Normat 61:2, 87–110 (2013) 87
Approximating cube roots of integers,
after Heron’s Metrica III.20
Trond Steihaug
a
and D. G. Rogers
b
a
Institutt for Informatikk
Universitetet i Bergen
PB7803, N5020
Bergen
trond.steihaug@ii.uib.no
b
dgrbgsu@gmail.com
For Christian Marinus Taisbak,
Institut for Græsk og Latin, Københavns Universitet, 1964–1994,
On his eightieth birthday, 17 February, 2014
Heron did not need any other corroboration than the fact that the method
works, and that the separate results are easily confirmed by multiplica-
tion.
C. M. Taisbak [28, §2]
1 Taisbak’s conjecture
How often, in the happy Chinese idiom, do we search high and low for our shoulder
pole, only at last to notice it again on our shoulder where we left it? For all that the
learned commentator might reassure us that some mathematician of the past could
not help but make some pertinent observation, just as surely we know, from our
own experience, that such acuity might e sc ape us for half a lifetime, before, all at
once, perhaps of a Summer’s night, the øre drops. This is, indeed, the story behind
Christian Marinus Taisbak’s conjecture in [28], as divulged in a recent letter [29].
So, we too were set thinking. We report here on some of our findings.
Heron, in Metrica III.20–22 , is concerned with the the division of solid figures —
pyramids, cones and frustra of cones — to which end there is a need to extract cube
roots [15, II, pp. 340–342] (see also [16, p. 430]). A case in point is the cube root
of 100, for which Heron obligingly outlines a method of approximation in Metrica
III.20 as follows (adapted from [15, 2, 28], noting that the addition in [28, p. 103,
fn. 1] appears earlier in [2, p. 69]; cf. [20, p. 191, fn. 124]):
88 Trond Steihaug and D. G. Rogers Normat 2/2013
Take the cube numbers nearest 100 both above and below, namely 125 and 64.
Then, 125 ≠ 100 = 25 and 100 ≠ 64 = 36.
Multiply 25 by 4 and 36 by 5 to get 100 and 180; and then add to get 280.
Divide 180 by 280, giving 9/14. Add this to the side of the smaller cube; this
gives 4
9
14
as the cube root of 100 as nearly as possible.
It seems short, unobjectionable work to turn this descriptive algorithm into a ge-
neral formula for approximating the cube root of some given integer N.Wefirst
locate N among the cubes of the integers:
m
3
<N<(m + 1)
3
.
Writing d
1
= N ≠m
3
and d
2
=(m+1)
3
≠N, Heron would then have us approximate
the cube root of N by
(1) m +
(m + 1)d
1
(m + 1)d
1
+ md
2
.
The text of Metrica as we have it today only came to light in the mid-1890s, with
a scholarly edition [24] published in 1903. How little was known for sure about
Metrica in the years immediately prior to this is suggested by [12]. Fragments were
known by quotation in other sources and Eutocius, in a commentary on the works of
Archimedes, reports that He ron used the same methods for square and cube roots
as Archimedes. But clearly this does not have the same cachet as a text — and we
still lack anything by Archimedes on finding cube roots. Gustave Wertheim (1843–
1902) proposed (1) in 1899 in [33], to be followed a few years latter by Gustaf
Hjalmar Eneström (1852–1923) in [9] with an exact (if tautological) expression,
given below in §5.2 as (22), for the cube root of N from which (1) follows on
discarding cubes of positive terms less than unity. (Besides work in mathematics
and statistics, Eneström had interests in the history of mathematics, as seen, for
example, in his note [8] on rules of convergence in the 1700s: he is perhaps best
remembered today for introducing the Eneström index to help identify the writings
of Leonhard Euler (1707–1783); but, while there seems to be little written about
him in English, the very first volume of Nordisk Matematisk Tidskrift carried a
centenary profile [13].)
To be sure, other formulae might fit Heron’s numerical instance in Metrica III.20 :
a nod is made to one in [18, pp. 137–138]:
(2) m +
d
1
Ô
d
2
N + d
1
Ô
d
2
.
At first sight, this gesture might see m pro forma, as it is conceded straightaway
that (2), when compared with (1), is both less easy to justify and not so accurate
for other values of N. But the record has not always been so clear-cut and it is (2),
not (1), that we find on looking back to [30, pp. 62–63], where reference is made
to an article [5] by Ernst Ludwig Wilhelm Maximilian Curtze (1837–1903) of 1897,
along with [33, 9]. Both Curtze’s tentative contribution (2) and another, similar
Normat 2/2013 Trond Steihaug and D. G. Rogers 89
formula,
(3) m +
(m + 1)d
1
N +(m + 1)d
1
,
had, in fact, been compared adversely for accuracy with (1) in 1920 by Josiah
Gilbart Smyly (1867–1948) in [26]; Smyly attributes to George Randolph Webb
(1877–1929; Fellow, Trinity College, Dublin) an estimate that the error in (1) is
of the order of 1/m
2
(see further §5.2, especially (33)). For the record, we might
note here that Smyly alludes to the work of Curtze, but not that of Wertheim
or Eneström; on the other hand, Heath [15] cites them, but not Curtze or Smyly
(truely the vagaries of citation are not easily explicable: in [25, p. 256, fn. 2], we find
Smyly footnoted as correcting Curtze, only for (3), rather than (1), to be printed).
