Generalizing about polygonal numbers in ancient Greek mathematics Ian Mueller

9
Generalizing about polygonal numbers
in ancient Greek mathematics
Ian Mu eller
Introduction
The main source for our information about the Greek handling of what are
called polygonal numbers is the Introduction to Arithmetic of Nicomachus
of Gerasa (c. 100 ce).1 Heath says of the Introduction that “Little or nothing
in the book is original, and, except for certain definitions and refinements of classification, the essence of it evidently goes back to the early
Pythagoreans.”2 I am not interested in this historical claim, the evidence
for which is very slight; indeed I am not interested in chronology at all but
only in certain features of Nicomachus’ treatment of polygonals, which I
discuss in Section 1, and in the general argumentative structure of a short
treatise by Diophantus called On Polygonal Numbers,3 which I discuss in
Section 2.
1. Nicomachus of Gerasa
In the Introduction Nicomachus makes a contrast between the standard
Greek way of writing numbers, in which, e.g., 222 is written σκβ, where σ
represents 200, κ 20, and β 2, and what he says is a more natural way:
ii.6.2 First one should recognize that each letter with which we refer to a number . . .
signifies it by human convention and agreement and not in a natural way; the
natural, direct (amethodos), and consequently simplest way to signify numbers
would be the setting out of the units in each number in a line side by side . . . :
1
2
3
Greek text: Hoche 1866; English translation: D’Ooge 1926; French translation: Bertier 1978.
There is material parallel to Nicomachus’ presentation in Theon of Smyrna (Hiller 1878). For
dates of individuals I use Toomer’s articles in The Oxford Classical Dictionary (Hornblower and
Spawforth 1996).
Heath 1921: i 99.
Greek text: T1893: 450,1–476,3; French translation: Ver Eecke 1926. I do not discuss the final
part of the treatise (476,4–480,2), a broken-off and inconclusive attempt to show how to find
how many kinds of polygonal a given number is. The Oxford Classical Dictionary locates
Diophantus in the interval between 150 bce and 280 ce. Heath 1921: ii 448 says that “he
probably flourished A.D. 250 or not much later.”
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unit
two
three
four
five
α,
αα,
ααα,
αααα,
ααααα,
and so on.
Nicomachus’ “natural” representation of numbers would seem to break
down the customary Greek contrast between the numbers and the unit, but
Nicomachus insists that it does not:
ii.6.3 Since the unit has the place and character of a point, it will be a principle
(arkhê) . . . of numbers . . . and not in itself (oupô) . . . a number, just as the point is
a principle of line or distance and not in itself a line or distance.
We find a close analog of Nicomachus’ “natural” representation of
numbers in the account of finitary number theory in Hilbert and Bernays’
great work Grundlagen der Mathematik, except that in the Grundlagen the
alphas are replaced by strokes. As that work makes clear, this representation provides a basis for developing all of elementary arithmetic, including
everything known to the Greeks. Much the most important feature of the
representation in this regard is the treatment of the numbers as formed
from an initial object (the unit or one) by an indefinitely repeatable successor operation which always produces a new number. This treatment validates definition and proof by mathematical induction, the core of modern
number theory. The finitary arithmetic of Hilbert and Bernays rests essentially on the intuitive manipulation of sequences of strokes (units) together
with elementary inductive reasoning.4 It is difficult for me to see any substantial difference between the manipulation of sequences of strokes or
alphas and the manipulation of lines and figures in what is frequently called
cut-and-paste geometry; the objects are different, but the reasoning seems
to me to be in an important sense the same.
I mention this modern form of elementary arithmetic only to provide a
contrast with its ancient forebears. Nicomachus relies heavily on the notion
of numbers as multiplicities of units and the representation of them as collections of alphas, but, after he has introduced his natural representation, it
by and large vanishes in favor of a much more clearly geometric or configurational representation in which three is a triangular number, four a square
number, and five a pentagonal number (Figure 9.1).
4
In this paper I use words like “inductive” and “induction” only in connection with
mathematical induction.
Polygonal numbers in ancient Greek mathematics
α
α
α
α
α
α
α
α
α
α
α
α
Figure 9.1 Geometric representation of polygonal numbers.
