Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970) — OCR (Apple Vision)

Source and licence

share: publicnone

Source: corpus/incoming/e-2/peet-rhind-1923/ — Apple Vision OCR (tools/e2_ocr.py, worker E-2, 2026-09-29) of corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf, 169 PDF pages; provenance and verification in corpus/incoming/e-2/STAGING.md

Attribution: Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970); public domain in Canada (published 1923; author T. Eric Peet (title page corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf pdf p. 1: "… by T. Eric Peet, Brunner Professor of Egyptology in the University of Liverpool … MCMXXIII — Kraus Reprint, Nendeln/Liechtenstein 1970"; pdf p. 2 reprint notice; preface signed Liverpool June 1923); death year UNVERIFIED — the PDF has no Author field and no in-folder authority record gives Peet's dates · note (PROPOSED-scan_copyright-E-2.csv): middle-egyptian-texts/README.md §3 states "died 1934" with no source — not counted (E-2 volume peet-rhind-1923) · Lead D authority lookup 2026-09-29 (outputs/REPORT-DEATH-YEARS-2.md, DEATH_YEARS_EVIDENCE.tsv): Peet d. 1934, BnF cb12504101w + IdRef); OCR text by the Kemetic project, Yousef Hanna 2026.

Licence: public domain in Canada (author died ≤ 1971, verified; LICENCES.md §7) — US position (recorded, does not gate the class; PD-CA 2026-09-26): PD-US (published ≤1930); the scan is the 1970 Kraus reprint, "Reprinted by permission of the original publisher" (no new authorship)

Jurisdiction note: this text is public domain under Canadian law (the project is published from Canada; LICENCES.md §7: author died in or before 1971). It may still be in copyright elsewhere — in countries with a life + 70 term, or where a later edition claims its own rights. The licence line above says what was verified. If you are outside Canada, check your own law before re-using it.

Full provenance and citation

The text — part 1 of 2

5,827 lines · a line number is a link to itself

ref	L0 witness (OCR)
p. 1
1	<0h
2	RHIND
3	INTRODUC'
4	THE
5	MATHEMATICAL PAPYR
6	BRITISH MUSEUM 10057 AND 10058
7	TION, TRANSCRIPTION, TRANSLATION AND COMMENTAR
8	BY
9	T. ERIC PEET
10	BRUNNER PROFESSOR OF EGYPTOLOGY IN THE UNIVERSITY OF LIVERPOOL;
11	LAYCOCK STUDENT IN EGYPTOLOGY AT WORCESTER COLLEGE, OXFORD:
12	FORMERLY CRAVEN • FELLOW IN THE UNIVERSITY OF OXFORD.
13	BIBLIO LEQUE
14	LILLE
15	UNIVERSITAIR
16	THE UNIVERSITY PRESS OF LIVERPOOL LIMITED
17	HODDER & STOUGHTON LIMITED, LONDON
18	MCMXXIII
19	KRAUS REPRINT
20	Nendeln/Liechtenstein
21	1970
22	22 504
23	JS
24	Y
p. 2
25	L/N
26	Reprinted by permission of the original publisher
27	KRAUS REPRINT
28	A Division of
29	KRAUS-THOMSON ORGANIZATION LIMITED
30	Nendeln/Liechtenstein
31	1970
32	Printed in Germany
33	Lessingdruckerei Wiesbaden
p. 3
34	PREFACE
35	nearly fifty years since Eisenlohr published his translation of and commentary on
36	the Rhind Mathematical Papyrus, and the mathematician and the Egyptologist are still almost
37	entirely dependent on this edition for their knowledge of the subject. Yet during these years
38	our knowledge of the Egyptian language has been doubled and
39	translations, guided though they were to some extent by mathematical necessity, must be very
40	seriously revised and modified. What is more, new documents have come to light: the New
41	York fragments, the Kahun fragments, the Moscow papyrus, the Berlin fragments, not to
42	mention later demotic, Coptic and Greek documents, must all be taken into account if a
43	correct view is to be obtained of the mathematical abilities of the Egyptian.
44	Every attempt has been made to render the book intelligible to the mathematician who
45	has no knowledge whatsoever of the Egyptian language. On the other hand, the Egyptologist
46	with little knowledge of mathematics may enter on it without fear: Egyptian mathematics
47	was a simple affair, and the author has tried throughout to deal with it in its own simple
48	terms without clothing it in a modern dress which is totally foreign to it.
49	The work was begun in 1911 and was well advanced in August 1914. Trom that time
50	it lay untouched until 1920, since when it has been practically rewritten, the changes being
51	mainly in the direction of greater simplicity of treatment. It was not until early in the
52	present year that I received the photographs of the Moscow Papyrus. These were sent to me
53	on the natural understanding that I should not publish their contents. I will only say of them
54	that in spite of their very high interest they have led me to modify practically nothing in
55	my volume, though I should have been very sorry to have had to publish it in ignorance of
56	them.
57	As usual my work contains many an admirable suggestion from Dr. Alan H. Gardiner,
58	unacknowledged, at his own request, and I owe some valuable points to Mr. Battiscombe
59	Gunn. I have also to thank Professor R. C. Archibald, of Brown University, Providence,
60	R.I., for the liberality with which he put at my disposal his admirable bibliography of the
61	Rhind Papyrus. As this bibliography is to be published in full in America in the near future,
62	I have thought it unnecessary to append to my volume what could at best be no more than
63	a selection from it. That I was able to obtain for study photographs of the Moscow Papyrus
64	is mainly due to Dr. Fritjof Nansen, whose sympathies in the work were kindly enlisted by
65	Professor D'Arcy Thompson, of St. Andrews.\| Professor Strouwé, of Petrograd, who is to
66	publish the papyrus, very unselfishly agreed to my having the photographs, and the Director
67	of the Museum of Fine Arts in Moscow, Dr. W. Ghiatzintoff, was kind enough to have them
68	made for me.
69	T. ERIC PEET
70	LIVERPOOL,
71	June 23rd, 1923.
p. 4
72	LIST OF ABBREVIATIONS.
73	A.Z. Zeitschrift für ägyptische Sprache.
74	В.M.Fаcs. Facsimile of the Rhind Mathematical Payyrus in the British Museum, London, 1898.
75	CANTOR CANTOR, M., Vorlesungen über Geschichte der Mathematil, 2nd ed., Vol. I, Leipzig, 1894.
76	BISENLOHR LISENLOHR, Ein mathematisches Handbuch der alten Agypter, übersetzt und erklärt,
77	Leipzig, 1877. (A second edition without plates, 1891.)
78	GRIFFITH, K.P. GRIFFITH, F. LL., Hieratic Papyri from Kahun and Gurob, London, 1898.
79	HEATE HEATH, SIr THOMAs, A History of Greek Mathematics, Oxford, 1921.
80	J.E.A. Journal of Egyptian Archaeology.
81	L., D. LEPSIUS, RICHARD, Denkmaeler aus Aegypten und Aethiopien, Berlin, n.d.
82	P.S.B.A. Proceedings of the Society of Biblical Archaeology.
83	SETRE, V.Z.Z SETHE, KURT, Von Zahlen und Zahlworten bei den alten Ägyptern, Strassburg, 1916.
84	Urk. Urkunden des ägyptischen Altertums, ed. Georg STEINDOrFF, Leipzig, various dates.
85	CONVENTIONAL SIGNS USED IN THE TRANSLATION.
86	Square brackets [ ] enclose restorations of damaged or lost passages in the papyrus. They are never
87	used in this volume as a mathematical symbol.
88	Pointed brackets ( enclose words or passages which never stood in the papyrus, but whose omission
89	there is due to an error on the part of the scribe.
p. 5
90	THE RHIND MATHEMATICAL PAPYRUS
91	INTRODUCTORY
92	PREVIOUS WORK ON THE PAPYRUS.
93	THe first scholar to study the papyrús would appear to have been Lenormant, who published a
94	note on it as early as 1867! He gave some account of its contents, naturally not altogether
95	accurate—he speaks of the determination of the volume of a pyramid—and dated the document
96	to the XIIth Dynasty. In the following year Dr. Birch described the papyrus with a few
97	quotations in hieroglyphs." The next student was Brugsch, who, in 1874, published a short
98	article giving some of the more important technical terms used in the document."
99	Little of importance appeared henceforward until in 1877 Eisenlohr published his volume
100	Ein mathematisches Handbuch der alten Aegypter. This was accompanied by a hieratic text
101	which was practically a reproductión of facsimile plates made by the Trustees of the British
102	Museum in 1869 and delayed in publication. A set of these plates had been lent to Eisenlohr
103	tuacing of thenia London in 1872, a courtesy which he seems to have repaid by publishing a without authority. The British Museum publication of the admirable and
104	almost perfect facsimile did not take place until 1898.*
105	when a series of brilliant articles from the pen of Griffith began to appear in the Proceedings
106	of the Society of Biblical Archaeology." These dealt not only with the Rhind Papyrus itself, but
107	with the subject of Egyptian weights and measures in general.
108	Since that time numerous articles treating of specific problems in the papyrus have
109	appeared, mainly by Borchardt and Schack-Schackenburg.
110	sections of the papyrus to which they refer.
111	mention here is F. Hultsch's Die Elemente der ägyptischen Teilungsrechnung in Abhandlungen
112	der. Kgl. Süchs. Gesellschaft der Wissenschaften, phil.-hist. Klasse, vol. 17, no. 1, Leipzig, 1895.
113	This is an exhaustive survey covering most of the ground of Egyptian fractional arithmetic as
114	exemplified in the Rhind Papyrus.
115	DESCRIPTION OF THE PAPYRUS.
116	The Rhind Papyrus now lies in the British Museum consists of two pieces
117	separately mounted between sheets of plate glass and numbered 10057 and 10058 respectively.
118	These two pieces once formed a single roll, and were probably separated in modern times by
119	an unskilful unroller. There is now a gap between them, and the Historical Society of
120	New York possesses a number of fragments which come from this gap. A glance at Plate E,
121	1 Note relative à un papyrus égyptien contenant un fragment d'un traité de géometrie appliquée à l'arpentage, Comptes
122	rendus de l'Acad. des Sciences, Paris, vol. 65, p. 903.
123	2 A.Z., 1868, 108-10. 3 Ä.Z., 1874, 147-49.
124	4 Facsimile of the Rhind Mathematical Papyrus, London, 1898.
125	5 Vols. XIII, 328 fl.; XIV, 26 ff.; XVI, 164 ff., 201 fl., 230 ff.
p. 6
126	RHIND MATHEMATICAL PAPYRUS
127	in which these fragments are arranged so far as possible in their proper places, will show that
128	it would have been impossible to cut the papyrus in two vertically at any point in this gap
129	without mutilating a problem. This tells strongly against Griffith's suggestion' that the
130	papyrus was cut in two by its owner in ancient times for reasons of convenience, and it is
131	far more likely that the damage was done after the finding in modern times, probably in an
132	ill-advised attempt at unrolling
133	There is no evidence of value as to how the fragments came to be separated from the
134	larger sheet or sheets. The two portions in the British Museum were bought by Mr. A. H
135	Rhind in Luxor in 1858, and said to have been found with others in a chamber in the ruins
136	of one of the small buildings near the Ramesseum." The entry in the Catalogue of the
137	Egyptian Collection of the New York Historical Society is as follows:—" 265. Fragments of two
138	or more papyri, containing numbers and quantities. Found with fragments of the Medical
139	Papyrus and No. 262." The Medical Papyrus in question is the now well-known Edwin Smith
140	Papyrus, and No. 262 is a tiny fragment of hieratic writing containing the name of Tuthmosis I.
141	The mathematical fragments came into the possession of the Society in 1907 as part of the
142	Edwin Smith Collection. It seems likely that the dates March 17/62 and Dec. 10/63, written
143	on the paper mounts by Edwin Smith, represent the time when he acquired the fragments ;
144	at least they were in his hands by those dates."
145	If these two last are indeed the dates of acquisition, and it is not easy to see what
146	else they could be, it would seem that the native finders of the roll attempted to open it in
147	or before 1858, when they sold the main portion to Mr. Rhind, but that they kept back the
148	fragments and sold them to Mr. Smith in two instalments on the dates given.
149	What is of still greater interest, if it be true, is the statement that the fragments were
150	scrap of the reign of
151	Tuthmosis I. If we could trust this statement we should infer that the Arabs found a cache
152	of scientific documents dating, like the Rhind and the Edwin Smith, from the Hyksos Period,
153	stored away not earlier than the reign of Tuthmosis I. No one, however, who knows the
154	habits of the native finder and dealer will be unwise enough to make any deduction at all.
155	The two sheets in the British Museum, Fig. 1, may be described as follows:—
156	Papyrus 10058.
157	Present length 206 cm., height 33 cm.
158	Recto. The recto consists of five pages of papyrus, each about 395 mm. broad, except
159	the first, and the last, which is incomplete.
160	hand end there is a blank space of about 10 cm., after which the title begins in vertical
161	columns. This is followed by a double vertical black line, and from this point leftwards the
162	sheet is ruled in black into six horizontal registers or bands throughout.
163	recto is devoted, with the exception of the title already referred to, to the division of 2 by
164	the odd numbers from 3 to 101.
165	Verso. This face is very heavily patched, not only at the blank left-hand end, but also
166	on the right: this patching was clearly done in ancient times, and where signs had disappeared
167	on a lost fragment they have been written in in very black ink on the patches by a later
168	hand. At 18 cm. from the right-hand end is a double vertical ruling in black, as on the
169	recto. Left of this the papyrus is ruled out into six horizontal registers. The writing begins
170	at the same end as on the other face. On the right, outside the double line, is the carelessly
171	1 P.S.B.A., XVI, 164-5. Griffith had not seen the fragments.
172	= B.M.Facs., Preface.
173	3 I owe this information to Mrs. C. Ransom Williams. For the very interesting details of Mr. Smith's stay
174	in Luxor see Breasted in Recueil d'Études Egyptologiques Champollion, Paris, 1922, 385 ff
p. 7
175	RHIND MATHEMATICAL PAPYRUS
176	PL XVII.
177	MENSURATION. BLANK ARITHMETICAL•
178	No.o/loati. 12 11 "No.ojleat
179	B.M. PAPYRUS 10,057.
180	Mo. of leaf. 14
181	LOST
182	Fig. 1 (after GrIFFITH, P.S.B.A., XVI, Pl. 1.) The plate numbers refer to the B.M.Face.
183	written No. 61. It is followed by Nos. 62 to 84, and to the left of this last the papyrus
184	is blank to its end (57 cm. away), except for the curious No. 85, written upside down near the
185	bottom and about half-way along the blank space.
186	Papyrus 10057.
187	Present length 319 cm., height 33 cm. or just over, the edge being actually under the
188	Recto. The pages are from 39 to 40cm. broad, and the gumming is very accurate.
189	The left-hand end is fragmentary. The whole face is ruled out in six horizontal registers.
190	Problem No. 1 begins the recto and we run without a break up to No. 40, after which there
191	is a blank of about 55 cm.: No. 41 begins a fresh page, and the problems continue up to
192	No. 60, which ends the recto.
193	Verso. This face is quite blank except for the calendrical entry No. 87, which is written
194	near the top about half-way along the right-hand end the papyrus has been patched
195	with a piece of another papyrus bearing the fragment of accounts numbered No. 86.
196	The whole papyrus is of a good bright colour. Iwo kinds of ink were used, a fairly
197	dark black and a bright red. This last is employed for headings, and also in order to bring
198	into prominence certain figures in the problems. The length of the papyrus before it was cut
199	in two was about 543 cm., and its recto showed 14 sheets, each from 383 to 400 mm. in width,
200	except the first and last which were narrower, unless this is due to damage. The pages of
201	writing are very variable in breadth.
202	DATE OF THE PAPYRUS.
203	The Rhind Papyrus is dated in the 33rd year of the Hyksos King Aauserre Apophis, who
204	some time between 1788 and 1580 B.c. There is no reason to doubt the
205	scribe's own statement' that it was a copy of an
206	Nemaré, Amenemmes III of the XIIth Dynasty, who was on the throne from about 1849 to
207	1801 B.C. It is thus the latest of our hieratic mathematical papyri, the Moscow Papyrus and
208	the Kahun and Berlin fragments all being definitely XIIth Dynasty documents: its prototype
209	clearly belonged to the same milieu as these.
210	1 Griffith, P.S.B.A., XIV, 436 note, doubted the truth of this in its entirety, on the ground that the double- and
211	quadruple-hekat found in some examples in Rhind were not in use in the XIIth Dynasty. He would hardly main-
212	tain this now, for his own decipherment of the Kahun fragments has shown that the double-hekat was regularly
213	used in the early Middle Kingdom, and in the case of the quadruple-hekat we ought not to argue ex silentio that
214	it was not in use as early as Nemarē.
215	в 2
p. 8
216	RHIND MATHEMATICAL PAPYRUS
217	CONTENTS OF THE PAPYRUS.
218	The Rhind Papyrus is not a mathematical treatise in the modern sense, that is to say
219	it does not contain a series of rules for dealing with problems of different kinds. It consists of
220	a number of examples, preceded by a table for the resolution of fractions whose numerator is
221	2 into the sum of two or more aliquot parts. A suggestion of a general rule occurs in No. 61B,
222	where the rule for finding 3 of an aliquot part is formulated. To get 3 of 1 we are told to
223	take the double and the 6-times of the aliquot part (sic), which would give as answer to + 30-
224	Then follow the words "Behold one does likewise in the case of any aliquot part which may
225	This is a general rule, but, if we except No. 66, it is the only one in the papyrus.
226	In content the papyrus does not stand alone, for none of those which we possess contain
227	general rules, but merely series of tables and of examples worked out by their aid, and we are
228	justified in doubting whether such things as theoretical general treatises existed in Egypt.
229	would be fully in keeping with what we know of the concrete nature of Egyptian thought had
230	their mathematicians failed to formulate general principles and confined themselves to the
231	working out of concrete instances.
