NATURE 



72> 



THURSDAY, NO\"EMBER 24, 1S98. 



EARLY MATHEMATICS. 

 Facsimile of the Rhind Mathematical Papyrus in the 

 British Museum. With an Introduction by E. A. 

 Wallis Budge, M.A., Litt.D., D.Lit., F.S.A., Keeper 

 of the Egyptian and Assyrian Antiquities. 21 Plates. 

 ( Printed by Order of the Trustees.) 



THE Rhind Mathematical Papyrus, a facsimile of 

 which the Truftees of the British Museum have 

 just issued, together with an introduction by Dr. Wallis 

 Budge, is the document from which we gather most of 

 what we know of the conception and use of mathematics 

 by the ancient Egyptians. The papyrus does not con- 

 tain a systematic treatise on mathematics, nor does it 

 attempt to deal with the subject from a scientific stand- 

 point. It consists rather of tables and sets of worked 

 out problems, such as would constantly require to be 

 solved by an Egyptian master-builder, land-owner, 

 farmer or estate-agent. In consequence of the inun- 

 dation, the area of an Egyptian farmer's holding was 

 constantly changing in extent, so that the need of some 

 practical method of measuring area was pressing. The 

 farmer after harvest would need some plan for estimating 

 the storage space required for his grain ; the cattle- 

 owner and employer of labour would constantly have to 

 face problems connected with the distribution of fodder 

 and provisions ; the builder would require some method 

 for estimating the angle of a pyramid to be erected upon 

 a given base. .Such problems as these were of every- 

 day occurrence, and they forced the ancient Egyptian to 

 employ his ingenuity in solving them. How far he was 

 successful, and to what extent he proved himself a 

 mathematician, we can gather from the Rhind Papyrus. 



The papyrus consists of a roll, now broken in two 

 pieces, which measure 10 feet 6 inches and 6 feet 9! 

 inches respectively. The te.xt is written throughout in 

 hieratic, but its actual date is not quite certain. Dr. 

 I Budge assigns it to a period not earlier than the begin- 

 Ining of the eighteenth dynasty, about 1700 B.C., but 

 I adds that the actual text goes back to an older period. 

 It was probably a copy of a papyrus written in the 

 [ Hyksos period, about 2000 i;.c., by a scribe Aah-mes, 

 jwho states hat he himself copied an original work of 

 jthe time of Amen-em-hat III., a king of the twelfth 

 I dynasty, about 2300 li.c. 



I Before treating of the contents of the papyrus it will 

 be well to indicate briefly the limits in their knowledge 

 ' of mathematics displayed by the ancient Egyptians, 

 \ whose system was not so perfectly developed as that of 

 I the old Sumerian inhabitants of Babylonia. They ap- 

 ] proached their subject from the practical and not from the 

 I theoretical side ; but in spite of numerous disadvantages 

 in their system of notation, it must be admitted that 

 I they showed great ingenuity in dealing with the mathe- 

 matical problems they attacked. With regard to integers 

 I the Egyptians used a decimal system of notation, but 

 I their system was inferior to the decimal system of the 

 I Arabs ; for while in the Arabic notation each power of 

 I 10 is indicated by simply adding or removing a cipher, 

 the Egyptians had a different name and symbol for 

 NO. I 5 17, VOL. 59] 



each power ; thus i, 10, 100, 1000, 10,000, and 100,000 

 were each expressed by a different figure. This fact, 

 however, did not prevent them from dealing without dif- 

 ficulty with very high numbers. In dealing with fractions, 

 howe\er, the case was different ; here they experienced 

 great difficulties, for, oddly enough, the Egyptian could 

 only express divisions of unity. Any number, in fact, 

 could be turned into a corresponding fraction by placing 

 before it the word re, e.g. i, J, \, &c. By this system 

 they could not express a fraction with a numerator 

 greater than i, though there was no limit to the divisor ; 

 in fact, in one of the sections on Plate x/ of this papyrus, 

 the fraction i 5432 occurs. There is one interesting 

 exception, however, for they were able to express |, but 

 still as a division of unity, for they represented the frac- 

 tion by a sign which may be rendered i li. The 

 Egyptians must have possessed multiplication tables, 

 and those for the lower numbers were probably com- 

 mitted to memory ; among these must have been in- 

 cluded a table of i 'lA values, for they could take 3 of a 

 number by a single operation, and if they wished to take 

 J of a number they halved its i/ii part. Besides adding 

 and subtracting they were able, by means of their tables, 

 without difficulty to halve and double, and to multiply 

 and divide by ten and five. Multiplication by other 

 numbers, however, they performed by repeated doubling 

 and then adding ; thus to multiply seven by six an 

 Egyptian would double 7=14; he would then double 

 the 14=28 ; and'he would then add 14 to 28 = 42. 



This brief sketch of the system of elementary Egyptian 

 arithmetic will serve to explain the interesting arith- 

 metical table which occupies the first six and a half plates 

 of the Rhind Papyrus. This table was evidently worked 

 out to help an Egyptian in his calculations with regard 

 to fractions ; it probably would not be committed to 

 memory, but merely used for consultation like a modern 

 table of logarithms. These six and a half plates contain 

 the working out of a table expressing in simple fractions, 

 with I for the numerator, the ratios of 2 to the odd 

 numbers from 3 to 99, i.e. the fractions 2/3, 2/5, 2/7, &c. 

 Plate i., for instance, gives the working out of these 

 fractions from 2/3 to 2/15, from which we get the follow- 

 ing table of results : — 



2 divided by 3= l/i.'. 



2 ,, ,, S = ''/3 +i/'5 



2 „ „ 7=1/4 +1/28 



2 ,, „ 9=1/6 -H/18 



2 „ ,, 11 = 1/6 -f 1/66 



2 ,, ,, 13=1/8 -H/52+1/104 



2 ,, ,, 15 = 1/10+1/30 



That this table was not due to the fancy of one scribe, 

 but was a recognised table of values in general use, is 

 proved by a fragment of papyrus found at Kahun in 

 April 1889, on which part of the same table is written, 

 and which shows the same values as the result of the 

 division of two by the odd numbers from 3 to 21. The 

 use of such a table is not at first sight very obvious, for 

 if it is necessary to express 25 in fractions with i as the 

 numerator, 1/5-H j is a simpler solution than i 3-M/15. 

 It has been suggested, however, that the Egyptians may 

 have used the table for reducing fractions with higher 

 numerators to fractions of unity ; thus 5 1 1 might of 

 course be expressed by the Egyptian as I'li-t-i 11 

 -I- I u 4- i/i I -t- 1 1 1, but by means of the table a shorter 



E 



