archimedes on the dimensions of the cosmos

Archimedes on the Dimensions of the CosmosAuthor(s): Catherine OsborneSource: Isis, Vol. 74, No. 2 (Jun., 1983), pp. 234-242Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/233105 .

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CATHERINE OSBORNE

ARCHIMEDES ON THE DIMENSIONS OF THE COSMOS

By Catherine Osborne*

In the fourth book of The Refutation of All Heresies, the early Christian writer Hippolytus of Rome (died A.D. 235) reports on some theories of mathematical astronomy ascribed to Aristarchus, Apollonius, Archimedes, and perhaps one other whose name is lost. The most extensive account concerns a theory of the distances of the heavenly bodies ascribed to Archimedes, but the corrupt state of the numerals in the sole surviving manuscript means that the material is difficult to handle.1 Here I offer a reconstruction of the data, followed by an examination of the rationale behind the numbers and of the question of their possible association with Archimedes.

I. THE DATA PROVIDED BY THE TEXT

Hippolytus's text provides the following material, all given in stades: (1) a series of numbers (here designated series A) for the intervals between the successive bodies, earth to moon, moon to sun, and so on, finishing with a distance from Saturn to the outermost sphere of the zodiac (4.8.6-7); (2) a number for the circumference of the zodiac (4.9.1) from which Hippolytus says it is possible to calculate the distance from the earth's surface to the zodiac by dividing by six (Tr = 3) to find the radius and subtracting (3) the number for the radius of the earth; (4) a series of numbers (series B) for the distance of each body from the surface of the earth (4.9.2), not including the moon or the zodiac; (5) some analysis of the numbers, including the repetition three more times of the distance of the moon from the earth (4.10.4, 4.10.5, 4.11.3), the repetition with some variation of six other numbers from series A with approximate proportions assigned to them (4.10.5-6), and the multiplication of the distance earth to moon by powers of two and three, with some errors (4.11). Hippolytus's ultimate source for these data is a Platonist critique; many of the numerals in 4.11 are deployed in reveal- ing the deficiencies of the system.

It is readily apparent that the distances in series B ought to be easily derivable from those in series A by simple addition. As the text stands, however, the two sets of numbers bear no clear relation (see Table 2 below).

The first to study the numbers was Paul Tannery, in 1883. Tannery declared them too corrupt for the basis of the scheme to be identified, concluding that they were "une fantaisie arithmetique" of no astronomical value and suggesting that they should not be associated with Archimedes. He was also the first to suggest that there was a relation between the numbers such that they could be rendered in the form

ma + nb

* King's College, Cambridge CB2 1ST, England. I am particularly grateful for the helpful suggestions of Dr. A. Barker, Dr. G. E. R. Lloyd, Prof.

G. E. L. Owen, Dr. P. Pattenden and Prof. G. C. Stead. 'Hippolytus, Refutatio 4.8.6-4.11.5, in P. Wendland, ed., Die griechischen christlichen

Schriftsteller, Vol. XXI (Leipzig, 1916). References in text and notes are to book, chapter, and section or, when referring to occurences of words and numerals, to page and line.

ISIS, 1983, 74:234-242

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ARCHIMEDES ON THE COSMOS

Table 1. The orders of the planets

Series A earth moon sun Venus Mercury Mars Jupiter Saturn zodiac

Series B earth moon Venus Mercury sun Mars Jupiter Saturn zodiac

(Mercury) (Venus)

NOTE: In other texts B usually runs as in parentheses.

where m and n are simple whole numbers and a = 5,000,000 stades (5 x 106) and b = 272,065.2 This system was also adopted in 1975 by Neugebauer, who analyzed the numbers in more detail and proposed considerable emendations to bring them into a more systematic relationship. Neither Tannery nor Neugebauer resolved the apparent contradictions between series A and B: Neugebauer de- tected only a hint that they bore some relation which made three of the intervals in A add up to the same as three equivalent intervals from B.3