232	The papyrus begins with the long table of resolution of fractions whose numerator is 2,
233	as mentioned above, preceded only by a very short title and by the name of the maker of the
234	copy and the date of its making. These together occupied the whole of the recto of Papyrus
235	10058, and the table extended over a portion of the gap between the two papyri from which
236	come the New York fragments.
237	This table is followed by a number of examples to which Fisenlohr has given the serial
238	numbers 1-84. Nos. 1-40 are purely arithmetical, and are in the main examples of the
239	multiplication and division of fractions. They are followed by a considerable blank in the
240	papyrus (see above, p. 2), after which follow a number of problems in the measurement of
241	areas, volumes and angles of slope, Nos. 41-60. These bring us to the end of the recto of
242	If we now turn the whole papyrus over on its longer axis we find at the beginning,
243	i.e. at the right-hand end of 10058, No. 61, which deals with a purely arithmetical matter,
244	the multiplication of fractions, and seems out of place here. It is in fact written outside the
245	ruling of the sheet (see above, p. 3), and was clearly not part of the scribe's original scheme
246	tor this portion of the papyrus. Possibly he added it afterwards in this very accessible blank
247	space because it contained a table of fractions to which he frequently needed to refer.
248	is followed by a group of miscellaneous problems, all purely arithmetical in type and couched
249	in concrete terms, numbered 62-84. These complete the mathematical portion of the papyrus,
250	but do not carry us even as far as the end of the verso of 10058, the rest of which is blank
251	except for the two curious columns of signs numbered 85, which are written upside down at
252	the bottom of the page about half-way along the blank space.
253	Passing across to the verso of 10057 we find that its right-hand end is patched with a
254	piece of papyrus (No. 86) bearing some accounts. The whole verso is blank except for the
255	so-called calendrical entries, No. 87, written at the top about half-way along.
256	The following is a synopsis of the contents:-
257	Table of resolution of fractions with numerator 2.
258	Book I. Arithmetic.
259	Tos. 1-6. Division of various numbers of loaves equally between 10 men Vos. 7-20. Hirst group of completion-calculations (sekem) involving multiplication o
260	fractions.
261	Nos. 21-23. Second group of completion-calculations, involving simple addition of
262	fractions.
p. 9
263	Bo
264	RHIND MATHEMATICAL PAPYRUS
265	Nos. 24-34. Arithmetical solution by trial of equations of the first degree
266	hau-calculations.)
267	Nos. 35-38. Similar equations involving the bushel or hekat.
268	Nos. 39 and 40. Division of loaves between men in unequal proportions
269	ok II. Mensuration.
270	Part I. Volumes and cubic content in corn.
271	Nos. 41-43. Cylindrical containers.
272	Nos. 44-46. Rectangular parallelopipedal containers.
273	No. 47. Expression in correct form' of to, do, up to oo of a hekat,
274	sum in cubic content.
275	Part II. Areas.
276	No. 48. Area of square and circle compared.
277	No. 49. Rectangle.
278	No. 50. Circle.
279	No. 51. Triangle.
280	No. 52. Truncated triangle.
281	No. 53. Trapezoid (?).
282	Nos. 54 and 55. Division of given area of land into equal-sized fields.
283	Part III. Batter, or angle of slope.
284	Nos. 56-59. Batter of pyramid.
285	No. 60. Slope of a cone (?).
286	ok III. Miscellaneous problems in arithmetic.
287	No. 61. Multiplication of fractions (probably out of place, see above).
288	No. 62. Proportionate values of precious metals.
289	No. 63. Division of loaves in unequal proportions.
290	No. 64. Division of barley into shares in arithmetical progression.
291	No. 65. Division of loaves in unequal proportions.
292	No. 66. Daily portion of a yearly ration of fat.
293	No. 67. Reckoning of livestock.
294	No. 68. Division of 100 hekat of corn in unequal proportions.
295	Nos. 69-78. So-called pefsu-reckonings. Conversion of grain into bread a eo ealte o aureokeonine.
296	No. 79. Geometrical progression.
297	No. 80-1. Conversion of fractions of the hekat (2, t, 3, etc.) into henu.
298	No. 82-3. Food estimate for a poultry yard.
299	No. 84. Estimate of food of an ox-stall.
300	litions.
301	No. 85. Unintelligible group of signs.
302	No. 86. Fragment of accounts.
303	No. 87. Calendrical entries.
304	1 See p. 25.
305	. (Eisenlohr's
306	lisguised as a
307	ad beer, and
p. 10
308	RHIND MATHEMATICAL PAPYRUS
309	DOCUMENTS AVAILABLE FOR THE STUDY OF EGYPTIAN MATHEMATICS.
310	A.—BARLY DOCUMENTS
311	The ancient Egyptian documents which deal with mathematics as such are not numerous.
312	They comprise the following papyri and tablets, all dating from the Middle Kingdom, with the
313	exception of the Rhind, which, while actually written down in its present form in Hyksos
314	times, is according to its own statement a copy of a Middle Kingdom original.
315	1. The Rhind Mathematical Papyrus, now in the British Museum, except for some fragments
316	in the possession of the Historical Society of New York.
317	2. The Moscow Mathematical Papyrus, in the Museum of Fine Arts of Moscow. This
318	document, which dates from the XIIth Dynasty, has lain unpublished at Moscow for many
319	years, and though by the kindness of the Director of the Museum of Fine Arts of Moscow and
320	it, I should be betraying my trust did I add anything to the short note in Ancient Egypt Professor Strouwé, who is to publish the papyrus, I have been allowed to have photographs of • hoogen pay ot
321	by the late Professor Turaiev. In this note was published one problem from the papyrus,
322	which seems to give the correct determination of the volume of a truncated pyramid on a
323	square base (p. 93). The writer also stated that the papyrus contained "19 problems, some of
324	which give us new types of calculation unknown till now, and therefore somewhat difficult to
325	comprehend. Four of these problems are geometrical ones. The first shows how to define
326	the length of the sides of a quadrilateral, when the relation of the sides and the area of the
327	quadrilateral are known. The two next give a method of calculating the area of a triangle:
328	a method already known to us." I will only add to this that though the papyrus is of the
329	highest interest owing to its early date and admirable state of preservation (in part at least)
330	it contains nothing, with the exception of the problem of the truncated pyramid, which will
331	greatly modify the conception of Egyptian mathematics given to us by the already published
332	papyri and fragments.
333	3. The Kahun fragments. These were found at Kahun in 1889 by Professor Flinders Petrie
334	and published by F. LI. Griffith in the volume called Hieratic Papyri from Kahun and Gurob
335	(London, 1898), pp. 15-18 of the text volume and Plate VIII. The mathematical contents of
336	the fragments are as follows:-
337	(a) Table of resolutions of all fractions whose denominator is odd and whose numerator is 2,
338	from } to ?i, into the sum of two or more fractions with numerator unity. PI. VIII, lines 1-10.
339	(b) Multiplication of $ + 1z by 9 (line I1), perhaps a fragment of some longer problem.
340	(c) The number 110 is divided by 8 and the result stated to be 13} + 12. From this } + ₺
341	is continuously subtracted nine times. (Line 12.)
342	(d) To find the content in khar of a cylinder of diameter 12 and height 8 cubits
343	(lines 13-14; see Ä.Z., 35, 150-2, and 37, 78-9).
344	(e) A list of eight very large numbers. The fragment is full of lacunae and the numbers
345	seem to have no connection one with another. Perhaps it was an addition. (Lines 15-22.)
346	Problem which can be expressed algebraically as follows: If {x - ‡x = 5, find x.
347	(Lines 23-28.)
348	(g) A difficult problem, with beginning lost, dealing with the
349	container or containers, parallelopipedal in form, the sides of whose bases are to one another
350	in a fixed ratio (lines 30-42). The solution involves the use of square root.
351	(h) Accounts of a poultry yard. (Lines 43-62).
352	4. Berlin Papyrus 6619, published by Schack-Schackenburg in Ä.Z., 38, 135 ff. with
353	Tafel IV: provenance not stated. The papyrus consists of four fragments reproduced on
354	Tafel IV under the numbers 1-4.
355	1 1917, 100-102.
p. 11
356	RHIND MATHEMATICAL PAPYRUS
357	No. 1 is a problem, To divide 100 square cubits into two squares whose sides are in the
358	proportion 1:3. It involves the use of square root.
359	No. 2. Fragment of a problem in exchanges of various kinds of grain. Too much is
360	lost to allow the real nature of the sum to be discerned.
361	No. 3. problem in numbers involving the correct determination of the square root
362	of 6f as 2t.
363	No. 4. Too fragmentary for diagnosis.
364	5. Two wooden tablets in the Cairo Museum. These bear the catalogue numbers
365	25367 and 25368. They are writing tablets of the usual stuccoed type and were found at
366	Akhmîm. One bears on one side a letter and a list of servants, and on the other certain
367	mathematical calculations. The other tablet bears on one side a list of servants and a
368	mathematical calculation, while the reverse is entirely occupied by calculations. The former
369	tablet is dated in year 28 of an unnamed king. The style of the script and the names of the
370	persons point to the Middle Kingdom.
371	The calculations, which are five in number, were first published,but badly misunderstood,
372	by Daressy. Möller next referred to them," stating that in them certain fractions in later
373	times used only to denote parts of the hekat or bushel were actually multiplied together, and
374	must therefore have been at this period pure fractions. Sethe has pointed out that the major
375	premise of this syllogism is false.
376	seventh, a tenth, an eleventh and In reality these cartali ona are noting mnre oe l a thirteenth respectively of a hekat or bushel in terms of the finding of a third, a
377	t pt te busl used in ordinr everyday transactions, namely the }, t, 3, t6, 35, 6z,
378	for each of which there existed a special sign (see p. 25), and the 3loth part, called the
379	ro. We ourselves are not accustomed to think in sevenths or thirteenths of a ton but
380	reduce these to hundredweights, quarters, pounds and ounces, weights of which we have a
381	fairly clear conception: so the Egyptian reduced these unwieldy fractions of the bushel to
382	certain fixed parts, and his system was superior to ours in that each part was exactly half of
383	the next larger.
384	A couple of examples will enable the reader to grasp the bearing of these tables. Thus
385	one seventh of a bushel is shown to be eguivalent to
386	(3+6t) bushel+(3+‡+ 1) ro,
387	while a third of a bushel works out to
388	(#+16+ 6t) bushel + 13 ro.
389	In addition to the above documents the various account papyri are of importance for
390	the study of mathematics and especially of weights and measures. The most important of
391	these so far published are Papyrus Bulag 18, see Ä.Z., 57, 51-68, and those dealt with by
392	Spiegelberg in his Rechnungen aus der Zeit Setis I.
393	B.—SOME LATER DOCUMENTS FROM EGYPT.
394	three later documents from Egypt which call for notice here, though in
395	considering them it must be remembered that owing to their late date we must make great
396	allowances for the possibility of contamination from Greek mathematics, and must not use
397	them to prove anything with regard the state of the science in the earlier periods in
398	Egypt. These are the Demotic Papyrus, the Byzantine and Coptic tables of fractions,
399	1 Recueil de Travaux, XXVIII, 62 ff. 2 A.Z., XLVIII, 99.
400	3 V.Z.Z., 74, note 2. 4 See my article in J.E.A., IX, 91 ff.
p. 12
401	RHIND MATHEMATICAL PAPYRUS
402	1. The Demotic Papyrus.' This is a stated to be in the Library of the
403	Egypt Exploration Society, and to have been dated by Griffith to the Roman period.
404	contains tables giving resolutions into aliquot parts? of fractions with various denominators,
405	which originally probably ran from 3 (Hultsch suggests 3!) to beyond 15. An example will
406	make this clear. The table dealing with the denominator 7 is as follows (partly restored) :-
407	1 ÷ 7 =
408	2÷ 7 = * + 78
409	3 ÷ 7 = 5+TE+7E
410	4 ÷ 7 =
411	5÷ 7 =
412	6 ÷ 7 = ++*+35
413	7÷ 7 =
414	In the same way in the ny denoninter tieo d partaidore te ton - l,
415	2. Several tables of fractions of Byzantine date from Egypt are known. The best of
416	them are those published by Sir Herbert Thompson in Ancient Egypt, 1914, 52 ff. These are
417	written on wood and are, as is clear from their numbering, the two last of a series of 16 such
418	tables; they deal with the division of whole numbers by 15 and 16 respectively, the results
419	being expressed in aliquot parts. Thus,
420	The 15th part of 1 =
421	2 = to +3o
422	3 =
423	4 = $+60
424	5 =
425	6 = } + 15 and so on.
426	The whole numbers divided in the 15-table run from 1 to 15, in the 16-table from 1 to 16.
427	The preceding tables, which are lost, doubtless dealt with division by the lower digits 2 to 14; the
428	first table perhaps dealt with multiplication by }.3
429	3. A Coptic ostracon published by Crum* contains a similar table, the full meaning of
430	which was first recognized by Sethe.® The divisor is here 31, and the numbers to be divided
431	run from 1 to 31 with the addition of } and §.
432	4. The Mathematical Papyrus of Akhmîm ® was discovered by natives in the cemeteries
433	of that town. It is in the form of a bound book with leather cover, and is in fairly good
434	burials in the cemetery and the style of the writing would indicate a date between the sixth preservation. There is no direct indication of date, but arguments from the nature of the the ninth centuries A.D. y o ditet The contents steni o ae the papyrus consist of a series of tables of
435	fractions followed by a number of problems partly illustrating these.
436	The tables give } and the various aliquot parts of each of a long series of
437	Thealiquot parts on down to As far as the one-tenth
438	table the whole numbers run by units from 1 to 10, then by tens up to 100, by hundreds
439	up to 1,000, and by thousands up to 10,000. In the remaining tables the whole numbers only
440	1 RÉVILIOUT, E., Mélanges sur la métrologie, l'économie politique et l'histoire de l'ancienne Égypte, Paris, 1895 ;
441	HuLTScH, F., Neue Beiträge zur ägyptischen Ieilungsrechnung, in Bibliotheca Mathematica, ser. 3, vol. II, 1901, 177-84.
442	2 } is, as usual, reckoned as an aliquot part. 3 See SETHE, V.Z.Z., 70-71.
443	6 BAILLET, J., Le papyrus mathématique d' Alhmim (Mémoires de la Mission Archéol. Franç. au Caire, vol. 9, fasc. 1),
444	See further LORIA, G., Un nuovo documento relativo alla logistica greco-egiziana in Bibliotheca Mathematica,
445	ser. 2, vol. VII, 1893, 79 ff.
p. 13
446	RHIND MATHEMATICAL PAPYRUS
447	go as high as the denominator of the aliquot part, e.g., in the table of it they run only from
448	1 to 11, as in the Byzantine and Coptic tables described above.
449	The problems are exactly 50 in number and range over various subjects, the volume of
450	various containers, division of earnings between several workmen, questions of interest on
451	money and so on. Baillet rightly remarks that as against the Rhind the Greek papyrus shows
452	less interest in the actualsolving ofits problems and more in the detail of their solution.
453	In the treatment of fractions it shows much that is new.! Yet here still, as in the Rhind, the
454	only fractions dealt with are aliquot parts (with the old exception of }). There is, however,
455	considerably more skill shown in their handling, and the calculator has undoubtedly acquired
456	greater power over them. Thus it is possible from the working to show that the resolution
457	of proper fractions into the sum of two or more aliquot parts was carried out according to
458	certain fixed formulae: it will be seen later that all attempts to show that this was the case
459	in the Rhind table of resolutions have failed.
460	The main result of the documents above described is to show that in Egypt, as elsewhere
461	in the east, the advanced mathematics of the Greeks had not even in this period succeeded in
462	replacing the cruder methods of earlier civilizations. Here at Akhmîm we have a system in
463	which all fractions except aliquot parts must be eschewed, at a time when the equivalent of
464	the modern notation of proper fractions had been known to the Greeks for some centuries.
465	DATE OF ORIGIN OF EGYPTIAN MATHEMATICS.
466	Our information on this point is sadly defective. The Rhind Papyrus dates from the
467	Hyksos Period, though it claims to be a copy of a document prepared in the XIIth Dynasty,
468	in the reign of Amenemhet III. This may well be, since both the Moscow Papyrus and the
469	Kahun fragments date from that Dynasty. But how much earlier must we go to find the
470	beginnings? Surely the complicated fabric of Egyptian mathematics can hardly have been built
471	in a century or even two, and it is tempting to suppose that the main discoveries of
472	mathematics should be dated to the Old Kingdom. There is a very definite tendency among
473	Egyptologists to put this period down as the Golden Age of Egyptian knowledge and wisdom.
474	There can be little doubt that some of the literary papyri have their roots in this era, as
475	for example the Proverbs of Ptabhotep, and the antiquated constructions of the medical papyri
476	make it possible that the science of medicine, such as it was, had its spring in the Old
477	Of definite evidence for this early date there is none. All we know is that by the
478	beginning of the First Dynasty the system of notation was complete up to the sign for 1,000,000
479	(see p. 11, note 1). In the IVth Dynasty we find in the tomb of Methen that the land measures
480	of the Rhind Papyrus are already in full development in a form which involves correct deter-
481	mination of the area of the rectangle, but not of necessity of the triangle or circle. There
482	appears to be no early evidence with of capacity,? though one may almost
483	take it for granted that with the measurement of the field on which the corn was grown.went
484	that of the containers in which it was stored and sold. That measurement by weighing was
485	practised can hardly be denied of various objects of Old Kingdom date which can
486	scarcely be anything but weights, as for example the stone weight of Khufu, formerly in the
487	Hilton Price collection, though the attempts to establish a standard from these objects have
488	been far from satisfactory.
489	1 E.g., the resolution of an aliquot part into the sum of two or more smaller ones, tt = ist85. Of.however
490	the Rhind resolution of TổI, p. 47.
