multitude is diminished by way of continued subtraction, is deprived of all number and takes an abiding position (μονήν) and rest.” If, after arriving at an unit in this way, we proceed to divide the unit itself into parts, we straightway have multitude again. (3) Some, according to Iamblichus (p. 11, 16), defined it as the “form of forms” (εἰδῶν εἶδος) because it potentially comprehends all forms of number, e.g. it is a polygonal number of any number of sides from three upwards, a solid number in all forms, and so on. (We are forcibly reminded of the latest theories of number as a “Gattung” of “Mengen” or as a “class of classes.”) (4) Again an unit, says Iamblichus, is the first, or smallest, in the category of how many (ποσόν), the common part or beginning of how many. Aristotle defines it as “the indivisible in the (category of) quantity,” τὸ κατὰ τὸ ποσὸν ἀδιαίρετον (Metaph. 1089 b 35), ποσόν including in Aristotle continuous as well as discrete quantity; hence it is distinguished from a point by the fact that it has not position: “Of the indivisible in the category of, and quâ, quantity, that which is every way (indivisible) and destitute of position is called an unit, and that which is every way indivisible and has position is a point” (Metaph. 1016 b 25). (5) In accordance with the last distinction, Aristotle calls the unit “a point without position,” στιγμὴ ἄθετος (Metaph. 1084 b 26). (6) Lastly, Iamblichus says that the school of Chrysippus defined it in a confused manner (συγκεχυμένως) as “multitude one (πλῆθος ἕν),” whereas it is alone contrasted with multitude. On a comparison of these definitions, it would seem that Euclid intended his to be a more popular one than those of his predecessors, δημώδης, as Nicomachus called Euclid’s definition of an even number.
The etymological signification of the word μονάς is supposed by Theon of Smyrna (p. 19, 7—13) to be either (1) that it remains unaltered if it be multiplied by itself any number of times, or (2) that it is separated and isolated (μεμονῶσθαι) from the rest of the multitude of numbers. Nicomachus also observes (I. 8, 2) that, while any number is half the sum (1) of the adjacent numbers on each side, (2) of numbers equidistant on each side, the unit is most solitary (μονωτάτη) in that it has not a number on each side but only on one side, and it is half of the latter alone, i.e. of 2.
DEFINITION 2.¶
᾿Αριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.
The definition of a number is again only one out of many that are on record. Nicomachus (I. 7, 1) combines several into one, saying that it is “a defined multitude (πλῆθος ὡρισμένον), or a collection of units (μονάδων σύστημα), or a flow of quantity made up of units” (ποσότητος χύμα ἐκ μονάδων συγκείμενον). Theon, in words almost identical with those attributed by Stobaeus (Eclogae I. 1, 8) to Moderatus, a Pythagorean, says (p. 18, 3—5): “A number is a collection of units, or a progression (προποδισμός) of multitude beginning from an unit and a retrogression (ἀναποδισμός) ceasing at an unit.” According to Iamblichus (p. 10) the description “collection of units” (μονάδων σύστημα) was applied to the how many, i.e. to number, by Thales, following the Egyptian view (κατὰ τὸ Αἰγυπτιακὸν ἀρέσκον), while it was Eudoxus the Pythagorean who said that a number was “a defined multitude” (πλῆθος ὡρισμένον). Aristotle has a number of definitions which come to the same thing: “limited multitude” (πλῆθος τὸ πεπερασμένον, Metaph. 1020 a 13), “multitude” (or “combination”) “of units” or “multitude of indivisibles” (ibid. 1053 a 30, 1039 a 12, 1085 b 22), “several ones” (ἕνα πλείω, Phys. III. 7, 207 b 7), “multitude measurable by one” (Metaph. 1057 a 3) and “multitude measured and multitude of measures,” the “measure” being unity, τὸ ἕν (ibid. 1088 a 5).
DEFINITION 3.¶
Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.
By a part Euclid means a submultiple, as he does in V. Def. 1, with which definition this one is identical except for the substitution of number (ἀριθμός) for magnitude (μέγεθος); cf. note on V. Def. 1. Nicomachus uses the word “submultiple” (ὑποπολλαπλάσιος) also. He defines it in a way corresponding to his definition of multiple (see note on Def. 5 below) as follows (I. 18, 2): “The submultiple, which is by nature first in the division of inequality (called) less, is the number which, when compared with a greater, can measure it more times than once so as to fill it exactly (πληρούντως).” Similarly sub-double (ὑποδιπλάσιος) is found in Nicomachus meaning half, and so on.
DEFINITION 4.¶
Μέρη δέ, ὅταν μὴ καταμετρῇ.
By the expression parts (μέρη, the plural of μέρος) Euclid denotes what we should call a proper fraction. That is, a part being a submultiple, the rather inconvenient term parts means any number of such submultiples making up a fraction less than unity. I have not found the word used in this special sense elsewhere, e.g. in Nicomachus, Theon of Smyrna or Iamblichus, except in one place of Theon (p. 79, 26) where it is used of a proper fraction, of which $\frac{2}{3}$ is an illustration.
DEFINITION 5.¶
Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.
The definition of a multiple is identical with that in V. Def. 2, except that the masculine of the adjectives is used agreeing with ἀριθμός understood instead of the neuter agreeing with μέγεθος understood. Nicomachus (I. 18, 1) defines a multiple as being “a species of the greater which is naturally first in order and origin, being the number which, when considered in comparison with another, contains it in itself completely more than once.”
DEFINITIONS 6, 7.¶
῎Αρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος.
Περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ [ὁ] μονάδι διαφέρων ἀρτίου ἀριθμοῦ.