II. RECONSTRUCTING THE DATA

The discordant relation between the two series of numbers is compounded by another discrepancy: each series has the planets in a different order (here designated by the same letter as the series: A or B-see Table 1).4 "Archimedes" is unlikely to have used two different orders in a single coherent theory; one or the other is probably incorrect. Several ancient writers mention the existence of two distinct schools of thought in this respect. Ptolemy implies that earlier astrono- mers used B and later ones A .5 In other texts B regularly turns up in Pythagorean contexts, while A is usually associated with Plato. In his commentary on Cic- ero's Dream of Scipio, Macrobius, who remarks on Cicero's using order B although Plato used A, claims that A is Egyptian, B Chaldean. Macrobius here mentions that Archimedes used order B. Later in the commentary Macrobius recounts a theory of Archimedes' that is clearly the same as the one given by Hippolytus; this is the only other extant reference to the theory.6 Although Ma- crobius gives no numbers, the planets are indeed in order B. Macrobius also mentions Platonist objections to the theory that are precisely the same as those Hip- polytus gives in his account. Both Macrobius and Hippolytus, therefore, clearly used a Platonist source, one unlikely to have changed the order from the Platonic to the Pythagorean order.7 We may therefore assume that the Pythagorean order, B, was the original one. Therefore, although the results would be equally

2 Paul Tannery, "Aristarque de Samos," Memoires de la Societe des Sciences Physiques et Na- turelles de Bordeaux, 2nd series, 1883, 5:237-258 = Memoires scientifiques, Vol. I (Paris, 1912), pp. 371-396, esp. 392-394, quoting p. 393. Tannery also discusses the numbers in Recherches sur I'histoire de l'astronomie antique (Paris, 1893), p. 333. On the possible relation of b with the circumference of the earth, see appendix below under principle (1).

3 0. Neugebauer, A History of Ancient Mathematical Astronomy (Berlin: Springer-Verlag, 1975), Vol. II, pp. 647-651, on p. 649. Neugebauer in fact uses a more complex system than Tannery: in order to account for certain of his numbers in series B he has to divide Tannery's b (= Neugebauer's d) into b = 27 x 104 and c = 2065.

4 Neugebauer did not attempt to resolve this contradication: "It remains a mystery why he spaced the planets so differently in (A) and (B)"; History, p. 650.

5 Ptolemy, Almagest 9.1. 6 Macrobius, In somnium Scipionis 1.19, 2.3.13. 7 Macrobius names Porphyry as his source; Hippolytus probably used the same or a closely related

source.

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CATHERINE OSBORNE

Table 2. Manuscript numbers

B B' A Series B (Distances derived

Series A (Distances from from numbers in A (Intervals) earth surface) by addition)

Earth Moon 554 4130 554 4130 Venus 5026 2065a 5081 5160 5580 6195

Mercury 5027 2065b 5268 8259 1 0607 8260 Sun 5081 7165 1 2160 4454 1 5689 5425 Mars 4554 4154C 1 3241 8581 2 0243 9579

Jupiter 2027 5065 2 0272 0646 2 2271 4644 Saturn 4037 2065 2 2269 2711 2 6308 6709 Zodiac 2008 0045 2 8316 6754

NOTE: Variants are given in Refutatio 4.10 for the following numbers (cf. Table 5): a5027 2065; b2027 2065; c4054 1108.

satisfactory if both series were adjusted either to A or to the usual Pythagorean order, I shall hereafter adjust the order where necessary to correspond with that in series B in Hippolytus.

Once this adjustment has been made, the two sets of numbers can be com- pared directly. Table 2 shows the numbers as they appear in the manuscript, and, in column B', the numbers that would be expected in series B if series A were correct. There appears to be no correlation between the numbers given and those required by the intervals listed in series A.