491	2 The hieratic signs for the dimidiated portions of the hekat occur in a VIth Dynasty papyrus (Berlin,
492	P 10500, unpublished). See Ä.Z., 18, 100.
493	3 P.S.B.A., XIV, 435-6, 442.
p. 14
494	10 RHIND MATHEMATICAL PAPYRUS
495	rom these feeble indications we pass straight to the fully developed mathematic:
496	system of the XIIth Dynasty, the early stages in the buildino up of which are entirel
497	concealed from us.
498	GENERAL CHARACTER OF EGYPTIAN MATHEMATICS
499	The outstanding feature of Egyptian mathematics is its intensely practical character.
500	This is not peculiar to mathematics, for it is typical of all the sciences in Egypt.
501	alone of the Greeks seems to have realized,' the Egyptians were essentially a "nation of shop-
502	and interest in or speculation concerning a subject for its own sake was totally
503	foreign to their minds.
504	To realize this we have only to take a glance through the problems of the Rhind
505	Here everything is expressed in concrete terms. The Egyptian does not speak or
506	think of 8 as an abstract number, he thinks of 8 loaves or 8 sheep? He does not work out
507	the slope of the sides of a pyramid because it interests him to know it, but because he needs
508	a practical working rule to give to the mason who is to dress the stones (see under No. 56).
509	If he resolves is into } + s + roz it is not because this fact in itself appeals in any way
510	to his curiosity, but simply because sooner or later he will come across the fraction is in a
511	sum, and since he has no machinery for dealing with fractions whose numerators are greater
512	than unity he will then urgently need the resolution above stated.
513	Perhaps it is in keeping with this attitude that there is in our papyrus practically no
514	instance of the use of a general formula, each case being worked out on its own merits, and
515	cases which to us seem analogous being sometimes dealt with by totally different methods.
516	In these facts we may see the cause why Egyptian mathematics stagnated, as they
517	undoubtedly did.. By the XIIth Dynasty the mathematician was already able to work out
518	any problem which he was liable to meet in ordinary life. He could measure a field or a
519	granary, and divide wages or booty in fixed proportions, and after all what more was needed?" civilization must present fresh problems for solution, or a genius must arise thirsting for 222235
520	Greece to do it.
521	of Thales, comparing him with other "philosophers" whose study was
522	practical politics, that "he carried his speculations beyond things of common utility." Heath*
523	is perhaps not too bold when he illustrates this with a passage from Proclus' Summary:—
524	"Thales ..... discovered many propositions himself, and instructed his successors in the
525	principles underlying many others, his method of attack being in some cases more general,
526	in others more empirical." Here undoubtedly lies the main difference between Greek and
527	Egyptian mathematics. When first the School of Pythagoras, somewhere round about
528	500 B.c., began to evolve the Theory of Numbers it had already taken a step which put
529	Greek mathematies on a different plane from Egyptian: for the Egyptian there could be
530	no theory of numbers, only a practice. Still less could an Egyptian have appreciated the
531	metaphysical subtleties of Plato's treatment of mathematics in the sixth and
532	of the Republic.
533	• Republic, iv, 436.
534	2 Nos. 61B and 66 are perhaps the only exceptions.
535	3 The reasons for the stagnation of medieine, however, are quite diffcrent. It could hardly be said that thi
536	medical knowledge of the XIIth Dynasty was sufficient to meet the normal need. What prevented development
537	was the fact that the science was permeated with magic, from which it never seemed able to break away.
p. 15
538	RHIND MATHEMATICAL PAPYRUS 11
539	1. SYSTEM OF NOTATION.
540	The Egyptian system as we find it at the beginning of the Dynastic Period, and as it
541	continued throughout history, was decimal." A unit was represented by a vertical stroke 1, two
542	by two strokes, and so on up to 9. Ten was represented by N, 20 by two such signs, and so
543	on up to 90. For 100 a new unit e appears, and this repeated the requisite number of times
544	served from 200 up to 900. For 1,000 qwas used, for 10,000 ), for 100,000 E, and for.
545	1,000,000g: Thus 143,257 would be written sIil! idfeemn !lI nnli
546	In the cursive ink-written script known as hieratic many of these numbers took on
547	ligatured and contracted forms, the four strokes, for example, being shortened into a horizontal
548	line. Hieratic forms for the numerals already existed as early as the First Dynasty,* and ran
549	through Egyptian history until replaced by the demotic in the Persian period.
550	Despite the fact that in historical times the system is definitely decimal it contains faint
551	traces of having originally been quinary. The evidence for this is too intricate to be discussed
552	here, but the main points of the latest pronouncement on the subject, that of Jéquier, are
553	as follows. The numbers from 1 to 5 have names resembling the African (Hamitic) names,
554	and are part of the African inheritance of the Egyptians. The numbers from 6-10 have names
555	offering some analogies with the Semitic names and are a later acquisition. The tens from
556	10 to 40 have special names which correspond neither to those of the Egyptian 1-5 nor to
557	those of the tens in either Hamitic or Semitic languages. The tens from 50 to 90 are formed
558	from the numbers 5-9, of which they are perhaps plural forms. These results must not be
559	regarded as final, and will doubtless meet with considerable criticism. What would appear
560	almost certain, however, is that there are remnants in the Egyptian system of a primitive
561	quinary system based on finger numbering (the number 5 was represented by the figure of a
562	hand), complicated by a later extension to a decimal system formed by the addition of the
563	second hand. Exactly what portions of this system are due to African and Semitic origins
564	respectively is still a matter of almost complete conjecture.
565	The defects of this system are obvious. In the first place it was cumbrous, for in
566	order to write such a number as 879 no fewer than 24 signs had to be made. This was to a
567	certain extent neutralized in hieratic, where almost every unit, ten, hundred, and thousand
568	developed a contracted form. The other defect of the system was the absence of anything
569	in the nature of value by position, a disadvantage which it shared with the Greek notation and
570	which was only cireumvented by the Arab mathematicians, who are said to have derived
571	positional notation from the Hindus and who passed it on to us. See, however, p. 28.
572	As against these defects the system had one virtue which the Greek could not claim:
573	it lent itself admirably to multiplication and division by 10, for in order to multıply 98 by 10
574	it was only necessary to turn the 8 units into ten-signs and the 9 ten-signs into hundred-
575	sıgns. The result of this was that multiplication and division by 10 played a large role in
576	the elementary processes of Egyptian reckoning.
577	2. THE SIMPLE ARITHMETICAL PROCESSES.
578	committed Thus, when I say 8 and 7 make 15 I am not performing a basic unhertul poes io friameic, tbat of constine o Wiat me
579	process, I am merely repeating a fact which I know from memory, and the child who says
580	1 In the reign of Narmer, Ist Dynasty or just before, we find the notation in full use up to 1,000,000; QUIBELL,
581	Hieraconpolis I, PI. XXVI, B.
582	2 See, however, Gunn in J.E.A., III, 280.
583	3 For the ring-sign in later times see ibidem.
584	4 PETRIE, Royal Iombs of the First Dynasty, I, PI. XIX, 11.
585	• Recueil d'Études Égyptologiques Champollion, 1922, 467 f. Cf. SETHE, V.Z.Z., 24-26.
586	c 2
p. 16
587	12 RHIND MATHEMATICAL PAPYRUS
588	8 and 7 are 14 is not making an error of calculation, but merely one of memory. If I wish
589	to prove that 8 and 7 really do make 15 I must count out 8 objects, then 7 more. I must
590	then count both lots together and I shall get 15. Just as addition—and therefore also sub-
591	traction—is a pure act of memory, except perhaps in simple cases such as 1 and 1 is 2, where
592	we may almost be said to count, so, too, multiplication and division are mere acts of memory.
593	When we say 9 multiplied by 6 is 54 we do not count, we merely repeat a fact learnt by
594	heart. To prove that it is so we must make 9 rows of 6 objects each and count right through
595	from 1 to 54.
596	The result is that the ability of a nation or an individual to make rapid arithmetical
597	calculations depends in a great measure on memory equipment. If I have all the multiplication
598	tables from 2 times to 19 times in my head I shall in general perform arithmetical calculations
599	with more speed and comfort than one whose equipment does not reach beyond 12 times 12.
600	How did the Egyptian stand in this respect? At the outset he possessed one slight
601	advantage over us in the matter of addition, for the very nature of his hieroglyphic notation
602	enabled him, if he so wished, to dispense with most of the memory work so familiar to us.
603	To write down 8 he had to make 8 strokes, and to write down 7 he had to make 7 strokes.
604	The consequence was that when he had to add 8 and 7 the 15 strokes were all actually there
605	before his eyes, and all he had to do was to count them. Similarly 80 and 70 could be added
606	by mere counting, and 800 and 700 and so on. In the same way subtraction was a mere
607	matter of counting. This must have been extremely pleasant for the Egyptian schoolboy, but
608	it must have reacted disastrously on the development of the arithmetician, for it is clear that
609	where there is little incentive to memorize little memorizing will be done. At the same time
610	it is certain that the power of adding numbers by memory was developed among the
611	professional mathematicians and accountants, otherwise they would never have evolved the
612	hieratic numerals, in which the separate strokes etc. are no longer to be discerned. We may
613	therefore credit the Egyptian with a certain facility for addition of simple numbers by memory.
614	The technical phrases for addition are mainly derived from the common use of the
615	preposition hr in the sense of "in addition to," doubtless a very early derivative from the
616	literal meaning of "on." Thus in No. 26 we read, (A number) fif Ir.f, "whose fourth part is
617	added to it": here we have a simple nominal sentence. More often, however, a verb is
618	added, either wih or dit, both meaning "to put" or "place" In No. 72 we find w:l-lr-k
619	100 hr-s, "You are to add 100 to it"; and in No. 22, 1 + 1o m wih Irf,"1 + 1o is what is
620	added to it," where wih is presumably a Neuter Passive Participle. The use of w;l in this
621	connection is doubtless very primitive, the meaning being almost literal, "to place upon.":
622	In Nos. 41 and 42 dit is used in place of wih.
623	The verb dmd, "to unite," can also be used of adding: in No. 52 we have
624	dmd-hr-k A lır B, "You are to unite A and B." The invariable noun for "total" is, as in
625	the account papyri, dmd.
626	The usual verb for to subtract is hbi, determined by the cross sticks which Grapow * has
627	i from 9." The word used for "remainder" is the familiar dit or wdit of the account papyri.®
628	In Nos. 21-23 a process amounting to simple subtraction is indicated by the verb skm,
629	1 Or an abstract noun. Cf. Pap. Bulaq 18; A.Z., 57, 55.
630	2 Or is it short for w3lı tp "count," for which
631	3 For a different use of lbż see Nos. 54 and 55.
632	4 Ä.Z., 49, 116 ff.
633	" hbt hft to in No. 82 means "to subtract at the rate of to." not "from tu.
634	• For a discussion of this word see SPIEGELBERG, Rechnungen aus der Zeit Setis I, Text, 40-41 and 49.
p. 17
635	RHIND MATHEMATICAL PAPYRUS 13
636	"to complete."1 That this is not specifically a word for subtraction is clear from the examples
637	Nos. 7-20, where it is used ot making one quantity up to another by the addition of aliquot
638	parts of itself.?
639	and is indeed Theeed apase nn to meultipliation and nitl epied eient, ya ie by shen actine n o try,
640	by any numbers except 2 and 10. The latter indeed did not even involve an act of memory,
641	for it is clear that in a decimal system with a notation of the Egyptian type all that has
642	to be done to multiply by ten is to turn unit-signs into ten-signs, ten-signs into hundred-signs,
643	and so on. Thus 45, nnnn'!!
644	process division by 10 could clearly be accomplished in a manner which was purely mechanical,
645	and which, it will easily be seen, was hardly made the less so by the development of the
646	hieratic numerals referred to above.
647	Multiplication by 2, however, was a true mnemory process with the Egyptian. He not
648	only had tables giving him its results, but he almost certainly knew them by heart. Further
649	than this, however, he did not go. He never multiplied by any higher number, except of
650	The result was that in order to perform multiplication by other numbers he had to
651	make what shift he could with 2 and 10. Thus 12 times a number was obtained by adding
652	To multiply 15 by 13 the following process was gone through :—
653	/1 x 15 = 15
654	2 x 15 = 30
655	4 x 15 = 60
656	8 × 15 = 120
657	Total 13 x 15 195
658	Here the Egyptian merely kept on doubling; he noticed that the multipliers 1, 4 and 8
659	added up to 13, and that therefore the products corresponding to these must amount to 13
660	times 15. To simplify his work he ticked off the multipliers in question and then ran down
661	the right column adding together the products opposite the ticks. It will be seen that in this
662	using no multiplier other than 2, any required multiplier could be arrived at.
663	process coula in many cases be shortened by the use of the mutipher 10, Which by doubling
664	so on, but in general it would seem that the mathematician of Rhind
665	preferred to work solely in powers of2.
666	Division was accomplished by reversing this process. Thus, to divide 77 by 7 we do as
667	/1x7 7
668	2x7 = 14
669	4x7 = 28
670	<8 ×7 56
671	Total 11 x 7 77
672	77. We therefore tick off the lines containing those products and add up the corresponding We note that the "taree products 1, 14 and56 add up precisely to the required
673	multipliers, 1, 2 and 8, giving the answer11.
674	Old and Middle Egyptian w:l tp is This process of multiplication or division ot dindin the lend ato the Bgap ibe glial i, up that
675	1 For a somewhat similar but perhaps intransitive use of km see No. 37 and notes thereto.
676	= In No. 28 addition and subtraction are indicated by the sign of the human legs facing to the right and
677	the left respectively. For the reading of these signs see the notes there.
678	3 E.g. Pap. Bulag 18 (MARIETTE), PI. XXVII, 2, 20. Cf. A.Z., 57, 61.
p. 18
679	14 RHIND MATHEMATICAL PAPYRUS
680	"nodding the head" was a primitive operation in the process counting, and so perhaps
681	became the phrase for "to count," the head being nodded, not necessarily at every digit, but
682	possibly at every five or ten counted off on the fingers. At any rate, wihtp (often irt' w3h-tp)
683	m 4 p° spw 5 means literally "count (make a counting) with 4 5 times," or "multiply 4
684	by 5." Similarly "divide 77 by 7" was rendered wih tp m 7 r gmt 77, "count with 7 to
685	find 77." This general sense "to count with" is well illustrated by such examples as No. 43,
686	where wsh tp m 8 "count with 8," or even more generally "operate on 8," is followed by
687	"you are to add to it one third of it; it becomes 10%." Here the specific translation
688	"multiply" or "divide" is impossible.
689	Division, as we have just seen, is usually expressed by wih tp. There is, however,
690	anotber technical term for the process, namely nis A hnt B, "divide A by B."* This can
691	be used whether A is greater or less than B, e.g. No. 66, nỉś 3200 hnt 365, "divide 3200 by
692	365: result 8}+to +z19" and No. 35, nỉs 1 hnt 31. The verb nis means to "call" or
693	"summon," and the preposition lnt must here have its usual sense of "out of" or "from
694	among." Possibly the original picture is that of one counting up to A and calling out at
695	every Bth number, or possibly the term is purely a metaphor. If the former be the correct
696	explanation the use in cases where B is greater than A must be a later extension. It is
697	worthy of notice that this term is used in the statement of the so-called resolutions of fractions
698	whosenumerator is2.Wherewesay "resolve?"the Egyptian merely said nis 2 hnt X,
699	"divide 2 by X" (see below).
700	Two other cases of niś must be noted here. In No. 44 we read tp n niś ší ifd,
701	"example of reckoning a cubical container," i.e. of finding its content in corn, and in No. 56
702	we have tp n nis mr, "example of working out a pyramid," the problem being to find the
703	slope of its sides, given its base and its vertical height. It is difficult to see how these more
704	complex uses of the term could be derived from its technical sense of divide, and they are
705	more probably metaphorical uses of its literal meaning of "call" or "summon."
706	The papyrus contains a few rare examples of multiplication or division direct by numbers
707	greater than 2. Thus in No. 37 we find one third of 30 given as 10, with no working or
708	explanation, and also one third of 90 as 30. The first of these is merely the reverse of a division
709	by 10, and as for the second it will be seen below that division by 3 was well within the powers
710	ot the reckoner, though it usually necessitated two steps. Facts of the type * x x = 1 also form
711	a frequent exception to the general rule, apparent only, since they do not in reality involve
712	division at all.
713	The result of the process of multiplication or division, and indeed the result of any mathe-
714	matıcal process, is expressed by the verb hpr "to become," generally followed by the prepo-
715	sition m. The most usual method is to use the śdm-f or sdm-hr.f form of the verb with the
716	Neuter (Masculine in form) 3rd Singular suffix pronoun.
717	hpr.hrf m 20, "it becomes 20." Sometimes the resulting figure is used as subject: hpr.hr 4,
718	"4 results" (No. 5). In Berlin Pap. 6619 we once find hprhr m 1f without apparent subject,
719	but this may be an error, though such an omission of subject is not at all unusual in Egyptian.®
720	expressing result is hprt im pw 1200 (Nos. 76 and 78;
721	of. Pap. Kah., PI. VIII, passim), a regular nominal clause in which hprt is the Neuter Active
722	Participle, " 1200 is what results therefrom."
723	1 In No. 30 irt alone without wih tp is used, in the form irhr-hi, of division. That this is not an accident
724	is clear from a second use in the passive in the same sum, and from Pap. Kah., PI. VIII, 27, 37 and 39.
725	2 r is often omitted.
726	3 In No. 57 we have niś lft, perhaps merely in error for lnt.
727	* The Egyptian for "take a quarter of it" is irt fif, literally "make its quarter." Cf. Nos. 26 and 44.
728	s Similarly, perhaps, in No. 62, though the following Relative Form didi-k may here have been regarded as
729	grammatical subject to lpr despite the intervention of m 4. See, however, p. 105.
p. 19
730	RHIND MATHEMATICAL PAPYRUS 15
731	3. FRACTIONS.
