Black Athena,Afro-Centrism,and the History of Science

Bernal Martin , Black Athena: The Afroasiatic roots of Classical civilization, i: The fabrication of ancient Greece 1785–1985 (New Brunswick, N.J., 1987); ii: The archaeological and documentary evidence (New Brunswick, N.J., 1991). For a very full listing of writings about these two volumes (hereafter BA, i and ii), see Levine Molly Myerowitz , American historical review, xcvii (1992), 440–64.

Bernal M. , “Animadversions on the origins of Western science”, in the special section, “The cultures of ancient science”, Isis, lxxxiii (1992), 596–607. I shall be responding to specific claims made by Bernal in this essay as the occasion arises.

For an account of some of these new approaches to the history of ancient science, see the contributions of Lloyd G. E. R. and von Staden Heinrich to the special section on “The cultures of ancient science” (ref. 2), 564–95.

Cohen Morris R. and Drabkin I. E. (eds), A source book in Greek science (Cambridge, Mass., 1958); Clagett Marshall (ed.), Ancient Egyptian science: A source book, i: Knowledge and order (2 tomes, Philadelphia, 1989). The next two promised volumes will deal, respectively, with astronomy and mathematics and with medicine and biology.

Neugebauer O. , A history of ancient mathematical astronomy (3 parts, New York, 1975), Part 2, p. 609.

Bilfinger Gustav , Der bürgerlicher Tag (Stuttgart, 1888), 10–16.

Neugebauer O. , “A Babylonian lunar ephemeris from Roman Egypt”, in Leichty E. (eds), A scientific humanist: Studies in memory of Abraham Sachs (Philadelphia, 1988), 301–4, p. 301. Neugebauer pursues this particular parameter into the Renaissance in “From Assyriology to Renaissance art”, Proceedings of the American Philosophical Society, cxxxiii (1989), 391–403.

Sigerist H. E. , Primitive and archaic medicine (New York, 1967; original edn, 1951), 357–8.

Sigerist, by the way, believes that “there can be no doubt whatsoever that the Greeks learned a great deal [in medicine] from Egypt” (ibid., 357). It is worth mentioning that a leading historian of Egyptian medicine accepts the Egyptian-Greek connection in this case ( Ghalioungui P. , “The relation of Pharaonic to Greek and later medicine”, Bulletin of the Cleveland Medical Library, xv/3 (1968), 96–107, p. 99).

There is an authoritative annotated English translation by Toomer G. J. Ptolemy's Almagest (New York, 1984). See also Pedersen Olaf , A survey of the Almagest (Odense, 1974). On all aspects of ancient mathematical astronomy the definitive work is Neugebauer , op. cit. (ref. 5).

Parker Richard A. , “Egyptian astronomy, astrology, and calendrical reckoning”, Dictionary of scientific biography , ed. by Gillispie Charles C. (New York, 1970–80), xv (Supplement I), 706–27, p. 706; Neugebauer , op. cit. (ref. 5), Part Two, p. 560.

Locher Kurt , “A further coffin-lid with a diagonal star-clock from the Egyptian Middle Kingdom”, Journal for the history of astronomy, xiv (1983), 141–4, p. 141.

Note the amusing opening sentences of Neugebauer, op. cit. (ref. 5), Part One, p. 1: “Many things are omitted here. The reader who wants to hear about Archimedes taking a bath or about the silver nose of Tycho Brahe can find innumerable books which dwell on these important biographical matters. Nor do I enumerate the pros and cons concerning the place or movement of the earth and the substance of the spheres.” Earlier, in his Preface, Neugebauer had written: “I have tried to come as close as possible to the astronomical problems themselves without hiding my ignorantia behind the smoke-screen of sociological, biographical and bibliographical irrelevancies” (p. vii).

See Hetherington Norriss S. , Ancient astronomy and civilization (Tucson, Arizona, 1987), 22.

Neugebauer O. , “The alleged Babylonian discovery of the precession of the equinoxes”, Journal of the American Oriental Society, lxx (1950), 1–8, p. 1; reprinted in Neugebauer O. , Astronomy and history: Selected essays (New York, 1983), 247–54.

Neugebauer O. , The exact sciences in Antiquity, 2nd edn (Providence, 1957), 71.

See Neugebauer's note on “The study of wretched subjects”, Isis, xlii (1951), 111; reprinted in Neugebauer, Astronomy and history (ref. 15), 3. “[T]he very foundations of our studies [are] the recovery and study of the texts as they are, regardless of our own tastes and prejudices.” (The term “wretched”, by the way, is not Neugebauer's).

Neugebauer O. , “On the orientation of pyramids”, Centaurus, xxiv (1980), 1–3, p. 1; reprinted in Neugebauer, Astronomy and history (ref. 15), 211–13. Bernal finds Neugebauer's proposal plausible but criticizes him for using the term “primitive” in characterizing the remarkably accurate alignment of the Great Pyramid (“Animadversions” (ref. 2), 601). It is, however, not the alignment that Neugebauer characterizes as primitive but a certain visual experience. Perhaps Bernal should concentrate more on real issues and worry less about words.

Diop Cheikh Anta , Civilization or barbarism: An authentic anthropology (Brooklyn, N.Y., 1991; French edn, 1981), 282.

Neugebauer , op. cit. (ref. 5), Part Two, p. 561.

The distinguished historian of early astronomy, Willy Hartner, is willing to speculate that Egyptian astronomers probably observed — without, for some reason, recording — lunar eclipses, constellations of stars, and planetary motions as early as the second millenium B.C. Hartner's really quite moderate claims are stimulating but ultimately unconvincing; see his comments on Giorgio de Santillana's “On forgotten sources in the history of science”, in Scientific change, ed. by Crombie A. C. (New York, 1963), 868–76.