Several steps are required to resolve this corruption. The first is to emend the Greek numerals. The principles on which the proposed restoration is based are summarized in the appendix below; Table 5 in the appendix reflects both these emendations and the assumptions explained here and in Section III. The resul- tant series of numbers appears in Table 3, expressed in Arabic numerals but grouped in fours to reflect the Greek orthography.

As emended, these numbers can be expressed in the form

ma + nb

where a = 107 and b = 272,065 (see Table 4). The distances in series B are in each case 4 stades less than they should be, so

that although the intervals between them are precisely those of series A, the whole sequence has moved in by 4 stades. An error of this sort could have occurred if the distances in series B were derived from distances measured from the center of the earth, and a number 4 stades too large was used for the radius of the earth. That the center of the earth was involved appears from Hip- polytus's instructions in 4.9.1, where he gives the circumference of the zodiac (447,310,000 stades) and tells us how to derive first the radius (center of earth to zodiac) and then the distance from the surface of the earth by subtracting the earth radius, which he gives as 40,000 stades. That the figure actually sub- tracted is too large can be explained as scribal error.8

8 40,000 in Greek is 8' myriads, 40,004 is /' myriads and 8' units. To subtract 40,004 the scribe

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Table 3. Emended numbers Table 4. Mathematical relationships

Series B Series A (distances from

(intervals) earth surface)

Earth Moon 54 4130 [54 4130] Venus 5027 2065 5081 6191 Mercury 2027 2065 7108 8256 Sun 5081 6195 1 2190 4451 Mars 4054 4130 1 6244 8581

Jupiter 4027 2065 2 0272 0646 Saturn 2027 2065 2 2299 2711 Zodiac [2081 6195] [2 4380 8910]

Series B Series A (distances from

(intervals) earth surface)

Earth Moon 2b [2b] Venus 5a+ b 5a + 3b-4 Mercury 2a + b 7a + 4b - 4 Sun 5a + 3b 12a + 7b- 4 Mars 4a + 2b 16a + 9b - 4

Jupiter 4a + b 20a + 10b - 4 Saturn 2a + b 22a + llb - 4 Zodiac [2a + 3b] [24a + 14b]

Hippolytus does not work out the result of the calculation he suggests for the zodiac, but the number he gives for its circumference must be incorrect in the manuscript: 447,310,000 stades gives a result for the radius (Or = 3) of 74,551,6662A stades, and hence a distance from the surface of the earth to the zodiac of 74,511,666% stades (earth radius = 40,000 stades). This is quite the wrong order of magnitude, being not much more than the distance to Mercury. The number for the distance from Saturn to the zodiac in series A is also corrupt, but neither Neugebauer's 20,272,065 (see the formula in note 3) nor my 20,816,195 (based on the Pythagorean musical theory examined below) gives a result for the circumference of the zodiac that bears any satisfactory relation to the number in the manuscript.9

III. THE RATIONALE BEHIND THE SEQUENCE

In order to reach a conclusion concerning the distance from the earth to the zodiac, we need to discover the basis of the whole system. As noted above, Hippolytus or his Platonist source criticizes the theory for being irrational and lacking the Platonic harmonies of the universe. The sequence does, however, bear a relation to Pythagorean musical theory. Ten ancient writers attribute to Pythagoras a theory of cosmic musical intervals whose sequence corresponds precisely to the sequence of intervals for the b factors in series A:

tone ?1 tone /2 tone 1/2 tones tone V/ tone /2 tone 1? tones [or ?1 tone] 2b b b 3b 2b b b ?3b [or b]

The last interval is doubtful: Pliny, Martianus Capella, and Hyginus report a tone and a half; Theo Smyrnaeus, Censorinus, and Favonius Eulogius report

substracts 4 from every unit column (a to 0) both in the myriads and in the units, instead of only in the case of the myriads: a simple, though foolish, error.