It is also worth observing that the effect of emendations is to move our understan-
ding of the received text in favour of the most accurate candidate, namely (1). As
it happens, in Metrica III.22 , Heron needs the estimate of another cube root, that
of 97050 according to [24], but in fact of 97804
4
5
, as pointed out in [3, pp. 338–340].
The approximation taken is 46, which cubes to 97336, so is not too far off either
way, suggesting that Heron did not allow himself to be blinded by science.
If the consensus on (1) is by now reasonably settled, there remains the question of
how Heron might have com e upon (1), as well as the somewhat different question
of how (1) might be justified. A formal derivation of (1) might well fail to satisfy
those who want some heuristic insight into the approximation; and Eneström may
have lost sight of the simplicity of his identity (26) in the manner he derives it (see
further §5.2). Taisbak strikes out on his own account in [28] from the constancy
of the third difference of the sequence of cubes of integers and builds up to the
observation that the gradient of the chord between m ≠1 and m is to the gradient
of the chord between m and m +1 approximately as m ≠ 1:m +1.Ineffect,
Taisbak sums up his thinking with a question [28, §3]: “Did the Ancients know and
use sequences of differences?”
As far as Taisbak’s mathematics goes, a rather similar argument was advanced some
thirty years ago by Henry Graham Flegg (1924– ) in a book [11, p. 137] (pleasingly
enough it was reissued in 2013). Others have bee n here, too: Oskar Becker (1889–
1964) in [2, pp. 69–71] in 1957; Evert Marie Bruins (1909-1990) in [3, p. 336] in
1964; Wilbur Richard Knorr (1945–1997) in [20, pp. 191–194] in 1986. It has also
been noted how (1) can be adapted for iterative use, although the accuracy of (1),
as remarked on by Smyly, coupled with the opportunity for rescaling it provides,
might make iteration otiose (cf. §5.4). But, in fairness, it might be remarked that
the main difference between these writers and Eneström is that their approximative
sleight of hand takes good care to wipe away small terms as they go, rather than
in one fell swoop at the end (we return to these comparisons in §5.2).
Our concerns are rather different. For a start, might there be more to discern in the
numerical instance Heron presents in Metrica? This prompts two further questions.
Why is no comparison made with the more straightforward cube root bounds (as in
(6) and (7)) analogous to those (as in (14) and (15)) seemingly in common use by
Archimedes, Heron and others for square roots? And, why do we not hear anything
90 Trond Steihaug and D. G. Rogers Normat 2/2013
like (1) in regard to square roots? Then, again, might there not be more to say
about (1) itself?
Our concerns in these regards are mathematical, not historical. Perforce, we respect
Taisbak’s stricture, as endorsed by Unguru [31], that we adopt as our epigraph.
Truly, the proof of the pudding is in the eating; and if, perhaps like Eutocius
commenting on the works of Archimedes, you have nothing more imaginative to
offer, arithmetical confirmation remains a safe recourse, if not always a sure one
(cf. [21, pp. 522, 540]). But we suspect that, if anything, others before us may have
been too abashed to descend our level of naiveté. Our excuse, if one is needed, is
that, even at this leve l, there is still much with which to be usefully engaged.
2 Heron’s example
The difference between successive cubes is
(4) (m + 1)
3
≠ m
3
=3m
2
+3m +1.
More generally, we may picture the difference between cubes by cutting up the lar-
ger cube into smaller cube with various other slabs and blocks, a three-dimensional
analogue of the pictures we might draw for the difference of two squares, perhaps
as an aide mémoire to our reading of Euclid’s Elements II (one traditional mode
of visualising the cube of a binomial expression is shown in Fig. 1; an alternative
dissection appears in Fig. 3 in conjunction with (16)).
Thus, as d
1
= N ≠m
3
and d
2
=(m + 1)
3
≠N sum to this difference, we can ensure
some cancellation in working with (1) if we arrange to take d
1
to be k(m+1)+1 for
some k with 0 Æ k Æ 3m. Perhaps Heron had something of this in mind in taking
an example in which d
1
=(2m ≠ 1)(m + 1) + 1 = m(2m + 1) and d
2
=(m + 1)
2
for m =4. At all events, generalising Heron’s example in this way, we obtain from
(1) a bound on the cube root of N = m
3
+ m(2m + 1) = m(m + 1)
3
≠ (m + 1)
2
:
(5) m +
2m +1
3m +2
= m +1≠
m +1
3m +2
.
It is a simple matter of ve rification to check that this is an upper bound.
But not only is this pleasing in itself, the form of these expressions suggests —
invites? — a comparison with the upper bounds obtained more straightforwardly
from binomial expressions analogous to those familiar for square roots (as in (14)
and (15)), of which Gerolamo Cardono (1501–1576) made celebrated use in Practica
Arithmetice (1539) [22, §2.4] (but cf. also (20)). Thus, for N = m
3
+ d
1
,thecube
root is bounded above by
(6) m +
d
1
3m
2
,
92 Trond Steihaug and D. G. Rogers Normat 2/2013
while for N =(m + 1)
3
≠ d
2
, the cube root is bounded above by
(7) m +1≠
d
2
3(m + 1)
2
.
So, in generalising Heron’s example, we have hit on a case where the upper bounds
in (6) and (7) als o come out rather neatly:
m +
2m +1
3m
; m +1≠
1
3
.