For an excellent brief introduction to Egyptian astronomy, see Parker's Richard A. essay (ref. 11). For a fuller account, see van der Waerden Bartel L. , Science awakening II: The birth of astronomy (Leyden and New York, 1974), chap. 1. For a compilation of Egyptian astronomical texts, see Neugebauer O. and Parker Richard A. , Egyptian astronomical texts (3 vols, Providence, 1960, 1964, 1969). This latter corpus of texts excludes “cosmogonic mythology, calendaric problems and time reckoning as well as astrology” ( Neugebauer , op. cit. (ref. 5), Part Two, pp. 566–7).

For a useful brief introduction to Babylonian astronomy, with full references, see van der Waerden B. L. , “Mathematics and astronomy in Mesopotamia”, in Gillispie , op. cit. (ref. 11), xv (Supplement I), 667–80.

Neugebauer , op. cit. (ref. 16), 91; and op. cit. (ref. 5), Part Two, p. 559.

Neugebauer , op. cit. (ref. 5), Part Two, p. 559. Elsewhere, Neugebauer points out that mathematical astronomy had two more independent origins: In the Chinese and Mayan civilizations.

Diop , op. cit. (ref. 19), 283.

Finch Charles S. , “Interview with Cheika Anta Diop”, Presence Africaine, June 1989, 366.

See Michael Eric Dyson's interview with Bernal , “On Black Athena”, Z, Jan. 1992, 56: “I have … sympathy for Afrocentricity, though I'm not an Afrocentrist myself.”

James George G. M. , Stolen legacy: The Greeks were not the authors of Greek philosophy, but the people of North Africa, commonly called the Egyptians (San Francisco, 1988; orig. edn, 1954). James's aim, he says, is “to establish better race relations in the world” (p. 7).

Pappademos John , “The Newtonian synthesis in physical science and its roots in the Nile Valley”, Journal of African civilizations, vi/2 (1984), 84–101. I discovered this publication in Bernal's bibliography in BA, i; I am uncertain as to whether Bernal accepts all its claims. According to a biographical note in Van Certima Ivan (ed.), Blacks in science: Ancient and modern (New Brunswick, 1983), 301, Pappademos is a nuclear physicist who teaches at the University of Illinois, Chicago Circle, and whose “research interests have shifted to the social aspects of physics, including its philosophy and history”.

Or do I misread Pappademos? I will quote: “Since the rise of slavery with its offspring the doctrine and practice of racism, the Black civilization of the Nile Valley has had its detractors. As recently as 1975, Otto Neugebauer, the well-known historian of ancient science, had this to say: ‘Egypt provides us with the exceptional case of a highly sophisticated civilization which flourished for many centuries without making a single contribution to the development of the exact sciences’” (Pappademos, “The Newtonian synthesis” (ref. 30), 95). Pappademos seems to have overlooked Neugebauer's phrase “highly sophisticated” — hardly an expression of detraction — and perhaps never even asked what Neugebauer meant by “the exact sciences”. Much worse, though, Pappademos overlooks Neugebauer's very next sentence: “In fact, however, this is not the exception but the rule” (Neugebauer, op. cit. (ref. 5), Part Two, p. 559). Even more explicitly, Neugebauer holds that, apart from ancient Babylonia and ancient Greece, “none of the other civilizations of antiquity, which have otherwise contributed so much to the material and artistic culture of the. world, have ever reached an independent level of scientific thought” (ibid., Part One, p. 6).

See Neugebauer , ibid., Part One, p. 3.

Pappademos , “The Newtonian synthesis” (ref. 30), 96.

Vercoutter Jean , chap. 1, “Egypt: Mathematics and astronomy”, in Taton René (ed.), Ancient and medieval science: From the beginnings to 1450 (New York, 1963), 35–36. Vercoutter is here following Neugebauer O. , “The origin of the Egyptian calendar”, Journal of Near Eastern studies, i (1942), 397–403; reprinted in Neugebauer , Astronomy and history (ref. 15), 196–203.

Neugebauer , op. cit. (ref. 16), 81. Even Diop quotes approvingly this remark by Neugebauer (whom he elsewhere abuses for his alleged anti-Egyptian prejudice). Diop's own prejudices are perhaps reflected in his erroneous remark that the Egyptian calendar is “the very one which, barely changed, regulates our life today” (Diop, op. cit. (ref. 19), 279). Diop refers in a note to our present (Gregorian) calendar, but omits to mention its methods for dealing with leap years, which make it vastly different from the Egyptian calendar — but not, of course, well adapted for astronomical purposes (ibid., 401–2, n. 63).

For a succinct account of Egyptian calendars, see Parker , “Egyptian astronomy” (ref. 11), 706–10. Bernal asserts that “the Greeks adopted an Egyptian rather than a Mesopotamian calendar” and that “this adoption is indicative of what seems to have been a wider Greek tendency to draw from nearby Egypt rather than more distant Mesopotamia” (“Animadversions” (ref. 2), 606). This is wrong on several counts. In the first place, there was never any single civil calendar adopted by the various Greek city-states (leading to the calendaric chaos which Classical scholars are still trying to dispel). There is evidence, however, of a “universal” Greek astronomical calendar based on the so-called Metonic cycle, i.e., on the assumption that 19 years contain 7 intercalary months (and therefore a total of 19 × 12 + 7 = 235 months). Such a cycle was known to the Babylonians from the early fifth century B.C. and there is some reason to believe that this was Meton's source. In any case, a calendar based on the Metonic cycle and improved by Callippus was used by Greek astronomers as late as Hipparchus (128 b.c.). Later Greek astronomers, including Ptolemy, preferred the Egyptian calendar for reasons already explained. For details, see Toomer G. J. , “Meton”, in Gillispie , op. cit. (ref. 11), ix, 337–9.

Vercoutter , op. cit. (ref. 34), 39.