Nicomachus (I. 7, 2) somewhat amplifies these definitions of even and odd numbers thus. “That is even which is capable of being divided into two equal parts without an unit falling in the middle, and that is odd which cannot be divided into two equal parts because of the aforesaid intervention (μεσιτείαν) of the unit.” He adds that this definition is derived “from the popular conception” (ἐκ τῆς δημώδους ὑπολήψεως). In contrast to this, he gives (I. 7, 3) the Pythagorean definition, which is, as usual, interesting. “An even number is that which admits of being divided, by one and the same operation, into the greatest and the least (parts), greatest in size (πηλικότητι) but least in quantity (ποσότητι)…while an odd number is that which cannot be so treated, but is divided into two unequal parts.” That is, as Iamblichus says (p. 12, 2—9), an even number is divided into parts which are the greatest possible “parts,” namely halves, and into the fewest possible, namely two, two being the first “number” or “collection of units.” According to another ancient definition quoted by Nicomachus (I. 7, 4), an even number is that which can be divided both into two equal parts and into two unequal parts (except the first one, the number 2, which is only susceptible of division into equals), but, however it is divided, must have its two parts of the same kind, i.e. both even or both odd; while an odd number is that which can only be divided into two unequal parts, and those parts always of different kinds, i.e. one odd and one even. Lastly, the definition of odd and even “by means of each other” says that an odd number is that which differs by an unit from an even number on both sides of it, and an even number that which differs by an unit from an odd number on each side. This alternative definition of an odd number is the same thing as the second half of Euclid’s definition, “the number which differs by an unit from an even number.” This evidently pre-Euclidean definition is condemned by Aristotle as unscientific, because odd and even are coordinate, both being differentiae of number, so that one should not be defined by means of the other (Topics VI. 4, 142 b 7—10).
DEFINITION 8.¶
᾿Αρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν.
Euclid’s definition of an even-times even number differs from that given by the later writers, Nicomachus, Theon of Smyrna and Iamblichus; and the inconvenience of it is shown when we come to IX. 34, where it is proved that a certain sort of number is both “even-times even” and “even-times odd.” According to the more precise classification of the three other authorities, the “even-times even” and the “even-times odd” are mutually exclusive and are two of three subdivisions into which even numbers fall. Of these three subdivisions the “even-times even” and the “even-times odd” form the extremes, and the “odd-times even” is as it were intermediate, showing the character of both extremes (cf. note on the following definition). The even-times even is then the number which has its halves even, the halves of the halves even, and so on, until unity is reached. In short the even-times even number is always of the form $2^n$. Hence Iamblichus (pp. 20, 21) says Euclid’s definition of it as that which is measured by an even number an even number of times is erroneous. In support of this he quotes the number 24 which is four times 6, or six times 4, but yet is not “even-times even” according to Euclid himself (οὐδὲ κατ᾽ αὐτόν), by which he must apparently mean that 24 is also 8 times 3, which does not satisfy Euclid’s definition. There can however be no doubt that Euclid meant what he said in his definition as we have it; otherwise IX. 32, which proves that a number of the form $2^n$ is even-times even only, would be quite superfluous and a mere repetition of the definition, while, as already stated, IX. 34 clearly indicates Euclid’s view that a number might at the same time be both even-times even and even-times odd. Hence the μόνως which some editor of the commentary of Philoponus on Nicomachus found in some copies, making the definition say that the even-times even number is only measured by even numbers an even number of times, is evidently an interpolation by some one who wished to reconcile Euclid’s definition with the Pythagorean (cf. Heiberg, Euklid-studien, p. 200).
A consequential characteristic of the series of even-times even numbers noted by Nicomachus brings in a curious use of the word δύναμις (generally power in the sense of square, or square root). He says (I. 8, 6—7) that any part, i.e. any submultiple, of an even-times even number is called by an even-times even designation, while it also has an even-times even value (it is ἀρτιάκις ἀρτιοδύναμον) when expressed as so many actual units. That is, the $\frac{1}{2^m}\text{th}$ part of $2^n$ (where $m$ is less than $n$) is called after the even-times even number $2^m$, while its actual value (δύναμις) in units is $2^{n-m}$, which is also an even-times even number. Thus all the parts, or submultiples, of even-times even numbers, as well as the even-times even numbers themselves, are connected with one kind of number only, the even.
DEFINITION 9.¶
᾿Αρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.
Euclid uses the term even-times odd (ἀρτιάκις περισσός), whereas Nicomachus and the others make it one word, even-odd (ἀρτιοπέριττος). According to the stricter definition given by the latter (I. 9, 1), the even-odd number is related to the even-times even as the other extreme. It is such a number as, when once halved, leaves as quotient an odd number; that is, it is of the form $2(2m+1)$. Nicomachus sets the even-odd numbers out as follows,
$$6,\ 10,\ 14,\ 18,\ 22,\ 26,\ 30,\text{ etc.}$$
In this case, as Nicomachus observes, any part, or submultiple, is called by a name not corresponding in kind to its actual value (δύναμις) in units. Thus, in the case of 18, the $\frac{1}{2}$ part is called after the even number 2, but its value is the odd number 9, and the $\frac{1}{3}\text{rd}$ part is called after the odd number 3, while its value is the even number 6, and so on.
The third class of even numbers according to the strict subdivision is the odd-even (περισσάρτιος). Numbers are of this class when they can be halved twice or more times successively, but the quotient left when they can no longer be halved is an odd number and not unity. They are therefore of the form $2^{n+1}(2m+1)$, where $n$, $m$ are integers. They are, so to say, intermediate between, or a mixture of, the extreme classes even-times even and even-odd, for the following reasons. (1) Their subdivision by 2 proceeds for some way like that of the even-times even, but ends in the way that the division of the even-odd by 2 ends. (2) The numbers after which submultiples are called and their value (δύναμις) in units may be both of one kind, i.e. both odd or both even (as in the case of the even-times even), or again may be one odd and one even as in the case of the even-odd. For example 24 is an odd-even number; the $\frac{1}{4}\text{th}$, $\frac{1}{12}\text{th}$, $\frac{1}{6}\text{th}$ or $\frac{1}{2}$ parts of it are even, but the $\frac{1}{3}\text{rd}$ part of it, or 8, is even, and the $\frac{1}{8}\text{th}$ part of it, or 3, is odd. (3) Nicomachus shows (I. 10, 6—9) how to form all the numbers of the odd-even class. Set out two lines ($a$) of odd numbers beginning with 3, ($b$) of even-times even numbers beginning with 4, thus:
$$\begin{array}{ll} (a) & 3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15\text{ etc.} \\ (b) & 4,\ 8,\ 16,\ 32,\ 64,\ 128,\ 256\text{ etc.} \end{array}$$
Now multiply each of the first numbers into each of the second row. Let the products of one of the first into all the second set make horizontal rows; we then get the rows
$$\begin{array}{l} 12,\ 24,\ 48,\ 96,\ 192,\ 384,\ 768\text{ etc.} \\ 20,\ 40,\ 80,\ 160,\ 320,\ 640,\ 1280\text{ etc.} \\ 28,\ 56,\ 112,\ 224,\ 448,\ 896,\ 1792\text{ etc.} \\ 36,\ 72,\ 144,\ 288,\ 576,\ 1152,\ 2304\text{ etc.} \end{array}$$
and so on.