732	The Egyptian fractional notation was a very simple one. With the sole exception of
733	} no fraction was ever written which had a numerator greater than unity,! or in other words,
734	with this same exception, all Egyptian fractions on paper were aliquot parts, }, 3, k, %, etc.
735	If in the course of a problem, owing to a multiplication by 2 (the only digit except 10 by
736	which the Egyptians multiplied directly) a fraction arose or threatened to arise whose numerator,
737	say, was 2, it was immediately resolved into the sum of two or more
738	were unity. In other words, the Egyptian never wrote the fraction 7r; if he
739	TT by 2 the result was f + 16-
740	How far this was a consequence and how far a cause of the very restricted notation of fractions
741	it would be difficult to say. To write one-thirteenth the Egyptian simply wrote the numeral
742	13 underneath the sign • (reduced in hieratic to a dot). This sign, as Sethe has shown,?
743	must here mean "a part." The only fractions not expressed in this way were ≥, $, f and }, for
744	which special hieratic signs existed. Just as doubling formed the basis of all multiplication, so
745	halving lay at the root of most division. Iwo of the most important measures in Egyptian
746	daily life, the acre (śt;t) and the bushel (hlc;t) were divided up into halves, quarters, eighths, etc.
747	These divisions may go back to a stage in reckoning even more primitive than that of simple
748	aliquot parts, a stage when only the } and its powers 4, 3, 16 and ss were used.
749	In this connection the parts of the acre are of special interest, for f-acre is written with
750	the sign →, which reads rmn "arm," sometimes " side," and which may be an earlier
751	word for in the general sense than the better known ==, gs. Still more important is the
752	sign for f-acre, which is a cross, x, or to be more exact a pair of sticks crossed. We know
753	that in late times this f-acre was called lısp, and Sethe " points out that this is the same as
754	the earlier lsb, a word meanung "to break." Combining this fact with the pictogram of the
755	that this old word for f was the fraction or "breaking" par excellence, a conjecture which is
756	borne out by the fact that hśb was in historical times the word for "to count" or "reckon."
757	In hieratic the crossed sticks remained throughout the sign for f, but in hieroglyphic they were
758	replaced at an early stage by the normal I except in the special senses of f-acre and
759	4-bushel.
760	Side by side with these early dimidiated fractions there must have existed in quite early
761	times another set based on division into three, since in this way we can best explain the
762	unique position of } among Egyptian fractions and the existence at all periods of a special
763	hieratic sign both for this and for }. Originally 3 was written in hieroglyphic thus iP,
764	with the old word r meaning "a part" and two equal strokes attached to it under its left-
765	hand end. Later the two strokes worked their way to the centre and one became longer
766	than the other, so that the sign came to be written T or #. The original form of
767	the sign leaves little doubt that the group was originally called by the Egyptians, as by most
768	nations, "two-parts." That it was the
769	Egyptian mind is clear from thefact that in mathematics one-third of a number or quantity
770	was invariably found by first obtaining two-thirds and then halving it. The curious writing of
771	1 in hieratic, which cannot possibly be brought into relation with the normal hieroglyphic ii,
772	is undoubtedly to be traced back to a primitive system of division into three, indeed Möller and
773	Sethe have suggested that the earliest form of the hieratic sign may well be derived from the
774	word "part," with a single vertical stroke attached to it under its left-hand end,*
775	meaning " one part," division into three being assumed.
776	The precise nature of the Egyptian fractional system and its methods of working has
777	1 There are very rare examples of , written as "three parts" analogously to }, "two parts" (see below).
778	= V.Z.Z., 85-87. 3 V.Z.Z., 75-78. * V.Z.Z., 82.
p. 20
779	16 RHIND MATHEMATICAL PAPYRUS
780	been analysed by Hultsch in a long study of considerable complexity, entitled Die Elemente
781	der ägyptischen Theilungsrechnung.! He rightly begins by pointing out that the result of any
782	divisional process in Egyptian must be arranged in whole numbers followed by a series of
783	aliquot parts in order of magnitude. Thus the result of dividing 2 by 13 was written
784	8 + 5g + To#• This is in fact a question of notation. Just as there is a notation for whole
785	numbers in tens, hundreds, etc., so there is a fixed notation for quantities less than unity,
786	namely the series }, ½, }, ł, }, etc.; after all this is not so unlike the modern decimal
787	notation, where the quantities lower than unity are expressed in terms of 1o, 100, rooo, etc.
788	Hultsch now goes on to say (p. 9), "Was nach ägyptischer Anschauung Vielheitstheilungen
789	oder noch nicht zu Ende geführte Divisionen waren, das sind für uns Brüche mit Zählern, die
790	grösser als 1 sind." "What the Egyptian looked upon as divisions of numbers greater than unity
791	or unaccomplished divisional processes are to us fractions with numerators greater than 1." If
792	this means that the Egyptian had no conception of a fraction whose numerator was greater
793	than unity, and that he would have regarded our fraction only as 5 divided by 8, and never
794	as 5 eighth-parts of unity, it is too sweeping a statement. It is true that he had no notation
795	for such quantities, but the argument from what he was capable of expressing in symbols
796	to what he was capable of conceiving is a non sequitur, and the suggestion that his notation
797	must surely have kept pace with his conception will fall on deaf ears in the case of those
798	acquainted with the amazing conservatism of the Egyptian mind in every branch of life.
799	What is more, there are at least certain cases in which it is obvious that a fraction
800	with numerator greater than 1 was conceived and that very clearly. Thus Sethe's researches
801	have shown from the earliest writings of the symbol for } that this fraction was originally
802	written "the 2 parts," ie. the 2 parts of a unit conceived as divided into three, just
803	as we now speak of "three parts," meaning }. Similarly the Egyptian reckoner, when
804	doubling fractional quantities, is wont without any discussion to replace twice 1z by ‡: in these
805	cases, where the unit in his mind is indisputably 1, it seems idle to deny that he reasoned
806	through 7z (though he could not write it) to , or to assert that his mind-process was "the
807	unaccomplished division 2 by 14 is the same as 1 by 7." The same is true of his method of
808	adding t7 and T4. His mental picture was not "1 divided by 14 added to 1 divided by 14
809	amounts to 2 divided by 14, which is the same thing as 1 divided by 7," but simply "If a
810	unit is divided into 14 parts and 2 of them are taken the result is one-seventh part of the
811	unit"; this is clear from his conception of aliquot parts and from his ability to double them.
812	If we are to suppose that } stood merely for the division of 2 units by 3 and not for 2 third-
813	parts of a unit, how are we to explain the Egyptian's ability to double } and obtain 1$, a
814	direct process which occurs over and over again in the papyrus ? If Hultsch's theory were
815	correct the result of doubling 2 divided by 3 could only be 4 divided by 3, whereas what the
816	Egyptian actually gets is a unit plus its third part.
817	It would thus appearthat the Egyptian had a perfectly clear conception of fractions
818	of the type n in the sense of two nth-parts of a unit, and not merely in the sense of 2
819	divided by n. There are not wanting suggestions which take us even farther. In No. 18 the
820	fractions f + 5 + ts are to be added, and the answer is set down without any working as f
821	How was this done, and why is the method of common denominator? or bloc extractif not
822	shown here as in the companion examples? Probably because f + is was, from the table
823	at the opening of the papyrus, seen to be equivalent to two-ninths and the addition of the
824	other t gave three-ninths or 3. The same process is employed in No.17.
825	Indications of this kind are not to be neglected, and we cannot therefore follow Hultsch
826	in his sweeping assertion. The conception of fractions with numerators greater than unity
827	1 Abhandl. der Kgl. Sächs. Gesellsch. der Wiss., phil.-hist. Classe, Band XVII. Leipzig, 1895.
828	2 See below, p. 17-18.
829	3 Surely the process &X‡= *, common in the papyrus, involves the conception of & as four twenty-fourth
830	parts of the unit.
p. 21
831	RHIND MATHEMATICAL PAPYRUS 17
832	seems to be inherent in some of the processes of Egyptian mathematies, but the notation did
833	not keeppace with it.
834	In order to appreciate the powers and limitations of the Egyptian mathematician in
835	dealing with fractions it is essential to bear this notation in mind throughout. What enabled
836	him to keep such a simple apparatus undeveloped throughout ages was the fact that he never
837	learned to multiply directly by any other number than 2. The result was that in his processes
838	he was rarely likely to be threatened with worse fractions than doubled aliquot parts, e.g.
839	"twice the thirteenth part." He had discovered that all such quantities could be resolved
840	into the sum of two or more aliquot parts, and had actually worked out tables for such reso-
841	lutions, running from twice a fifth-part to twice a 101st-part. This simple apparatus, a copy
842	of which begins our Rhind Papyrus, saved him the trouble of evolving a more complicated
843	fractional notation.
844	It is apparent from the Egyptian notation and general conception of fractions that their
845	addition and subtraction could only play a limited part in mathematics. Since none but
846	aliquot parts were used an answer in the form 1% + † + I's was in no way repugnant to the
847	occasions Egyptian mind, and even if it had been the notation offered no remedy. There were, however, when additions had to be made, generally in sums where some fractional quantity
848	had been multiplied by some other quantity, whole or fractional, and it was required to show
849	that the result was equal to some whole number or to some quantity involving only very
850	simple fractions. Thus in the proof of No. 32 we have toshow that (14+t=+T17+328)
851	x (1} + 4) = 2. The working is as follows:—
852	+T= + TI#+ 228
853	+18+ 35+312+=87
854	+*++is+=3+t2
855	Remainder 1
856	1ettteette+*+t*++**+=+512
857	Total 228 ie. ł
858	In the first line the multiplicand is set out with the multiplier 1 before it. In the
859	second line it is multiplied by }, and in the third by 4. These three products have now to
860	be added. The simpler quantities 16 +, + clearly give 1f + *, since - + ] is }, a com-
861	bination well known to the Egyptian and frequently used. We have now only to show that
862	the sum of the more complicated fractions to the right of the vertical line amounts to the
863	remaining 4. Here the Egyptian employs a method which at first sight appears to be that of
864	a common denominator. All the fractions or aliquot parts seem to be reduced to terms of the
865	highest aliquot part,' namely the 912th part: under each fraction is placed in red (here repre-
866	sented by italics) the number of 912ths contained in that fraction, a number which, it will be
867	observed, is not in all cases a whole number. This step must involve a certain amount of
868	rough working, which is always omitted in the papyrus. The red figures are now added and
869	seen to come to 228, which is 1 of 912. Therefore the sum of all these fractions is the
870	required 4, and adding on the already obtained 1} + f we get 2 for the product of the two
871	original quantities.
872	This method small details from the modern method of common
873	For instance, we always choose as our denominator the smallest number into
874	which all the separate denominators will divide integrally. The Egyptian, unfamiliar with
875	the principle of factors, often used a denominator which was smaller than the L.C.M., and
876	1 In No. 33 the number chosen is not the highest aliquot part.
877	D
p. 22
878	18 RHIND MATHEMATICAL PAPYRUS
879	consegurn to med fra tion fuquet ente rarely int olvie numarltorfactione frati os tt ie hardly.
880	were all unity.
881	These, however, are distinctions of mere detail, and despite them the general principle
882	involved might be the This is denied by both Hultsch' and Rodet. The former
883	would describe the process given above as follows? The unit employerwas originally 1, and
884	this unit (Stammeinheit) was kept so long as addition in terms of it proved leasible, ie. up
885	to the point where the fractions had been added up to 1f + 1. Then, according
886	to Hultsch, to facilitate the addition of the complicated fractions to the right of the vertical
887	line a new unit is chosen (Hülfseinheit), namelyt= =1. The aliquot parts to be added are
888	now multiplied each by 912, that is to say, they "are transformed into multiples or parts
889	of the Hülfseinheit." The results
890	us to "return to the Stammeinheit," which in this case is unity.
891	But here again the question is surely one merely of notation. What is done is in
892	what happens in our own method of
893	We may clothe this in whateverwords or signs we like; the factsremain We
894	may say that the Egyptian uses a new unit, namely ałz, but what else do we do when
895	we choose a common denominator 912? We have only to look at Hultsch's attempt to word
896	the problem in order to see this. For instance, on p. 112 he describes the following addition:—
897	30 258
898	9 18 24 1
899	His concluding words are "Hierauf folgt im Texte die Summe mit den Worten 'zusammen
900	1 72. Das soll bedeuten 'zusammen 72 Hülfseinheiten (deren jede = v53 ist), das ist ‡ der
901	Stammeinheit.'" It is clear that if the numbers which are added to give 72 are all Hülfsein-
902	heiten, ie. 288th parts of unity, then the process is precisely the modern method of common
903	denominator, and the Egyptian is simply replacing J≥ by nine-288ths, a conception which,
904	incidentally, Hultsch himself has denied to him (above, p. 16).
905	Rodet adopts an entirely different view. He says, "il est bien certain qu'Aahmesu
906	ne réduisait pas ses fractions à un dénominateur commun, mais que, comme on l'a fait après
907	lui pendant vingt-six et trente siecles encore, il choisissait un nombre, bloc extractif, fonds
908	commun ou comme on voudra l'appeler, d'où il puisse tirer toutes ces fractions, soit, comme
909	ses successeurs, à l'état d'entiers, soit, comme il s'en contentait, à l'état d'ù peu près entiers,
910	mais, dans ce cas, avec une fraction d'expression simple; et c'est sur les substituts ainsi
911	obtenus pour ses fractions qu'il opérait." Rodet supports this view by references to
912	eastern mathematicians of the Middle Ages and later, by whom the method which he describes
913	appears to have been practised. The number, called by Rodet bloc extractif, seems to have
914	been called môré by Aben-ezra (12th century A.D.) and mokhrag by Mahmid of Herat. The
915	mokhrag of a group of fractions is an integer which by multiplication will turn each of the
916	fractions into an integer. It is in fact a common denominator viewed from a slightly different
917	Thus if we wish to add 1 and ! we reduce them to the common denominator 20: we
918	that is 5-twentieths and that \| is 4-twentieths, total 9-twentieths. The method of
919	mokhrag is accoiding to Rodet different. Here, unable to add and! in terms of the unit l,
920	the reckoner takes the mokhrag and adds 1 and , of that, result 9; answer
921	difference is purely one of notation. The Arab mathe-
922	matician avoids actually saying that 1 is equivalent to 5-twentieths, but he is bound to admit
923	it tacitly, for in the end his mokhrag must become a denominator and he confesses it when
924	he uses the word "twentieths."
925	1 Op. cit., 9-10. = Op. cit., 112.
926	3 Journal Asiatiyue, 1881, 196-215.
p. 23
927	RHIND MATHEMATICAL PAPYRUS 19
928	The fact is that both Hultsch and Rodet have been deceived by notation. There is
929	and can be only one way of adding fractions, though there may be several ways of writing
930	down the process. The fractions f and , are quite irreconcilable as they stand, and we can
931	only combine them by reducing them to some smaller part of unity of which they are both
932	multiples. We may do this in the modern way by means of the common denominator 20, or we
933	may do it in the Arab way by means of the mokhrag 20. Avoid the notation zo as we may,
934	we cannot in the end escape the fact that the 20 really stands for the twentieth part of some
935	unit, and that the 5 is 5-twentieths of that unit. The process as seen in the Rhind papyrus
936	is particularly deceptive since all the complicated additions there used are in the nature of proofs,
937	i.e. the result is known to be some very simple aliquot part, e.g. f or %, and though our
938	denominator or mokhrag may be 960, the fact that we are really working in 960ths is apt to
939	be overlooked or forgotten when the addition comes to 240 or 120, and the 960 drops out of
940	sight, leaving only a simple f or .
941	The ease and accuracy with which the Egyptian dealt with these very complicated-
942	looking fractions often compel our admiration. At the same time, in matters of everyday
943	occurrence they were probably not very frequent. As will be seen below, the measures of
944	capacity and of area were largely based on the principle of halving, with the result that the
945	fractions involved were very easily added. Thus 3s + 35 at once gives 1%, which can be
946	taken up with another i giving }, which will combine with another } to give ‡, and so on.
947	The papyrus contains no definite example of the subtraction of fractions except such
948	simple cases as 2 less 11 + ‡ = 1. See, however, p. 58.
949	As in the case of whole numbers, so in the case of fractions the only multipliers in general
950	use were 10 and 2, though, as will shortly be seen, the anomalous } opened up possibilities
951	of division by 3 which were not neglected.
952	it is obvious that a fraction could at once be divided by 2 or by 10 by simply multiplying the
953	denominator, as we should call it, by 2 or 10.4 Thus the Egyptian had no difficulty in seeing
954	that \| x to was so, or, in other words, that if you divided a tenth aliquot part of a whole
955	into 5 the result would be 50th aliquot parts. Multiplication, however, was a much more
956	notation adopted.
957	ex-
958	sum of two or more fractions whose numerator was 1. He had even drawn up a table of such
959	resutns fom 3 3, ฿ up to 3g and rổi, the fractions with even denominators being of
960	course omitted, since he saw that they reduced at once to aliquot parts (62 = 3r). Since he
961	never multiplied by a number greater than 2, it is clear that this table sufficed for any
962	fractional operations arising in the course of multiplication.
963	Thus to multiply (1,+ 4) by 24 we proceed as follows:—
964	13 +
965	-2 3/+/(3being/+=)
966	++*r
967	The second line is got by doubling the first and resolving the twice by table into #+ 2's.
968	The third line is 1 of the first. We now tick off the required multipliers and add the corre-
969	sponding products. The two as give and the two zgs give d. Result 3}+3+ t*.
970	1 A fifth was obtained by doubling a tenth.
971	- Fractions are practically never multiplied by I0 exeept when their denominator is a multiple of 10. For
972	an exception see No. 4, where the simple 3+! is multiplied by 10 and gives 7. In No. 56 tu of (1} + Ts) is
973	stated, without proot, to be ro + ds.