I suspect Pappademos picked up this allusion to stellar azimuths from one of his other sources; see Sarton George , A history of science: Ancient science through the Golden Age of Greece (Cambridge, 1952), 30: “the combination of a plumb line with a forked rod … enabled [the Egyptians] to determine the azimuth of a start [sic].” But no records of such azimuth measurements have been found. On this question of angle measurement, Neugebauer remarks: “The coverage of the sky with picturesque configurations of stars” — which the Egyptians certainly accomplished — “is not the equivalent of the use of mathematically defined spherical coordinates” ( Neugebauer , op. cit. (ref. 5), Part Two, p. 577).

Vercoutter , op. cit. (ref. 34), 42.

Krupp E. C. , “Astronomers, pyramids, and priests”, in Krupp E. C. (ed.), In search of ancient astronomies (New York, 1977), 186–218, p. 203.

ibid., 233.

Heath Thomas L. , Greek astronomy (New York, 1932), p. xv.

Thus, Vercoutter writes that “Representations of the sky on certain tombs have enabled scholars to identify some of the constellations known to Egyptians — for example, the Great Bear …, Bootes …, Cygnus …, Orion …, Cassiopeia …, and Draco, the Pleiades, Scorpio and Aries, each represented by characteristic figures” ( Vercoutter , op. cit. (ref. 34), 37). Vercoutter gives no specific reference, but his opinion here is an exception to his usual reliance on Neugebauer for the details of Egyptian astronomy.

Neugebauer , op. cit. (ref. 16), 89.

Neugebauer , op. cit. (ref. 5), Part Two, p. 566.

“Ptolemy refers nowhere to Egyptians for astronomical observations or theories …, only to calendaric concepts and to some astrological doctrines …” (Neugebauer, op. cit. (ref. 5), Part Two, p. 562, n. 14).

Van der Waerden , op. cit. (ref. 22), 38.

Neugebauer O. , Egyptian planetary texts, Transactions of the American Philosophical Society, n.s., xxxii (1942).

ibid., 239.

Van der Waerden , op. cit. (ref. 22), 39.

Parker Richard A. , “Egyptian astronomy” (ref. 11), 719.

For translation and analysis of these two texts, see Neugebauer , op. cit. (ref. 48).

Abetti Giorgio , The history of astronomy (New York, 1952), 21.

Neugebauer , op. cit. (ref. 5), Part Two, p. 568. Earlier , Neugebauer had recognized “one doubtful [Egyptian] reference to a partial eclipse of 610 B.C.” (Neugebauer, op. cit. (ref. 16), 95).

Aristotle , De caelo, ii.12, 292a8, cited by Thomas Heath, Aristarchus of Samos: The ancient Copernicus (Oxford, 1913), 220.

Heath , op. cit. (ref. 55), 48; idem, op. cit. (ref. 42), p. xxvi.

Heath , op. cit. (ref. 55), 130–1.

Pappademos , “The Newtonian synthesis” (ref. 30), 97.

Sarton , op. cit. (ref. 38), 200.

Van der Waerden , op. cit. (ref. 22), 83. Cf. Neugebauer , op. cit. (ref. 5), Part Two, p. 593: “The Almagest and all ancient and medieval mathematical astronomy uses orthogonal ecliptic coordinates for its coordinate system … this system is of Babylonian origin.” It is uncertain, however, whether these ecliptic coordinates were ever used much beyond the zone of the ecliptic and the ‘Normal Stars’ (a set of 31 reference stars in the vicinity of the ecliptic).

Lockyer Norman , The dawn of astronomy (London, 1894; reprinted Cambridge, Mass., 1964, with a laudatory introduction by Giorgio de Santillana).

Krupp , op. cit. (ref. 40), 222–3.

See the sensitive éloge by Sivin N. in Isis, lxvii (1976), 439–43.

de Santillana Giorgio and von Dechend Hertha , Hamlet's mill: An essay on myth and the frame of time (Boston, 1969), 59.

I am quoting from Lynn White Jr's not unsympathetic review in Isis, lxi (1970), 541. White attributes to von Dechend the book's more far-fetched notions, which he characterizes as “arrogant oversimplification”.

Pappademos , “The Newtonian synthesis” (ref. 30), 98.

Neugebauer , op. cit. (ref. 5), Part One, p. 104, n. 4.

Heath , op. cit. (ref. 55), 313.

Neugebauer , op. cit. (ref. 5), Part Two, p. 658.

Van der Waerden , op. cit. (ref. 22), 37.

The version of the passage from Archimedes used by Pappademos is in Heath, op. cit. (ref. 55), 302.

Pappademos , “The Newtonian synthesis” (ref. 30), 98.

It is arguable that heliocentricity is of no great significance in the history of mathematical astronomy until Copernicus. As Neugebauer puts it: “Without the accumulation of a vast store of empirical data and without a serious methodology for their analysis the idea of heliocentricity was only a useless play on words” (op. cit. (ref. 5), Part Two, p. 698).

Pappademos , “The Newtonian synthesis” (ref. 30), 98.

Ibid. Bernal adds that Ptolemy was called “the Upper Egyptian” in early Arabic writings; see “Animadversions” (ref. 2), 606.

Pappademos , “The Newtonian synthesis” (ref. 30), 93.

One indication of this is the fact that the last of the Ptolemaic rulers (Cleopatra) was the first to speak Egyptian. As a recent study of the two cultures puts it: “Ptolemaic Egypt … remained throughout its history a land of two cultures which did coexist but, for the most part, did not coalesce or blend. … We … discern the manifestations of the two discrete cultures in every aspect of their coexistence. … It would be difficult … to exaggerate the significance of the fact that, except for some local designations of places, measures, and so on, no native Egyptian word made its way into Greek usage in the thousand years that Greek endured as the language of Ptolemaic, Roman, and Byzantine Egypt” ( Lewis Naphtali , Greeks in Ptolemaic Egypt: Case studies in the social history of the Hellenistic world (Oxford, 1986), 154–5). See also, Goudriaan Koen , Ethnicity in Ptolemaic Egypt (Amsterdam, 1988).