Now, says Nicomachus, you will be surprised to see (φανήσεταί σοι θαυμαστῶς) that ($a$) the vertical rows have the property of the even-odd series, 6, 10, 14, 18, 22 etc., viz. that, if an odd number of successive numbers be taken, the middle number is half the sum of the extremes, and if an even number, the two middle numbers together are equal to the sum of the extremes, ($b$) the horizontal rows have the property of the even-times even series 4, 8, 16 etc., viz. that the product of the extremes of any number of successive terms is equal, if their number be odd, to the square of the middle term, or, if their number be even, to the product of the two middle terms.
Let us now return to Euclid. His 9th definition states that an even-times odd number is a number which, when divided by an even number, gives an odd number as quotient. Following this definition in our text comes a 10th definition which defines an odd-times even number; this is stated to be a number which, when divided by an odd number, gives an even number as quotient. According to these definitions any even-times odd number would also be odd-times even, and, from the fact that Iamblichus notes this, we may fairly conclude that he found Def. 10 as well as Def. 9 in the text of Euclid which he used. But, if both definitions are genuine, the enunciations of IX. 33 and IX. 34 as we have them present difficulties. IX. 33 says that “If a number have its half odd, it is even-times odd only”; but, on the assumption that both definitions are genuine, this would not be true, for the number would be odd-times even as well. IX. 34 says that “If a number neither be one of those which are continually doubled from 2, nor have its half odd, it is both even-times even and even-times odd.” The term odd-times even (περισσάκις ἄρτιος) not occurring in these propositions, nor anywhere else after the definition, that definition becomes superfluous. Iamblichus however (p. 24, 7—14) quotes these enunciations differently. In the first he has instead of “even-times odd only” the words “both even-times odd and odd-times even”; and, in the second, for “both even-times even and even-times odd” he has “is both even-times even and at the same time even-times odd and even-times even.” In both cases therefore “odd-times even” is added to the enunciation as Iamblichus had it; the words cannot have been added by Iamblichus himself because he himself does not use the term odd-times even, but the one word odd-even (περισσάρτιος). In order to get over the difficulties involved by Def. 10 and these differences of reading we have practically to choose between (1) accepting Iamblichus’ reading in all three places and (2) adhering to the reading of our MSS. in IX. 33, 34 and rejecting Def. 10 altogether as an interpolation. Now the readings of our text of IX. 33, 34 are those of the Vatican MS. and the Theonine MSS. as well; hence they must go back to a time before Theon, and must therefore be almost as old as those of Iamblichus. Heiberg considers it improbable that Euclid would wish to maintain a pointless distinction between even-times odd and odd-times even, and on the whole concludes that Def. 10 was first interpolated by some ignorant person who did not notice the difference between the Euclidean and Pythagorean classification, but merely noticed the absence of a definition of odd-times even and fabricated one as a companion to the other. When this was done, it would be easy to see that the statement in IX. 33 that the number referred to is “even-times odd only” was not strictly true, and that the addition of the words “and odd-times even” was necessary in IX. 33 and IX. 34 as well.
DEFINITION 10.¶
Περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.
The odd-times odd number is not defined as such by Nicomachus and Iamblichus; for them these numbers would apparently belong to the composite subdivision of odd numbers. Theon of Smyrna on the other hand says (p. 23, 21) that odd-times odd was one of the names applied to prime numbers (excluding 2), for these have two odd factors, namely 1 and the number itself. This is certainly a curious use of the term.
DEFINITION 11.¶
Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.
A prime number (πρῶτος ἀριθμός) is called by Nicomachus, Theon, and Iamblichus a “prime and incomposite (ἀσύνθετος) number.” Theon (p. 23, 9) defines it practically as Euclid does, viz. as a number “measured by no number, but by an unit only.” Aristotle too says that a prime number is not measured by any number (Anal. post. II. 13, 96 a 36), an unit not being a number (Metaph. 1088 a 6), but only the beginning of number (Theon of Smyrna says the same thing, p. 24, 23). According to Nicomachus (I. 11, 2) the prime number is a subdivision, not of numbers, but of odd numbers; it is “an odd number which admits of no other part except that which is called after its own name (παρώνυμον ἑαυτῷ).” The prime numbers are 3, 5, 7 etc., and there is no submultiple of 3 except ⅓rd, no submultiple of 11 except ⅟11th, and so on. In all these cases the only submultiple is an unit. According to Nicomachus 3 is the first prime number, whereas Aristotle (Topics VIII. 2, 157 a 39) regards 2 as a prime number: “as the dyad is the only even number which is prime,” showing that this divergence from the Pythagorean doctrine was earlier than Euclid. The number 2 also satisfies Euclid’s definition of a prime number. Iamblichus (p. 30, 27 sqq.) makes this the ground of another attack upon Euclid. His argument (the text of which, however, leaves much to be desired) appears to be that 2 is the only even number which has no other part except an unit, while the subdivisions of the even, as previously explained by him (the even-times even, the even-odd, and odd-even), all exclude primeness, and he has previously explained that 2 is potentially even-odd, being obtained by multiplying by 2 the potentially odd, i.e. the unit; hence 2 is regarded by him as bound up with the subdivisions of even, which exclude primeness. Theon seems to hold the same view as regards 2, but supports it by an apparent circle. A prime number, he says (p. 23, 14—23), is also called odd-times odd; therefore only odd numbers are prime and incomposite. Even numbers are not measured by the unit alone, except 2, which therefore (p. 24, 7) is odd-like (περισσοειδής) without being prime.