Of course, the former is not so good as the latter, reflecting the closer proximity of
this N to (m + 1)
3
than to m
3
. Rather more strikingly neither of these bounds is
as good as that in (5) obtained from (1); indeed,
2m +1
3m +2
<
2
3
<
2m +1
3m
.
It is possible to squeeze (6) further by increasing the denominator in the fraction,
and some writers in Arabic in the early 1000s worked with 3m
2
+1 in place of 3m
2
(cf. [22, §3.2]. But this still does not give an improvement over (5).
Whether or not Heron may have indulged himself in such exercises, a few numerical
instances like this would surely convey to any impressionable mind that (1) cannot
be completely without merit. Trouble might spring more from the opposite corner,
not to run away with too favourable an endorsement based only on evidence of
this sort. However, as we show in §5.3, an approximate construction of two mean
proportionals examined by Pappus early in Synagogue III allows us to improve on
(5), indicating that it is by no means the best the Greeks could have done, had
they put their minds to it.
3 Square roots
3.1 Elementar y theory of proportions
When we look at the formulation of (1), it would seem that it is a recipe we could
write down for other functions be sides cubes and cube roots; and, if for cubes and
cube roots, why not before that for squares and square roots? In fact, we might
recognize (1) in the setting of the elementary theory of proportions that was well-
articulated by the Greeks. For, given a/b > c/d > 0, an early result in that theory
gives
c
d
<
a + c
b + d
<
a
b
,
and, more generally, for weights w
1
and w
2
,
(8)
c
d
<
aw
1
+ cw
2
bw
1
+ dw
2
<
a
b
.
94 Trond Steihaug and D. G. Rogers Normat 2/2013
the chord going on from n to m +1, that is, d
2
/(n +1≠m), where for our present
purposes in this sec tion we write d
1
= n
2
≠m
2
and d
2
=(m + 1)
2
≠n
2
in analogy
with the notation for (1). But, if
(10)
d
1
n ≠ m
<
d
2
m +1≠ n
,
then it follows that, for 0 Æ d
1
Æ 2m +1,
(11) n>
(m + 1)d
1
+ md
2
d
1
+ d
2
= m +
d
1
2m +1
.
Equality would hold here if the two gradients were equal, in which case the common
value would be the gradient of the chord from m to m +1, confirming that this
lower bound on n is the ordinate ¯n at which N is attained on this chord (as in
Fig. 2).
Of course, in this case, d
1
and d
2
are just differences of squares,
d
1
= n
2
≠m
2
=(n ≠ m)(n + m); d
2
=(m + 1)
2
≠n
2
=(m +1≠ n)(m +1+n),
so
d
1
n ≠ m
= n + m;
d
2
m +1≠ n
= m +1+n.
Hence, (10) holds trivially:
n + m<m+1+n.
But, looking at this last inequality, we see that it is readily reversed by judicious
counterpoised weighting, mutiplying the le ft-hand side by m+1 and the right hand
side by m:
(m + 1)(n + m) >m(m +1+n).
So, in addition to (10), we also have
(12)
(m + 1)d
1
n ≠ m
>
md
2
m +1≠ n
from which we deduce in turn the upper bound
(13) n<
(m + 1)
2
d
1
+ m
2
d
2
(m + 1)d
1
+ md
2
= m +
(m + 1)d
1
(m + 1)d
1
+ md
2
,
thereby providing easy confirmation that the analogue of (1) for square roots.
But the algebra here is such that conversely, if a upper bound of the form (13)
holds, then the weighted gradients stand as in (12), a point to be ar in m ind when
considering (1).
Normat 2/2013 Trond Steihaug and D. G. Rogers 95
3.3 Square root bounds
However, the sad fact of the matter is that (13) is not much help because we already
do better with one or other of the standard upper bounds for square roots obtained
from binomial expressions that complement the lower bound (11); the implicit use
of all the bounds (11), (14) and (15) in antiquity is examined in extenso in [14,
pp. lxxvii–xcix] (cf. [12, pp. 53–57]). We recall that, for N = m
2
+ d
1
,
(14) n =
Ô
N<m+
d
1
2m
while, for N =(m + 1)
2
≠ d
2
,
(15) n =
Ô
N<m+1≠
d
2
2(m + 1)
.
We work with (14) for 0 <d
1
Æ m, switching to (15) for 0 <d
2
Æ m +1.
Notice that (14) and (15) also follow from the iterative scheme that He ron sketches
by example for N = 720 in Metrica I.8 :
m
1
=
1
2
3
N
m
0
+ m
0
4
,
with m
0
= m for (14) and m
0
= m +1 for (15). Whether Heron recognised (15)
explicitly depends in large part on what inference can be drawn from the way
fractions are recorded (cf. [15, II, p. 326]). There are other puzzles in relation
to Heronian iteration. For instance, samplings in [14, p. lxxxii] and [7, p. 6] of
estimates used by Heron for square roots includes that for
Ô
75 as 8
11
16
(cf. (14)),
rather than 8
2
3
(cf. (15); and see further [3, pp. 10–11]), which is simpler, as well
as more accurate; and a further example is raised in §5.4.