The scholia occupy some fourteen pages in the first full publication, edited by Casini Paolo : “Newton's classical scholia”, History of science, xxii (1984), 1–58.

ibid., 15.

See Dobbs Betty Jo Teeter , The Janus faces of genius: The role of alchemy in Newton's thought (Cambridge, 1991), 193–212.

Tompkins Peter , Secrets of the Great Pyramid (New York, 1971), 30–33. Tompkins seems to think that once Newton had the correct value for the Earth's radius his task was effectively completed — the law of gravitation had been discovered — a travesty of Newton's actual route to the law. But, then, Tompkins also believes that “[W]hoever built the Great Pyramid … knew the precise circumference of the planet, and the length of the year to several decimals … [and] may well have known the mean length of the earth's orbit round the sun, the specific density of the planet, the 26,000 year cycle of the equinoxes, the acceleration of gravity and the speed of light” (pp. xiv–xv)! For a recent account of Newton's “moon test” of the theory of gravitation, see Hall A. Rupert , Isaac Newton: Adventurer in thought (Oxford, 1992), 59–64.

For a transcription of the three accounts with detailed analysis, see Herivel John , The background to Newton's Principia (Oxford, 1965), 65ff.

Westfall Richard S. , “Newton's theological manuscripts”, in Bechler Zev (ed.), Contemporary Newtonian research (Dordrecht, 1982), 129–43, p. 131.

See Herivel , op. cit. (ref. 82), 183–91.

See Westfall Richard S. , Never at rest: A biography of Isaac Newton (Cambridge, 1980), 344–8.

Ghalioungui Paul , The House of Life: Per Ankh. Magic and medical science in ancient Egypt , 2nd edn (Amsterdam, 1973), 51; Bianchi Robert Steven , “Pyramidiots”, Archaeology, xl/6 (Nov./Dec. 1991), 84.

Bernal too thinks that the existence of advanced (but as yet unrevealed) Egyptian mathematics follows from the fact that Greek and Roman building techniques required such mathematics and the fact that Egyptian architecture was not technically inferior to Greek and Roman architecture; see “Animadversions” (ref. 2), 605. Bernal should ask workers in the building trades (or even architects) how much advanced mathematics they know! (By advanced mathematics I mean anything in Archimedes or the later books of Euclid's Elements.) On Egyptian building techniques, see Clarke Somers and Engelbach R. , Ancient Egyptian construction and architecture (New York, 1990; original edn, 1930); on Greek building techniques, see Coulton J. J. , Ancient Greek architects at work (Ithaca, 1977). Coulton is properly sceptical about the idea that the Parthenon architects deliberately used conic section curves (parabolas and hyperbolas) in their design; among the reasons for his scepticism is the fact that the first geometrical analysis of such curves (by Menaichmos) was not formulated until a century later (pp. 107–8, and 175, n. 22).

van der Waerden B. L. , Science awakening (New York, 1961), 35.

Gillings Richard J. , Mathematics in the time of the Pharaohs (New York, 1982), 234. Gillings's book is the fullest and most up-to-date account in English; he has also written a briefer but still fairly detailed account: “The mathematics of ancient Egypt”, in Gillispie (ed.), op. cit. (ref. 11), xv (Supplement I), 681–705.

In his “The mathematics of ancient Egypt” (ref. 89), 704–5, Gillings provides a list of all of these texts, with approximate dates and other useful details. One recent publication on Egyptian mathematics should be singled out: A new English version of the RMP including beautiful colour facsimiles of the original papyrus, Robins Gay and Shute Charles (eds), The Rhind Mathematical Papyrus: An ancient Egyptian text (New York, 1987).

Neugebauer , op. cit. (ref. 16), 5.

The manipulation of unit fractions gives rise to interesting and difficult problems in number theory, which have attracted the attention of modern mathematicians; see Gardner Martin , Fractal music, hypercards, and more…: Mathematical recreations from Scientific American magazine (New York, 1992), chap. 7, “Egyptian fractions”. There is, of course, no reason to believe the Egyptians were even capable of formulating such problems. Mott Greene has recently suggested that the unit fraction notion originated in the unit weights of Egyptian pan balances; see his Natural knowledge in Preclassical Antiquity (Baltimore, 1992), chap. 2.

Gillings , Mathematics in the time of the Pharaohs (ref. 89), 3.

Ibid ., 166–70. Van der Waerden does not mention summation of progressions in his exposition of Egyptian mathematics (op. cit. (ref. 88), chap. 1).

Gillings , Mathematics in the time of the Pharaohs (ref. 89), 216.

Ibid ., 214, 217. Diop is, then, clearly mistaken when he says (referring only to the Berlin Papyrus) that “the Egyptians knew how to rigorously extract the square root, even of the most complicated whole or fractional numbers” (Diop, op. cit. (ref. 19), 258). It is interesting to note that calculations of square roots by Greek mathematicians are also only imperfectly understood by modern scholars, and in this case also plausible guesses have been made as to just how certain approximations, say, to the square root of 3, were calculated. Greek computational techniques must have been highly sophisticated; Archimedes, for instance, assumes (without explanation) that the square root of 3 lies between 265/153 and 1351/780, i.e., between 1.7320261 and 1.7320512. (The correct value, to seven decimal places, is 1.7320508.) See Heath Thomas L. , A manual of Greek mathematics (New York, 1963; orig. edn, 1931), 309–10.

See van der Waerden , Science awakening (ref. 88), 77, or “Mathematics and astronomy in Mesopotamia” (ref. 23), 670.

Neugebauer , op. cit. (ref. 16), 34.

Gillings , Mathematics in the time of the Pharaohs (ref. 89), 140; idem, “The mathematics of ancient Egypt” (ref. 89), 696.

