Approximating cube roots of integers, after Heron's Metrica III.20
AApproximating cube roots of integers,after Heron’s
Metrica III.20
Trond Steihaug and D. G. Rogers
Institutt for Informatikk, Universitetet i BergenPB7803, N5020, Bergen, Norge [email protected]
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 methodworks, and that the separate results are easily confirmed by multiplication.
C. M. Taisbak [28, § How often, in the happy Chinese idiom, do we search high and low for our shoulderpole, only at last to notice it again on our shoulder where we left it? For all that thelearned commentator might reassure us that some mathematician of the past couldnot help but make some pertinent observation, just as surely we know, from ourown experience, that such acuity might escape us for half a lifetime, before, all atonce, perhaps of a Summer’s night, the øre drops. This is, indeed, the story behindChristian 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 cuberoots [15, II, pp. 340–342] (see also [16, p. 430]). A case in point is the cube rootof 100, for which Heron obligingly outlines a method of approximation in
MetricaIII.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]):
Take the cube numbers nearest 100 both above and below, namely 125 and 64.Then, −
100 = 25 and −
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; thisgives as the cube root of 100 as nearly as possible. a r X i v : . [ m a t h . HO ] M a y t seems short, unobjectionable work to turn this descriptive algorithm into a generalformula for approximating the cube root of some given integer N . We first locate N among the cubes of the integers: m < N < ( m + 1) . Writing d = N − m and d = ( m + 1) − N , Heron would then have us approximatethe cube root of N by m + ( m + 1) d ( m + 1) d + md . (1)The text of Metrica as we have it today only came to light in the mid-1890s, witha 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 wereknown by quotation in other sources and Eutocius, in a commentary on the worksof Archimedes, reports that Heron used the same methods for square and cube rootsas Archimedes. But clearly this does not have the same cachet as a text — andwe 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 GustafHjalmar Enestr¨om (1852–1923) in [9] with an exact (if tautological) expression, givenbelow in § N from which (1) follows on discardingcubes of positive terms less than unity. (Besides work in mathematics and statistics,Enestr¨om had interests in the history of mathematics, as seen, for example, in hisnote [8] on rules of convergence in the 1700s: he is perhaps best remembered todayfor introducing the Enestr¨om index to help identify the writings of Leonhard Euler(1707–1783); but, while there seems to be little written about him in English, thevery 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]: m + d √ d N + d √ d . (2)At first sight, this gesture might seem pro forma , as it is conceded straightaway that(2), when compared with (1), is both less easy to justify and not so accurate forother 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 anarticle [5] by Ernst Ludwig Wilhelm Maximilian Curtze (1837–1903) of 1897, alongwith [33, 9]. Both Curtze’s tentative contribution (2) and another, similar formula, m + ( m + 1) d N + ( m + 1) d , (3)had, in fact, been compared adversely for accuracy with (1) in 1920 by Josiah GilbartSmyly (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 of1 /m (see further § 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 understandingof the received text in favour of the most accurate candidate, namely (1). As ithappens, in