Nicomachus also mentions hexagonal, heptagonal, and octagonal
numbers, and there is no question that he has the idea of an n-agonal
number, for any n, but he only expresses this with words like “and so on
forever in the direction of increase” (aei kata parauxêsin houtôs; ii.11.4). It
is clear that Nicomachus intends to make some kind of generalization, but
it is not at all clear what, if any, theoretical or mathematical ideas underlie
it. Any connection between what he says and the natural representation
of numbers is at best indirect. Nicomachus is relying on the idea that the
numbers go on forever, but much more central to his account of polygonal
numbers is the geometric fact that an n-agon is determined by the n points
which are its vertices. If induction lies behind the reasoning, it is not made
at all explicit.
I turn now to some further features of what Nicomachus says. The first
sentence of his description of triangular numbers is quite opaque, but it is
clearly intended to bring out their configurational aspect. I quote it in the
translation of d’Ooge:
II.8.1 A triangular number is one which, when it is analyzed into units, shapes into
triangular form the equilateral placement of its parts in a plane. Examples are 3, 6,
10, 15, 21, 28, and so on in order. For their graphic representations (skhêmatographiai) will be well-ordered and equilateral triangles . . . .
Here again we have the thought of continuing indefinitely. Nicomachus
now indicates the arithmetical procedure for generating these triangular
numbers, again insisting on the distinction between the unit and a number
even though leaving it aside would simplify his description.
And, proceeding as far as you wish, you will find triangularization of this kind,
making the thing which consists of a unit first of all most elementary, so that
the unit may also appear as potentially a triangular number, with 3 being actually
the first.
ii.8.2 The sides <of these numbers> will increase by consecutive number, the side
of the potentially first being one, that of the actually first (i.e., 3) two, that of the
actually second (i.e., 6) three, that of the third four, of the fourth five, of the fifth six,
and so on forever.
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If we ignore the distinction between a unit and a number,5 we may express
Nicomachus’ claim here as:
The side of the nth (actual or potential) triangular number is n.
Nicomachus now turns to deal more explicitly with the question of the
relationship between the sequence of triangular numbers and the “natural”
numbers:
ii.8.3 Triangular numbers are generated when natural number is set out in
sequence (stoikhêdon) and successive ones are always added one at a time
starting from the beginning, since the well-ordered triangular numbers are
brought to completion with each addition and combination. For example,
from this natural sequence
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
if I take the very first item I get the potentially first triangular number, 1:
α
then, if I add to it the next term, I get the actually first triangular number,
since 3 is 2 and 1, and in its graphic representation it is put together as
follows: two units are placed side by side under one unit and the number is
made a triangle:
α
α
α
And then, following this, if the next number, 3, is combined with this and
spread out into units and added, it gives and also graphically represents 6,
which is the actually second triangular number:
α
α
α
α
α
α
Nicomachus continues in this vein for the first seven (potential and actual)
triangular numbers, essentially showing that:
The nth triangular number is the sum of the first n “natural” numbers.
5
As I shall sometimes do, without – I hope – introducing any confusion or uncertainty.
Polygonal numbers in ancient Greek mathematics
315
He proceeds to show in the same way that:
The nth square number is the sum of the first n odd numbers and its side is n.
but in this case the odd numbers are added so as to preserve the square
shape (Figure 9.2).
α
α
α
α
α
α
α
α
α
α
1
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
1+3
1+3+5
1 + 3 + 5 + 7.
Figure 9.2 The generation of square numbers.
The formulation corresponding to the presentation of the pentagonal
numbers is:
The nth pentagonal number is the sum of the first n numbers x1, x2, . . . , xn which
are such that xi+1 = xi + 3, and its side is n.
The first three are represented below (Figure 9.3).
α
1
α
α
α
α
α
α
1+4
α
α
α
α
α
α
α
α
α
α
α
1+4+7
Figure 9.3 The generation of the first three pentagonal numbers.
We are not given a graphic representation of the the next pentagonal number 22, but its representation would certainly be the following (Figure 9.4):
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α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
1 + 4 + 7 + 10
Figure 9.4 The graphic representation of the fourth pentagonal number.
Nicomachus proceeds through the octagonal numbers without figures,
making clear that:
[Nic*]. The sum of the first n numbers x1, x2, . . . , xn which are such that xi+1 = xi + j
is the nth j+2-agonal number and its side is n.