Neugebauer , op. cit. (ref. 16), 47. The value 3 for π is also found, in effect, in the Old Testament (1 Kings, vii.23 and 2 Chronicles, iv.2).

Gillings reproduces what seem to be re-drawn sketches of the three diagrams (Mathematics in the time of the Pharaohs (ref. 89), 139–40), but one can examine photographs of the originals in Robins and Shute, op. cit. (ref. 90), Plates 14–16.

Robins and Shute (op. cit. (ref. 90), 44–45) unfortunately provide no interpretation of Problem 48, and their own hypothesis as to how the circle area formula was derived appeals to the Pythagorean Theorem, which, as we shall see, the Egyptians almost certainly did not know.

For some speculation, see Gillings , op. cit. (ref. 89), 189–93.

Archimedes , On the sphere and cylinder, preface to Bk I; see The works of Archimedes , transl. by Heath T. L. (New York, n.d.; original edn, 1912), 2. Archimedes repeats this claim, and adds that Democritus was the first to state the result, in the preface to his Method; see ibid., Supplement, The Method of Archimedes, 13.

Bernal , “Animadversions” (ref. 2), 603. Bernal's “many years” for the length of Eudoxus's stay in Egypt is perhaps exaggerated; Huxley G. L. writes “more than a year” in “Eudoxus of Cnidus”, Gillispie (ed.), op. cit. (ref. 11), iv, 465–7, p. 466. Bernal also follows Giorgio de Santillana in assuming that Eudoxus could only have visited Egypt in order to improve his mathematical and astronomical knowledge and that Eudoxus's translation of texts from the Egyptian Book of the dead suggests that “Egyptian religious and mystical writings and drawings may well contain esoteric mathematical and astronomical wisdom” (“Animadversions” (ref. 2), 603). This ignores the fact that Eudoxus had many scholarly interests beyond mathematics and astronomy — and hence other possible motives for visiting Egypt — as shown by the remaining fragments of his geographical treatise, in which “[B]eginning with remote Asia, Eudoxus dealt systematically with each part of the known world in turn, adding political, historical, and ethnographic detail and making use of Greek mythology” (Huxley, “Eudoxus of Cnidus”, 467).

On Archytas, only fragments of whose writings have survived, see the enthusiastic but critical assessment of van der Waerden , op. cit. (ref. 88), 149–59.

Bernal , “Animadversions” (ref. 2), 599.

Burkert Walter , Lore and science in ancient Pythagoreanism (Cambridge, 1972; German edn, 1962), 10. Bernal ignores Burkert but accuses of “Aryan ingenuity” (BA, i, 105) a Belgian scholar (writing in 1922), who is sceptical about the Pythagorean tradition.

ibid., 408ff. Many of the ancient references to Pythagoras (including those in Herodotus and Isocrates) are conveniently assembled in Barnes Jonathan , Early Greek philosophy (London, 1987), chap. 5.

ibid., 426.

Burkert Walter , The Orientalizing Revolution: Near Eastern influence on Greek culture in the early Archaic Age (Cambridge, Mass., 1992; German edn, 1984). In his introduction Burkert explains the role of anti-Semitism in delaying the acceptance by certain scholars of massive Near Eastern influence on the Greeks, and in this connection he refers to Bernal's work as “provocative” (p. 154, n. 5).

Gillings , Mathematics in the time of the Pharaohs (ref. 89), 194.

ibid., 197–8. Gillings presumes that Dinostratus — about whom we actually know very little — was the first Greek mathematician to find the circumference of a circle.

ibid., 199–200.

ibid., 200.

Neugebauer , op. cit. (ref. 16), 80. Beatrice Lumpkin objects to what she calls Neugebauer's “highly prejudiced statement” as “an unhistoric judgment … much like faulting the inventor of the crystal radio for not inventing solid state television first”, and she also asserts that “[T]hese same Egyptian fractions were used by scientists for thousands of years after their invention, right on up to the modern period” (“Mathematics and engineering in the Nile Valley”, Journal of African civilizations , vi (1984), 102–119, pp. 103, 106). But, in the first place, since Neugebauer has in mind a comparison with contemporary Babylonian numerical procedures, there is nothing “unhistoric” about his judgement; and, secondly, it is an historical fact that, though Egyptian arithmetic “probably influenced the Hellenistic and Roman administrative offices and thus spread further into other regions of the Roman empire” ( Neugebauer , op. cit. (ref. 16), 72), complicated computations (such as those in mathematical astronomy) have never been carried out, to my knowledge, using the Egyptian technique of unit fractions. Indeed, according to Karl Menninger, even in practical affairs, the Romans and medievals only “[N]ow and then … expressed a fraction in the Egyptian fashion, as the sum of certain standard fractions” (Number words and number symbols: A cultural history of numbers (Cambridge, Mass., 1969; German edn, 1958), 158.

Neugebauer , op. cit. (ref. 16), 72.

According to Bernal, Neugebauer had an “early passion for ancient Egypt” (“Animadversions” (ref. 2), 600), which he presumably lost by the time he rejected the hemisphere interpretation of MMP 10 in favour of “a much more primitive interpretation which is preferable” ( Neugebauer , op. cit. (ref. 16), 78). Unfortunately for Bernal's (undocumented) interpretation of Neugebauer's changing attitudes toward ancient Egypt, the comment Bernal cites, with its (for Bernal) objectionable term “primitive”, occurs in the same book in which Neugebauer says such glowing things about ancient Egypt (see above, Section 2).

Neugebauer , op. cit. (ref. 16), 48.

Diop , op. cit. (ref. 19), 237. The translation must be faulty here but Diop's meaning seems clear enough: In the quoted text “Platonic bodies” is followed by the phrase “of work”, which makes no sense.

Bernal , “Animadversions” (ref. 2), 598–9.