Metrica III.22 , Heron needs the estimate of another cube root, that of97050 according to [24], but in fact of 97804 , as pointed out in [3, pp. 338–340].The approximation taken is 46, which cubes to 97336, so is not too far off eitherway, 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 ofhow Heron might have come upon (1), as well as the somewhat different question ofhow (1) might be justified. A formal derivation of (1) might well fail to satisfy thosewho want some heuristic insight into the approximation; and Enestr¨om may havelost sight of the simplicity of his identity (26) in the manner he derives it (see further § m − m is to the gradient of the chordbetween m and m + 1 approximately as m − m + 1. In effect, Taisbak sums uphis thinking with a question [28, § Did the Ancients know and use sequences ofdifferences? ”As far as Taisbak’s mathematics goes, a rather similar argument was advanced somethirty years ago by Henry Graham Flegg (1924– ) in a book [11, p. 137] (pleasinglyenough it was reissued in 2013). Others have been here, too: Oskar Becker (1889–1964) in [2, pp. 69–71] in 1957; Evert Marie Bruins (1909-1990) in [3, p. 336] in1964; Wilbur Richard Knorr (1945–1997) in [20, pp. 191–194] in 1986. It has alsobeen 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. § § 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 byArchimedes, Heron and others for square roots? And, why do we not hear anythinglike (1) in regard to square roots? Then, again, might there not be more to sayabout (1) itself?Our concerns in these regards are mathematical, not historical. Perforce, we respectTaisbak’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 com-menting 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 beentoo abashed to descend our level of naivet´e . Our excuse, if one is needed, is that,even at this level, there is still much with which to be usefully engaged.3
Heron’s example
The difference between successive cubes is( m + 1) − m = 3 m + 3 m + 1 . (4) ArÃthmbticae Liber i i;• !•h%Bfnomium autem illud contra dum ad cubum, ccrnÃtur fub>iftis partÃculÃs compofÃtionÃs. X^j ч Wb±S9±\lCopofituro Ãgit regula de extraftt'onib» cubÃcts ex bÃnomÃjs& refiduis,refpiciat ad diftraaioné partÃcula^ pofitamA fciareflê .pportionalitatê continua Ãnter 4 fupremas partÃculas,fci*licet interV%8 di г 16,funt 1 iS¿ A*S9 Muo mcdia »pportionaÃœa,fciatCK partÃculas mediaksad fe addita s,perficere portionêК rj binomrji- G e n e r a t e d o n - - : G M T / h tt p :// h d l . h a n d l e . n e t / / u c m . P u b li c D o m a i n , G oo g l e - d i g i t i z e d / h tt p :// . h a t h i t r u s t . o r g / a cc e ss _ u s e pd - g oo g l e Figure 1: Picture of a cubed binomial from
Arithmetica Integra (1544)More generally, we may picture the difference between cubes by cutting up the largercube into smaller cube with various other slabs and blocks, a three-dimensionalanalogue of the pictures we might draw for the difference of two squares, perhapsas an aide m´emoire to our reading of Euclid’s
Elements II (one traditional modeof visualising the cube of a binomial expression is shown in Fig. 1; an alternativedissection appears in Fig. 3 in conjunction with (16)).Thus, as d = N − m and d = ( m + 1) − N sum to this difference, we can ensuresome cancellation in working with (1) if we arrange to take d to be k ( m + 1) + 1 forsome k with 0 ≤ k ≤ m . Perhaps Heron had something of this in mind in takingan example in which d = (2 m − m + 1) + 1 = m (2 m + 1) and d = ( m + 1) 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 + m (2 m + 1) = m ( m + 1) − ( m + 1) : m + 2 m + 13 m + 2 = m + 1 − m + 13 m + 2 . (5)4t is a simple matter of verification 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 straightforwardlyfrom binomial expressions analogous to those familiar for square roots (as in (14)and (15)), of which Gerolamo Cardono (1501–1576) made celebrated use in PracticaArithmetice (1539) [22, § N = m + d , the cuberoot is bounded above by m + d m , (6)while for N = ( m + 1) − d , the cube root is bounded above by m + 1 − d m + 1) . (7)So, in generalising Heron’s example, we have hit on a case where the upper boundsin (6) and (7) also come out rather neatly: m + 2 m + 13 m ; m + 1 − . Of course, the former is not so good as the latter, reflecting the closer proximity of this N to ( m + 1) than to m . Rather more strikingly neither of these bounds isas good as that in (5) obtained from (1); indeed,2 m + 13 m + 2 < < m + 13 m . 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 3 m + 1 in place of 3 m (cf. [22, § only on evidence ofthis sort. However, as we show in § Synagogue III allows us to improve on(5), indicating that it is by no means the best the Greeks could have done, had theyput their minds to it.