9 The closest result is attained by supposing that the radius of the earth is not 40,000 but 45,334'/6 (from a circumference of 272,065 7r = 3), and the distance Saturn to zodiac 20,816,195. This gives a circumference of zodiac of approximately 1,463,100,000 while (by principle [3], see appendix below) we could read 147,310,000 as the figure in Hippolytus' text. The number is still too small by a power of 10, which cannot be an error of transmission but would have to be an arithmetical error.

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CATHERINE OSBORNE

a semitone; the remaining four are unclear. We therefore have equal testimony for either of the two alternatives.10

As regards the musical aspect, the scale with a semitone at the top is uncon- vincing in view of the asymmetry of the two tetrachords. With a tone and a half at the top, however, the scale spans a ninth instead of an octave. The late Neo- platonist musical theorist Aristides Quintilianus does record a scale spanning a ninth as among the modes used by "the ancients," as distinct from the tetra- chordic divisions in use in his own day; he includes an account of an ancient Dorian mode with the sequence tone, quarter tone, quarter tone, ditone, tone, quarter tone, quarter tone, ditone. The scale is quoted in an enharmonic form (Aristides specifies that the diesis is the enharmonic diesis) but if the enharmonic sequence quarter tone, quarter tone, ditone were replaced with the chromatic sequence, semitone, semitone, tone and a half, to produce a chromatic version of the ancient Dorian mode, the scale would be precisely that recounted by Pliny. Thus the version with a tone and a half at the top could be an ancient chromatic Dorian scale, or a later reconstruction of such a scale."1

This sequence of harmonies has significant results when applied to the spheres. There is an octave between the second note (the moon) and the last note. Effectively the tonic is not the first note, but the second, and a pros- lambanomenos is (as the term indicates) added below.12 Since on a geocentric theory the earth does not revolve and hence should not produce a sound, the lowest note actually sounding will be that of the moon, and thus the planets will produce a total range of an octave, as Pliny reported, apparently erroneously.13 This result would not occur if the distance from the earth to the zodiac were only twelve semitones. The results are similar for the perfect harmonies of fourth and fifth: several of the texts with a semitone at the top report that the sun makes a harmony of a fifth with the earth and a fourth with the zodiac.14 In fact, if the earth makes no sound, the fifth will not be produced; but with an overall span of a ninth both harmonies will sound, since there is a perfect fifth from the sun to the zodiac and a perfect fourth from the moon to the sun. The sun is theoretically a perfect fifth from the earth, which places it exactly central, a feature that Chalcidius and Theo claim was important for Pythagoras.

The musical evidence thus argues in favor of 3b in the top interval, and hence for 20,816,195 stades as the distance from Saturn to the zodiac.

The b factors thus seem to derive from a Pythagorean harmony theory. The a factors, on the other hand, display no such clear rationale: again they divide,

10 Pliny, Naturalis historia 2.20.84; Martianus Capella, De nuptiis Mercurii et Philologiae 2.169-196; Plutarch, De animae procreatione in Timaeo, pp. 1028-1029; Heraclides Ponticus, Al- legoriae Homeri 12.3; Theo Smyrnaeus, Expositio, pp. 138ff; Censorinus, De die natali 13; Chal- cidius, In Timaeum 72.140; Achilles Tatius, Commentariorum in aratum reliquiae 17, ed. E. Maass (Berlin, 1898), p. 43; Hyginus, Astronomica 4.14, p. 117; Favonius Eulogius, Disputatio de somnio Scipionis 25.

11 Aristides Quintilianus, De musica 1.9, p. llj; see A. Barker, "Aristides Quintilianus and Con- structions in Early Music Theory," Classical Quarterly, 1982, 32:184-197. There is no clear evidence that such a version existed as an ancient scale, but it is not impossible that it should have been invented by analogy with the later chromatic genos. Deliberate archaizing to create a theory that could be attributed to Pythagoras may well be the origin.

12 Cf. Plutarch, De animae procreatione, 1028f, where this term is used of the position of the earth; see Walter Burkert, "Hellenistische Pseudopythagorica," Philologus, 1961, 105:16-43, on p. 33. On Greek musical terminology see Solon Michaelides, The Music of Ancient Greece (London: Faber, 1978).