Neugebauer , op. cit. (ref. 16), 35, 48. The Babylonian way of putting it would have been (as Neugebauer suggests): The irrationality of √2 means that p² = 2q² has no integral solutions in p and q.

ibid., 35, and Plate 6a for a picture of the tablet.

ibid., 33–34. As Neugebauer explains, irregular numbers are those containing prime numbers not contained in 60 (i.e., prime numbers different from 2, 3, and 5).

Gillings , Mathematics in the time of the Pharaohs (ref. 89), 208.

On the same evidence, Diop concludes that “the Egyptians knew the theorem attributed to Pythagoras perfectly well” ( Diop , op. cit. (ref. 19), 260).

Lumpkin Beatrice , “The Egyptians and Pythagorean triples”, Historia mathematica, vii (1980), 186–7, p. 186.

Lumpkin , “Mathematics and engineering in the Nile Valley” (ref. 116), 109.

Cantor Moritz , Vorlesungen über Geschichte der Mathematik, i (Leipzig, 1880), 56.

See, e.g., Appendix 5, “The Pythagorean Theorem in ancient Egypt”, in Gillings , Mathematics in the time of the Pharaohs (ref. 89).

See Arnold Dieter , Building in Egypt: Pharaonic stone masonry (New York, 1991), 14–15.

Ibid., 15.

Neugebauer , op. cit. (ref. 16), 40.

Gillings , “The mathematics of Ancient Egypt” (ref. 89), 690.

For an account of the method and its history, see van der Waerden , op. cit. (ref. 88), 184–7, 216–25.

Heath , op. cit. (ref. 104), 93.

ibid., pp. lxxx–lxxxiv.

Diop , op. cit. (ref. 19), 243.

Ibid., 231.

Heath , op. cit. (ref. 104), 43.

Diop , op. cit. (ref. 19), 237, 242. Later, when Diop comes to discuss MMP 10, he insists that the problem is about the surface of a hemisphere (p. 251).

ibid., 242.

ibid., 246.

Exactly the same discussion of Lauer appears in Bernal , “Animadversions” (ref. 2), 600. The golden section and Pythagorean triangles are usually defined geometrically but it is the numerical expression of the geometrical relationships that has to be compared with the results of length measurements. The golden section refers to a ratio of two line segments, say, in a rectangle, of , which is approximately 0.618. For a discussion of the golden section in the context of ancient Greek mathematics, see Cohen and Drabkin (op. cit. (ref. 4), 50–51). Pythagorean triangles, as we have already seen, are right triangles whose sides can be expressed numerically by triples of integers, such that the sum of the squares of two of them is equal to the square of the third (e.g., {3,4,5}, {5,12,13}, {8,15,17}). For a discussion of Pythagorean triangles in the context of ancient Greek mathematics, see ibid., 21–23.

Robins Gay and Shute Charles C. D. , “Mathematical bases of ancient Egyptian architecture and graphic art”, Historia mathematica, xii (1985), 107–22, p. 112. The authors, as already noted, are the editors of the recent edition of the Rhind Mathematical Papyrus (ref. 90); Robins is an Egyptologist in the Department of Art History at Emory University and Shute once held a chair of medical biology at Cambridge University.

Robins and Shute , “Mathematical bases” (ref. 145), 108. Robins and Shute have invented what seems to be a reliable photographic procedure which obviates the need for directly measuring the pyramids themselves (pp. 108–9).

Robins and Shute (ibid., 108) make much of the fact that the Egyptian measure of inclination of a pyramid face, the ‘seked’, was defined in terms of the horizontal displacement in ‘palms’ per vertical drop in ‘royal cubits’ (where a royal cubit was equal to seven palms). They remind us that the seked, unlike our own measure of inclination, is a function of the cotangent of the angle of inclination. For the discussion here the distinction is irrelevant.

ibid., 112.

The history, Herodotus, transl. by Grene David (Chicago, 1987), 186.

I am following the figures in Markowsky George , “Misconceptions about the golden ratio”, The college mathematics journal , xxiii/1 (Jan. 1992), 2–19, p. 6. Markowsky's essay contains a valuable bibliography as well as excellent mathematical and critical discussions of various applications of the golden section in art and literature, from the Parthenon to Virgil's Aeneid to the Building U.N. in New York.

Gillings rejects the emendation, and following him, so does Markowsky , who cites the original Greek as well as a word-by-word literal translation (ibid., 17).

For an elaboration of this idea to truly bizarre lengths, see the appendix, “Notes on the relation of ancient measures to the Great Pyramid”, by Stecchini Livio Catullo , in Tompkins , op. cit. (ref. 81), 287–382. Bernal finds “some plausibility” in Stecchini's work (BA, i, 275).

To illustrate the uncertainty of pyramidal measurements I cite those of a contemporary pyramidologist, who recently published “the result of a 15-year study involving, among other research, three trips to Egypt, where I resided for nearly one year, investigating these monuments firsthand on a daily basis” ( Lepre J. P. , The Egyptian pyramids: A comprehensive illustrated reference (Jefferson, N.C., 1990), p. vii). Lepre gives 762.24 feet for the base and 485.5 feet for the height of the Great Pyramid; these figures differ from our previous ones by almost 1%. Lepre's compendium of basic information on some one hundred pyramids, as well as his bibliography of some two hundred items, may be useful as a guide for anyone wishing to study pyramidology more deeply. For a recent brief and reliable account of the pyramids by the Director General of the Giza Pyramids and Saqqara, see Hawass Zahi A. , The pyramids of ancient Egypt (Pittsburgh, 1990). It is worth noting that even such a sober student of the pyramids as Hawass feels called upon to begin his essay with the “spiritual aspect” of the pyramids and the claim that they “have made a mockery of death; they cannot be killed. Their physical presence defies the limitations of time.” In the very next paragraph, however, we learn that some of the pyramids “are now barely distinguishable from the sand and rubble that surround them” (p. 1)!.

Gillings , op. cit. (ref. 89), 238.