When we look at the formulation of (1), it would seem that it is a recipe we couldwrite down for other functions besides cubes and cube roots; and, if for cubes andcube roots, why not before that for squares and square roots? In fact, we might5ecognize (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 theorygives cd < a + cb + d < ab , and, more generally, for weights w and w , cd < aw + cw bw + dw < ab . (8)In particular (cf. (11), (13), (22) and (26)) m = m m < ( m + 1) w + m w ( m + 1) w + mw < ( m + 1) m + 1 = m + 1 , (9)where the central expression can then be rewritten as (cf. (1))( m + 1) w + m w ( m + 1) w + mw = m + ( m + 1) w ( m + 1) w + mw . This is pudding that anyone can eat, but it might not always satisfy WinstonChurchill’s demand that pudding have a theme . For, how to explain the choiceof weights for different functions? m nn_ m+1m N(m+1) Figure 2: Approximating square roots from belowFor any increasingly increasing function, such as squaring or cubing, chords lieabove the curve, so a particular height N will be encountered on the chord before itis encountered on the curve, giving a simple means of finding a lower bound on theordinate for which N is attained, after the manner of solution traditionally known6s “ double false position ” (a brief introduction to the history of which is recentlyto hand in [17]). Let us illustrate the thinking here rather naively in the case ofsquares. So, suppose now that we are given N , with m < N < ( m + 1) , and we are interested in the square root n = √ N . Then we expect that the gradientof the chord between m and n , that is, d / ( n − m ), to be less than the gradient ofthe chord going on from n to m + 1, that is, d / ( n + 1 − m ), where for our presentpurposes in this section we write d = n − m and d = ( m + 1) − n in analogywith the notation for (1). But, if d n − m < d m + 1 − n , (10)then it follows that, for 0 ≤ d ≤ m + 1, n > ( m + 1) d + md d + d = m + d m + 1 . (11)Equality would hold here if the two gradients were equal, in which case the commonvalue would be the gradient of the chord from m to m + 1, confirming that this lowerbound on n is the ordinate ¯ n at which N is attained on this chord (as in Fig. 2).Of course, in this case, d and d are just differences of squares, d = n − m = ( n − m )( n + m ); d = ( m + 1) − n = ( m + 1 − n )( m + 1 + n ) , so d n − m = n + m ; d 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 judiciouscounterpoised weighting, mutiplying the left-hand side by m + 1 and the right handside by m : ( m + 1)( n + m ) > m ( m + 1 + n ) . So, in addition to (10), we also have( m + 1) d n − m > md m + 1 − n (12)from which we deduce in turn the upper bound n < ( m + 1) d + m d ( m + 1) d + md = m + ( m + 1) d ( m + 1) d + md , (13)thereby providing easy confirmation that the analogue of (1) for square roots. Butthe algebra here is such that conversely, if a upper bound of the form (13) holds, thenthe weighted gradients stand as in (12), a point to bear in mind when considering(1). 7 .3 Square root bounds However, the sad fact of the matter is that (13) is not much help because we alreadydo better with one or other of the standard upper bounds for square roots obtainedfrom binomial expressions that complement the lower bound (11); the implicit useof 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 + d , n = √ N < m + d m (14)while, for N = ( m + 1) − d , n = √ N < m + 1 − d m + 1) . (15)We work with (14) for 0 < d ≤ m , switching to (15) for 0 < d ≤ m + 1.Notice that (14) and (15) also follow from the iterative scheme that Heron sketchesby example for N = 720 in Metrica I.8 : m = 12 (cid:18) Nm + m (cid:19) , with m = m for (14) and m = m + 1 for (15). Whether Heron recognised (15) ex-plicitly depends in large part on what inference can be drawn from the way fractionsare recorded (cf. [15, II, p. 326]). There are other puzzles in relation to Heronianiteration. For instance, samplings in [14, p. lxxxii] and [7, p. 6] of estimates used byHeron for square roots includes that for √
75 as 8 (cf. (14)), rather than 8 (cf.(15); and see further [3, pp. 10–11]), which is simpler, as well as more accurate; anda further example is raised in § d and d for (14) and (15),( m + 1) d + md ≤ m ( m + 1) , with equality if and only if d = m and d = m + 1. Hence (13) is only as goodas (14) or (15) in the case where d = m and d = m + 1, when all three boundscome out the same, namely m + (but see § Metrica III.20 . Clearly, we need to examine how thearguments leading to (11) and (3) for square roots go over to cube roots, especiallyas it is the innocent use of counterpoised weighting in shifting from (10) to (12) thatlies at the heart of Taisbak’s musings in [28].