A variety of other names were applied to prime numbers. We have already noted the curious designation of them as odd-times odd. According to Iamblichus (p. 27, 3—5) some called them euthymetric (εὐθυμετρικός), and Thymaridas rectilinear (εὐθυγραμμικός), the ground being that they can only be set out in one dimension with no breadth (ἀπλατὴς γὰρ ἐν τῇ ἐκθέσει ἐφ᾽ ἓν μόνον διστάμενος). The same aspect of a prime number is also expressed by Aristotle, who (Metaph. 1020 b 3) contrasts the composite number with that which is only in one dimension (μόνον ἐφ᾽ ἓν ὤν). Theon of Smyrna (p. 23, 12) gives γραμμικός (linear) as the alternative name instead of εὐθυγραμμικός. In either case, to make the word a proper description of a prime number we have to understand the word only; a prime number is that which is linear, or rectilinear, only. For Nicomachus, who uses the form linear, expressly says (II. 13, 6) that all numbers are so, i.e. all can be represented as linear by dots to the required amount placed in a line.
A prime number was called prime or first, according to Nicomachus (I. 11, 3), because it can only be arrived at by putting together a certain number of units, and the unit is the beginning of number (cf. Aristotle’s second sense of πρῶτος “as not being composed of numbers,” ὡς μὴ συγκεῖσθαι ἐξ ἀριθμῶν, Anal. Post. II. 13, 96 a 37), and also, according to Iamblichus, because there is no number before it, being a collection of units (μονάδων σύστημα), of which it is a multiple, and it appears first as a basis for other numbers to be multiples of.
DEFINITION 12.¶
Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ.
By way of further emphasising the distinction between “prime” and “prime to one another,” Theon of Smyrna (p. 23, 6—8) calls the former “prime absolutely” (ἁπλῶς), and the latter “prime to one another and not absolutely” or “not in themselves” (οὐ καθ᾽ αὑτούς). The latter (p. 24, 8—10) are “measured by the unit [sc. only] as common measure, even though, taken by themselves (ὡς πρὸς ἑαυτούς), they be measured by some other numbers.” From Theon’s illustrations it is clear that with him as with Euclid a number prime to another may be even as well as odd. In Nicomachus (I. 11, 1) and Iamblichus (p. 26, 19), on the other hand, the number which is “in itself secondary (δεύτερος) and composite (σύνθετος), but in relation to another prime and incomposite,” is a subdivision of odd. I shall call more particular attention to this difference of classification when we have reached the definitions of “composite” and “composite to one another”; for the present it is to be noted that Nicomachus (I. 13, 1) defines a number prime to another after the same manner as the absolutely prime; it is a number which “is measured not only by the unit as the common measure but also by some other measure, and for this reason can also admit of a part or parts called by a different name besides that called by the same name (as itself), but, when examined in comparison with another number of similar character, is found not to be capable of being measured by a common measure in relation to the other, nor to have the same part, called by the same name as (any of) those simply (ἁπλῶς) contained in the other; e.g. 9 in relation to 25, for each of these is in itself secondary and composite, but, in comparison with one another, they have an unit alone as a common measure and no part is called by the same name in both, but the third in one is not in the other, nor is the fifth in the other found in the first.”
DEFINITION 13.¶
Σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος.
Euclid’s definition of composite is again the same as Theon’s definition of numbers “composite in relation to themselves,” which (p. 24, 16) are “numbers measured by any less number,” the unit being, as usual, not regarded as a number. Theon proceeds to say that “of composite numbers they call those which are contained by two numbers plane, as being investigated in two dimensions and, as it were, contained by a length and a breadth, while (they call) those (which are contained) by three (numbers) solid, as having the third dimension added to them.” To a similar effect is the remark of Aristotle (Metaph. 1020 b 3) that certain numbers are “composite and are not only in one dimension but such as the plane and the solid (figure) are representations of (μίμημα), these numbers being so many times so many (ποσάκις ποσοί), or so many times so many times so many (ποσάκις ποσάκις ποσοί) respectively.” These subdivisions of composite numbers are, of course, the subject of Euclid’s definitions 17, 18 respectively. Euclid’s composite numbers may be either even or odd, like those of Theon, who gives 6 as an instance, 6 being measured by both 2 and 3.
DEFINITION 14.¶
Σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ τινι μετρούμενοι κοινῷ μέτρῳ.
Theon (p. 24, 18), like Euclid, defines numbers composite to one another as “those which are measured by any common measure whatever” (excluding unity, as usual). Theon instances 8 and 6, with 2 as common measure, and 6 and 9, with 3 as common measure.
As hinted above, there is a great difference between Euclid’s classification of prime and composite numbers, and of numbers prime and composite to one another, and the classification found in Nicomachus (I. 11—13) and Iamblichus. According to the latter, all these kinds of numbers are subdivisions of the class of odd numbers only. As the class of even numbers is divided into three kinds, (1) the even-times even, (2) the even-odd, which form the extremes, and (3) the odd-even, which is, as it were, intermediate to the other two, so the class of odd numbers is divided into three, of which the third is again a mean between two extremes. The three are:
the prime and incomposite, which is like Euclid’s prime number except that it excludes 2;
the secondary and composite, which is “odd because it is a distinct part of one and the same genus (διὰ τὸ ἐξ ἑνὸς καὶ τοῦ αὐτοῦ γένους διακεκρίσθαι) but has in it nothing of the nature of a first principle (ἀρχοειδές); for it arises from adding some other number (to itself), so that, besides having a part called by the same name as itself, it possesses a part or parts called by another name.” Nicomachus cites 9, 15, 21, 25, 27, 33, 35, 39. It is made clear that not only must the factors be both odd, but they must all be prime numbers. This is obviously a very inconvenient restriction of the use of the word composite, a word of general signification.
is that which is “secondary and composite in itself but prime and incomposite to another.” The actual words in which this is defined have been given above in the note on Def. 12. Here again all the factors must be odd and prime.