974	D 2
p. 24
975	20 RHIND MATHENATICAL PAPYRUS
976	Division, as in the case of whole numbers, is simply the reversal of this process, trial
977	multipliers being taken until in the products-column products can be seen which add up to the
978	required dividend.
979	It is clear that by the introduction of fractions the bounds of Egyptian multiplication
980	and division have been greatly widened. They were still further widened by the ingenious use
981	made of the anomalous fraction two-thirds, "the two parts." Strange as it may seem to us,
982	tlıe Egyptian was accustomed to take two-thirds of a number by a single process. No doubt
983	he used tables for the purpose, but the mere fact that tables existed is one more testimony to
984	the fundamental nature of the concept of "the two parts" in the Egyptian mind. Most
985	of 5 would do it by taking one third and doubling it.
986	So far was the Egyptian from doing this that his sole means of finding one-third of a quantity
987	to take two-thirds and then halve it.' In the case of fractions he made use of the
988	equation 3 = } + đ, see No. 61B, so that all he had to do was to multiply the "denominator "
989	of the fraction by 2 and then by 6, thus:-
990	3ot,=ot3a
991	third, which in its turn could be halved to give one sixth; one twelfth, and so on; in other This ability to take two-thirds opened up a new series of divisions, for halving it gave one- o toiudh bc h vi o mo
992	words,it facilitated division by 3,6, 12, 24, ete.: No. 32 is a good example of this.
993	As a general rule the operations performed in the papyrus are accomplished before our
994	Thus in No. 37
995	} of } is stated to be 1, and } of ft is given as 1 +, the necessary two-thirds line
996	being omitted. Other cases would seem to involve a conception which comes near that of the
997	improper fraction. Thus in of 3] is stated to be ! in No. 35, and in No. 38 dz of 3} is said
998	to be 4, but the daring nature of this statement is felt to demand an apologetic explanation,
999	which is solemnly given.
1000	4. OTHER MATHEMATICAL PROCESSES KNOWN TO THE EGYPTIANS.
1001	a. Square and square root.
1002	That the conception of squaring was familiar is known to us from the problem of the
1003	truncated pyramid from the Moscow Papyrus.* Here the phrase used for "square 4" is
1004	irlır-k 4 pn m A," "You are to make this 4 in square." No conjecture can be hazarded as to
1005	the reading of the sign here used for "square": from the point view of mathematical
1006	clarity it is unfortunate that the same sign should be used in Rhind No. 28 for addition.
1007	No example of square root occurs in Rhind, but Pap. Berlin 6619, Pap. Kahun PI. VIII,
1008	1. 40, and Pap. Moscow (unpublished) show that the idea of square root existed and that the
1009	technical term for it was lenbt, literally "corner" or "angle," the idea presumably being that
1010	the original number, say 16, represented the area of a square, while the length of each of the
1011	two sides containing any corner of it was its square root, ‡. The Egyptians were even
1012	capable of taking the root of quantities involving simple fractions, the quantities whose root
1013	is taken in Berlin 6619 being 61 and 11 + 1-
1014	6. Solution of equations.
1015	It will be seen below that the problems Nus. 24-38 involve the solution of equations
1016	of the first degree with one unknown by means of a simple method of trial.
1017	Equations of the second degree where there is virtually only one unknown were also
1018	understood. the Berlin Papyrus 6619 (see above, p. 6-7) we have to divide 100 square
1019	1 There is an apparent exception in No. t2. where oath of a quantity is got direct from 6th of it instead of
1020	through Ith (i.e. * ot (tl).
1021	* Ancient Egypl, 1917, 100-102. * Here facing as in the hieratic.
p. 25
1022	RHIND MATHEMATICAL PAPYRUS 21
1023	cubits into two squares whose sides are to one another in the ratio l to 4.
1024	problem y in the Kahun Papyrus (see above, p. 6) involves a similar equation.!
1025	c. Progression.
1026	No. 64 contains the solution an arithmetical progression, given the sum and the
1027	common difference. In No. 40 we are given the number of terms (5), the sum, and the fact
1028	that the sum of the two lowest terms is one-seventh of the sum of the three highest,
1029	we are asked to find the common difference. The word here used for common difference is
1030	sight it would appear to be a technical term used of arithmetical pro-
1031	This, however, can hardly be the case, for it is used in a somewhat different sense
1032	in No. 39, where there is no progression, and we must therefore suppose that the use in No. 40
1033	is a specialized employment of a more general term.
1034	No. 79 contains a geometrie progression whose first term is unity. The solution (g.v.)
1035	that the Egyptian had a very clear grasp of the nature of a series of this particular
1036	type, which he sums neatly and accurately, but does not tell us whether he was equally at
1037	home with a series in which the first term was not unity.
1038	5. GEOMETRY.
1039	The areas of the square and rectangle were correctly estimated; that of the circle
1040	was found by squaring 5 of its diameter, a very fair approximation. triangle
1041	see pp. 51 f. The volumes of cube and rectangular parallelopiped were known, while the
1042	Moscow Papyrus gives the correct solution for a frustrum of a regular pyramid on a square
1043	The volume of a cylinder was got by multiplying the area of its base
1044	obtained as alove by the height. F'or the determination of an angle by its cotangent
1045	METHOD OF SETTING OUT THE SUMS.
1046	Each sum consists of: 1. Title; 2. Statement of problem; 3. Working out; and 4. Proof.
1047	The title begins with the words tp n "Example? of." Thus in No. 52 the title
1048	reads "Example of calculating a trapezoid of land." Very typical of these titles is the use
1049	of the verb int "to make" or "to do" in the sense of "calculating" or "dealing with."
1050	An extreme case is seen in Nos. 2-6, where irt t: X replaces the "Example of dividing (psš)
1051	X loaves" of No. 1.
1052	The statement of the problem introduces the data and should begin with the
1053	words mi dd-(w0) nk, "If they say to you," ie. "If you are asked." In No. 52 this
1054	section runs, "If you are asked: a trapezoid of land, 10 khet in its mryt, 6 in its base and
1055	4 in its cut side, what is its land-content ?" In some cases, e.g. No. 68, we have ir dd nk sš,
1056	"If the scribe says to you," in place of mi dd•(v) nk. This is followed by śdm-f, "Let him
1057	Of the working vut little need be said here except that the directions given
1058	generally in the verbal form śdmlyrk," which we may perhaps render in English by the slightly
1059	antiquated form, "You are to do so and so."
1060	1 See further Sımow, Geschichle der Mathematik im Altertum, 41-2.
1061	2 This meaning scems to be established by the last words of No. G6, "You shall do likewisc in the case
1062	of anything asked of you similar to this example.
1063	firstly to express a polite command or direction. " The use of this form is typical of all the mathematical papyri as of the medical. and secondly to express the It has two distinet uses,
1064	operation, lprlrfm I, "It becomes X. In No. 55 it is preceded in this last sense by the particle ler.
1065	of tlis unusual and archaic vorbal form is doubtless due to the lact that the diction of the sciences of
1066	mathematics and modicine was probably fixed in the Old Kingdom, and continued almost umaltered into the Middle
p. 26
1067	22 RHIND MATHEMATICAL PAPYRUS
1068	4. The proof consists, as with us, in showing that the result which has been obtained
1069	bythe working actually satisfies the conditions of the problem. In some cases it concludes
1070	with the words mitt pu or nt pu, "That is the same" or "That is it," indicating
1071	figure arrived at in the proof by performing upon the answer found the operations prescribed
1072	in the setting out is actually that originally set; in other words, the phrase is equivalent, to
1073	The Moscow truncated pyramid problem "See, there you have it, 56, you have
1074	found it successfully (gmn-k nfr)," and Pap. Kahun, PI. VIII, II. 55-62, concludes witl the formula
1075	of the literary papyri, iwf pw, "It has come to an end."
1076	When we come to deal with the technical terms used in parts 3 and 4 above a great difficulty
1077	faces us,for the Egyptians were very apt to confuse proof and working in their sums. This
1078	doubtless arose from the fact that many of their problems were solved empirically; in other
1079	words the mathematician often knew the answer and then set the sum. In cases of this kind
1080	it is clear that there will be no working out but only a proof. Thus in the bread sums
1081	Nos. 1-6 the problem is never solved at all: the question is asked, the answer is at once
1082	stated, and a proof is given which consists in multiplying this by 10 and showing that the
1083	result is the number of loaves to be divided. In such cases it is easy to see how what is in
1084	reality only the proof of a guessed answer assumes the appearance of the working, for other
1085	working there is none. Hence it came about that the Egyptian was hazy about the application
1086	to the various parts of the sum of the correct technical terms for working and proof.
1087	The difficulty does not end here. The Rhind papyrus is only a copy, probably by a
1088	poor mathematician, of a document which was doubtless differently arranged. The attempt to
1089	compress problems into the narrow horizontal registers into which the papyrus was ruled off
1090	has led in many cases to the misplacement of the headings of the various parts of the sum.
1091	With these difficulties in mind we may now attempt to disentangle the three terms
1092	used as headings in the working and the proof. These are śšmt, irt mi lpr, and tp n sity.
1093	The last of these may be dismissed first, for its meaning is clear from its use. Philologically
1094	there is little to be said concerning it. Outside Rhind the phrase occurs only in Pap. Kahun,
1095	PI. VIII, 1. 29, and the word śity is unknown in published Egyptian literature. Griffith
1096	suggests that it may be a causative from the same root as the obscure 1° of Siut,
1097	Tomb IV, line 130. If sity means "proof," as appears certain from the Rhind examples, the
1098	whole phrase must mean "method of proof," or more likely "section of proof," ie. section
1099	containing the proof. It occurs, if we except No. 21, where it is obviously out of place, only
1100	in the set of problems dealing with equations of the form *+%=c. It is used once in
1101	Nos. 32, 33, 34, and twice in each of Nos. 35, 37, and 38. In every case its meaning is
1102	perfectly clear. Each sum consists of two parts, the first of which contains the solution of the
1103	equation. It is this second part which is headed tp n sity, and the meaning can only be
1104	suggests that the various sections of Rhind were originally borrowed from different sources, the
1105	diction of each original being carried over with the sums into the new surroundings, and no
1106	attempt being made to establish uniformity of expression throughout the whole collection.
1107	The term śšmt,' clearly an abstract or semi-abstract noun from the verb sšm "to lead,"
1108	the papyrus, in addition to its occurrences in the table of the division
1109	where it is to be understood in every sum. In two of the other instances, Nos. 65 and
1110	66, it forms a heading to the whole of the working, following at once on the enunciation of
1111	the problem. In the seven other cases it occurs in various forms, ki n sšmt "form of
1112	1 samt in the literal sense of "guidance " is well known, e.l). Sinai, 53, 1. 15. I cannot find any other examples
1113	of it in a technical sense. The Masculine abstract săm (4.Z., 40, 114, Pap. Bulag 18, ed. Marietre, XXVII, 2),
1114	"disposal" of the income in kind of a temple or office among the various lawful recipients,
1115	priests, officials, ete.
p. 27
1116	RHIND MATHEMATICAL PAPYRUS
1117	working" in Nos. 41-3, tp n xšmt, "section (?) of working" in No. 41, and išmt simply in
1118	In all these cases the meaning is the same, the word being applied
1119	not to the whole working but only to what we call the "rough working."
1120	words, the whole working is divided into two parts: the first, with no heading, merely states
1121	the actual operations performed and gives their results: the second part, headed sšmt, ki n ≤smt
1122	or tp n xšmt, gives the laborious detail of the actual multiplications and divisions used.
1123	Whether the confining of the śšmt to this latter part only of the working is merely due
1124	to the mis-handling of the original arrangement of the sums by our scribe, owing to his ignorance
1125	and the exigencies of his narrow registers, it is impossible to say. It may be so, and the word
1126	may in reality be the title of the whole working out. We can, however, only go by what we
1127	see before our eyes, and in the papyrus as we have it sšmt is a word for the "working out,"
1128	either the whole or the rougher part of it.
1129	More difficult is the remaining phrase irt mi lpr. This we must approach first from the
1130	philological side. irt can only be the Infinitive or the Impersonal Passive, i.e. it can only
1131	mean " the doing" or "one does (it)." hpr must be either an Impersonal śdm-f, "it happens,"
1132	or a Neuter Participle Active with Masculine form,' "that which happens" or "has happened."
1133	The fact that a title is likely to be a nou in form is in favour of taking irt as an Infinitive,
1134	and in this case the grammatical construction of lpr will scarcely affect the meaning,
1135	will be "The doing as it happens" or "The doing according to that which happens."
1136	the phrase should denote some portion of the sum in which an application of
1137	some number or quantity to the actual facts of a case takes place."
1138	With this in mind we must now examine the uses of the phrase in the papyrus.
1139	occurs 26 times, generally in groups of sums of the same type. Its use appears to fall under
1140	four separate heads :—
1141	of a proof in which the answer found is subjected to the conditions laid
1142	in the problem and shown to satisfy them and thus to be correct: Nos. 1-6, 24-25,
1143	62-64, and 75-77. In Nos. 1-6 it should be noted that though the phrase covers all the
1144	working given it must not be rendered "working" as opposed to "proof," for in these sums
1145	the answer is guessed and the only "working" supplied is in reality a proof of this answer.
1146	outline of the working has been given: Nos. 50, 52, 66. Here it corresponds exactly to the 2. As title of detailed rough work (multiplications and divisions) added after a general e mot ceudti ise ios aal dis
1147	more common use of sšmt.
1148	3. In a perfectly literal sense similar to that of 1 above, but in application to a step
1149	in the working, not to a proof. The best instance is the first step of No. 40, where we may
1150	translate freely, "What would actually happen supposing that the difference of share were 52"
1151	The other case, No. 35, is less obvious, the step consisting in the application of the number 1
1152	to the actual facts of the problem as set. It is necessary to remark, however, that in both
1153	these cases the step headed irt mi lpr. is the first, and in consequence it is possible that the
1154	phrase applied to the whole working and not to this step alone, despite the fact that the
1155	present arrangement of the papyrus gives the latter impression.
1156	4. As title of the whole working. Here it stands immediately after the setting out of
1157	but the instances of it are somewhat unsatisfactory. Thus we have it in 43,
1158	while in the precisely similar 41 and 42 it is omitted. Similarly 51 has it, but the parallel
1159	It occurs in 49, a very inaceurate and unreliable piece of copying. In
1160	67 it precedes the whole of the work, which, however, is all of the nature of rough work.
1161	however, the Feminine lpit im pu of Nos. 76 and 78.
1162	=For nửhpr meaning "as it actually happens" or "happened " see Pap. Ebers, 67, 5, BruascH, Wörterbuch
1163	1341, line 3, and Urk., IV, 121, In the last case irt precedes, but the sense is difficult to seize.
1164	* Baillet (Pap. Math. Alchmim, 60) is certainly wrong when he says "la formule irt må lpr se retrouve dans
1165	le ourw moíeL" (" proceed as follows") of the Akhmim papyrus. The Egyptian cannot mean this.
p. 28
1166	24 RHIND MATHEMATICAL PAPYRUS
1167	Two other examples are difficult to classify. No. 28 is an incomplete sum ending in
1168	irt mi hpr, and in No. 69 it is clearly out of place, occurring before a single step in the
1169	The conclusion to be drawn from these uses is as follows. The safest translation of
1170	irt mi lpr in the papyrus is the literal, if clumsy, "The doing as it occurs" or "happens."
1171	As a technical term in mathematics it appears to have been somewhat fluid in meaning.
1172	main use is as a heading for a proof by substitution, that is to say a proof which consists of
1173	performing on the answer to be tested the operations prescribed in the problem and showing
1174	that it conforms with the data. It may further be applied to such parts of the work as may
1175	be looked upon as rough work in contradistinction to the general exposition of the working.
1176	Whether it ought to be used as a general heading for the working of a problem would seem
1177	doubtful.
1178	EGYPTIAN WEIGHTS AND MEASURES
1179	Though weights and measures form in some sense an essential portion
1180	no attempt to treat them in full is made in this volume, which. would be vastly increased in
1181	size by such a discussion. An even better reason for the abstention is the fact that the
1182	subject has already been admirably dealt with in an epoch-making article by Griffith."
1183	Despite that writer's modest disclaimer of finality there is practically nothing to be added to this
1184	treatment, which was based on a careful reading of all previous work on the subject backed
1185	by what is even more important, a first-hand acquaintance with the Egyptian texts.
1186	We shall therefore take this article for granted and only repeat such parts of it as are
1187	essential for the understanding of the present papyrus.
1188	1. MEASURES OF LENGTH.
1189	The unit used in the papyrus is the cubit (ml). Griffith has very ingeniously shown that
1190	this is the "royal cubit" of 20•6 inches, from the fact, evident from Nos. 41 ff. of this papyrus,
1191	that a khar, which is 5 quadruple hekat or 50 quadruple henu, is two-thirds of the cubic cubit.
1192	Now the capacity of various inscribed henu-measures which have survived is on the average
1193	just over 29 cubic inches, which demands that the cubit used in the above equation should be
1194	roughly 20•6 inches or about 523 mm. The "short cubit" marked on the Egyptian cubit
1195	measures would not give this result."
1196	The royal cubit is in this papyrus divided into 7 palms (Nos. 56 ff.), a palm being the
1197	breadth of the four fingers, and the palm again into 4 fingers (Nos. 58 and 59).
1198	In themeasurement of landthe unit of length was the khet (ht) o, of 100 cubits,
1199	the full name of which was lt nt nw!!, or "reel (?) of cord," a measure which may well be
1200	compared with our chain. For the use of this unit see below.
1201	2. MEASURES OF AREA.
1202	The commonest unit of area is the setat ({t:t) or square khet, whieh contained 10,000
1203	This was divided into dimidiated fractions, d, t, ¿, ete., each of which was
1204	distinguished by a special sign and doubtless a special name. Thus :
1205	setat = rmn,
1206	i setat = liśb (later lisp), written x, later &o»
1207	1 P.S.B.A., XIV, 403-450; XV, 301-316.