But before leaving this discussion of square roots it may be instructive in comparisonwith the derivation of Enestr¨om’s identity (26) to take a brief look at a formula8eveloped by Elcie Edouard Leon Mellema (1544–1622) as a baroque example of themethod 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 ) − q, rearranging this equation to make ( n + p ) the subject yields Mellema’s formula:( n + p ) = ( a + p ) f ( b ) − ( b + p ) f ( a ) f ( b ) − f ( a ) . In contrast with (26), from which (1) follows as an approximation, the best thatcan be said of Mellema’s formula is that it is a trick on him, if not also on any whomight be taken in by it, as it just recomputes q , which we might suppose would beknown more swiftly on completing the square in the quadratic. So, let us now return to cube roots and our initial supposition that we are given N ,with m < N < ( m + 1) , and write d = N − m ; d = ( m + 1) − N. If n is the cube root of N , so n = N , then, possibly calling to mind Heron’s accountof frustra of pyramids and cones in Metrica II.6, 9 (cf. [15, II, pp. 332–334]; thatthe formulae Heron provides were not always used with sufficient care is suggestedin [27, pp. 107–108]), d = n − m = ( n − m )( n + nm + m ) , (16)so that d n − m = n + nm + m . (17)Similarly d m + 1 − n = ( m + 1) + ( m + 1) n + n . (18)It follows that, on the lines of (10), we have d m + 1 − n − d n − m = 2 m + n + 1 > , (19)from which we deduce, in perfect analogy with (11), the lower bound m l = ( m + 1) d + md d + d = m + d m ( m + 1) + 1 , (20)9 n-mn-m n mm n-mm n-m n m nnn-mm m mn-mn-m Figure 3: Difference of cubes dissected according to (16)and then, iterating the argument, the further refined lower bound m l + N − m l m l ( m + 1) + ( m + 1 − m l ) . By way of illustration, in Heron’s example with N = 100, neither the the lowerbound m l = 4 obtained from (20) nor the refined one, which involves much heaviercomputation, are as close to the cube root of 100 as Heron’s upper bound 4 . Yet,as a matter of historical record, Leonardo Pisano (Fibonacci; 1170?–1250?), in LiberAbaci (1202) [22, § De Practica Geometrie (1223) [19, pp. 260–262], approximates cube roots by means of (20), sometimes in sequence with itsimprovement, knowing to ignore the term ( m + 1 − m l ) in the denominator of thefraction in the latter and even the analogous 1 in the denominator of the last fractionin (20) if it suits the calculation (the textual problem raised in [22, p. 92, fn. 7] asto 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 aboutapplying Taisbak’s hunch on counterpoised weightings to (17) and (18) that, aswe have seen in the previous section, does lead in the case of square roots to theanalogue (13) of (1)?Thus, in place of (19), we shall need to consider:( m + 1) d n − m − md m + 1 − n = n − m ( m + 1) . (21)Now, with (21), we see the contingent nature of the expression in (1) as a bound onthe cube root of N . For, if N > m ( m + 1) , as is certainly the case when N > m + ) , 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 < m ( m + 1) ,(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 alwayslarger than its counterpart in (20):( a p + b q )( p + q ) ≥ ( ap + bq ) . In this latter case, let us take by way of illustration N = 85, so d = 21 and d = 40;the two lower bounds then come out as 4 , for (20), and 4 , for (1).Of course, we can always up the ante by further loading the weights. Moving upfrom (21), we find that( m + 1) d n − m − m d m + 1 − n = (2 m + 1) n + m ( m + 1) n > , so at least we have the upper bound n < ( m + 1) d + m d ( m + 1) d + m d , (22) throughout the range m < N < ( m + 1) , for what it is worth. But, in the test case N = m + m (2 m + 1) considered in §
2, (22) gives the upper bound m + 2 m + 13 m + 1 . Thus, (22) loses the advantage we found (1) has over (7) for such N (even if itremains better than (6)). All comparisons, it is has often been said, are odious, but, as an anonymous reviewerwryly rejoined in the
Edinburgh Review [1, p. 400] for September, 1818:
No man, when he learns that the three angles of every triangle are equal totwo right angles, ever thought of saying, that the series of comparisons bywhich that truth is demonstrated was invidious; neither has the fate of thoseinteresting portions of space ever been deemed particularly hard, for havingbeen subjected to such an investigation.