He then turns to showing that his presentation of polygonal numbers is in
harmony with geometry (<hê> grammikê <didaskalia>), something which
he says is clear both from the graphic representation and from the following
considerations:
ii.12.1 Every square figure divided diagonally is resolved into two triangles and
every square number is resolved into two consecutive triangulars and therefore is
composed of two consecutive triangulars. For example, the triangulars are:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc.,
and the squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
If you add any two consecutive triangulars whatsoever you will always produce
a square, so that in resolving any square you will be able to make two triangulars
Polygonal numbers in ancient Greek mathematics
from them. And again if any triangle is joined to any square figure6 it produces
a pentagon, for example if the triangular 1 is joined to the square 4, it makes the
pentagonal 5, and if the next <triangular>, that is 3, is added to the next <square>
9 it makes the pentagonal 12, and if the following <triangular> 6 is added onto the
following <square> 16, it gives the following <pentagonal> 22, and 25 added to 10
gives 35, and so on forever.
Nicomachus states similar results for adding triangulars to pentagonals to
get hexagonals, to hexagonals to get heptagonals, and to heptagonals to get
octagonals, “and so on ad infinitum.” He introduces a table (Table 9.1) as an
aid to memory:
Table 9.1:
Triangles
1
3
6
10
15
21
28
36
45
54
Squares
1
4
9
16
25
36
49
64
81
100
Pentagons
1
5
12
22
35
51
70
92
117
145
Hexagons
1
6
15
28
45
66
91
120
153
190
Heptagons
1
7
18
34
55
81
112
148
189
2357
and describes some of the relevant sums, results which we might formulate
as:
The n+1th square number is the nth triangular number plus the n+1th triangular
number;
The n+1th pentagonal number is the nth triangular number plus the n+1th square
number,
or, generally,
The n+1th k+1-agonal number is the nth triangular number plus the n+1th k-agonal
number.
At this point I would like to introduce some of Heath’s remarks about
Nicomachus’ Introduction:
It is a very far cry from Euclid to Nicomachus. Numbers are represented in Euclid
by straight lines with letters attached, a system which has the advantage that, as in
algebraical notation, we can work with numbers in general without the necessity of
giving them specific values . . . . Further, there are no longer any proofs in the proper
sense of the word; when a general proposition has been enunciated, Nicomachus
regards it as sufficient to show that it is true in particular instances; sometimes we
are left to infer the proposition by induction from particular cases which are alone
given. . . . probably Nicomachus, who was not really a mathematician, intended his
6
7
Here some exaggeration, since the triangle and the square have to “fit together.”
Apparently the octagons are missing.
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Introduction to be, not a scientific treatise, but a popular treatment of the subject
calculated to awaken in a beginner an interest in the theory of numbers . . . . Its
success is difficult to explain except on the hypothesis that it was at first read by
philosophers rather than mathematicians . . . , and afterwards became generally
popular at a time when there were no mathematicians left, but only philosophers
who incidentally took an interest in mathematics.8
Heath’s remarks here are aimed at the whole of the Introduction, but I
wish only to consider them in relation to Nicomachus’ treatment of polygonal numbers. There is no question that, as Heath also notes, Nicomachus’
flowery and imprecise language is a “far cry” from Euclid’s sparse, formal
formulations. But the representation of polygonal numbers by straight lines
would obliterate their configurational nature. Nicomachus shows how triangular configurations of units can be generated as the series 1, 1+2, 1+2+3,
etc. But I do not see what he could do to “prove” this fact and, therefore,
how he could “prove” any fact about polygonal numbers as configurations.
Of course, we know how to prove things about polygonal numbers, namely
by eliminating all geometric content and transforming Nic*, which for
Nicomachus expresses an arithmetical fact about configurations, into an
arithmetical definition in which the geometrical terminology is at most a
convenience, perhaps as follows:
[Defgeo/arith]. p is the nth j + 2-agonal number with side n if and only if p = x1 + x2 +
⋅⋅⋅ + xn, where xi+1 = xi + j and x1 = 1.
I assume that Fowler had something of this kind in mind when he
advanced the hypothesis that lying behind Nicomachus’ presentation
were ancient proofs using mathematical induction.9 I doubt this very
much, but the more important point for me is that, unless something like
Defgeo/arith is used to eliminate the configurational aspect of polygonal
numbers, anything like a Euclidean foundation for the theory of them lies
well beyond the scope of Greek mathematics.