1208	2 For the markings on Egyptian cubit-wands see LepsIus, Über die altägyptische Elle und ihre Eintheilung,
p. 29
1209	RHIND MATHEMATICAL PAPYRUS 25
1210	setat = s: (?). Sor (late writings)
1211	ta setat=swS (late)
1212	a setat =rm:. 30 (late)
1213	Of these parts only the 1, and 3 occur in our papyrus, Nos. 53 and 54. The whole of
1214	this system of division of the setat is referred by Sethe to a very early stage in Egyptian
1215	civilization.?
1216	For practical purposes in land-measuring there was a tendency for a unit called the
1217	"cubit-of-land" and a 1000-fold multiple of it called the "thousand-of-land" to be used in
1218	preference to the setat. A cubit-of-land is a narrow strip of land 100 cubits long and 1 cubit broad
1219	and is expressed in Rhind by (the ordinary cubit sign, see No. 55): this unit is clearly
1220	one-hundredth part of a setat. The thousand-of-land was written in early times ld
1221	in Rhind, however, it is regarded as a unit and indicated simply by a vertical stroke.
1222	in No. 53 we find setat doubled and the result given as \|Ti, ie. 1 thousand-of-land
1223	and 4 setat. Fractions of the setat are expressed in the dimidiated fractions described
1224	above, so far as possible, and the small remainders in cubits-of-land (ż.e. hundredths of the
1225	setat). In No. 54 we have, quite exceptionally, a special hieratic sign for 10 cubits-of-land
1226	resembling the numeral 30.
1227	3. MEASURES OF CAPACITY.
1228	The unit of capacity is the hekat or bushel, containing 10 henu, each henu being about
1229	29-2 cubic inches according to the examples which have survived. It would seem, however,
1230	that despite this numerical relation the henu and the hekat may have independent origins, for
1231	the henu was not used in actual reckoning as a part of the hekat but as a totally distinct unit.
1232	The hekat was divided into $, 4, 4, 1w s and "*. each of which had a separate sign and
1233	doubtless a separate name. The hekat was further divided into 320 ro (•, the word had
1234	here probably its early sense of "part"), so that 1 henu contained 32 ro. Fractions of
1235	hekat other than those enumerated above (} to (t) were not tolerated, but were reduced to
1236	terms of these and of the ro. Thus hekat = (1+to +=4) hekat + 13 ro. above, p.7.
1237	The notation used in hieroglyphie for these dimidiated parts of the hekat was as follows,
1238	written from left to right:-
1239	} hekat <* th hekat •+
1240	Möller was the first scholar to perceive that these signs can all be arranged together
1241	to form the figure of the wdit, or sacred eye, Fig. 2. It must not be inferred, however, that
1242	Fig. 2.
1243	1 The short hieratie form for is used in Rhind.
1244	* SETHE, V.Z.Z., 74 f. " E.g. L., D., II, "
1245	Both signs appear to me to be reversed in MöLLER, Paläographie, Nos. 708, 711. Contrast A.Z., 48, 101.
1246	line 10, and bear in mind that all the signs are there written in the opposite direction.
1247	E
p. 30
1248	26 RHIND MATHEMATICAL PAPYRUS
1249	this was their origin: these measures were in constant use in Egyptian medieine, and it may
1250	have been an ingenious discovery of some scribe that they could be arranged in such a way
1251	as to form the sacred eye so closely connected in Egyptian eyes with healing power.!
1252	For multiples of the hekat we find a special notation in the papyri of this period.
1253	A tall stroke standing in front of the hekat sign thus \|* stands for 100 hekat, two
1254	strokes for 200 and so on. 50 hekat is expressed by the ordinary sign for * placed after the
1255	hekat sign, "=, 25 by the sign for ‡.*x The two signs together would indicate
1256	75 hekat. Ten hekat are shown by a tall stroke after the hekat sign, and 5, 6, 7 and 8 by
1257	special hieratic signs to which we do not know the hieroglyphic equivalents." Single hekat
1258	up to 4 are indicated by dots, after the hekat sign, and fractions of a hekat follow in the usual
1259	Horus-eye notation.
1260	In No. 82 we are introduced to a double-hekat unit, written?))'. This made
1261	its appearance, so far as we know, in the Middle Kingdom, for it is first met with in the
1262	Kahun Papyri. It is equipped with its dimidiated parts just as is the single hekat, each
1263	being presumably double the corresponding part of the single heliat. This double-hekat, written
1264	also continued in use into the XVIIIth Dynasty, for it is used in pfiw reckonings
1265	on a stela from Tell el-'Amarneh.*
1266	Nos. 41 to 47 make use of a still larger unit, the quadruple-hekat or "great quadruple-
1267	hekat" (for the various writings see the problems referred to). This too is divided by halves
1268	after the manner of the single hekat, and continued to be used, especially for measuring grain,
1269	until well on into the New Kingdom.
1270	The only other measure of capacity used in Rhind is the khar (hir) , which,
1271	as is clear from Nos. 41 ff., contained 5 quadruple-hekat. It is also the capacity of two-thirds
1272	of a cubic cubit. This same measure is written 23→ Al in the Westcar Papyrus
1273	(12, 4), but seems to die out in the New Kingdom, when it is replaced for bulky substances
1274	by a f consisting of 4 quadruple-hekat. The reading of this new measure is unknown,
1275	since it is nowhere written out, but it is not impossible continued to be read hir,
1276	though its value had fallen from 5 quadruple-hekat to 4.
1277	The $, } and } of this measure, which correspond to 1, 2, and 3 quadruple-hekat
1278	respectively, are expressed by dots, one, two or three as the case may be. The notation
1279	described by SPIEGELBErG, Rechnungen aus der Zeit ySetis I, Text, p. 49, is probably incorrect.
1280	Louvre Pap. 3226 is perfectly clear on the point. Some laterpapyri at Turin (unpublished)
1281	are, however, less clear, and need fresh collation and working out.
1282	1. WEIGHTS.
1283	The complicated system of weights used in Egypt plays but little part in this papyrus.
1284	Suffice it to say that in No. 62 we have the deben (dbn, formerly read uten) of which the kite
1285	(Içdt) is the tenth part. For the "ring" as a definite weight see below under No. 62. The
1286	weight established for the deben by actual weighing of New Kingdom specimens is between
1287	1400 and 1500 grains."
1288	1 The hieratic formıs, which show but slight resemblance to the parts of the eye, are doubtless older than the
1289	hicroglyphic, none of which have yet been found earlier than the XVIIIth Dynasty, some not before the XXth. It was
1290	perhaps in establishing the hieroglyphic forms from the hieratic that the sacred eye myth was brought into play.
1291	" In No. 82 there is a unique instanee of the fraction } preceded by * standing for 33) helat, a clumsy
1292	ixpedient since it leaves us with } hekat which must be resolved into correct Horus-eye notation.
1293	3 See MöLLer, Palüographie, I, Nos. 699-702.
1294	* P.S.B.A., XV, 306. Cf. too Dümıchen, Kalender-Inschriften, PI. XXXIX.
1295	" See Petrie's article Eyyptian Weights and Measures in Encyclopedia Britannica; also HULTscH, Die Gewichte des
1296	Altertums, EISENLOHR,
1297	poids égyptiens, in Annales du Service, XIII, 125-160.
p. 31
1298	RHIND MATHEMATICAL PAPYRUS 27
1299	COMPARISON OF EGYPTIAN MATHEMATICS WITH BABYLONIAN.
1300	As early as 3000 B.C. it would seem that the world was in possession of two separate
1301	and highly developed systems of mathematies, the Egyptian and the Babylonian.
1302	may never take us far enough back into the roots of civilization to enable us to decide whether
1303	these two systems had entirely independent origins, or, if not, what was their common source,
1304	and how much each owed to it and to the other.
1305	when they first come under our observation and compare them.
1306	For the modern the mathematies of Babylonial possess aninterest which those of Egypt
1307	cannot rival, for in, them lies, it would seem, the origin of the modern division of the circle
1308	into 360 degrees and of the day into 24 hours. The term Babylonian is however here a
1309	complex one, for we must distinguish between two elements in Babylonia, the non-Semitic
1310	Sumerians and the Semitic Akkadians with their later Semitic followers. These Semites of
1311	Babylon appear to have used a system which was purely decimal, like the Egyptian, but
1312	which very soon became contaminated by the sexagesimal system of the Sumerians."
1313	the extent to which Babylonian mathematics is dominated by the sexagesimal notation it
1314	is clear that most of it owed its origin to the Sumerians, and we are therefore justified in
1315	going back to the earliest times in the enquiry which follows.
1316	1. C'ONTRAST BETWEEN EGYPTIAN AND SUMERIAN MATHEMATICAL RECORDS.
1317	There is an essential difference between the mathematical material which has come
1318	down to us from the two lands. From Egypt we have two complete papyri of problems
1319	and considerable fragments of others, in other words we have mathematics in use.
1320	Sumeria we have little except the means employed, consisting of numerous tablets containing
1321	tables of multiplication and division, of squares, of square roots and of cube roots.
1322	for what can be gathered from astronomical reckonings and from calculations in weights and
1323	measures, mostly very unfruitful, we have no pictures of these tables in action. This wide
1324	in the nature of the material makes it difficult to institute comparisons
1325	of any value between Egypt and Sumeria.
1326	2. NOTATION.
1327	The Egyptian notation is decimal, its units being 1, 10, 100 and So on. The Sumerian
1328	is essentially sexagesimal, that is to say its main units are 1, 60, 3600, etc. At the same
1329	time this is not entirely primitive, and there clearly must have been a time when the
1330	Sumerians could not count up to 60. Such a time is marked by the survival in the sexagesimal
1331	system of a subsidiary unit 10. We may go back even farther than this, for the names of
1332	the numbers from 6 to 10 appear to be secondary formations from those for 1 to 5, which
1333	points to a stage where the fingers of one hand were used to count up to five (for a similar
1334	possibility in Egyptian see above, p. 11).
1335	In the Sumerian sexagesimal system this unit 10 occupied a subordinate place, which
1336	can be best illustrated by the fact that 100, 1000, etc. were not units in the system* and had
1337	1 The latest works on the subject are THUREAU-DANGIN, Nuération et métrologie sumériennes,
1338	Mathematical, metrological and chronoloyical tablets from the temple library
1339	of Nippur (The Babylonian Expedition of the University of Pennsylvania, Vol. XX, Part 1); THUREAU-DANGIN,
1340	L'u, le ga et la mine, leur mesure et leur rapport, in Journal Asiatique, 1909, 79-112.
1341	3 See THUREAU-DANGIx, Orientalische Literaturzeituny, 1909, 383, n1. 2.
1342	* Perhaps astrological would be i more correct term, sce HiLpRECHr, op. cit., 34, and quotation there given
1343	* They were used by the Akkadians, bowever, and even survived the Sumerianizing of the Akkadian arithmetic.
p. 32
1344	28 RHIND MATHEMATICAL PAPYRUS
1345	no special signs : here we have an essential difference from Egyptian. The number 10 was
1346	however used to form new units, for just as the unit 1 could be multiplied by 10 to give a
1347	unit with special sign, so the sexagesimal units 60, 3600, etc. could each be
1348	multiplied by 10 to give 600, 36,000, and 2,160,000, a series of fresh units each with a
1349	special name.
1350	numbers, tor 60 cubed was already 216,000. This may be compared with the fully developed The highness unit 60 had tor its consequence a tendency to lead
1351	system of high numbers in the Egyptian system, reached by quite a different route.
1352	In one important point the
1353	the purely mathematical tablets we find a system, imperfect it must be admitted, of positional
1354	notation. Supposing that in our decimal notation we write 365. We read this (3x 10°) plus
1355	(6 x 10) plus (5 x 1). Our unit being 10 we can represent each portion of it by a single
1356	digit, 3, 6 and 5. In a system with a main unit 60 unit 10 we should
1357	have to represent each portion by two digits, thus 32.12.43 would stand for (32 x 60°) plus
1358	(12 x 60) plus (43 x 1), or 115,963. This is precisely what the Sumerians did, and it was
1359	here that their secondary unit 10 served them in such good stead. In the mathematical
1360	tablets two signs alone are used in this notation, the sign for 1 and the sign for 10. The
1361	number above referred to would be two units; a ten and two
1362	units; four tens and three units: the fact that the first group is to be multiplied by 60°, the
1363	second by 60, and the last by 1 is taken for granted, just as the multiplication of 3, 6 and 5
1364	by 100, 10 and 1 respectively is in our decimal notation for 365.
1365	The system was still further perfected by the use of a sign for zero in cases where
1366	middle term was missing, e.g. in 12.0.33, which stood for (12 x 60°) plus (0 x 60) plus (33 x 1).
1367	Here we have all the elements of positional notation with one exception: there was nothing
1368	to correspond to our decimal point. In other words, if I write in modern terms 365 I know
1369	that the 5 is units, the 6 tens, and the 3 hundreds; and if I write 36•5 I know that the
1370	5 is tenths, the 6 units, and the 3 tens. When, however, the Sumerian wrote 12.25.33 he
1371	had no means of showing whether the lowest unit, that to be multiplied by 33 was 3600, 60,
1372	1, 6o, or some other sexagesimal unit. All that was fixed was that whatever the lowest of
1373	these units (to be multiplied by 33) was, the next (to be multiplied by 25) was 60 times greater,
1374	and the highest (to be multiplied by 12) 3600 times as great. An interesting corollary of
1375	this will be noticed below in connection with the tables of division and multiplication.
1376	3. FRACTIONS.
1377	In the true sexagesimal system the highest fractional unit was naturally 6o and the
1378	of thesecondary unit 10,hwerer,gave also(610)and t
1379	(3600 × 10). In practical everyday reckonings the fraction assumed a paramount position
1380	both in the Sumerian system and in the later Babylonian systems derived from it, and from
1381	a comparatively early period we find a notation for 6, 3, 2, and . As early as the First
1382	Dynasty of Babylon we find a notation for } as well as for and }.
1383	Thus in Babylonia we have up to the present no instances of the complicated fractional
1384	reckonings to which the Egyptian mathematician seems to have been •perfectly accustomed.
1385	This may be the merest accident, and probably is, for in a system whose first fractional units
1386	are 6o and Joow calculations involving
1387	been avoided. The use of the fraction is paralleled in Egypt, but the existence of shows
1388	us that the Babylonians had in some cases at least passed beyond the Egyptian convention
1389	which demanded that all fractions except } should be aliquot parts.!
1390	1 SETHE, V.Z.Z., 103, is not inclined to ascribe to Babylonian influence the appearance of a sign for 2 in Egypt
1391	in Ptolemaic times.
p. 33
1392	RHIND MATHEMATICAL PAPYRUS 29
1393	4. TABLES OF MULTIPLICATION AND DIVISION.
1394	In early Egypt nothing has yet been discovered in the way of tables except those for
1395	dividing 2 by the various odd numbers. It would be rash to say that multiplication tables
1396	did not exist, but at the same time there is a reason why they should never have needed
1397	a very high development. The Egyptian system of multiplication by doubling was a slow one,
1398	but it was sure ; it is indeed astonishing how simply and even how quickly the most com-
1399	plicated multipliers can be constructed by this method, especially when we remember that
1400	the purely mechanical process of multiplication by 10 can be introduced as an aid at any
1401	moment. The fact is that the Egyptian's apparatus for multiplication
1402	he could well dispense with tables, we must be prepared for the possibility that he
1403	actually did, though excavation may prove this to be wrong.
1404	The Sumerian, on the other hand, was a lover of tables. Unfortunately we have no
1405	idea whatsoever how his multiplications were accomplished. One might, however, hazard the
1406	guessthat at quite an early period he had acquired considerable power over the process,
1407	otherwise he could hardly have faced the multiplicational difficulties involved in a notational
1408	system based on so high a figure as 60. The third unit in his system was no less than 3600!
1409	The multiplication tables known to us come mostly from the library at Nippur.
1410	table multiplies by a certain multiplier the numbers from 1 to 20 inclusive, then 30, 40 and
1411	50 ; it will be observed that in this way any multiplicand from 1 to 59
1412	either directly or by a simple addition. The reason why these multiplicands stop at 59 is
1413	obvious.
1414	alter the figures in the multiplicand, but merely the units. Thus to multiply by 60 was
1415	point or insert a cypher, leaving the digits of the multiplier as they are. He too left his
1416	neither point nor final cypher he could not
1417	use them, and presumably trusted to his memory to remind him that his lowest unit was
1418	The multipliers comprised in these tables are forty-six in number; they begin with 2
1419	to 6, 8, 9, 12, 18, and end with 3000, 160,000, 162,000 and
1420	series, the choice of which it is not easy to see. Hilprecht has attempted, not very con-
1421	vincingly, to derive them from the division by certain numbers of
1422	or 12,960,000, which he believes to be Plato's geometric number.'
1423	Side by side with these multiplication tablets must be considered four tablets which,
1424	•according to Hilprecht, give the division of this large number 12,960,000 by various numbers
1425	from 12 to 81. These tables are certainly divisions, but Thureau-Dangin has rightly pointed
1426	out? that the number divided need not be that supposed by Hilprecht, for owing to the absence
1427	of a point ("sexagesimal point" we should have to call it) in the Babylonian system we
1428	cannot determine what is the unit. Thus the number translated by Hilprecht as 1,080,000 is
1429	on the tablet actually
1430	might equally well be 5 x any other power of 60, such as 60°, 60, 1, 6o or o Indeed, here
1431	lies the very • beauty of these tables. Just as our 12-times table will serve us to multiply 12,
1432	or 12,000 or •000012, so this table will divide for us not only 12,960,000, which is 60*, but also
1433	60°, 6oa, and so on. If ths remarkable elasticity of unit was really intended by the Sumerians,
1434	they had certainly far surpassed the Egyptians in the matter of notation.
1435	Tables of squares may be regarded as extracts from the multiplication tables. Nippur
1436	has furnished three running from 1 to 50, and an earlier discovered tablet runs from 1 to 60.