The Greeks did debate the propriety of geometrical procedures — we turn to oneexample in § did not need any other corroboration than the fact that themethod works . 11n contrast, for us today proposal of an approximative method is incomplete unlessaccompanied by examination of how well it performs against both rivals and the tar-get. So, in this section, we first look at an instance where Heron provides, not onlya demonstration, but compares the resulting bound with an older rule of thumb;we then make a more thorough investigation of Enestr¨om’s identity; and we go onto show how a geometric scheme considered by Pappus can be adapted to improveon (1) for the family of numerical cases in §
2. We conclude by observing how theimproving accuracy of (1), as revealed by (33), allows us to make good effect ofrescaling (returns to scale). The Newton-Raphson and Halley methods of approx-imating cube roots in (29) and (31), in contrast, do not guarantee such improvingaccuracy, even if some juggling may be possible (a rather more obvious distinctionis that (1) is exact when N is the cube of an integer). Metrica I.27–32 : Area of a circular segment
Heron, in
Metrica I.27–32 , is concerned with formulae for the area of a circularsegment (see [15, II, pp. 330–331]). Let AB be the arc of a circle subtending asegment less than a semicircle and let C be the midpoint of the arc. Then Heronasserts that the area subtended by AB is greater than four thirds the area of thetriangle (cid:52) ABC ; that is, if the arc AB has sagitta h and subtended chord b , thesubtended segment between arc and chord has area at least43 (cid:18) hb (cid:19) . (23)But, rather out of character for him, Heron goes further, proving (23) in a mannerreminiscent of Archimedes’ De quadratura parabolae, Prop. 24 . However, despitebeing game to take on this task, Heron does not seem entirely sure of himself: hesets 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. § Metrica I.30, 31 , Heron volunteers comparisonof (23) with a more traditional approximation, namely h ( b + h )2 , (24)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 theattention of commentators. Wilbur Knorr, in particular, has made much of thepassage, 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 booksbuild. 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 notalways 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 criterion, but ne can see how it results from considering where the two rules yield the sameresult, namely, bh/ h ( b + h ) / , whence b = 3 h . . . .The [former] rule, by virtue of its association with that for the parabolic seg-ment, suggests an Archimedean origin. One suspects that the rather sophis-ticated effort reported by Hero to assess the relative utility of these two rulesfor the circular segments is also due to an Archimedean insight. Now, there is no doubt that inequalities are more tricky to handle than equalitiesfor pupils today, no less than in the past; and we all resort to simple means ofreassurance that we have them right. But, if Knorr’s comments here arrest ourattention, it is because of the incongruity between the supposed Archimedean ori-gin of the comparison and the method advanced for seeing that it holds. PerhapsKnorr is empathising too much with the difficulty Heron might have encounteredin understanding some abstruse Archimedean proto-text. Comparison of (23) and(24) would surely present little challenge to those, such as Archimedes, if not alsoHeron, 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 morethan the area of the triangle (cid:52)
ABC , in keeping with the way the proof presentedby Heron runs. So, in place of (23), we might write the bound as hb (cid:18) hb (cid:19) = h ( b + b/ . (25)Our areal intuition then suggests seeing in (24) and (25) triangles with commonheight h and bases b + h ; b + b , respectively. Which triangle has the larger area is simply a matter of which base islonger, leading to the conclusion that (25) is a better lower bound when the latterbase is the larger, that is, when b/ h , as Heron claimed.But, with Taisbak’s stricture as our epigraph, the point to remember here — andthe point of this excursus — is that this is only our intutition, not necessarily thatof 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. 