2. The argument of Diophantus’ On Polygonal Numbers
In Tannery’s edition of On Polygonal Numbers there are four propositions.
The propositions are purely arithmetical and in none of them is there a
mention of polygonals.10 I quote them and give algebraic representations
8
9
10
Heath 1921: i 97–9.
Fowler 1994: 258.
When I say that these propositions are purely arithmetical, I only mean to point out the
absence of the notion of polygonality from the formulations and proofs of the propositions.
Polygonal numbers in ancient Greek mathematics
of their content; in footnotes I give simple algebraic proofs of the results
in order to show that the results are correct. Diophantus’ arguments are
cumbersome and roundabout.
[452,2] [Dioph 1] If three numbers exceed one another by an equal amount, then
eight times the product of the greatest and the middle one plus the square on the
least produces a square the side of which is equal to the sum of the greatest and
twice the middle one.
if x = y + k and y = z + k, then 8xy + z2 = (x + 2y)2.11
[454,6] [Dioph 2] If there are numbers in any multitude in equal excess, <the
excess> of the greatest over the least is their excess multiplied by one less than the
multitude of numbers set out.
if x1, x2, . . . xn+1 are such that xi+1 = xi + j then xn+1 − x1 = nj.12
[456,2] [Dioph 3] If there are numbers in any multitude in equal excess, the sum
of the greatest and least multiplied by their multitude makes a number which is
double of the sum of the numbers set out.
if x1, x2, . . . xn are such that xi+1 = xi + j then
(xn + x1)n = 2(x1 + x2 + . . . + xn).13
[460,5] [Dioph 4] If there are numbers in any multitude in equal excess starting
from the unit, then the sum multiplied by eight times their excess plus the square of
two less than their excess is a square of which the side minus 2 will be their excess
11
I do not mean to suggest, nor do I believe, that Diophantus’ reasoning does not include
geometric elements of the kind we find in the so-called geometric algebra of Book 2 of Euclid’s
Elements. But discussion of that issue would require a detailed examination of Diophantus’
proofs, a task which I cannot undertake here.
Proof: Let x = z + 2k and y = z + k. Then we should prove that:
8(z + 2k)(z + k) + z2 = ((z + 2k) + 2(z + k))2.
But
8 ( z + 2k ) ( z + k ) + z 2 = 8 ( z2 + 3zk + 2k2 ) + z2 = 9z2 + 24zk + 16k2 =
( 3z + 4k) 2 = ( ( z + 2k ) + 2z + 2k ) 2 = ( ( z + 2k ) + 2 (z + k ) ) 2.
12
13
Dioph 2 is sufficiently obvious that there is really nothing to prove, the basic idea being that
x2 = x1 + j, x3 = x2 + j = x1 + 2j, x4 = x3 + j = x1 + 3j, and so on.
We give an inductive proof of Dioph 3. For n = 1 the theorem says that x2 − x1 = 1 . j. Suppose
(x1 + xn)n = 2(x1 + x2 + . . . + xn). We wish to show that:
(x1 + xn+1)(n + 1) = 2(x1 + x2+ . . . + xn+1).
But:
(x1 + xn+1)(n + 1) = (x1 + xn + j)(n + 1)
= (x1 + xn)n + x1 + xn + (n + 1)j = 2(x1 + x2+ . . . + xn) + x1 + nj + xn + j
= 2(x1 + x2 + . . . + xn) + xn+1 + xn+1 = 2(x1 + x2 + . . . + xn+1).
(That xn+1 = x1 + nj is a trivial reformulation of Dioph 2.)
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multiplied by a certain number which, when a unit is added to it, is double of the
multitude of all the numbers set out with the unit.
if p = x1 + x2 + . . . + xn, where xi + 1 = xi + j and x1 = 1, then
p8j + (j − 2)2 = ((2n − 1)j + 2)2 [= ((n + n − 1)j + 2)2].
It is easy to prove Dioph 4 using Dioph 2 and 3,14 as Diophantus does,
although his argument is cumbersome. Here I wish only to present the
very beginning of his argument and a diagram, provided by me, representing it.