Now, in these ranges for d
1
and d
2
for (14) and (15),
(m + 1)d
1
+ md
2
Æ 2m(m + 1),
with equality if and only if d
1
= m and d
2
= m +1. Hence (13) is only as good
as (14) or (15) in the case where d
1
= m and d
2
= m +1, when all three bounds
come out the same, namely m +
1
2
(but see §5.4 for a reprieve of sorts for (9)). This
points up the altered situation for cub e roots, where the evidence of the previous
section shows that (1) does better than (4) and (5), at least in a family of instances
generalizing Heron’s example in Metrica III.20 . Clearly, we need to examine how
the arguments leading to (11) and (3) for square roots go over to cube roots,
especially as it is the innocent use of counterpoised weighting in shifting from (10)
to (12) that lies at the heart of Taisbak’s musings in [28].
3.4 Mellema’s formula for quadratics
But before leaving this discussion of square roots it may be instructive in compari-
son with the derivation of Eneström’s identity (26) to take a brief look at a formula
96 Trond Steihaug and D. G. Rogers Normat 2/2013
developed by Elcie Edouard Leon Mellema (1544–1622) as a baroque example of
the method of false position (cf. [17]). Suppose that a function f(x) has a root at
n with a<n<b, then, trivially,
(f(n) ≠ f (a))f(b)=(f(n) ≠ f(b))f (a).
However, in the case of a quadratic function where the square has been completed,
that is, where
f(x)=(x + p)
2
≠ q,
rearranging this equation to make (n + p)
2
the subject yields Mellema’s formula:
(n + p)
2
=
(a + p)
2
f(b) ≠ (b + p)
2
f(a)
f(b) ≠ f (a)
.
In contrast with (26), from which (1) follows as an approximation, the bes t that
can be said of Mellema’s formula is that it is a trick on him, if not also on any who
might be taken in by it, as it just recomputes q, which we might suppose would be
known more swiftly on completing the square in the quadratic.
4 Cube roots
So, let us now return to cube roots and our initial suppos ition that we are given
N,with
m
3
<N<(m + 1)
3
,
and write
d
1
= N ≠ m
3
; d
2
=(m + 1)
3
≠ N.
If n is the cube root of N ,son
3
= N, then, possibly calling to mind Heron’s account
of frustra of pyramids and cones in Metrica II.6, 9 (cf. [15, II, pp. 332–334]; that
the formulae Heron provides were not always used with sufficient care is suggested
in [27, pp. 107–108]),
(16) d
1
= n
3
≠ m
3
=(n ≠ m)(n
2
+ nm + m
2
),
so that
(17)
d
1
n ≠ m
= n
2
+ nm + m
2
.
Similarly
(18)
d
2
m +1≠ n
=(m + 1)
2
+(m + 1)n + n
2
.
98 Trond Steihaug and D. G. Rogers Normat 2/2013
fraction in (20) if it suits the calculation (the textual problem raised in [22, p. 92,
fn. 7] as to the use of the improved bound is resolved on cross-reference with [19,
p. 262]). A version of (20) appears again in use in the 1500s (cf. [25, p. 255, fn. 4];
[17]).
So far, so good, although this is entirely as we might expect. But what about
applying Taisbak’s hunch on counterpoised weightings to (17) and (18) that, as
we have seen in the previous section, does lead in the case of square roots to the
analogue (13) of (1)?
Thus, in place of (19), we shall need to consider:
(21)
(m + 1)d
1
n ≠ m
≠
md
2
m +1≠ n
= n
2
≠ m(m + 1).
Now, with (21), we see the contingent nature of the expression in (1) as a bound
on the cube root of N. For, if N
2
>m
3
(m + 1)
3
, as is certainly the cas e when
N>(m +
1
2
)
3
, then the right-hand side of (21) is positive, and, as, in the previous
section, it follows that (1) gives an upper bound. On the other hand, if N
3
<
m
3
(m + 1)
3
, (1) will give another lower bound along with (20), although one that
improves on (20), as it is a matter of easy algebra to check that the expression in
(1) is always larger than its counterpart in (20):
(a
2
p + b
2
q)(p + q) Ø (ap + bq)
2
.
In this latter case, let us take by way of illustration N = 85,sod
1
= 21 and
d
2
= 40; the two lower bounds then come out as 4
21
61
, for (20), and 4
21
53
, for (1).
Of course, we can always up the ante by further loading the weights. Moving up
from (21), we find that
(m + 1)
2
d
1
n ≠ m
≠
m
2
d
2
m +1≠ n
=(2m + 1)n
2
+ m(m + 1)n>0,
so at least we have the upper bound
(22) n<
(m + 1)
3
d
1
+ m
3
d
2
(m + 1)
2
d
1
+ m
2
d
2
,
throughout the range m
3
<N<(m + 1)
3
, for what it is worth. But, in the test
case N = m
3
+ m(2m + 1) considered in §2, (22) gives the upper bound
m +
2m +1
3m +1
.
Thus, (22) loses the advantage we found (1) has over (7) for such N (even if it
remains better than (6)).
Normat 2/2013 Trond Steihaug and D. G. Rogers 99
5 Comparisons
All comparisons, it is has often been said, are odious, but, as an anonymous reviewer
wryly rejoined in the Edinburgh Review [1, p. 400] for September, 1818:
No man, when he learns t hat the three angles of every triangle are equal to
two right angles, ever thought of saying, that the series of comparisons by
which that truth is demonstrated was invidious; neither has the fate of those
interesting portions of space ever been deemed particularly hard, for having
been subjected to such an investigation.