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 openingsection. For example, Curtze includes a list of quadratic approximations. Wertheimanticipates the spirit of Taisbak in [28], providing a foundation on which Enestr¨ombuilds in [9]. Indeed, as Taisbak [29] playfully observes of any purported “ newinsight ,” 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 havesaid it. = d − ( n − m ) ; ∆ = d − ( m + 1 − n ) , then Enestr¨om, in [9], goes through a series of algebraic manipulations that brings n out in this notation as n = m + ( m + 1)∆ ( m + 1)∆ + m ∆ . (26)Clearly, if we ignore terms that are cubes of positive numbers less than unity, theright-hand side of (26) is just (1). But (26) must hold as an identity, so going througha routine of solving for n , as Enestr¨om does, might seem somewhat artificial. Whynot proceed more simply by direct computation with ∆ and ∆ ? We have∆ = 3 mn ( n − m ); ∆ = 3( m + 1) n ( m + 1 − n ) , (27)expressions already familiar from [28] as approximations for d and d . So, it readilyfollows that ( m + 1) i ∆ + m i ∆ = 3 m ( m + 1) n i , i = 1 , . Hence (cf. (9), (11), (13) and (22)), n = ( m + 1) ∆ + m ∆ ( m + 1)∆ + m ∆ = m + ( m + 1)∆ ( m + 1)∆ + m ∆ , (28)as desired.Looked at in this way, we see both that there is less mystery about Enestr¨om’s exactexpression (26), but also less difference between him and later writers whose strategyis to get in early with the approximations for d and d given by (27), rather thanwaiting to the end. Either way, while it is apparent that (1) is an approximationfor the cube root of N , because we are modifying both numerator and denominatorin the fraction we form in (28), we are left uncertain how good an approximation itis, or even whether we obtain an upper bound or a lower bound. As Taisbak drawsinspiration 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 needed here can be seenfrom [22, § a and b not necessarily integerswe have a < N < ( a + b ) and we obtain the approximation a + b (cid:48) after the mannerof (1), as Knorr has us imagine, then certainly, at the next round of the iteration,we substitute for a + b (cid:48) 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 (cid:48) 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 theapproximations in (27) in greater conformity with the Greek style by replacing the14hree terms on the left-hand side of (17) and (18) by three times their respectivemiddle 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-Raphsonapproximation for the cube root of N , such as N + 2 m m (29)obtained by similarly replacing the same three terms by three times the last term,as Knorr also remarks. −1.5−1−0.500.511.52 x 10 −3 ≤ N ≤ m + ( m + ) d / (( m + ) d + m d ) − N / Error in equation (1)
Figure 4: Damped oscillation exhibited by error in (1), as given by (32)For that matter, we could take this line of discussion further, by replacing the samethree terms by three times the first term to obtain an approximation for the square of the cube root of N , 2 N + m m , (30)and then cap this cleverness, by observing that an improved approximation for thecube root of N proposed by Edmund Halley is given as the ratio of the expressionsin (29) and (30): m (cid:18) N + m N + 2 m (cid:19) . (31)Halley’s approximation in (31) does at least serve to remind us that in (1) we arealso involved with a ratio, a ratio moreover, as (28) makes clear, of two blends of15he approximations in (27). Strangely enough, Knorr seems distracted from thesignificance of these differences between (1) and, say, (29), even while digressing atlength on discoveries in approximation theory.It may also be worth remembering that the statement of a result for illustrativepurposes by way of a succinct algorithmic description, such as suits Heron’s purposein Metrica III.20 might not be the formulation used were the result recast as amore formal proposition. It is natural that historians of mathematics should wishto adhere to the text as they understand it, that is, to (1) as encapsulating thenumerical instance in