[460,13] For let AB, CD, EF be numbers in equal excess starting from the unit.15
I say that the proposition results. For let there be as many units in GH as the
numbers set out with the unit. And since the excess by which EF exceeds a unit
is the excess by which AB exceeds a unit multiplied by GH minus 1,16 if we set
out each unit, AK, EL, GM, we will have that LF is KB multiplied by MH. So LF
is equal to the product of KB, MH. And if we set out KN as 2, we will investigate
whether, if the sum is multiplied by 8 KB (which is their excess) and the square of
NB (which is less than their excess by 2) is added, the result is a square of which the
side minus 2 produces a number which is their excess (KB) multiplied by GH,HM
together (Figure 9.5).
A
K
1
N
B
j −2
2
K
E
B
L
F
M
1
H
n−1
Figure 9.5 Diophantus’ diagram, Polygonal Numbers, Proposition 4.
14
Since:
((2n −1)j + 2)2 − ( j −2)2
= 4n2j2 − 4nj2 + j2 + 8nj − 4j + 4 − j2 + 4j − 4
= 4n2j2 − 4j2n + 8nj = 4j(n2j − jn + 2n),
to prove Dioph 4 we need only prove:
2(x1 + x2 + . . . + xn) = n2j − jn + 2n,
or, by Dioph 3:
(xn + x1)n = n2j − jn + 2n, that is xn+ x1 = nj − j + 2.
15
16
j
xn
xn − 1
1
G
x2
Note that AB, CD, and EF are numbers, not the unit.
Cf. Dioph 2.
n
Polygonal numbers in ancient Greek mathematics
As I have said, the material described thus far in this section is purely arithmetical. However, if one accepts Defgeo/arith, what Diophantus has shown is
that:
[Dioph 4geo/arith] if p is the nth j + 2-agonal number, then
p8j + (j − 2)2 = ((2n − 1)j + 2)2.
It is clear from Diophantus’ initial less specific statement of what he will
show that he does think that he can establish this :
[450,11] Here it is established (edokimasthê) that if any polygonal is multiplied by a
certain number (which is a function (kata tên analogian) of the multitude of angles
in the polygonal) and a certain square number (again a function of the multitude of
angles in it) is added, the result is a square.
if p is a j + 2-agonal number, there are functions f and c such that
f( j + 2)p + c( j + 2) is a square number.
After announcing this result Diophantus states the goal of the treatise:
[450,16] We will establish this and indicate how one can find a prescribed polygonal
with a given side and how the side of a given polygonal can be taken.
That is,
how to find (1) the j + 2-agonal p with side n and (2) the side of a j + 2-agonal p.
This last subject is the concern of the final part of the treatise (472,21–
476,3). Nic* allows one to solve these in a slightly cumbersome mechanical
way, but what Diophantus proves enables him to give what amounts to
formulae for the solutions:
(1) p =
(2) n =
1
2
2
2
((2n −1) j + 2) −( j −2)
8j
冉
2
p8j + ( j −2) – 2
j
冊
+1
,
.
This last material is quite mundane, and I shall not discuss it. My major
concern will be with the material immediately following the presentation
of the arithmetical results Dioph 1 to 4. For those four propositions are
purely arithmetical; they do not say anything about polygonal numbers and
certainly do not establish anything about spatial configurations of units.
It is in the remainder of the treatise that Diophantus tries to establish a
general truth corresponding to Nic*, but as I have indicated, I believe that it
is impossible to prove this truth within the confines of Greek mathematics.
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What I will try to explain is the specific reason why Diophantus’ attempt
to do so fails, emphasizing that, although his reasoning is mathematically
much more elaborate than Nicomachus’, his handling of generalization is
essentially the same, namely the presentation of examples which make the
general truth “obvious.”
Before turning to that material I want to signal the very first statement in
On Polygonal Numbers, which concerns the first (actual) polygonal number
of each kind:
[450,1] Each of the numbers starting from three which increase by one is a first
polygonal after the unit. And it has as many angles as the multitude of units in it.
Its side is the number after the unit, i.e., 2. 3 is triangular, 4 square, 5 pentagonal,
and so on.
This is, I think, the only application of Defgeo/arith that Diophantus takes for
granted, i.e., he takes for granted that:
the first j+2-agonal (after 1) has side 2 and is j + 2.