Besides the inconvenience of restricting the term composite to odd numbers which are composite, there is in this classification the further serious defect, pointed out by Nesselmann (Die Algebra der Griechen, 1842, p. 194), that subdivisions (2) and (3) overlap, subdivision (2) including the whole of subdivision (3). The origin of this confusion is no doubt to be found in Nicomachus’ perverse anxiety to be symmetrical; by hook or by crook he must divide odd numbers into three kinds as he had divided the even. Iamblichus (p. 28, 13) carries his desire to be logical so far as to point out why there cannot be a fourth kind of number contrary in character to (3), namely a number which should be “prime and incomposite in itself, but secondary and composite to another”!
DEFINITION 15.¶
Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις.
This is the well known primary definition of multiplication as an abbreviation of addition.
DEFINITION 16.¶
Ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.
The words plane and solid applied to numbers are of course adapted from their use with reference to geometrical figures. A number is therefore called linear (γραμμικός) when it is regarded as in one dimension, as being a length (μῆκος). When it takes another dimension in addition, namely breadth (πλάτος), it is in two dimensions and becomes plane (ἐπίπεδος). The distinction between a plane and a plane number is marked by the use of the neuter in the former case, and the masculine, agreeing with ἀριθμός, in the latter case. So with a square and a square number, and so on. The most obvious form of a plane number is clearly that corresponding to a rectangle in geometry; the number is the product of two linear numbers regarded as sides (πλευραί) forming the length and breadth respectively. Such a number is, as Aristotle says, “so many times so many,” and a plane is its counterpart (μίμημα). So Plato, in the Theaetetus (147 E—148 B), says: “We divided all numbers into two kinds, (1) that which can be expressed as equal multiplied by equal (τὸν δυνάμενον ἶσον ἰσάκις γίγνεσθαι), and which, likening its form to the square, we called square and equilateral; (2) that which is intermediate, and includes 3 and 5 and every number which cannot be expressed as equal multiplied by equal, but is either less times more or more times less, being always contained by a greater and a less side, which number we likened to the oblong figure (προμήκει σχήματι) and called an oblong number…. Such lines therefore as square the equilateral and plane number [i.e. which can form a plane number with equal sides, or a square] we defined as length (μῆκος); but such as square the oblong (here ἑτερομήκης) [i.e. the square of which is equal to the oblong] we called roots (δυνάμεις) as not being commensurable with the others in length, but only in the plane areas (ἐπιπέδοις), to which the squares on them are equal (ἃ δύνανται).” This passage seems to make it clear that Plato would have represented numbers as Euclid does, by straight lines proportional in length to the numbers they represent (so far as practicable); for, since 3 and 5 are with Plato oblong numbers, and lines with him represent the sides of oblong numbers (since a line represents the “root,” the square on which is equal to the oblong), it follows that the unit representing the smaller side must have been represented as a line, and 3, the larger side, as a line of three times the length. But there is another possible way of representing numbers, not by lines of a certain length, but by points disposed in various ways, in straight lines or otherwise. Iamblichus tells us (p. 56, 27) that “in old days they represented the quantuplicities of number in a more natural way (φυσικώτερον) by splitting them up into units, and not, as in our day, by symbols” (συμβολικῶς). Aristotle too (Metaph. 1092 b 10) mentions one Eurytus as having settled what number belonged to what, such a number to a man, such a number to a horse, and so on, “copying their shapes” (reading τούτων, with Zeller) “with pebbles (ταῖς ψήφοις), just as those do who arrange numbers in the forms of triangles or squares.” We accordingly find numbers represented in Nicomachus and Theon of Smyrna by a number of α’s ranged like points according to geometrical figures. According to this system, any number could be represented by points in a straight line, in which case, says Iamblichus (p. 56, 26), we shall call it rectilinear because it is without breadth and only advances in length (ἀπλατῶς ἐπὶ μόνον τὸ μῆκος πρόεισιν). The prime number was called by Thymaridas rectilinear par excellence, because it was without breadth and in one dimension only (ἐφ᾽ ἓν μόνον διστάμενος). By this must be meant the impossibility of representing, say, 3 as a plane number, in Plato’s sense, i.e. as a product of two numbers corresponding to a rectangle in geometry; and this view would appear to rest simply upon the representation of a number by points, as distinct from lines. Three dots in a straight line would have no breadth; and if breadth were introduced in the sense of producing a rectangle, i.e. by placing the same
number of dots in a second line below the first line, the first plane number would be 4, and 3 would not be a plane number at all, as Plato says it is. It seems therefore to have been the alternative representation of a number by points, and not lines, which gave rise to the different view of a plane number which we find in Nicomachus and the rest. By means of separate points we can represent numbers in geometrical forms other than rectangles and squares. One dot with two others symmetrically arranged below it shows a triangle, which is a figure in two dimensions as much as a rectangle or parallelogram is. Similarly we can arrange certain numbers in the form of regular pentagons or other polygons. According therefore to this mode of representation, 3 is the first plane number, being a triangular number. The method of formation of triangular, square, pentagonal and other polygonal numbers is minutely described in Nicomachus (II. 8—11), who distinguishes the separate series of gnomons belonging to each, i.e. gives the law determining the number which has to be added to a polygonal number with $n$ in a side, in order to make it into a number of the same form but with $n + 1$ in a side (the addend being of course the gnomon). Thus the gnomonic series for triangular numbers is 1, 2, 3, 4, 5…; that for squares 1, 3, 5, 7…; that for pentagonal numbers 1, 4, 7, 10…, and so on. The subject need not detain us longer here, as we are at present only concerned with the different views of what constitutes a plane number.