The Greeks did debate the propriety of geometrical procedures — we turn to one
example in §5.3. But their practical arithmetical competence was more pragmatic
it seems. Approximations tend to be stated blankly, without supporting argument,
but also without comparison with other methods, as though truly, as Taisbak has
it with (1), the Greeks did not need any other corroboration than the fact that the
method works.
In contrast, for us today proposal of an approximative method is incomplete unless
accompanied by examination of how well it performs against both rivals and the
target. So, in this section, we first look at an instance where Heron provides, not
only a demonstration, but compares the resulting bound with an older rule of
thumb; we then make a more thorough investigation of Eneström’s identity; and
we go on to show how a geometric scheme considered by Pappus can be adapted to
improve on (1) for the family of numerical cases in §2. We conclude by observing
how the improving accuracy of (1), as revealed by (33), allows us to make good
effect of rescaling (returns to scale). The Newton-Raphson and Halley methods
of approximating cube roots in (29) and (31), in contrast, do not guarantee such
improving accuracy, even if some juggling may be possible (a rather more obvious
distinction is that (1) is e xact when N is the cube of an integer).
5.1 Metrica I.27–32: Area of a circular segment
Heron, in Metrica I.27–32, is concerned with formulae for the area of a circular
segment (see [15, II, pp. 330–331]). Let AB be the arc of a circle subtending a
segment less than a semicircle and let C be the midpoint of the arc. Then Heron
asserts that the area subtended by AB is greater than four thirds the area of the
triangle —ABC; that is, if the arc AB has sagitta h and subtended chord b,the
subtended segment between arc and chord has area at least
(23)
4
3
3
hb
2
4
.
But, rather out of character for him, Heron goes further, proving (23) in a manner
reminiscent of Archimedes’ De quadratura parabolae, Prop. 24 . However, despite
being game to take on this task, Heron does not seem entirely sure of himself: he sets
up his diagram as if intending to argue in one way, but then heads off in another;
and underlying this dithering is a certain uneasiness in handling inequalities (at
issue, in a sense, are returns to scale resulting from the circle’s convexity, cf. §5.4).
100 Trond Steihaug and D. G. Rogers Normat 2/2013
So, it may be some surprise to find that, in Metrica I.30, 31 , Heron voluntee rs
comparison of (23) with a more traditional approximation, namely
(24)
h(b + h)
2
,
even stating, but without further comment, when one is to be preferred to the
other.
This is all rather remarkable, and not unnaturally Metrica I.27–32 has caught the
attention of commentators. Wilbur Knorr, in particular, has made much of the
passage, returning to tease it out several times, as for example, in his books [20,
pp. 168–169] and [21, pp. 498–501], as well as in earlier papers on which the books
build. Knorr adjudicates the comparison of (23) and (24) in a footnote [20, p. 168,
fn. 63] (in a further footnote [21, p. 501, fn. 34], he reports how advantage was not
always taken of the improved bound):
[Hero] adds that one should use this rule when b is less than three times h,
but the former rule when b is greater. He does not explain this c riterion, but
one can see how it results from considering where the two rules yield the same
result, namely, 2bh/3=h(b + h)/2,whenceb =3h....
The [former] rule, by virtue of its association with that for the parabolic seg-
ment, suggests an Archimedean origin. One suspects that the rather sophisti-
cated effort reported by Hero to assess the relative utility of these two rules
for the circular segments is also due to an Archimedean insight.
Now, there is no doubt that inequalities are more tricky to handle than equalities
for pupils today, no less than in the past; and we all resort to simple means of
reassurance that we have them right. But, if Knorr’s comments here arrest our
attention, it is because of the incongruity between the supposed Archimedean origin
of the comparison and the method advanced for seeing that it holds. Perhaps
Knorr is empathising too much with the difficulty Heron might have encountered
in understanding some abstruse Archimedean proto-text. Comparison of (23) and
(24) would surely present little challenge to those, such as Archimedes, if not also
Heron, for whom thinking in terms of areas was stock-in-trade.
In terms of areas, (23) tells us that the area of the subtended segment is a third
more than the area of the triangle —ABC, in keeping with the way the proof
presented by Heron runs. So, in place of (23), we might write the bound as
(25)
hb
2
+
1
3
3
hb
2
4
=
h(b + b/3)
2
.
Our areal intuition then suggests seeing in (24) and (25) triangles with common
height h and bases
b + h; b +
b
3
,
respectively. Which triangle has the larger area is simply a matter of which base is
longer, leading to the conclusion that (25) is a better lower bound when the latter
base is the larger, that is, when b/3 is bigger than h, as Heron claimed.
Normat 2/2013 Trond Steihaug and D. G. R ogers 101
But, with Taisbak’s stricture as our epigraph, the point to remembe r here — and
the point of this excursus — is that this is only our intution, not necessarily that
of Heron or Archimedes, however plausible we fancy it to be. On the other hand,
they were clearly not in want of competence of their own.
