fractions through double fractions 213 + 213 + 213 + 213 + 113 = 4(18 + 152 + 1104) + 113 = 12 + 213 + 126 = 12 + 18 + 126 + 152 + 1104; this operation was performed in the head, only the result being written down, and to facilitate it tables were drawn up of the division of 2 by odd numbers. With integers, besides adding and subtracting, it was easy to double and to multiply by 10: multiplying and dividing by 5 and finding the 112 value were also among the fundamental instruments of calculation, and all multiplication proceeded by repetitions of these processes with addition, e.g. 9 × 7 = (9 × 2 × 2) + (9 × 2) + 9. Division was accomplished by multiplying the divisor until the dividend was reached; the answer being the number of times the divisor was so multiplied. Weights and measures proceeded generally on either a decimal or a doubling system or a combination of the two. Apart from a few calculations and accounts, practically all the materials for our knowledge of Egyptian mathematics before the Hellenistic period date from the Middle Kingdom.
The principal text is the Rhind Mathematical Papyrus in the British Museum, written under a Hyksos king c. 1600 B.C.; unfortunately it is full of gross errors. Its contents fall roughly into the following scheme, but the main headings are not shown in the original:—
I. Arithmetic.—A. Tables and rule to facilitate the employment of fractions.
- (a) Table of the divisions of 2 by odd numbers from 3 to 99 (e.g. 2 ÷ 11 = 16 + 166), see above.
- (b) Conversions of compound fractions (e.g. 23 × 13 = 16 + 118), with rule for finding 23 of a fraction.
B. The “bread” calculation—a division by 10 of the units 1 to 9.
C. “Completing” calculations.
- (a) Adding multiples of a fraction to produce a more convenient fraction (perhaps connected with the use of palms and cubits in decoration in a proportion based on the number 8).
- (b) Finding the difference between a given fraction and a given whole number.
D. Ahe[1] or “mass”-problems (of the form x + xn = a, to find the ahe x).
E. Tooun-problems (tooun, “rising,” seems to be the difference between the shares of two sets of persons dividing an amount between them on a lower and a higher scale).
II. Geometry.—A. Measurement of volume (amounts of grain in cylindrical and rectangular spaces of different dimensions and vice versa).
B. Measurement of area (areas of square, circular, triangular, &c., fields).
C. Proportions of pyramids and other monuments with sloping sides.
III. Miscellaneous problems (and tables) such as are met with in bread-making, beer-making, food of live-stock, &c. &c.
The method of estimating the area of irregular fields and the cubic contents of granaries, &c., is very faulty. It would be interesting to find material of later date, such as Pythagoras is reported to have studied.
See A. Eisenlohr, Ein mathematisches Handbuch der alten Ägypter (Leipzig, 1877); F. Ll. Griffith, “The Rhind Mathematical Papyrus” in Proceedings of the Soc. of Biblical Archaeology, Nov. 1891, March, May and June 1894.
Astronomy.—The brilliant skies of day and night in Egypt favoured the development of astronomy. A papyrus of the Roman period in the British Museum attributes the invention of horoscopes to the Egyptians, but no early instance is known. Professor Petrie has indeed suggested, chiefly on chronological grounds, that a table of stars on the ceiling of the Ramesseum temple and another in the tomb of Rameses VI. (repeated in that of Rameses IX. without alteration) were horoscopes of Rameses II. and VI.; but Mahler’s interpretation of the tables on which this would rest appears to be false. Astronomy played a considerable part in religious matters for fixing the dates of festivals and determining the hours of the night. The titles of several temple books are preserved recording the movements and phases of the sun, moon and stars. The rising of Sothis (Sirius) at the beginning of the inundation was a particularly important point to fix in the yearly calendar (see below, § “Chronology”). The primitive clock[2] of the temple time-keeper (horoscopus), consisting of a ὡρολόγιον καἱ φοίνικα (Clemens Alex. Strom., vi. 4. 35), has been identified with two inscribed objects in the Berlin Museum; these are a palm branch with a sight-slit in the broader end, and a short handle from which a plummet line was hung. The former was held close to the eye, the latter in the other hand, perhaps at arm’s length. From the above-mentioned tables of culmination in the tombs of Rameses VI. and IX. it seems that for fixing the hours of the night a man seated on the ground faced the horoscopus in such a position that the line of observation of the Pole-star passed over the middle of his head. On the different days of the year each hour was determined by a fixed star culminating or nearly culminating in it, and the position of these stars at the time is given in the tables as “in the centre,” “on the left eye,” “on the right shoulder,” &c. According to the texts, in founding or rebuilding temples the north axis was determined by the same apparatus, and we may conclude that it was the usual one for astronomical observations. It is conceivable that in ingenious and careful hands it might give results of a high degree of accuracy.
See L. Borchardt, “Ein altägyptisches astronomisches Instrument” in Zeitschrift für ägyptische Sprache, xxxvii. (1899), p. 10; Ed. Meyer, Ägyptische Chronologie, p. 36. Besides the sun and moon, five planets, thirty-six dekans, and constellations to which animal and other forms are given, appear in the early astronomical texts and paintings. The zodiacal signs were not introduced till the Ptolemaic period. See H. Brugsch, Die Ägyptologie (Leipzig, 1891), pp. 315 et seqq., for a full account of all these.
Medicine.—Except, that splints are sometimes found on the limbs of bodies of all periods, at present nothing is known, from texts or otherwise, of the existence of Egyptian surgery or dentistry. For historical pathology the examination of mummies and skeletons is yielding good results. There is little sign of the existence of gout or of syphilitic diseases until late times (see Mummy). A number of papyri have been discovered containing medical prescriptions. The earliest are of the XIIth Dynasty from Kahūn, one being veterinary, the other gynaecological. The finest non-religious papyrus known, the Ebers Papyrus, is a vast collection of receipts. One section, giving us some of “the mysteries of the physician,” shows how lamentably crude were his notions of the constitution of the body. It teaches little more than that the pulse is felt in every part of the body, that there are vessels leading from the heart to the eyes, ears, nose and all the other members, and that “the breath entering the nose goes to the heart and the lungs.” The prescriptions are for a great variety of ailments and afflictions—diseases of the eye and the stomach, sores and broken bones, to make the hair grow, to keep away snakes, fleas, &c. Purgatives and diuretics are particularly numerous, and the medicines take the form of pillules, draughts, liniments, fumigations, &c. The prescriptions are often fanciful and may thus bear some absurd relation to the disease to be cured, but generally they would be to some extent effective. Their action was assisted by spells, for general use in the preparation or application, or for special diseases. In most cases several ingredients are prescribed together: when the amounts are indicated it is by measure not by weight, and evidently no very potent drugs were employed, for the smallest measure specified is equal to about half of a cubic inch. Little has yet been accomplished in identifying the diseases and the substances named in the medical papyri.
See G. A. Reisner, The Hearst Medical Papyrus (Leipzig, 1905), (XVIIIth Dynasty), and for a great magical text of the Roman period (3rd century A.D.) with some prescriptions, F. Ll. Griffith and H. Thompson, The Demotic Magical Papyrus of London and Leiden (London, 1904).
Literature.—The vast mass of writing which has come down to us from the ancient Egyptians comprises documents of almost every conceivable kind, business documents and correspondence, legal documents, memorial inscriptions, historical, scientific, didactic, magical and religious literature; also tales and lyrics and other compositions in poetical language. Most of these classes are dealt with in this article under special headings. In addition there should be mentioned the abundant explanatory inscriptions attached to wall-scenes as a secondary element in those compositions. As early as the Middle Kingdom, papyri are found containing classified lists of words, titles, names of cities,