Of plane numbers in the Platonic and Euclidean sense we have seen that Plato recognises two kinds, the square and the oblong (προμήκης or ἑτερομήκης). Here again Euclid’s successors, at all events, subdivided the class more elaborately. Nicomachus, Theon of Smyrna, and Iamblichus divide plane numbers with unequal sides into (1) ἑτερομήκεις, the nearest thing to squares, viz. numbers in which the greater side exceeds the less side by 1 only, or numbers of the form $n(n + 1)$, e.g. $1 \cdot 2$, $2 \cdot 3$, $3 \cdot 4$, etc. (according to Nicomachus), and (2) προμήκεις, or those whose sides differ by 2 or more, i.e. are of the form $n(n + m)$, where $m$ is not less than 2 (Nicomachus illustrates by $2 \cdot 4$, $3 \cdot 6$, etc.). Theon of Smyrna (p. 30, 8—14) makes προμήκεις include ἑτερομήκεις, saying that their sides may differ by 1 or more; he also speaks of parallelogram-numbers as those which have one side different from the other by 2 or more; I do not find this latter term in Nicomachus or Iamblichus, and indeed it seems superfluous, as parallelogram is here only another name for oblong. Iamblichus (p. 74, 23 sqq.), always critical of Euclid, attacks him again here for confusing the subject by supposing that the ἑτερομήκης number is the product of any two different numbers multiplied together, and by not distinguishing the oblong (προμήκης) from it: “for his definition declares the same number to be square and also ἑτερομήκης, as for example 36, 16 and many others: which would be equivalent to the odd number being the same thing as the even.” No importance need be attached to this exaggerated statement; it is in any case merely a matter of words, and it is curious that Euclid does not in fact use the word ἑτερομήκης of numbers at all, but only of geometrical oblong figures as opposed to squares, so that Iamblichus can apparently only have inferred that he used it in an unorthodox manner from the geometrical use of the term in the definitions of Book I. and from the fact that he does not give the two subdivisions of plane numbers which are not square, but seems only to divide plane numbers into square and not-square. The argument that ἑτερομήκεις numbers are a natural, and therefore essential, subdivision Iamblichus appears to found on the method of successive addition by which they can be evolved; as square numbers are obtained by successively adding odd numbers as gnomons, so ἑτερομήκεις are obtained by adding even numbers as gnomons. Thus $1 \cdot 2 = 2$, $2 \cdot 3 = 2 + 4$, $3 \cdot 4 = 2 + 4 + 6$, and so on.
DEFINITION 17.¶
Ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.
What has been said of the two apparently different ways of regarding a plane number seems to apply equally, mutatis mutandis, to the definitions of a solid number. Aristotle regards it as a number which is so many times so many times so many (ποσάκις ποσάκις ποσόν). Plato finishes the passage about lines which represent the sides of square numbers and lines which are roots (δυνάμεις), i.e. the squares on which are equal to the rectangle representing a number which is oblong and not square, by adding the words, “And another similar property belongs to solids” (καὶ περὶ τὰ στερεὰ ἄλλο τοιοῦτον). That is, apparently, there would be a corresponding term to root (δύναμις)—practically representing a surd—to denote the side of a cube equal to a parallelepiped representing a solid number which is the product of three factors but not a cube. Such is a solid number when numbers are represented by straight lines: it corresponds in general to a parallelepiped and, when all the factors are equal, to a cube.
But again, if numbers be represented by points, we may have solid numbers (i.e. numbers in three dimensions) in the form of pyramids as well. The first number of this kind is 4, since we may have three points forming an equilateral triangle in one plane and a fourth point placed in another plane. The length of the sides can be increased by 1 successively; and we can have a series of pyramidal numbers, with triangles, squares or polygons as bases, made up of layers of triangles, squares or similar polygons respectively, each of which layers has one less in the side than the layer below it, until the top of the pyramid is reached, which of course is one point representing unity. Nicomachus (II. 13—16), Theon of Smyrna (p. 41—2), and Iamblichus (p. 95, 15 sqq.), all give the different kinds of pyramidal solid numbers in addition to the other kinds.
These three writers make the following further distinctions between solid numbers which are the product of three factors.
First there is the equal by equal by equal (ἰσάκις ἰσάκις ἴσος), which is, of course, the cube.
The other extreme is the unequal by unequal by unequal (ἀνισάκις ἀνισάκις ἄνισος), or that in which all the dimensions are different, e.g. the product of 2, 3, 4 or 2, 4, 8 or 3, 5, 12. These were, according to Nicomachus (II. 16), called scalene, while some called them σφηνίσκοι (wedge-shaped), others σφηκίσκοι (from σφήξ, a wasp), and others βωμίσκοι (altar-shaped). Theon appears to use the last term only, while Iamblichus of course gives all three names.
Intermediate to these, as it were, come the numbers “whose planes form ἑτερομήκεις numbers” (i.e. numbers of the form $n(n + 1)$). These, says Nicomachus, are called parallelepipedal.
Lastly come two classes of such numbers each of which has two equal dimensions but not more.
If the third dimension is less than the others, the number is equal by equal by less (ἰσάκις ἴσος ἐλαττονάκις) and is called a plinth (πλινθίς), e.g. $8 \cdot 8 \cdot 3$.
If the third dimension is greater than the others, the number is equal by equal by greater (ἰσάκις ἴσος μειζονάκις) and is called a beam (δοκίς), e.g. $3 \cdot 3 \cdot 7$. Another name for this latter kind of number (according to Iamblichus) was στηλίς (diminutive of στήλη).
Lastly, in connexion with pyramidal numbers, Nicomachus (II. 14, 5) distinguishes numbers corresponding to frusta of pyramids. These are truncated (κόλουροι), twice-truncated (δικόλουροι), thrice-truncated (τρικόλουροι) pyramids, and so on, the term being used mostly in theoretic treatises (ἐν συγγράμμασι μάλιστα τοῖς θεωρηματικοῖς). The truncated pyramid was formed by cutting off the point forming the vertex. The twice-truncated was that which lacked the vertex and the next plane, and so on. Theon of Smyrna (p. 42, 4) only mentions the truncated pyramid as “that with its vertex cut off” (ἡ τὴν κορυφὴν ἀποτετμημένη), saying that some also called it a trapezium, after the similitude of a plane trapezium formed by cutting the top off a triangle by a straight line parallel to the base.
DEFINITION 18.¶
Τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ [ὁ] ὑπὸ δύο ἴσων ἀριθμῶν περιεχόμενος.
A particular kind of square distinguished by Nicomachus and the rest was the square number which ended (in the decimal notation) with the same number as its side, e.g. 1, 25, 36, which are the squares of 1, 5 and 6. These square numbers were called cyclic (κυκλικοί) on the analogy of circles in geometry which return again to the point from which they started.