5.2 Eneström’s identity
It would be wrong to give the impression that the papers of Curtze [5] and Wertheim
[33] are confined to the elaboration of Heron’s text as discussed in the opening
section. For example, Curtze includes a list of quadratic approximations. Wertheim
anticipates the spirit of Taisbak in [28], providing a foundation on which Eneström
builds in [9]. Indeed, as Taisbak [29] playfully observes of any purported “new
insight,” on comparing Wertheim’s contribution with his own,
If someone else said the same, it must be true. If not, it is high time to have
said it.
Now, if we write
1
= d
1
≠ (n ≠ m)
3
;
2
= d
2
≠ (m +1≠ n)
3
,
then Eneström, in [9], goes through a series of algebraic manipulations that brings
n out in this notation as
(26) n = m +
(m + 1)
1
(m + 1)
1
+ m
2
.
Clearly, if we ignore terms that are cubes of positive numbers less than unity, the
right-hand side of (26) is just (1). But (26) must hold as an identity, so going
through a routine of solving for n, as Eneström does, might seem somewhat arti-
ficial. Why not proce ed more simply by direct computation with
1
and
2
?We
have
(27)
1
=3mn(n ≠ m);
2
= 3(m + 1)n(m +1≠ n),
expressions already familiar from [28] as approximations for d
1
and d
2
. So, it readily
follows that
(m + 1)
i
1
+ m
i
2
=3m(m + 1)n
i
,i=1, 2.
Hence (cf. (9), (11), (13) and (22)),
(28) n =
(m + 1)
2
1
+ m
2
2
(m + 1)
1
+ m
2
= m +
(m + 1)
1
(m + 1)
1
+ m
2
,
as desired.
Looked at in this way, we see both that there is less mystery about Eneström’s
exact expression (26), but also less difference between him and later writers whose
strategy is to get in e arly with the approximations for d
1
and d
2
given by (27),
102 Trond Steihaug and D. G. Rogers Normat 2/2013
rather than waiting to the e nd. Either way, while it is apparent that (1) is an
approximation for the cube root of N , because we are modifying both numerator
and denominator in the fraction we form in (28), we are left uncertain how good an
approximation it is, or even whether we obtain an upp er bound or a lower bound. As
Taisbak draws inspiration from the gradient of chords between successive integers
and their cubes, his approach inherently sets up the expectation of an upper bound.
Naturally, a version of (28), and so of (1), can be developed for general intervals,
as in [20, p. 192] and [6, p. 29, (1)] (that thoroughness is nee ded here can be seen
from [22, §2.1]). But Knorr’s description in [20, p. 192] of a prospective iterative
application of such an extension of (1) also appears to be written in the expectation
that the result gives an upper bound. If, for some a and b not necessarily integers
we have a
3
<N<(a+b)
3
and we obtain the approximation a+b
Õ
after the manner
of (1), as Knorr has us imagine, then certainly, at the next round of the iteration,
we s ubstitute for a + b
Õ
for a + b, but only if this approximation is an upper bound.
In view of (21), we shall need to check this. If, in the event, it turns out that a + b
Õ
is a lower bound, we shall have to substitute it for a, not a + b, at the next round.
Knorr rightly goes on to question the authenticity of wiping away of small quan-
tities, whenever in the scheme of things it happens, noting that we can reach the
approximations in (27) in greater conformity with the Greek style by replacing the
three terms on the left-hand side of (17) and (18) by three times their respective
middle terms, rather than being tied to versions of the binomial expansion (4) (see
[20, p. 193]). So far as this approach goes, it is on a par with a Newton-Raphson
approximation for the cub e root of N , such as
(29)
N +2m
3
3m
2
obtained by similarly replacing the same three terms by three times the last term,
as Knorr also remarks.
For that m atter, we could take this line of discussion further, by replacing the same
three te rms by three times the first term to obtain an approximation for the square
of the cube root of N,
(30)
2N + m
3
3m
,
and then cap this cleverness, by observing that an improved approximation for the
cube root of N proposed by Edmund Halley is given as the ratio of the expressions
in (29) and (30):
(31) m
3
2N + m
3
N +2m
3
4
.
Halley’s approximation in (31) does at least serve to remind us that in (1) we are
also involved with a ratio, a ratio moreover, as (28) makes clear, of two blends
of the approximations in (27). Strangely enough, Knorr seems distracted from the
significance of these differences between (1) and, say, (29), even while digressing at
length on discoveries in approximation theory.
104 Trond Steihaug and D. G. Rogers Normat 2/2013
Thus, starting from (21), we find that
(32)
(m + 1)
2
d
1
+ m
2
d
2
(m + 1)d
1
+ md
2
≠ n =
(n
2
≠ m(m + 1))(n ≠ m)(m +1≠ n)
(m + 1)d
1
+ md
2
.
To bound the absolute value of the left-hand side of (32) without going into too
much fine detail, we note, first of all, that
|n
2
≠ m(m + 1)|Æm + 1;
secondly, by the inequality between geometric and arithmetic means (cf. Elements
VI.27 )
(n ≠ m)(m +1≠ n) Æ
1
4
,
with equality if and only if n = m +1/2; and thirdly
(m + 1)d
1
+ md
2
>m(d
1
+ d
2
) Ø 3m
2
(m + 1).
Hence, putting these ingredients together, we conclude that
(33)
-
-
-
-
(m + 1)
2
d
1
+ m
2
d
2
(m + 1)d
1
+ md
2
≠ n
-
-
-
-
<
1
12m
2
,
of comparable order of magnitude to the bound 3/(80m
2
) that Smyly tells us in [26]
had been obtained by Webb. Another elementary bound is proved in [6, Theorem
3], but on the interval (m, m + 1) is is weaker than (33).