13 Pliny lists 14 semitones and then states that the total is an octave (12 semitones), Naturalis historia 2.20.84.

14 Theo Smyrnaeus, Expositio; Censorinus, De die natali 13; Plutarch, De animae procreatione 1028f. Favonius Eulogius has the intervals reversed (incorrectly).

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with the sun exactly central and with 12a on either side (which could be musical). If we took a as a semitone again, the sequence would be fourth, tone, fourth (= octave), ditone, ditone, tone, tone; but while the lower three make some sense, the upper four do not. The combination of a and b factors seems to make even less sense: it divides with the sun central at 12a + 7b, and a few other distances are related,15 but otherwise it seems to have no basis, either scientific or fantastic.

IV. THE ATTRIBUTION TO ARCHIMEDES

That Archimedes would offer this set of numbers as a serious contribution to astronomy seems somewhat implausible: there appears to be no reason to as- sociate him with a theory that implies harmony in the spheres. Although it would seem natural to attribute such a theory to the Pythagoreans, there are problems with this suggestion as well.

Pythagorean theory predicated that harmony was based on ratios of mag- nitudes, which produced harmonic intervals. The scale of bs found here does not work on that basis, but depends on assuming that intervals can be measured in terms of equal linear distances, which is not Pythagorean but Aristoxenean. Further, the theory as reported by Hippolytus would not in fact produce any music in the spheres. The harmonies exist in the series of bs alone, but the actual distances, comprising a and b factors, do not form any harmonies. A Pythago- rean could not have devised such a system.

Nevertheless it appears that a musical scale has been behind these numbers at some stage. If it originated as a Pythagorean sequence formulated in terms of tones and semitones, it might subsequently have been incorrectly expressed with equal linear distances for equal intervals. The sequences of bs alone, when interpreted as equal distances with a factor of 272,065, would result in an absurdly small universe. The a factors, which destroy the harmonies, might then be explained as the additions of someone whose concern it was not to "preserve the harmonies" but to bring the distances up to a reasonable order of magnitude: some sort of scientist, but no Pythagorean.

Could this operation be the work of Archimedes? The Sandreckoner provides evidence that it might be. There Archimedes' aim is to calculate an absolute maximum size for the universe, which he does by taking the largest estimate of each distance made by his predecessors and multiplying it by substantial factors to be sure he has not underestimated the size. The method is justified by the aim: not to determine a correct figure for the size of the universe, but consciously to overestimate it. It is possible that some other purpose might have justified him in employing a traditional Pythagorean harmony theory in a similar way and adding multiples of 107; again, as in the Sandreckoner, he need not have intended to make a serious contribution to accurate mathematical astronomy.

Two peculiarities in the terminology of the passage also argue in favor of an association with Archimedes. In the section reporting the theory of Archimedes, the names 11vpoei; and 2z1i3tov occur for Mars and Mercury instead of "ApI;q and 'Ep#r;I, which the Platonist commentator uses in Chapter 11.16 Franz Cu- mont showed that these nondivine names for the planets are an invention of the Hellenistic period and little used later except in antiquarian contexts. This is not conclusive evidence for Archimedes' authorship of the theory reported here,

15 Venus to Mercury = Jupiter to Saturn; Venus to Mercury + Jupiter to Saturn = sun to Mars; earth to Venus = Mercury to sun; earth to Mercury = Venus to sun.

16 See Hippolytus, Refutatio, p. 41, line 18; p. 42, lines 12, 14.

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CATHERINE OSBORNE

since it may only reflect the use of the names in the Pythagorean theory on which the numbers may be based: five of the ten sources for the musical theory described above use these names.'7 That the Pythagorean theory may be ascribed a Hellenistic date means that Archimedes could have known and used it.