Gardner Martin , Fads and fallacies in the name of science , chap. 15, “The Great Pyramid” (New York, 1957), 173–85, p. 179.

Panofsky Erwin , “The history of the theory of human proportions as a reflection of the history of styles”, in Meaning in the visual arts (New York, 1955), 55–107, p. 91.

See Elkins James , “The case against surface geometry”, Art history, xiv (1991), 143–74, p. 143.

ibid., 143.

ibid., 154.

Bernal is deliberately following these scholars when he remarks that “I should like to take it as given that Steuer R. O. Saunders J. B. de C. M. , and Ghalioungui Paul have established not merely that Egyptian medicine contained considerable ‘scientific’ elements long before the emergence of Greek medicine, but that Egyptian medicine played a central role in the development of Greek medicine” (“Animadversions” (ref. 2), 599).

Newsome Frederick , “Black contributions to the early history of Western medicine”, in Van Sertima (ed.), op. cit. (ref. 30), 127–39, p. 128. The liver is notably missing from the list but this seems to be a simple oversight, since Newsome's source, Gardiner's Egyptian grammar, does in fact list signs for the liver. (I owe this information to my colleague Martha Risser.) It may be recalled that, unlike the Mesopotamians, Etruscans, Greeks, and Romans, the Egyptians did not practise hepatoscopy (divination through examination of the livers of animals). Burkert sees “[T]he spread of hepatoscopy [as] one of the clearest examples of cultural contact in the orientalizing period [750–650 b.c.]” (op. cit. (ref. 111), 51). It is interesting that Cicero — whom Burkert cites in this connection — erroneously attributes hepatoscopy also to the Egyptians ( Cicero XX , transl. by Falconer William Armistead (Cambridge, Mass., 1929), De divinatione (II.12), 402–3).

For all of these claims, see Finch Charles S. , “The African background of medical science”, in Van Sertima (ed.), op. cit. (ref. 30), 140–56 passim.

Newsome , “Black contributions” (ref. 30), 135.

At least, this is the view of Smith Wesley D. , The Hippocratic tradition (Ithaca, 1979); see chap. 2, “Galen's Hippocratism”. As Smith puts it: “Galen's version of Hippocratic science and its tradition is in large part his own, a projection of his concerns onto history. While his medical system was put together out of Hellenistic medical developments, his peculiar Hippocratism was fashioned largely as rhetorical and ideological patina for it. His claims about Hippocrates' original philosophical and scientific system were put forth for the circle of intellectuals in Rome, phrased in terms relevant to them” (p. 175). For an introduction to the problem of identifying the genuine writings of Hippocrates, see Lloyd G. E. R. , “The Hippocratic question”, Classical quarterly , n.s., v (1975), 171–92. The only “complete” edition of Hippocrates (Greek texts with facing French translations) was published over a century and a half ago, but remains useful — some would even say, indispensable — today: Littré Emile , Oeuvres completes d'Hippocrate (10 vols, Paris, 1839–61). For a discussion of Littré and his edition of Hippocrates, see Smith, op. cit., 31–36.

Von Staden's analysis takes the form of an Introduction to his edition of all the texts bearing on the medical ideas of the great Alexandrian physician, Herophilus (330/320–260/250 B.C.), and of his followers: Herophilus: The art of medicine in early Alexandria (Cambridge, Mass., 1989). Since none of Herophilus's writings has survived, what von Staden's edition amounts to is a compilation of several hundred passages (in Greek or Latin, with English translations) from over fifty later medical writers (more from Galen than from anyone else) who mention Herophilus or cite his writings.

ibid., 4.

ibid., 162–4.

May Margaret Tallmadge , Galen, On the usefulness of the parts of the body (Ithaca, 1968); Furley David J. and Wilkie J. S. , Galen, On respiration and the arteries (Princeton, 1984); Brain Peter , Galen on bloodletting: A study of the origins, development and validity of his opinions, with a translation of the three works (Cambridge, Mass., 1986); Hankinson R. J. , Galen on the therapeutic method, Books I and II (Oxford, 1991).

Majno Guido , The healing hand: Man and wound in the ancient world (Cambridge, Mass., 1975), 73. The Grapow volumes — there are nine, not ten — were published between 1954 and 1962 in Berlin; Hildegard von Deines and Wolfhart Westendorf were Grapow's co-editors. Von Staden tells us that he has based his English versions of Egyptian medical texts on the German of the Grapow edition ( von Staden , op. cit. (ref. 165), 6, n. 15).

Majno , op. cit. (ref. 169), 73.

Ghalioungui , op. cit. (ref. 86), 38.

Sigerist , op. cit. (ref. 8), 307.

Majno , op. cit. (ref. 169), 92. Given the known chemical composition and bactericidal action of honey, Majno explains that in recommending it to dress the wound (which was standard practice) Egyptian physicians “happened to choose an ingredient that was practically harmless to the tissues, aseptic, antiseptic, and antibiotic. I should say the ingredient: Nothing else, in ancient Egypt, could have begun to match these properties of honey” (p. 118; italics in original). As for grease, Majno's experiments showed that beef fat, vaseline, and butter were either benign or actually favoured the healing of wounds. Without benefit of modern experimental method, though, the Egyptians probably selected the honey-grease mixture simply because it prevented bandages from sticking, possessed a soothing consistency, and did not readily spoil (pp. 118–20).

ibid., 128.

Von Staden , op. cit. (ref. 165), 8.

Ghalioungui , op. cit. (ref. 86), 38.

Lloyd G. E. R. (ed.), Hippocratic writings (Harmondsworth, Middlesex, 1978), 237.

Von Staden , op. cit. (ref 165), 8–9, 7.

Ghalioungui , op. cit. (ref. 86), 57. Since there is only scanty evidence for vowels in the Egyptian language, most authors write mtw.

For a colour illustration of the routes of the mtw, see Majno , op. cit. (ref. 169), Plate 3.8. This wildly wrong anatomy makes no appearance in Greek medicine.