5.3 Synagogue III: Two mean propor tionals
Pappus musters in Synagogue III a collection of constructions of two mean pro-
portionals between two line segments by non-planar means. Perhaps by way of
cautionary prologue, he also describes a geome trical solution, purportedly by plane
considerations only, from some unnamed source, specifically with a view to showing
that it fails. The flaws in the construction are fairly transparent, and Pappus’ demo-
lition of them is not especially edifying. However, for all the imperfections Pappus
would have us see in it, the construction is not without other merits. Knorr offers a
sensitive geometrical re-appraisal at some length in [21, pp. 64–70]; more recently,
Serafina Cuomo has returned to the cons truction in a study [4, §4.1] of Pappus’
mathematics in the setting of Late Antiquity. Earlier attempts at rehabilitating
the construction tended to recast it as an iterative scheme of approximation to
the mean proportionals, using an algebraic notation alien to the spirit of Pappus’
Synagogue. Nevertheless, what we might notice about this algebra for our pre-
sent purposes is how well it meshes with the family of numerical examples in §2
generalising Heron’s case, N = 100,inMetrica III.20.
In this regard, the pioneering effort was made by Richard Pendlebury (1847–1902;
Senior Wrangler, 1870) in a note [23] published in 1873, as reported in [15, I,
pp. 268–270] (see further [21, p. 64, fn. 8]; [4, p. 130]). Suppose that N = m
3
≠lm
2
,
Normat 2/2013 Trond Steihaug and D. G. R ogers 105
for some l and m, then Pendlebury shows that iteration of the construction faulted
by Pappus in Synagogue III can be generalised as a recursive computation,
(34) n
i+1
= m ≠
(m ≠ n
i
)lm
2
m
3
≠ n
3
i
,
for some given n
0
,withthen
i
successively better approximations to the cube root
of N, giving upper bounds when n
0
is bigger than this cube root, and lower bounds
when it is smaller.
Now, the family of N in §2 generalising Heron’s example is given by taking l =1.
If we start with our Heronian uppe r bound (5),
n
0
= m ≠
m
3m ≠ 1
= m(1 ≠
1
3m ≠ 1
),
then (34) gives the improved upper bound
(35) n
1
= m ≠
(3m ≠ 1)
2
3(3m ≠ 1)(3m ≠ 2) + 1
.
In particular, for Heron’s example, N = 100 is the case m =5, when (35) yields
(36) n
1
=5≠
196
547
=4
351
547
,
an improvement on Heron’s upper bound 4
9
14
for the cube root of 100.
In this exercise, we may be scrabbling after crumbs, waiting for a spark from heaven
to fall. This particular construction never seems to have attracted much attention
until analysed by Pendlebury, although Leonardo Pisano and Gerolamo Cardano
retained geometrical accounts of second mean proportionals in their discussions
of cube root extraction. But, over the course of countless Greek lives, there was
presumably time for many other failed constructions and, in amongst them, some
near-misses, possibly the occasional success — after all, we still have Archime des’
On the Measurement of a Circle.
5.4 Rescaling
None of the ingredients we use in producing (33) could reasonably be said to be
beyond the compete nce of the ancient Greek mathematicians, and yet we would
naturally hesitate when it comes to an error bound like (33) itself. Nevertheless,
if we do have a sense that the going gets better, however we might come by it,
we can always try rescaling. Thus, to estimate the cube root in Heron’s example,
N = 100, we might divide the estimate from (1) for the cube roots, say, of 800 or
2700 by 2 or 3 respectively to get
4
322
502
;4
7328
11421
;
106 Trond Steihaug and D. G. Rogers Normat 2/2013
the first of these estimates is a lower bound not as close to the cube root of 100 as
the upper bound in (36) while the s ec ond is an upper bound improving on that in
(36).
Of course, (1) is most in error for some small values of N. About the worst offender
proportionately is N =5, when the estimate from (1) is 1
8
11
, with a cube greater
than 5.153. It is here that we can use rescaling to good advantage. Amusingly
enough, if we divide the estimates from (1) for 40 or 135 by 2 or 3 respectively, we
come out with the same lower bound for the square root of 5, namely 1
22
31
,witha
cube greater than 4.997. Going further and dividing the estimate from (1) for 320
by 4 gives the upper bound 1
615
866
, with a cube now less than 5.002.
Maybe there is some redemption to be found here, too, for the comparatively weak
upper bound for square roots in (13), because, if we continue with the algebra there,
we find that the diminution in the error is on the order of 1/m. For example, Heron,
in Metrica I.9 , wants to compute
Ô
1575 and notes he can get at this as
10
2
Ô
63,
offering the upper bound 7
15
16
for
Ô
63, either by Heronian iteration as in Metrica
I.8 or possibly as an application of (15) (cf. Stereometrica I.33 ). Of course, if we
stick with the same method and use it to approximate
Ô
1575 directly we come out
with the same estimate either way. However, as it so happens, Heron also alludes
to
Ô
1575 in passing as “the square root of the fourth part of 6300 ” (cf. [3, p. 203]).