After making a remark about the ordinary conception of square
numbers,17 Diophantus gives (450,11 and 16) the announcement of what
he is going to prove, which I have already quoted, and proves his four arithmetical propositions. It is at this point that he first reintroduces the notion
of a polygonal number in his announcement of what he intends to prove
next, which is tantamount to Defgeo/arith:
[468,14] These things being the case, we say that if there are numbers starting from
the unit in any multitude and in any excess, the whole is polygonal. For it has as
many angles as the number which is greater than the excess by 2, and the number
of its sides is the multitude of the numbers set out with the unit.
He now invokes Dioph 4:
[470,1] For we have shown that the sum of all the numbers set out multiplied by 8
KB plus the square of NB produces the square of PK.
Here Diophantus is working with a figure in which the line AKNB of Figure 6
for Dioph 4 is extended to the right so that PK is a representation of
(2n − 1)j + 2.18 But to get a representation of j + 2 he also extends AKNB to
the left (Figure 9.6):
[470,4] But also if we posit AO as another unit, we will have KO as two, and KN is
similarly two.
17
18
[450,9] “It is immediately clear that squares have arisen because they come to be from some
number being multiplied by itself.”
This specification of PK occurs at 466,1–2.
Polygonal numbers in ancient Greek mathematics
O
A
1
O
K
1
A
N
2
j−2
B
K
N
B
K
N
B
P
Figure 9.6 Diophantus’ diagram, Polygonal Numbers.
But now Diophantus is only interested in OB (j + 2 = 1 + (1 + j)), KB (j), and
BN (j − 2), and, in his only application of Dioph 1, he says:
[470,6] Therefore OB, BK, BN will exceed one another by an equal amount.
Therefore, 8 times the product of the greatest OB and the middle BK plus the square
of the least BN makes a square the side of which is the sum of the greatest OB and
2 of the middle BK. Therefore OB multiplied by 8 KB plus the square of NB is equal
to the square of OB and 2KB together.
(j + 2)8j + (j − 2)2 = (j + 2 + 2j)2.
This is, of course, just the special case of Dioph 4 in which n = 2. To make
this point clear Diophantus argues that j + 2 + 2j = (2·2 − 1)j + 2:
[470,13] And the side minus two (OK) leaves 3 KB, which is KB multiplied by three.
But three plus one is 2 multiplied by 2.
Diophantus underlines the analogy with Dioph 4 and then points out that
OB (j + 2) is the first j + 2-agonal number:
[470,17] . . . the sum of the numbers set out with the unit produces (poiei) the same
problem as OB, but OB is a chance number and is the first polygon {of its kind}
after the unit (since AO is a unit and the second number is AB), and {OB} has two
as side.
So, in addition to proving Dioph 4, Diophantus has proved a special case
of it in which n = 2, a case for which he has asserted that p is the first
j + 2-agonal number. These two propositions by themselves do not imply
that whenever the conditions of Dioph 4 hold, p is a j + 2-agonal number
with side n. But this is precisely what Diophantus asserts:19
[470,21] Therefore also the sum of all the numbers set out is a polygon with as many
angles as OB and having as many angles as it is greater by 2 (i.e., by OK) than the
excess, KB; and it has as side GH, which is the number of the numbers set out with
the unit.
19
Commentators have standardly approved this “reasoning,” or at least not raised any doubts
about it. See Poselger 1810: 34–5; Schulz 1822: 618; Nesselmann 1842: 475; Heath 1885: 252;
Wertheim 1890: 309; Massoutié 1911: 26; and Ver Eecke 1926: 288.
323
(2n − 1)j + 2 + 2
j+2
j
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There is no more basis for this generalization than for the generalizations
we have seen in Nicomachus; indeed, in a sense there is even less since
Diophantus has considered only first polygonals and shown that they
satisfy Dioph 4. Like Nicomachus, he clearly could show the same thing for
any particular example, but that hardly proves his claim or the next one:
[470,27] And what is said by Hypsicles in a definition20 has been demonstrated,
namely:
If there are numbers in equal excess in any multitude starting from the unit,
then, when the excess is one the whole is triangular, when it is two, square,
three, pentagonal. The number of angles is said to be greater than the excess
by two, and its sides are the multitude of numbers set out with the unit.