Metrica III.20 ; and that is what we find, with proposed proofsin which the manipulations of ratios closely follows the form of (1). But, considering(13), (22), and now (28), in the general setting provided (8) and (9), we mightsuspect that it is these more symmetric equivalents of (1) that lend themselves morereadily both to proof and to further examination. −3 Error in (1) and Webb’s boundN m + ( m + ) d / (( m + ) d + m d ) − N / errorUpper boundLower bound Figure 5: Heron’s Wave: error in (1) with Ward’s bound superimposedThus, starting from (21), we find that( m + 1) d + m d ( m + 1) d + md − n = ( n − m ( m + 1))( n − m )( m + 1 − n )( m + 1) d + md . (32)To bound the absolute value of the left-hand side of (32) without going into toomuch fine detail, we note, first of all, that | n − m ( m + 1) | ≤ m + 1;secondly, by the inequality between geometric and arithmetic means (cf. Elements I.27 ) ( n − m )( m + 1 − n ) ≤ , with equality if and only if n = m + 1 /
2; and thirdly( m + 1) d + md > m ( d + d ) ≥ m ( m + 1) . Hence, putting these ingredients together, we conclude that (cid:12)(cid:12)(cid:12)(cid:12) ( m + 1) d + m d ( m + 1) d + md − n (cid:12)(cid:12)(cid:12)(cid:12) < m , (33)of comparable order of magnitude to the bound 3 / (80 m ) that Smyly tells us in [26]had been obtained by Webb. Another elementary bound is proved in [6, Theorem3], but on the interval ( m, m + 1) is is weaker than (33). Synagogue III : Two mean proportionals
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 ofcautionary prologue, he also describes a geometrical solution, purportedly by planeconsiderations only, from some unnamed source, specifically with a view to showingthat 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 Pappuswould have us see in it, the construction is not without other merits. Knorr offers asensitive geometrical re-appraisal at some length in [21, pp. 64–70]; more recently,Serafina Cuomo has returned to the construction in a study [4, § Synagogue .Nevertheless, what we might notice about this algebra for our present purposes ishow well it meshes with the family of numerical examples in § N = 100, in Metrica 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 − lm , forsome l and m , then Pendlebury shows that iteration of the construction faulted byPappus in Synagogue III can be generalised as a recursive computation, n i +1 = m − ( m − n i ) lm m − n i , (34)for some given n , with the n i successively better approximations to the cube rootof N , giving upper bounds when n is bigger than this cube root, and lower boundswhen it is smaller.Now, the family of N in § l = 1.If we start with our Heronian upper bound (5), n = m − m m − m (1 − m − , n = m − (3 m − m − m −
2) + 1 . (35)In particular, for Heron’s example, N = 100 is the case m = 5, when (35) yields n = 5 − , (36)an improvement on Heron’s upper bound 4 for the cube root of 100.In this exercise, we may be scrabbling after crumbs, waiting for a spark from heavento fall. This particular construction never seems to have attracted much attentionuntil analysed by Pendlebury, although Leonardo Pisano and Gerolamo Cardanoretained geometrical accounts of second mean proportionals in their discussionsof cube root extraction. But, over the course of countless Greek lives, there waspresumably time for many other failed constructions and, in amongst them, somenear-misses, possibly the occasional success — after all, we still have Archimedes’ On the Measurement of a Circle . None of the ingredients we use in producing (33) could reasonably be said to bebeyond the competence of the ancient Greek mathematicians, and yet we wouldnaturally hesitate when it comes to an error bound like (33) itself. Nevertheless, ifwe do have a sense that the going gets better, however we might come by it, we canalways 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 2or 3 respectively to get 4 322502 ; 4 732811421 ;the first of these estimates is a lower bound not as close to the cube root of 100 asthe upper bound in (36) while the second 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 offenderproportionately is N = 5, when the estimate from (1) is 1 , with a cube greaterthan 5.153. It is here that we can use rescaling to good advantage. Amusinglyenough, if we divide the estimates from (1) for 40 or 135 by 2 or 3 respectively, wecome out with the same lower bound for the square root of 5, namely 1 , with acube greater than 4.997. Going further and dividing the estimate from (1) for 320by 4 gives the upper bound 1 , with a cube now less than 5.002.Maybe there is some redemption to be found here, too, for the comparatively weakupper 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 √ √ for √
63, either by Heronian iteration as in
MetricaI.8 or possibly as an application of (15) (cf.