DEFINITION 19.¶
Κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ [ὁ] ὑπὸ τριῶν ἴσων ἀριθμῶν περιεχόμενος.
Similarly cube numbers which ended with the same number as their sides, and the squares of those sides also, were called spherical (σφαιρικοί) or recurrent (ἀποκαταστατικοί). One might have expected that the term spherical would be applicable also to the cubes of numbers which ended with the same digit as the side but not necessarily with the same digit as the square of the side also. E.g. the cube of 4, i.e. 64, ends with the same digit as 4, but not with the same digit as 16. But apparently 64 was not called a spherical number, the only instances given by Nicomachus and the rest being those cubed from numbers ending with 5 or 6, which end with the same digit if squared. A spherical number is in fact derived from a circular number only, and that by adding another equal dimension. Obviously, as Nesselmann says, the names cyclic and spherical applied to numbers appeal to an entirely different principle from that on which the figured numbers so far dealt with were formed.
DEFINITION 20.¶
Ἀριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ δευτέρου καὶ ὁ τρίτος τοῦ τετάρτου ἰσάκις ᾖ πολλαπλάσιος ἢ τὸ αὐτὸ μέρος ἢ τὰ αὐτὰ μέρη ὦσιν.
Euclid does not give in this Book any definition of ratio, doubtless because it could only be the same as that given at the beginning of Book V., with numbers substituted for “homogeneous magnitudes” and “in respect of size” (πηλικότητα) omitted or altered. We do not find that Nicomachus and the rest give any substantially different definition of a ratio between numbers. Theon of Smyrna says, in fact (p. 73, 16), that “ratio in the sense of proportion (λόγος ὁ κατ᾽ ἀνάλογον) is a sort of relation of two homogeneous terms to one another, as for example, double, triple.” Similarly Nicomachus says (II. 21, 3) that “a ratio is a relation of two terms to one another,” the word for “relation” being in both cases the same as Euclid’s (σχέσις). Theon of Smyrna goes on to classify ratios as greater, less, or equal, i.e. as ratios of greater inequality, less inequality, or equality, and then to specify certain arithmetical ratios which had special names, for which he quotes the authority of Adrastus. The names were πολλαπλάσιος, ἐπιμόριος, ἐπιμερής, πολλαπλασιεπιμόριος, πολλαπλασιεπιμερής (the first of which is, of course, a multiple, while the rest are the equivalent of certain types of improper fractions as we should call them), and the reciprocals of each of these described by prefixing ὑπό or sub. After describing these particular classes of arithmetical ratios, Theon goes on to say that numbers still have ratios to one another even if they are different from all those previously described. We need not therefore concern ourselves with the various types; it is sufficient to observe that any ratio between numbers can be expressed in the manner indicated in Euclid’s definition of arithmetical proportion, for the greater is, in relation to the less, either one or a combination of more than one of the three things, (1) a multiple, (2) a submultiple, (3) a proper fraction.
It is when we come to the definition of proportion that we begin to find differences between Euclid, Nicomachus, Theon and Iamblichus. “Proportion,” says Theon (p. 82, 6), “is similarity or sameness of more ratios than one,” which is of course unobjectionable if it is previously understood what a ratio is; but confusion was brought in by those (like Thrasyllus) who said that there were three proportions (ἀναλογίαι), the arithmetic, geometric, and harmonic, where of course the reference is to arithmetic, geometric and harmonic means (μεσότητες). Hence it was necessary to explain, as Adrastus did (Theon, p. 106, 15), that of the several means “the geometric was called both proportion par excellence and primary…though the other means were also commonly called proportions by some writers.” Accordingly we have Nicomachus trying to extend the term “proportion” to cover the various means as well as a proportion in three or four terms in the ordinary sense. He says (II. 21, 2): “Proportion, par excellence (κυρίως), is the bringing together (σύλληψις) to the same (point) of two or more ratios; or, more generally, (the bringing together) of two or more relations (σχέσεων), even though they be subjected not to the same ratio but to a difference or some other (law).” Iamblichus keeps the senses of the word more distinct. He says, like Theon, that “proportion is similarity or sameness of several ratios” (p. 98, 14), and that “it is to be premised that it was the geometrical (proportion) which the ancients called proportion par excellence, though it is now common to apply the name generally to all the remaining means as well” (p. 100, 15). Pappus remarks (III. p. 70, 17), “A mean differs from a proportion in this respect that, if anything is a proportion, it is also a mean, but not conversely. For there are three means, of which one is arithmetic, one geometric and one harmonic.” The last remark implies plainly enough that there is only one proportion (ἀναλογία) in the proper sense. So, too, says Iamblichus in another place (p. 104, 19): “the second, the geometric, mean has been called proportion par excellence because the terms contain the same ratio, being separated according to the same proportion (ἀνὰ τὸν αὐτὸν λόγον διεστῶτες).” The natural conclusion is that of Nesselmann, that originally the geometric proportion was called ἀναλογία, the others, the arithmetic, the harmonic, etc., means; but later usage had obliterated the distinction.
Of proportions in the ancient and Euclidean sense Theon (p. 82, 10) distinguished the continuous (συνεχής) and the separated (διῃρημένη), using the same terms as Aristotle (Eth. Nic. 1131 a 32). The meaning is of course clear: in the continuous proportion the consequent of one ratio is the antecedent of the next; in the separated proportion this is not so. Nicomachus (II. 21, 5—6) uses the words connected (συνημμένη) and disjoined (διεζευγμένη) respectively. Euclid regularly speaks of numbers in continuous proportion as “proportional in order, or successively” (ἑξῆς ἀνάλογον).
DEFINITION 21.¶
Ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς.