It is not clear exactly what the definition of Hypsicles was.21 In Diophantus’
representation he said something about the first three polygonals, but it
seems reasonable to suppose that he at least intended a generalization and
so can be credited with Defgeo/arith.22 But we have no information about how
he used it – if he did. In any case Diophantus would have been on firmer
footing had he made the definition the basis of his treatise rather than purporting to do the impossible, namely demonstrate it. Had he done this he
would not have had to worry about Dioph 1 and the special case of it which
he invokes to deal with first j + 2-agonal numbers.
Diophantus now applies Hypsicles’ definition and his own results to triangular numbers.
[472,5] Hence, since triangulars result when the excess is one and their sides are the
greatest of the numbers set out, the product of the greatest of the numbers set out
and the number which is greater by one than it is double the triangular indicated.
If p = x1 + x2 + . . . + xn with xi+1 = xi + 1 and x1 = 1, then p is a triangular
with side xn and xn(xn + 1) = 2p.23
Diophantus returns again to first polygonal numbers. He recalls the application of Dioph 1 at 470,6.
[472,9] And since OB has as many angles as there are units in it, if it is multiplied
by 8 multiplied by what is less than it by two (that is by the excess; that will be
20
21
22
23
D’Ooge 1926: 246 endorses Gow’s (1884: 87) suggestion that en horôi might mean “in a book
called Definition.” In itself this suggestion seems to me unlikely, but the recurrences of the
word horos in 472,14 and especially 472,20 seem to me to rule it out completely.
Standard floruit: c. 150 bce.
Contrast Nesselmann 1842: 463.
Proof: It follows from Hypsicles’ definition that x1 + x2 + . . . + xn is a triangular number p with
side n. But by Dioph 2 xn = (n − 1) . 1 + 1 = n. And by Dioph 3 2(x1 + x2 + . . . + xn) =
n(xn + x1) = xn(xn + 1).
Polygonal numbers in ancient Greek mathematics
8 × KB) <and> the square of what is less than it by 4 is added (that is NB), it produces a square.
j + 2 is a j + 2-agonal and (j + 2)8j + (j − 2)2 is a square.
This, too, is immediately generalized with no justification.
[472,14] And this will be a definition (horos) of polygonals:
Every polygonal multiplied by 8 multiplied by what is less by two than the
multitude of its angles plus the square of what is less than the multitude of
angles by 4 makes a square.
If p is j + 2-agonal, p8j + (j − 2)2 is a square.
[472,20] In this way we have demonstrated simultaneously this definition of polygonals and that of Hypsicles.
In this case the truth which Diophantus purports to establish as a definition is not a definition in the standard sense at all, since n8j + (j − 2)2 can be
a square even when n is not a j + 2-agonal; 2·8·3 + (3 − 2)2 = 72, but 2 is not
pentagonal.24 And his claim to have demonstrated it is just as weak as his
claim to have established Defgeo/arith.
Conclusion
It is certainly not surprising that Diophantus’ treatise on polygonal numbers
shows great mathematical skill. And it is perhaps also not surprising that its
sense of logical rigor is at times not superior to that of Nicomachus. Within
the limits of Greek mathematics there can be no mathematical demonstration of an arithmetical characterization of configurationally conceived
polygonal numbers. Within those limits Aristotle (Posterior Analytics 1.6
(Ross)) was correct to insist that the generic difference between arithmetic
and geometry cannot be breached.
24
This shortcoming is already pointed out in the editio princeps of the Greek text of Diophantus
(Bachet 1621: 21 of the edition of On Polygonal Numbers). What Diophantus says at 472,14
could serve as a definition for triangulars and squares. For, ignoring complications that would
arise if one tried to avoid “numbers” less than 1, it is easy to prove that:
p = 1 + 2 + . . . + n (i.e., p is a triangular) if and only if p8·1 + (3 – 4)2 is a square (i.e., if and only
if 8p + 1 is a square);
p = 1 +3 + . . . + 2n − 1 (i.e., p is a square) if and only if p8·2 + (4 – 4)2 is a square (i.e., if and
only if 16p is a square, i.e., if and only if p is a square).
It is tempting to think, although it cannot be proved, that Diophantus was misled by the
truth of these biconditionals to the false notion that Dioph 4 was the basis of a definition of
polygonality in general.
325
326
ian mueller
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