Steuer Robert O. , Aetiological principle of pyaemia in ancient Egyptian medicine, Supplement 10, The bulletin of the history of medicine (Baltimore, 1948), 11.

Herodotus (II, 77), writes of the Egyptians: “…for three days in succession in each month they physic themselves, hunting health with emetics and purges, because they think that from the food that nourishes mankind come all their diseases” (op. cit. (ref. 149), 163).

Ghalioungui , op. cit. (ref. 86), 124. On the “enormous pharmacopoeia” of the Egyptians, Majno comments that “if there was one effect it could definitely induce, that was probably diarrhea” ( Majno , op. cit. (ref. 169), 129)!.

Saunders J. B. de C. M. , The transitions from ancient Egyptian to Greek medicine (Lawrence, Kansas, 1963), 31.

It may be worth quoting a formulation of the humoral theory (from the Hippocratic Nature of man): “The body of man has in itself blood, phlegm, yellow bile and black bile; these make up the nature of his body, and through these he feels pain or enjoys health. Now he enjoys the most perfect health when these elements are duly proportioned to one another in respect of compounding, power and bulk, and when they are perfectly mingled” ( Hippocrates , transl. by Jones W. H. S. (Cambridge, Mass., 1931), “Nature of man”, iv, 11).

Saunders , op. cit. (ref. 184), 25. Steuer , on the other hand, in his monograph on whdw, refers to “views in the Corpus Hippocraticum and in the writings of Aristotle, which appear similar in some respects although not in the fundamental approach to the aetiology of suppurative conditions in particular” (op. cit. (ref. 181), 30–31; italics in original). Ten years later, Steuer discovered in the Cnidian school of medicine (fifth-fourth centuries b.c.) an aetiological theory intermediate between Egyptian mtw theory and Hippocratic humoral theory. This Cnidian theory identified the causes of disease as putrefaction of phlegm and bile rather than imbalance of the (Hippocratic) humours; see Steuer Robert O. and Saunders J. B. de C. M. , Ancient Egyptian & Cnidian medicine (Berkeley and Los Angeles, 1959), 35–36. Using as evidence their interpretation of selected texts from the Papyrus Anonymus Londinensis (an Egyptian papyrus of the second century a.d. containing excerpts from a lost history of medicine by Aristotle's pupil, Menon) and from the Hippocratic Collection, Steuer and Saunders conclude that “the most immediate connecting link between Ancient Egyptian and Cnidian aetiology is the belief in the rising of fecal excrements in the body as the primary cause of disease …” (ibid., 54). Suffice it to say that I find their argument laboured and unconvincing. Also, it is curious that Saunders himself does not even cite this earlier work in support of his claims for Egyptian medicine (op. cit. (ref. 184)).

Ghalioungui , op. cit. (ref. 86), 75.

See Kenyon F. G. (ed.), Greek papyri in the British Museum (5 vols, London, 1893), i, 48, and Sudhoff Karl , Ärztliches aus griechischen Papyrus-Urkunden (Leipzig, 1909), 260–1; and for a photograph of the payprus, Scott Edward , Greek papyri in the British Museum, facsimiles (London, 1893), Papyrus XLIII.

Majno , op. cit. (ref. 169), 317.

See Littré , op. cit. (ref. 164), vii, 562–3. (I owe this reference to Lesley Jones of the Classics Department, University of Texas at Austin).

Burkert , op. cit. (ref. 108), 293.

ibid., 217.

Steuer , op. cit. (ref. 181), 31.

Von Staden , op. cit. (ref. 165), 13.

Wine (both red and white) and vinegar do, in fact, possess germicidal properties. The efficacious agent in the wine is not, however, as one might suppose, the alcohol (which is too dilute to kill germs), but rather a certain organic compound (chemically, a polyphenol) present in all wines. See Majno , op. cit. (ref. 169), 186–8.

Saunders , op. cit. (ref. 184), 28. Von Staden cites four different Hippocratic treatises as evidence for Greek treatment of wounds; see op. cit. (ref. 165), 14, n. 46.

Majno , op. cit. (ref. 169), 192.

Riddle John M. , Dioscorides on pharmacy and medicine (Austin, Texas, 1985), p. xviii.

Von Staden , op. cit. (ref. 165), 17, 19.

Hanson Ann Ellis , “Papyri of medical content”, Yale classical studies , xxviii: Papyrology, ed. by Lewis Napthali (New Haven, 1985), 25–47, p. 27.

Von Staden , op. cit. (ref. 165), 21.

Ghalioungui , “The relation of Pharaonic to Greek and later medicine” (ref. 9), 105.

Ibid ., 101; idem, op. cit. (ref. 86), 112.

Von Staden , op. cit. (ref. 165), 22.

ibid., 30.

Ghalioungui , op. cit. (ref. 86), 46.

Lloyd G. E. R. , The revolutions of wisdom: Studies in the claims and practices of ancient Greek science (Berkeley, 1987), 3.

ibid., 89 (n. 143) and 124.

ibid., 55.

Bernal , “Animadversions” (ref. 2), 597. Bernal refers only to a very early work by Lloyd, with no mention of The revolutions of wisdom.

Referring to the (eventually successful) Greek struggle to develop a concept of rigorous proof by deduction from clearly identified premises, Lloyd comments: “There is … no call whatsoever in this respect (or indeed in any other) to speak of the Greeks as endowed with some special natural characteristic, some distinctive mental ability, as those who fantasised about the ‘Greek miracle’ liked to do” ( Lloyd , op. cit. (ref. 203), 75). On the other hand, Jonathan Barnes, in an enthusiastic review of Lloyd's book, writes: “It is unfashionable to speak of a Greek ‘miracle’…. But let the pendulum of fashion swing as it may, the Greeks invented science and philosophy” (Times literary supplement, 16–22 December 1988, 1392).