Theon of Smyrna remarks (p. 36, 12) that, among plane numbers, all squares are similar, while of ἑτερομήκεις those are similar “whose sides, that is, the numbers containing them, are proportional.” Here ἑτερομήκης must evidently be used, not in the sense of a number of the form $n(n + 1)$, but. as synonymous with προμήκης, any oblong number; so that on this occasion Theon follows the terminology of Plato and (according to Iamblichus) of Euclid. Obviously, if the strict sense of ἑτερομήκης is adhered to, no two numbers of that form can be similar unless they are also equal. We may compare Iamblichus’ elaborate contrast of the square and the ἑτερομήκης. Since the two sides of the square are equal, a square number might, as he says (p. 82, 9), be fitly called ἰδιομήκης (Nicomachus uses ταὐτομήκης) in contrast to ἑτερομήκης; and the ancients, according to him, called square numbers “the same” and “similar” (ταὐτούς τε καὶ ὁμοίους), but ἑτερομήκεις numbers “dissimilar and other” (ἀνομοίους καὶ θατέρους).
With regard to solid numbers, Theon remarks in like manner (p. 37, 2) that all cube numbers are similar, while of the others those are similar whose sides are proportional, i.e. in which, as length is to length, so is breadth to breadth and height to height.
DEFINITION 22.¶
Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.
Theon of Smyrna (p. 45, 9 sqq.) and Nicomachus (I. 16) both give the same definition of a perfect number, as well as the law of formation of such numbers which Euclid proves in the later proposition, IX. 36. They add however definitions of two other kinds of numbers in contrast with it, (1) the over-perfect (ὑπερτελής in Nicomachus, ὑπερτέλειος in Theon), the sum of whose parts, i.e. submultiples, is greater than the number itself, e.g. 12, 24 etc., the sum of the parts of 12 being $6 + 4 + 3 + 2 + 1 = 16$, and the sum of the parts of 24 being $12 + 8 + 6 + 4 + 3 + 2 + 1 = 36$, (2) the defective (ἐλλιπής), the sum of whose parts is less than the whole, e.g. 8 or 14, the parts in the first case adding up to $4 + 2 + 1$, or 7, and in the second case to $7 + 2 + 1$, or 10. All three classes are however made by Theon subdivisions of numbers in general, but by Nicomachus subdivisions of even numbers.
The term perfect was used by the Pythagoreans, but in another sense, of 10; while Theon tells us (p. 46, 14) that 3 was also called perfect “because it is the first number that has beginning, middle and extremity; it is also both a line and a plane (for it is an equilateral triangle having each side made up of two units), and it is the first link and potentiality of the solid (for a solid must be conceived of in three dimensions).”
There are certain unexpressed axioms used in Book VII. as there are in earlier Books.
The following may be noted.
If A measures B, and B measures C, A will measure C.
If A measures B, and also measures C, A will measure the difference between B and C when they are unequal.
If A measures B, and also measures C, A will measure the sum of B and C.
It is clear, from what we know of the Pythagorean theory of numbers, of musical intervals expressed by numbers, of different kinds of means etc., that the substance of Euclid Books VII.—IX. was no new thing but goes back, at least, to the Pythagoreans. It is well known that the mathematics of Plato’s Timaeus is essentially Pythagorean. It is therefore a priori probable (if not perhaps quite certain) that Plato πυθαγορίζει even in the passage (32 A, B) where he speaks of numbers “whether solid or square” in continued proportion, and proceeds to say that between planes one mean suffices, but to connect two solids two means are necessary. This passage has been much discussed, but I think that by “planes” and “solids” Plato certainly meant square and solid numbers respectively, so that the allusion must be to the theorems established in Eucl. VIII. 11, 12, that between two square numbers there is one mean proportional number, and between two cube numbers there are two mean proportional numbers[1].
HISTORICAL NOTE¶
It is no less clear that, in his method and line of argument, Euclid was following earlier models, though no doubt making improvements in the exposition. The tract on the Sectio Canonis, κατατομὴ κανόνος (as to the genuineness of which see above, Vol. I., p. 17) is in style and in the form of the propositions generally akin to the Elements. In one proposition (2) the author says “we learned (ἐμάθομεν) that, if as many numbers as we please be in (continued) proportion, and the first measures the last, the first will also measure the intermediate numbers”; here he practically quotes Elem. VIII. 7. In the 3rd proposition he proves that no number can be a mean between two numbers in the ratio known as ἐπιμόριος, the ratio, that is, of $n + 1$ to $n$, where $n$ is any integer greater than unity. Now, fortunately, Boethius, De institutione musica, III. 11 (pp. 285—6, ed. Friedlein), has preserved a proof by Archytas of this same proposition; and the proof is substantially identical with that of Euclid. The two proofs are placed side by side in an article by Tannery (Bibliotheca Mathematica, $\text{VI}_3$, 1905/6, p. 227). Archytas writes the smaller term of the proportion first (instead of the greater, as Euclid does). Let, he says, $A, B$ be the “superparticularis proportio” (ἐπιμόριον διάστημα in Euclid). Take $C, DE$ the smallest numbers which are in the ratio of $A$ to $B$. [Here $DE$ means $D + E$: and in this respect the notation is different from that of Euclid who, as usual, takes a line $DF$ divided into two parts at $G$, $GF$ corresponding to $E$, and $DG$ to $D$, in Archytas’ notation. The step of taking $C, DE$, the smallest numbers in the ratio of $A$ to $B$, presupposes Eucl. VII. 33.] Then $DE$ exceeds $C$ by an aliquot part of itself and of $C$ [cf. the definition of ἐπιμόριος ἀριθμός in Nicomachus, I. 19, 1]. Let $D$ be the excess [i.e. $E$ is supposed equal to $C$]. “I say that $D$ is not a number but an unit.”
For, if $D$ is a number and a part of $DE$, it measures $DE$; hence it measures $E$, that is, $C$. Thus $D$ measures both $C$ and $DE$, which is impossible; for the smallest numbers which are in the same ratio as any numbers are prime to one another. [This presupposes Eucl. VII. 22.] Therefore $D$ is an unit; that is, $DE$ exceeds $C$ by an unit. Hence no number can be found which is a mean between two numbers $C, DE$. Therefore neither can any number be a mean between the original numbers $A, B$ which are in the same ratio [this implies Eucl. VII. 20].
We have then here a clear indication of the existence at least as early as the date of Archytas (about 430—365 B.C.) of an Elements of Arithmetic in the form which we call Euclidean; and no doubt text-books of the sort existed even before Archytas, which probably Archytas himself and Eudoxus improved and developed in their turn.
BOOK VII. PROPOSITIONS.