1437	Hilprecht's argument (op. cit., 24, end of note 3 continued from p. 23) that squares of numbers
1438	from 31 to 60 were obtained by the formula (a + b)° = a° + 2ab + b°, where a is 30, seems to me
1439	1 Op. cit., 20 ft. : Revue d'Assyrioloyie, XVIII, 124.
p. 34
1440	30 RHIND MATHEMATICAL PAPYRUS
1441	simple non seguitur. Bn ae durus in tahle of sguares haruy et beee prundein of tht, thougit a techamad
1442	(see p. 20).
1443	oots and one oi cube roots. Ugypt has vielded nothing of this kind ide by side with the tables of squares we have from Babylonand thoucls the concestion
1444	of square root was well known there that of cube root has not yet been found.
1445	5. WEIGHTS AND MEASURES.
1446	With regard to Babylonian weights and measures we have a mass of evidence which
1447	can scarcely be glanced at here. Suffice it to say that Babylonia had a complete system of
1448	measurements for length, area, volume, capacity, liquid content and weight. In general these
1449	oftrs gu poito ot orparion vito tie eent an tao huiracteriatio eerpeian d liaten d
1450	the sexagesimal notation are more usual.
1451	The Babylonian unit of length, the cubit, as determined by the measurements
1452	of Gudea, patesi of Lagash, and by the dimensions of the great tower at Babylon,
1453	was just under 496 mm. in length. This compares fairly closely with the Egyptian royal
1454	cubit of 523 mm., but no conclusion must be drawn from this, for the arm, or more exactly
1455	the forearm, is an obvious primitive unit of measurement for any people. In Sumerian the
1456	finger-breadth is one-thirtieth of the cubit; in Egyptian it is one-twenty-eighth. The Sumerian
1457	measure of the foot of 20 fingers, or of a cubit, does not occur in Egyptian.
1458	Thureau-Dangin has sought to prove that a definite relation existed between
1459	of length, capacity and weight in Babylonia.! He finds that the ga,* the unit of capacity, is
1460	Tiz of the cubic cubit, and that the mina" is the weight of a volume of water of zta of the
1461	cubic cubit. This will certainly need more convincing proof than has been given so far, and it
1462	would seem a priori a little unlikely that so artificial a system should have been adopted,
1463	though the modern decimal system affords an obvious parallel. However this may be,
1464	evidence is as yet available for determining whether any relation of this kind exists between
1465	the various units in the Egyptian system.*
1466	6. GEOMETRY.
1467	Our knowledge of Babylonian geometry is slight. For the determination of areas there
1468	seems to be but a single document, a tablet of the second Dynasty of Ur, now in the Museum
1469	at Constantinople. On this are plans of fields accompanied by certain lengths and areas,
1470	from which it would appear that the following determinations of area were correctly known to
1471	the Babylonians : the area of the rectangle (and as a special case that of a square), the area
1472	of a right-angled triangle determined by half the product of the two sides enclosing the right-
1473	angle, and the area of a trapezoid, namely the product of the altitude and half the sum of
1474	the parallel sides.
1475	Of these the first was known to the Egyptians, and indeed must be known to all peoples
1476	who have evolved the conception of square measure. Whether the second and third were
1477	correctly known in Egypt depends on the interpretation of the word mryt in Nos. 51 and 52,
1478	see pp. 91 ft.
1479	1 Revue d'Assyriologie, XVIII, 127-132 ; Journal Asiatique, 1909, 79 tt.
1480	= Found from the silver vase of Entemena and other evidenee to be about 404 millilitres.
1481	3 The ancient mind was about 404 grammes, the later 505.
1482	* THUREAU-DANGIN, op. cit., 107-8, comes to this conclusion.
1483	5 THUREAU-DANGIN, Revue dl' Assyriologie, IV, 16 ff.; OppeRr, op. For later literature
1484	HILPRECHT, op. cil., 11, note 9. On this tablet an irregular area is divided for purposes of measurement into a
1485	series of right-angled triangles and approximate rectangles.
p. 35
1486	RHIND MATHEMATICAL PAPYRUS 31
1487	If Thureau-Dangin is correct with regard to the relation between
1488	and the cubic cubit, it follows that the Babylonians were capable of estimating the volume
1489	We might indedhave inferredthifom the existence of a fully developed
1490	system of measures of volume, which obviously presupposes the determination of the volume of
1491	a parallelopiped.'
1492	THE GREEKS ON EGYPTIAN MATHEMATICS.
1493	The Greeks looked up to the Egyptians as the originators of their mathematics and
1494	more particularly of geometry. Herodotus' tells us a story on which perhaps all later refer-
1495	a combination of Ramesses II of the XIXth Dynasty with at least one of the great Senusrets
1496	of the XIIth, he says: "This king divided up the land among the Egyptians, giving an equal
1497	square plot to each man; from this he derived his revenue, imposing a rent to be paid each
1498	year. But if the river carried away a portion of any man's lot he would come to him and
1499	report what had happened. And the king would send men to examine and to measure by
1500	how much the land had been diminished, in order that he might pay only a proportionate
1501	amount of the rent. It appears to me that geometry was discovered in this way and that it
1502	afterwards came over into Greece."
1503	e ith re nedo tare nud dition oflail bemen dhs opopl dat paibabl
1504	Greek geometry from Egypt. Geometry is an intensely practical science and would naturally
1505	first appear in a country where land was of very great value, in other words, in a highly
1506	agricultural country. The two obvious places of origin are Mesopotamia and Egypt.
1507	Herodotus' story with regard to the carrying away of land by the river is not altogether
1508	convincing, and it is probable that there is here a confusion with the fact that the Nile when
1509	it rises year by year to some extent obliterates the boundaries between field and field, a fact
1510	which in Strabo's Geography and in the Summary of Proclus is given as the origin of geometry
1511	in Egypt. As Strabo remarks," the land "had to be measured again and again."
1512	Later Greek writers do not add much to Herodotus' story.* Diodorus,' however, tells
1513	us that the Egyptians themselves claimed astronomy and geometry as Egyptian discoveries,
1514	and Plato in the Phaedrus® makes Socrates say that he has heard that the god Thoth was the
1515	inventor of arithmnetic, calculation, geometry and astronomy. Aristotle," however, diverges
1516	from the common story in attributing the discovery of these sciences in Egypt not to the
1517	fact that there was need for land-measuring there, but to the fact that there was a leisured
1518	class of priests who had time to spare for such pursuits. In reference to this be it said that
1519	there is no particle of evidence that in early times Egyptian mathematics were in any sense
1520	in the hands of the priests, whatever may have been true of Aristotle's day.
1521	The actual introduction into Greece of Egyptian geometry is attributed by the writer
1522	of the Summary of Proclus to Thales, "who first went to Egypt and from there introduced
1523	the study into Greece." This dependence upon Egypt is made very clear by lamblichus'
1524	Lite of Pythagoras, where Thales, after teaching his young pupil all he knew, advised him to
1525	go and study with the Egyptian priests, which indeed he did, spending no fewer than 22
1526	years in the temples of Egypt learning astronomy and geometry.
1527	In this connection an interesting problem is raised by Democritus' reference to the
1528	cylinder. 1 HILPRECHT, op. cit., 36-38, draws attention to a possible case of the determination of the volume of a
1529	= II, 109. 3 XVII, 3.
1530	* The two letters of Rhabdas of Smyrna, 1341 A.D., are perhaps of greater interest than value in this connection.
1531	* Metaphysics, A. 1, 981 b, 23.
p. 36
1532	32 RHIND MATHEMATICAL PAPYRUS
1533	harpedonaptai of Egypt.' The philosopher boasted that no one of his time had surpassed him
1534	in constructing figures from lines and in proving their properties, not even the so-called har-
1535	pedonaptai of Egypt. Who were these harpedonaptai? More than one historian of mathematics
1536	has supposed The literal meaning of the word is
1537	stretchers, " and it is suggested? that they were acquainted with the fact that a triangle whose
1538	sides were 3, 4 and 5 contained a right-angle, and that they constructed right-angles accord-
1539	ingly,
1540	For this last statement I can find no foundation whatsoever: nothing in Egyptian
1541	mathematics suggests that the Egyptians were acquainted even with special cases of Pythagoras
1542	theorem concerning the squares on the sides of a right-angled triangle. That the harpedonaptai
1543	were land-measurers on the other hand is most probable, indeed we can even see such persons
1544	at work in the pictures on the walls of Egyptian tombs. In the tomb of Khaemhet at Thebes
1545	we see a number of men equipped with ropes and writing material measuring a field, and
1546	there is a similar scene in the tomb of Menena." In either case the persons engaged in the
1547	work might most suitably be described as "rope-stretchers"; the very unit by which fields
1548	were measured was a "reel of rope" of 100 cubits in length (see p. 24).
1549	This process of land-measuring with a rope, the Egyptian name for which is not known,
1550	has been confused by historians of mathematics with the ceremony called pd šs "the stretching
1551	of the cord." This was one of the initial ceremonies at the foundation of a temple.* The
1552	king or whoever represented him took a sighting of the pole-star through a cleft stick, another
1553	person standing north of him with a plumb-bob attached to a wooden arm. Each then drove
1554	a stake into the ground in front of him, and a cord stretched between the two gave a true
1555	north and south line and enabled the four corners of the temple to be fixed. Here the stretching
1556	of the cord is used not necessarily in measurement, but in the fixing of the orientation.®
1557	Possibly it was something of this kind which Democritus had in mind when he spoke of the
1558	skill of the harpedonaptai.
1559	1 Clemens Alexandrinus, Stromata, ed. Potter, I, 357.
1560	2 HEATH, I, 122.
1561	3 WRESZINSKI, Aflas zur altägyptischen Kulturgeschichte, Taf. 191 and 232. For a statue of an "overseer of
1562	land" holding a measuring-rope see A.Z., 42, 72.
1563	4 I have to thank Dr. Blackman for putting at my disposal his admirable unpublished notes on this ceremony.
1564	• Examples of these two instruments,
1565	Borchardt in Ä.Z., 37, 10 f.
1566	in Clemens Alexandrinus, Stromata, VI, 4, 35.
1567	ûpтєбоváттs but ¿pookóтos.
1568	• The cord may have been used in the subsequent measurements.
p. 37
1569	RHIND MATHEMATICAL PAPYRUS
1570	TRANSLATION AND COMMENTARY
1571	TITLE. AND INTRODUCTION. Plate A.
1572	"Rules for enquiring into nature, and for knowing all that exists, [every] mystery, .... Titleand
1573	every secret. Behold this roll was written in Year 33, month 4 of the inundation season, .....
1574	[under the majesty of the King of Upper] and Lower Egypt Aauserré, endowed with life, in
1575	the likeness of a writing of antiquity made in the time of the King of Upper and Lower Egypt
1576	Nemarē. It was the scribe Ahmöse who wrote this copy."
1577	The small black cross under the word hit indicates that the words mht in the second
1578	column are to be inserted here. Eisenlohr through failing to observe this has missed the sense
1579	of the whole passage.
1580	The papyrus begins with a formal title and indication of its contents which has unfor-
1581	tunately been damaged. This title is written in vertical lines and begins in red ink. It is
1582	followed by the dating and the name of the scribe who made the copy.
1583	tp hśb n hit m ilt. The concluding words hit m iht must mean "going into" or
1584	"probing" things. Ilt probably has the same general sense which it bears in the phrase
1585	"a wise man," one who knows facts or things. I can find no good parallel to the figurative use
1586	of hit m. In Nos. 35-38 hit r appears to be used in the literal sense of " go (down) into," but
1587	no attempt should be made to contrast a metaphorical use with m and a literal with r, for both
1588	prepositions are used with hit in its literal sense in the Old and Middle Kingdoms. It now
1589	remains for us to fix the meaning of tp lśb. The best known examples of the phrase occur in the
1590	Eloquent Peasant, B 1, Il. 98, 148, 161, 274, 311, 325, and B 2, l. 94. Vogelsang's notes to
1591	these passages should be consulted.' From a comparison of the various uses of the phrase in
1592	this text it is clear that it means in general "accuracy," not only in actual reckoning but also
1593	in conduct, that is to say moral as well as intellectual. It also occurs in a rather difficult
1594	sentence? in Prisse 5, 7, and again in 8, 5. The most frequent use of the phrase is in the title
1595	sš iler n tp luśb (Urk., IV, 122, 964), "Excellent scribe of correct reckoning," where of course it
1596	is used in the more concrete sense, meaning that the scribe is accurate in copying down and in
1597	the computation of accounts. There can be little doubt that this intellectual sense is to be given
1598	to tp hisb in the present passage and that it simply means "accuracy," or still more concretely,
1599	"accurate method of," ie. "rules of." The whole phrase" must therefore mean "Rules for" or
1600	"Correct method of enquiring into facts" or "into nature."
1601	snkt. This reading is certain. The word is a slightly unusual one for "dark." See
1602	BRUGSCH, Würterbuch, 379 and 1255. For the noun sukt "darkness " see LACAU, Textes Religreux,
1603	LXXXIX; we seem to have a masculine form in A.Z., 1864, 2. snkt oceurs in the meta-
1604	phorical sense of obscurity of speech in J.E.A., IV, 34, PI. VIII, line 7. The following word was
1605	doubtless nbt "every."
1606	DIVISION OF 2 BY ODD NUMBERS. Plates A—E.
1607	Plates I to VI and part of VII in B.M. Facs. are occupied, after the title, by a problem Division
1608	which has attracted mathematicians of all ages, namely the expression of each of the various of 2
1609	• VOGRLSANO, Koumender zu den Kluyen des udgment-hall works by rule (ep lil), all ita dealings are aceording " The passage 8, 5 would seem to mean 94, 212, and 219.
1610	to the measuring tape, " ie. strictly accurate.
1611	3 tp hób, if Griffith is right in restoring the words so, must have a similar sense in R.P., PI. VIII, 30. Cf.
1612	also Louvre stela C 14.
1613	F
p. 38
1614	34 RHIND MATHEMATICAL PAPYRUS
1615	Division fractions whose numerator is 2 and whose denominator is one of the odd numbers from 3 to 101
1616	of 2.
1617	The purpose of this process is not hard to find. The Egyptian, as we have seen above,
1618	had no notation disliked dealing with fractions other than aliquot parts, with the
1619	single exception of }. When brought face to face with such fractions he reduced them at
1620	once to a more workable form. Now any proper fraction can be reduced at once to the sum
1621	of a number of fractions whose numerators are either 1 or 2. Thus, to take a simple example,
1622	TT=TT+3(11).
1623	sum of a series of aliquot parts.
1624	1=3==$→=
1625	For this purpose the Egyptian reckoner was accustomed to keep by him a table of
1626	"the division of 2," or, in modern terms, of the resolution of fractions whose numerators are 2.
1627	Among the Kahun Papyri is one (PI. VIII) which gives a series of such resolutions from } to
1628	The Rhind Papyrus goes much farther and carries us up to ổr.
1629	By what process did the Egyptian arrive at his results? This may best be understood
1630	by examining a typical specimen of a resolution, namely that of 7, which may be paraphrased
1631	in modern terms as follows:—
1632	Problem: Express 2 ÷ 7 sum of a series of aliquot parts.
1633	Answer: 1+*=\|th of 7; = 2gth of 7.
1634	2.e·=1+78
1635	Proof:
1636	-4 (Proof of this step) 7
1637	28
1638	There is no doubt as to what takes place here. The 2 is broken up into two parts,
1639	namely (1z + t) and i. The first of these is then shown to be f of 7 by the simple process of
1640	dividing 7 by 2 and then by 2 again, while the second is shown to be 2s of 7 by multiplying
1641	7 by 4 and obtaining 28. Thus 2÷7=1+2
1642	As a proof this is satisfactory, but it does not throw the slightest light on the one
1643	feature of interest in these problems, namely the manner in which the Egyptian obtained
1644	his answer, which, be it noted, is not worked out at all, but merely assumed and then proved.
1645	To arrive at the method by which the answer was obtained it is necessary to examine
1646	the whole series. of resolutions from } to rör, and to try to discern in them any signs of the
1647	employment of a general formula.
1648	The fraction } is not resolved, since for the Egyptian reckoner it presented no difficulties
1649	(see above, p. 15). In the case of all other fractions whose denominator is a multiole of 3
1650	a very obvious resolution presented itself, for the numerator 2 could be broken up into 1}
1651	and }, and since 1½ divides exactly into 3 and all its multiples the problem was at once
1652	solved. Thus: 3=1+=6+1s*
1653	1 Cantor treats this portion of the papyrus at great length, 24-33.
1654	* Strictly speaking, we ought to write 2÷9 and not 3, since the Egyptian had no notation to correspond
1655	to the latter.
1656	3 For the formula here used, vir.x}= 2u t ta, compare No. G1B.
p. 39
1657	RHIND MATHEMATICAL PAPYRUS 35
1658	Similarly those fractions whose denominator was 5 or a multiple thereof could be dealt Division
1659	withby breaking up the numerator 2 into 13 and1 Thus 2 = *+ = 15 + 75. In this
1660	the denominators 5,25, 85 were dealt with, 15, 45 and 75 having been
1661	already treated as multiples of 3, 35 being treated irregularly, 55 as a multiple of 11, and
1662	When the denominator was divisible by 7 the 2 was broken up into 13 (or 1 +}+ *
1663	as the Egyptian called it) and 1
1664	resolved 7, 49 and 77; 21 and 63 were treated as multiples of 3, and 35 and 91 were dealt
1665	with irregularly.