Stereometrica I.33 ). Of course, if we18tick with the same method and use it to approximate √ √ the square root of the fourth part of 6300 ” (cf. [3, p. 203]).But, if we divide the estimate of √ . Similarly, when Heron wants an approximation for √
720 in
Metrica I.8 , his first estimate is the upper bound 26 , whereas working(13) with 72 ,
000 improves this to 26 .Then, again, in any practical example, the convenience of working with an estimatemay 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 4-6-8triangle, Heron seems to show some awareness that gains can be made from delayin the taking of square roots, initially proposing a , an upper bound with N = 4 (cid:114) <
11 23 = a , but then, on rewriting N by multiplying into the square root, observing that we cando better using a , with N = √ <
11 1321 = a . Typically, nothing is said about the derivation of these bounds. Interestingly enoughthough, Heronian iteration, as in (15), applied to N gives 11 , which falls in betweenthe two bounds, a = 11 23 >
11 58 >
11 1321 = a ; (37) a results on applying Heronian iteration, or (15), to √
136 = 4 (cid:113) ; and a improveson a precisely by Heronian iteration, a = 12 (cid:18) a + a (cid:19) . (38)A possible alternative derivation of a , in line with Heron’s handing of √
75 notedin § N , giving aless good upper bound 11 , which, however, encourages nudging up to the simplerfraction a . But all of this is speculative, and those who enjoy numerical coincidenceswill be amused to see the early Fibonacci numbers showing up in (37), still moreperhaps to learn that these bounds are the 4th, 6th and 8th convergents of thecontinued fraction for √ , which does improve on a , 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) mayrun slightly counter to the view in [15, II, p. 326] on Heron’s own use of Heronianiteration, while leaving it a mystery as to how he obtained bounds that improve ona first instance of the method. Something similar might be at work in the handlingof √
28 as discussed in [3, p. 309]. In this case, we might expect the bound 5 (cf.(14)), but the weaker bound 5 (cf. (15)) lends itself more easily to improvementby Heronian iteration, giving 5 . However, what might require us to rethink, or19t least re-express, the matter is the observation that rescaling combined with (15)does allow us to give the supposedly improved bounds in both cases more directly: √
28 = 13 √ < (cid:18) − (cid:19) = 13 (cid:18)
15 78 (cid:19) = 5 724 ; (39) √
135 = 13 √ < (cid:18) − (cid:19) = 13 (cid:18)
34 67 (cid:19) = 11 1321 . (40)Fortunately, under Taisbak’s dispensation, we are not so pressed to account for therather weak estimates Heron also uses on occasion, as, for example, 43 for √ for √ N of the order of 10 in com-parison with tables of seven-figure logarithms, and Knorr [20, p. 192], in dilatingon iterative use of (1), possibly overlook this simple trick of rescaling to obtainimproved estimates for smaller N . Scaling, in the elementary sense of the law ofindices, is one thing; the notion of returns to scale another, rather more subtle.Some accounts of Greek approximations for √ √ §
3] asks in regard to his conjecture whether the Ancients knew andused sequences of differences. With an eye to (39) and (40), we follow suit: did theAncients know and use rescaling?
Numerical corroboration, of course, might not be to everyone’s taste. Bartel Leen-dert van der Waerden (1903–1996), for one, in the original Dutch edition of
Ontwak-ende Wetenschap ( Science Awakening ) [32, p. 306], in 1950, places Heron in heavilyweighted scales.
Laten we blij zijn, dat we de meesterwerken van Archimedes en Apollonioshebben, en niet treuren om het verlies van talloze rekenboekjes `a la Heron.[Let us rejoice in the masterworks of Archimedes and of Apollonius and notmourn the loss of numberless little accounting books after the manner ofHeron.]
The translation in English in 1954 is less pointed, but, recalling Heron’s own math-ematical outlook as expressed in the preface to
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