The second example reinforces the similarity to the Sandreckoner. The largest numbers, which run to over eight digits, are given in the form of a number of &6erepoi dpiO oi (multiples of 108), followed by the other eight digits. Archimedes devised this system for expressing large numbers and demonstrated it in the

Sandreckoner.l8 It is not found elsewhere, and in Hippolytus's text it appears only in the section reporting Archimedes (4.9), while the author of the Platonic criticisms expresses these numbers as uvpid6e;Q uopidSwcv (4.11). The theory Hip- polytus reports might derive from another work that, like the Sandreckoner, Archimedes wrote to demonstrate his number system. Archimedes himself refers to a work in which he had explained the system to Zeuxippus.19

Together these points strongly suggest that Hippolytus was justified in as- sociating the theory with the name of Archimedes.

APPENDIX: EMENDATION OF THE GREEK NUMERALS

The restoration is primarily based on three principles. 1. A reduction of the distance of the moon by 5 million stades: this means

omitting the initial (p in order to read 544,130 instead of 5,544,130. The number occurs four times in the text as 5,544,130 and is correctly multiplied as such for the numbers in 4.11, but it is quite possible that the numbers were already corrupt when Hippolytus used them. (p is the figure assigned to Apollonius immedi- ately before the first occurrence of the distance given by "Archimedes" for the moon (p. 41, line 13). This might have originated the corruption.

Is this plausible as a suggestion for the distance of the moon? Less than a tenth of the number in the manuscript, it produces a very different set of proportions. There are some observational difficulties, including problems of parallax resulting from a moon so close to the earth. These do not seem to have been a concern in the formulation of the Pythagorean theory, which has a far smaller distance for the moon, 126,000 stades. This is half the number 252,000 frequently given as the circumference of the earth (normally ascribed to Eratosthenes) and it might be that the factor 272,065 in "Archimedes" ' theory was also related to the circumference of the earth.20

The emendation also has implications for the ratio of the sizes of the sun and moon, for that ratio was thought to be equivalent to the ratios of their distance from the earth, since the angular distance was taken to be the same. The ratio of 1:220 resulting from the emended figures is well outside the highest value (1:30) envisaged by Archimedes in Sandreckoner 1.9, but Archimedes adopted this value on the basis of previous estimates, for the purpose of a simple calculation

17 Franz Cumont, "Les noms des planetes et l'astrolatrie chez les Grecs," Antiquite classique, 1935, 4:5-43, esp. pp. 13-43. Cumont also notes the appearance of these Hellenistic names in the reports on the Pythagorean theory given in Plutarch, Censorinus, Theo Smyrnaeus, Achilles Tatius, and Martianus Capella (see n. 10 above).

18 The system provided for units of 108 (second numbers), 1016 (third numbers), 1024 (fourth numbers) and so on, with units of 108X108 as "first numbers" of the "second period," continuing up to the (108)th period, i.e., 10 to the power of 80,000 million million.

19 Archimedes, Arenarius 1.3, 3.1. It was perhaps called dpxai, "first principles," ibid., 1.7. 20 See Tannery, Memoires scientifiques, Vol. VII, pp. 155ff., and the computations in note 9

above.

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Table 5. Greek text

Manuscript Emended Planet Page & line numerals numerals Basis of emendation

Intervals

Moon 41:13-14 v8 ,Sph' va ,6pA' (1)

Venus 41:15 ,eK,/3S E' ,K ,EK ,lE'

43:18 ,EK ,?fE'

Mercury 41:16 ,eK; ,fl:E' ,flK& ,3E'

43:20 ,PK4 ,E'

Sun 41:17 ,ra pe ,ICr'a , -pge' (2) (MS 4 doubtful)

Mars 41:18 ,8pv8 ,pv8' , 8v ,8pX' ,6pv8' perhaps a

43:23 8v ,ap7' repetition of ,8pv8; ,app7' perhaps from a marginal gloss "Apq; = Mars; ? (3); -r for X.