But, if we divide the estimate of
Ô
6300 from (13) by 10, we obtain a (slightly)
improved upper bound: 7
1183
1262
. Similarly, when Heron wants an approximation for
Ô
720 in Metrica I.8 , his first estimate is the upper bound 26
5
6
, whereas working
(13) with 72, 000 improves this to 26
30002
36023
.
Then, again, in any practical example, the convenience of working with an estimate
may outweigh its accuracy, so such gains are largely a matter of theory. Moreover,
elsewhere, in Geometrica 53, 54 (cf. [15, II, p. 321], when dealing with the 4-6-8
triangle, Heron seems to show some awareness that gains can be made from delay
in the taking of s quare roots, initially proposing a
1
, an upper bound with
N =4
Ú
8
7
16
< 11
2
3
= a
1
,
but then, on rewriting N by multiplying into the square root, observing that we
can do better using a
2
,with
N =
Ô
135 < 11
13
21
= a
2
.
Typically, nothing is said about the derivation of these bounds. Interestingly enough
though, Heronian iteration, as in (15), applied to N gives 11
5
8
, which falls in be-
tween the two bounds,
(37) a
1
= 11
2
3
> 11
5
8
> 11
13
21
= a
2
;
Normat 2/2013 Trond Steihaug and D. G. R ogers 107
a
1
results on applying Heronian iteration, or (15), to
Ô
136 = 4
Ò
8
1
2
; and a
2
im-
proves on a
1
precisely by Heronian iteration,
(38) a
2
=
1
2
3
135
a
1
+ a
1
4
.
A possible alternative derivation of a
1
, in line with Heron’s handing of
Ô
75 noted
in §3.3, might be to stick with Heronian iteration in the form (14) for N, giving
a less good upper b ound 11
14
22
, which, however, encourages nudging up to the
simpler fraction a
1
. But all of this is speculative, and those who e njoy numerical
coincidences will be amused to see the early Fibonacci numbers showing up in (37),
still more perhaps to learn that these bounds are the 4th, 6th and 8th convergents
of the continued fraction for
Ô
135. Notice, however, that Heronian iteration with
the middle bound in (37) yields 11
307
496
, which does improve on a
2
, if only just.
Thus, it is uncertain whether the improvement Heron notes here derives from his
rescaling per se or from a change in the method of approximation. Indeed, (38) may
run slightly counter to the view in [15, II, p. 326] on Heron’s own use of Heronian
iteration, while leaving it a mystery as to how he obtained bounds that improve on
a first instance of the method. Something similar might be at work in the handling
of
Ô
28 as discussed in [3, p. 309]. In this case, we might expect the bound 5
3
10
(cf.
(14)), but the weaker bound 5
1
3
(cf. (15)) lends itself m ore easily to improvement
by Heronian iteration, giving 5
7
24
. However, what might require us to rethink, or
at least re-express, the matter is the obse rvation that rescaling combined with (15)
does allow us to give the supposedly improved bounds in both case s more directly:
(39)
Ô
28 =
1
3
Ô
252 <
1
3
3
16 ≠
4
32
4
=
1
3
3
15
7
8
4
=5
7
24
;
(40)
Ô
135 =
1
3
Ô
1215 <
1
3
3
35 ≠
10
70
4
=
1
3
3
34
6
7
4
= 11
13
21
.
Fortunately, under Taisbak’s dispensation, we are not so pressed to ac count for the
rather weak estimates Heron also uses on occasion, as, for e xample, 43
1
3
for
Ô
1875
or 14
1
3
for
Ô
207, the former squaring to more than 1877, the latter to less than
206 (see [15, II, pp. 326, 328]).
Smyly [26, p. 67], in extolling the virtues of (1) for N of the order of 10
6
in
comparison with tables of seven-figure logarithms, and Knorr [20, p. 192], in dilating
on iterative use of (1), possibly overlook this simple trick of rescaling to obtain
improved estimates for smaller N. Scaling, in the elementary sense of the law of
indices, is one thing; the notion of returns to scale another, rather more subtle.
Some accounts of Greek approximations for
Ô
2 and
Ô
3 would have us believe that
the Greeks were great self-improvers, working their way to better estimates through
solutions of Pell equations or the convergents of continued fractions which might be
seen as implicitly involving a form of rescaling (indeed, not unlike (39) and (40)).
Taisbak [28, §3] asks in regard to his conjecture whether the Ancients knew and
used sequences of differences. With an eye to (39) and (40), we follow suit: did the
Ancients know and use rescaling?
108 Trond Steihaug and D. G. Rogers Normat 2/2013
6 A last reckoning
Numerical corroboration, of course, might not be to everyone’s taste. Bartel Le-
endert van der Waerden (1903–1996), for one, in the original Dutch edition of
Ontwakende Wetenschap (Science Awakening) [32, p. 306], in 1950, places Heron
in heavily weighted scales.
Laten we blij zijn, dat we de meesterwerken van Archimedes en Apol lonios
hebben, en niet treuren om het verlies van talloze rekenboekjes à la Heron.
[Let us rejoice in the masterworks of Archimedes and of Apollonius and not
mourn the loss of numberless little accounting books after the manner of Her-
on.]
The translation in English in 1954 is less pointed, but, recalling Heron’s own mat-
hematical outlook as expresse d in the preface to Metrica, it is likely that he could
at least hold his ow n (cf. [10]).
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