Jupiter 41:19 ,K4 ,Efe' ,8K ,flE' First parts originally reversed due to simi-

Saturn 41:20 ,SAX ,/3e' ,Kf ,fe' larity; e for P; X for K.

Zodiac 41:22 ,f83rl e' ? ,fl3ra ,'pgE' Uncertain: ? (2); see text.

Distances from center of earth

Venus 42:16 ,eLTa ,Ep' ,ErTa ,p9a' (2); final a lost at end of chapter.

Mercury 42:15 ,EO?-CT ,c-rvO' ,4p- ,lo-vg" (2); first part inex- plicable.

Sun 42:13 [1]a ,f3p ,SvvS' [1] ,f3p ,Svva' (2); (3)

Mars 42:12 [2]b ,ya/oa ,qiozra' [1] ,o-T ,rYfra' (2); (3)

Jupiter 42:11 [3]1 oo8/3 Xg-' [2] aoof X.S'

Saturn 42:9 [4]d ,3ot ,flita' [2] ,Pfl-9, ,3ta' (2)

NOTE: The symbol [1] stands for 108, the symbol [2] for 2 x 108. In the manuscript various expressions are used for these Sevrepol dptlfoi: a 8Etep(TpOV ap,I(Vw /Lova I.uav b 8evrepw, ddptJv Ovav fuiova ptav c '

adipOluv utova ft' d 8errepwv api,ptx/v .. /.ov&Sac; 8o. [In the system of notation used above for the Greek numerals, the diacritical mark ', added to the last sign in a

series, has been used to indicate numerals (as opposed to the overbar).-EDs.]

founded on the premise that the moon was (moderately) smaller than the earth, and in the Sandreckoner he characteristically works with estimates by past think- ers and does not attempt to correct estimates. Archimedes might well have experimented with a smaller and nearer moon here. The ratio 1:220 would of course be nearer to the true value than any previous estimate. The reduction of the size and distance of the moon does not affect the size of the orbit of the sun, which remains well within the limit envisaged in the Sandreckoner.

2. The restoration of the letters digamma and koppa (and sampi) throughout where they have been replaced by more familiar letters, mainly zeta and xi (and

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CATHERINE OSBORNE

tau) respectively. Digamma occurs three times in the manuscript when it should occur seven or eight times; koppa never occurs but should occur seven times (sampi never occurs but is restored by the editors four times in 4. 11). The restoration of koppa in five places is of fundamental importance.

3. The restoration of alpha for delta and vice versa on two or three occasions. This corruption was recognized by Neugebauer, History, pp. 648-649.

In addition, unsystematic errors are identified in the table, with the supposed corrupt numeral preceding the proposed emendation.

HENRY SCHUMAN PRIZE

The competition for the annual award of $500, established in 1955 by Ida and Henry Schuman of New York City for an original prize essay in the history of science and its cultural influences, is open to graduate and undergraduate students in any college, university, or institute of technology. Papers submitted for the prize competition should be in English, approximately 8,000 words in length (exclusive of footnotes), and thoroughly documented. It is hoped that the prize-winning essay will merit publication in Isis.

It was the wish of the donors that "history of science and its cultural influences" be interpreted very broadly. The papers may deal with the ideas and accomplishments of scientists in the past; they may trace the evolution of particular scientific concepts; or study the historical influences of one branch of science upon another. The phrase "cultural influences" is taken to include studies of the social and historical conditions that have influenced the growth of science, or the effects of scientific development upon society in the realms of philosophy, religion, social thought, economic progress, art and literature. Essays dealing with medi- cal subjects are not acceptable, although papers dealing with the relations between medicine and the natural sciences are welcome.

Essays may be submitted to the Chairman, Schuman Prize Com- mittee, through the Isis Editorial Office. It is requested that three copies of each essay be sent and that the names and institutions of the contributors be placed on a separate title page so that they may be removed before being read by members of the committee. The announcement of the prize-winning essay is normally made at the annual meeting of the History of Science Society, in December.

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