1666	In the case of 11 and its multiple 55 the 2 was resolved into 13 + 6 (i.e. 1f) and 6
1667	Up to this point it may be said that the method has been marked by considerable
1668	regularity. We are now left with the prime numbers between 13 and 97. A modern mathe-
1669	matician would probably treat these, as indeed all numbers, by some such formula as that
1670	suggested by Griffith,! namely:-
1671	3 = at na, where a ="+),
1672	which has the advantage of resolving each 2-fraction into two aliquot parts only, but the
1673	bound by no fetters of this kind, for the simple reason that he reached his results not by disadvantage of giving a second fraction with a very high denominator. The Egyptian was
1674	formula but by trial. An inspection of them is sufficient to show this. Even in the treat-
1675	mentof multiples of the lower primenumbers we have already seen that therewas
1676	irregularity, and this is only emphasized when we come to the higher prime numbers.
1677	Eisenlohr has attempted to embrace the Egyptian results under a series of rules which
1678	he enunciates as follows :—
1679	1. Resolution into three fractions was preferred to resolution into four.
1680	2. If a resolution existed (i.e. could be found) in which the denominator of the first
1681	root-fraction was the product of factors which when separately multiplied by the denominator
1682	of the original 2-fraction give the denominators of the remaining root-fractions, if, that is to
1683	say, 3 could be broken up into ao t ẩn + ön or into abc t ấn + ổm + ổn, then this resolution was
1684	Otherwise that in which the denominator of the first root-fraction was not ah
1685	but% ab afora was adopted.
1686	3. High factors of the original denominator were avoided.
1687	It is true that as a matter of actual fact the resolutions given by the Egyptian do to
1688	extent conform to these rules. Thus the resolutions of 17, 31, 37, 43, 47, 59, 67, 73
1689	and 97 conform to the simple formula given above; in the case of 19, 41, 71, 79 and 83
1690	the denominator of the first root-fraction is aj, in the case of 53 it is g in the case
1691	of 13, 29 and 89 it is % and inthe case of6lit isab But even here Eisenlohr's
1692	rules are by no means consistently carried out. Thus in dealing with 13 the denominator
1693	of the first root-fraction is # where the rule would prescribe to; in 19 the simple formula
1694	is disregarded and one in which the first denominator is 1≥ is used; 29, 71 and 89 are similar
1695	exceptions; in the case of 73 a resolution into three fractions is preferred to that into two,
1696	while 91, instead of obeying the formula, is resolved into only two fractions.
1697	The fact is that Eisenlohr is here employing a method of analysis which ought not to
1698	be applied to Egyptian mathematics. Even could we show that all the results corresponded
1699	• Badly mixprinted in P.s.B.4., XVI, 202 and 203.
1700	F 2
p. 40
1701	36 RHIND MATHEMATICAL PAPYRUS
1702	Division method in all cases. He had indeed observed that where the denominator of the 2-fraction
1703	of 2.
1704	was a multiple of 3 the same resolution could be used as for 3 itself, and similarly for 5, 7,
1705	and even 11; but when he came to the higher prime numbers he had no formula to help him.
1706	His method was undoubtedly that of trial. He had grasped the fact that the problem
1707	consisted in breaking up 2 into the sum of several quantities each of which would divide
1708	without remainder into the given denominator. The first of these quantities was always
1709	greater than unity, preferably greater than 1½, and was such that when reduced to an im-
1710	proper fraction (to use a modern term) its numerator was precisely the given denominator.
1711	Thus to resolve 19 the 2 is broken up into quantities the first of which is 1 + } + te or t3.
1712	If we can now find one or more aliquot parts which when added on will make this up to
1713	2 the problem is solved, for 13 must divide exactly into 19, and any aliquot part when divided
1714	by a whole number, namely the given denominator, remains an aliquot part. In the present
1715	case the obvious fractions to be added are kand 1.
1716	There are two points here which call for explanation. If the Egyptian had no con-
1717	ception' of the quantity 1 + ½ + 7» under the form 13, how did he obtain it, and how did he
1718	know it to be a twelfth of 19? Undoubtedly by trial divisions of the original denominator
1719	19, making use of his favourite processes of successive division by 2 or of multiplication by 3,
1720	followed by halvings of this, i.e. of division by 3. Thus the first step of the proof in the case
1721	of 19 gives us a correct idea of the method by which the result was originally arrived at:—
1722	1 19
1723	12%
1724	Here on arriving at ta of the original denominator he finds it to be lf + 1½, a very
1725	suitable number for the first of his parts of 2, and of course t› of the denominator 19. The
1726	other parts, which must be aliquot parts, are easily found, for the simple reason that the
1727	reckoner has a set of tables, constructed perhaps for this very purpose, namely the śkm-tables,
1728	Nos. 21-23 below. He there finds that to complete 2 from 1! + te we must add ‡ + %.
1729	Twoshort multiplications show that fis tath of 19 and f isitath of it.
1730	In this case then the proof not only shows the correctness of the result but gives us
1731	a clue as to how the result was actually obtained. The method was to write down the denomi-
1732	nator, to begin by halving or taking two-thirds of it, and then continuing to halve until a
1733	number greater than unity (often greater than 1}) but less than 2 was arrived at. If this
1734	could by the use of śkm-tables be made up to 2 by the addition of two or three aliquot
1735	parts, the problem was solved; if not, another trial must be made, starting this time by
1736	taking } instead of ½, or vice versa. Neither Cantor nor Eisenlohr in their elaborate analysis
1737	of this table has realized how much the Egyptians were here at the mercy of their inability
1738	to divide by numbers other than 2, 10 and 3, this last only through multiplication by }.
1739	If the denominators of the first aliquot parts into which the 2-fractions with prime denomi-
1740	nators from ll to 97 are resolved be examined it will be found that with two exceptions,
1741	42 and 56, they contain only 2, 3 and 10 as factors. The two multiples of 7, namely 42
1742	and 56, merely serve to emphasize the complete dependence of the table on trial, and its
1743	lack of regularity. Cantor is doubtless right in insisting that it was a gradual empirical
1744	accumulation.
1745	The following table will enable both the results and the method employed in obtaining
1746	them to be seen at a glance. In the first column is the fraction to be resolved, in the second
1747	are the parts into which its numerator 2 is resolved, and in the third the solution.
1748	1 See, however, p. 16.
p. 41
1749	2÷ 5
1750	7
1751	9
1752	11
1753	13
1754	15
1755	17
1756	19
1757	21
1758	23
1759	25
1760	27
1761	29
1762	31
1763	33
1764	35
1765	37
1766	39
1767	41
1768	43
1769	45
1770	47
1771	49
1772	51
1773	53
1774	55
1775	57
1776	59
1777	61
1778	63
1779	65
1780	67
1781	69
1782	71
1783	73
1784	75
1785	77
1786	79
1787	81
1788	83
1789	RHIND MATHEMATICAL
1790	RESOLUTION OF 2.
1791	13 •
1792	13
1793	cã NỐ N CTO N ot at. • . кр /-5f- •
1794	•
1795	1 +++. + tz A/- tof • oj-A→
1796	1
1797	1 N/M KÂ- GHN R~ COO CUINO ++ c-o- •
1798	1 • •
1799	+ 24
1800	+ cr. •
1801	to
1802	1 •
1803	• to
1804	+..
1805	•
1806	1} 11 + + • To
1807	•
1808	12 . to
1809	•
1810	15 + = •
1811	13 13 • •
1812	13+10+20. $
1813	15+10 . ÷+30
1814	1}
1815	1 •
1816	snazava L8
1817	710
1818	{1 +
1819	+
1820	+ &L.
1821	+ 22 .
1822	101+ 38 +
1823	ef + lI
1824	82 + 18 + 8,1
1825	{11 + 24 +31
1826	* + E
1827	243 + 31
1828	+
1829	+ 398+E4I+
1830	+ sqI +.
1831	+ 2 2.
1832	+
1833	+ + 262
1834	+
1835	+ 272 +
1836	+ 6ộI+ IŞE +
1837	+
1838	0.E + e4E + I7I
1839	+ 26I
1840	+ 30I
1841	+ 964 + 8L€
1842	+ eșe
1843	8,€
1844	28 +
1845	*7e + + 012 + 88E.
1846	+ 281
1847	GE + S6I
1848	ef + açk + sfE
1849	+ 88L
1850	+ . 322 +
1851	99e +263+ 012 + 92
1852	+ 29I.
1853	+ BQ€.
1854	26k+ 215 + 48e + 12
1855	39E + S1* + 288 + 22
1856	99e + 49
1857	E41 + 88
1858	29E + 92 268 + *$s +
1859	oşI + ft
1860	281 + 32
1861	045 + 985 + 22
1862	94k+ 842 + 22
1863	8Q1 + 22
1864	292 + =9E + 302 +1ęI
p. 42
1865	38 RHIND MATHEMATICAL PAPYRUS
1866	Division In the sums which follow there are endless scribe's errors, the most usual being the
1867	of 2. omission of the tick which denotes that a certain line in a multiplication is used in the
1868	addition by which the answer is obtained (see above, p. 13), and the incorrect omission or
1869	insertion of the fractional dot. This last is a common error throughout the papyrus, and
1870	in order to save space it has been corrected without further comment in these division sums.
1871	It is further to be noted that in the earlier sums the number by which 2 is divided
1872	23, indicating that it is to be used again as the first
1873	23. After the sum 2 ÷ 29 this dot is almost
1874	always omitted, but it is undoubtedly to be understood, and for the sake of clearness we have
1875	inserted it in rendering.
1876	DIvIDE 2 by 3. (Pl. A.)
1877	2 is erds.
1878	No proof is necessary or indeed possible.
1879	DIVIDE 2 by 5. (Pl. A.)
1880	13 is frd, } is r'sth.
1881	Working out:
1882	5
1883	3%
1884	1%1
1885	As this sum lies in the top register of the papyrus the rubrics nis ... hnt "Divide ...
1886	by" and 55mt "Working out" are inserted. Throughout the table they are omitted except in
1887	such sums as begin a new page. They must be regarded as repeated with every sum.
1888	The answer to the sum is * + t's, which is stated by implication in the second line.
1889	Note that of 5 is reached through ?, as always. For the division see above, pp. 13 and 20.
1890	DIvIDE 2by 7. (PI. A.)
1891	1 is th, *is zgth.
1892	Working out :
1893	1 7
1894	1
1895	2 14
1896	28
1897	The figure 4 in front of the multiplier zs indicates that 4 is the number by which ?
1898	must be multiplied to produce the required 28. This is proved in theshort piece of working
1899	on the right.
1900	DIvIDE 2 by 9. (PI. A.)
1901	11 is fth, } is tsth.
1902	Working out:
1903	1 9
1904	6
1905	3
1906	The papyrus is damaged, but the reading certain.
1907	1 Wrongly given as 1} in the plate.
1908	= Actually 28. The Egyptian wasveryinconsistent with regard to the fractional dot in these cases: two
1909	different statements of the same fact were in his mind, viz. 4xi= 28 and the resulting \| is agth of T.
p. 43
1910	RHIND MATHEMATICAL PAPYRUS 39
1911	DIvIDE 2 by 11. (Pl. A.) Division
1912	13 tề is =th, 6 is đ5th. of 2.
1913	Working out :
1914	1]1
1915	[2 2]2
1916	33 [4 4]4
1917	13 + ÷] [6]6
1918	This restoration is practically certain and fits the traces ou the torn edges of the papyrus
1919	)erfectlv. Oddly enouch Griffith (P.S.B.A., XVI, 207) restores only the right-hand portion c
1920	the above working, and notes "this leaves the first fraction t 13 stupidly unexplained.
1921	The very position of this right-hand portion (left in the papyrus) shows that it was preceded
1922	by something else, viz.the working out of 1%+ * is fth. (Cf. the division of 2 by 7, or
1923	indeed any which involve the use of fractions with high denominators.)
1924	DIVIDE 2 by 13. (Pl. A.)
1925	14 + * is gtb, · is 5ond 8 iS Tozth.
1926	Working out:
1927	13
1928	15+$
1929	Tốt
1930	For the 4 and the 8 in the last two lines of. the division of 2 by 7 and remarks there.
1931	Note the tacit assumption that 1% +} added to t and gives 2 Here we see an example of
1932	the usefulness of the completion tables Nos. 21-23.
1933	DIvIDE 2 by 15. (PI. A.)
1934	1f is to, →is soth.
1935	Working out:
1936	1 15
1937	/30
1938	• DIvIDE 2 by 17. (PI. A.)
1939	1a t tz is Tath, * is srst, # is osth.
1940	Working out:
1941	11g
1942	}
1943	21+$
1944	_ Remainder $ + 1
1945	1
1946	2
1947	}
1948	Here we have a slight change in method. After the first ticked line the worker sums
1949	accounted only for the first. Aphisposition.Ofthethaeetermsiserandiofwaichtheaseriscomposedeheba The ‡ and he describes as "remainder" and proceeds to dea.
1950	with them separately. In the last multiplication he has turned the multiplicands from whole
p. 44
1951	40 RHIND MATHEMATICAL PAPYRUS
1952	Division uumbers into fractions by dotting them: strictly speaking this is not
1953	on the left are multipliers and not divisors (see footnote to 2÷7 an
1954	Note the tacit assumption that 1# + * is equivalent to 13
1955	DIvIDE 2 by 19. (PI. A.)
1956	15+is fath, $ is feth, 1 is rtath.
1957	Working out:
1958	1 19
1959	123
1960	Remainder * + /
1961	1 19
1962	2 38
1963	76
1964	Remainder f
1965	1 19
1966	2 38
1967	4 76
1968	-6 114
1969	For themethod comparethe preceding sum.
1970	DIVIDE 2 by 21. (PL.A.)
1971	If is izth, t is tand
1972	Working out:
1973	1 21
1974	14
1975	42
1976	Note that the Egyptian is well aware that the inverse of 3
1977	DIvIDE 2 by 23. (Pl.A.)
1978	13+fis rath, to is stath.
1979	Working out:
1980	1 23
1981	15%
1982	73
1983	12 +÷÷
1984	Remainder t2
1985	1 23
1986	>10 230
1987	46
1988	Total
1989	Note the assumption1+*+=13+4
1990	DIVIDE 2 by 25. (PI. B.)
1991	13 is fsth, } is fsth.
1992	Working out:
1993	1 25
1994	quite correct, for the digits
1995	d cf. next sum).
1996	+Te.
1997	is 1g
p. 45
1998	RHIND MATHEMATICAL PAPYRUS 41
1999	DIvIDE 2by 27. (PI. B.) of 2. Division
2000	1f is r'sth, ț is sath.
2001	Working out:
2002	1 27
2003	DIVIDE 2 hy 29. (PI. B.)
2004	1+= is Jth, } is s'sth, f is T7#th, $ is atand.
2005	Working out:
2006	1 29
2007	> 1 (sic)
2008	T7E
2009	The multiplier 1 (a dot) with a tick is obviously an error, for these digits on the left
2010	are those by which 29 must be multiplied to give the denominators of the fractions whiehi
2011	respectively follow them. No working is given for the step against which it stands.
2012	DIVIDE 2 by 31. (PI. B.)
2013	1ttt is toth, * is rlzth, ‡ is rhsth.
2014	Working out:
2015	31
2016	1 (sic) 20 15+20
2017	T}5
2018	The rot 1 in front of do is an error (cf. last sum). The working is highly condensed
2019	as in many of the sums from here onward.
2020	DIVIDE 2 by 33. (PI. B.)
2021	lf is zond, ¿ is teth
2022	Working out:
2023	1 33
2024	DIvIDE 2by • 35. (PI. B.)
2025	1 is soth, 3 + * is z'ynd.
2026	7 5
2027	Working out:
2028	1 35
2029	3+*
2030	The digits 6 (in red), Tand 5 are unexpected here. The 7 and the 5 clearly indicate
2031	the number ofsixths in l ss nuater e egichi in o telie cottli negpectivet and , ha ic, to bhatendh eot uendet the that these are actually
2032	G
p. 46
2033	42
2034	Division DIVI
2035	of 2.
2036	DIvI
2037	DIv
2038	DIVI
2039	RHINDMATHEN
2040	DE 2 by 37. (Pl. B.)
2041	1t + 2t is zith, } is rirth,
2042	Working out:
2043	1
2044	Te
2045	12+
2046	1
2047	2
2048	-3
2049	1
2050	2
2051	DE 2 by 39. (PI. B.)
2052	1} is 2tth, } is tsth.
2053	Working out :
2054	1
2055	2
2056	DE 2 by 41. (PI. B.)
2057	13 + 2z is 24th, fis ztath,
2058	Working out:
2059	1
2060	t
2061	/27
2062	1
2063	Total
2064	DE 2 by 43. (PI. B.)
2065	172 is zand }is sith,
2066	Working out:
2067	1
2068	Found
2069	3
2070	The sign here translated "found" is th
2071	•
2072	ATICAL PAPYRUS
2073	is 29t
2074	37
2075	243
2076	12%
2077	31'2
2078	12 + zz
2079	Remainder } + }
2080	37
2081	74
2082	111
2083	Remainder $
2084	37
2085	74
2086	148
2087	296
2088	39
2089	gis 37gth.
2090	•
2091	41
2092	133
2093	63+ 1
2094	33+1z
2095	13 +÷
2096	Remainder & + ‡
2097	41
2098	82
2099	164
2100	246
2101	328
2102	is Thath, } is s0rst.
2103	43
2104	e gui-bird, an abbreviation for some part of the
p. 47
2105	RHIND MATHEMATICAL PAPYRUS 43
2106	the result of which has been derived from some outside source. Thus it is inserted here betore
2107	the division of 43 by 42, and later before that of 53 by 30, the result of which is 13 + to.
2108	Grittith notes that it is sometimes omitted when we most expect it, e.g. in 25 ÷ 15 = 1} (division
2109	of 2 by 25), in 29 ÷ 24 = 1% + z= (division of 2 by 29). Conversely we are surprised to find
2110	it in 62 ÷ 93 = 3 and 66 ÷ 99 = }, for to the Egyptian the taking of } is as straightforward
2111	a process as taking an aliquot part. In these cases the symbol perhaps indicates that recourse
2112	has been had to tables.
2113	DIvIDE 2 by 45. (PI. B.)
2114	1f is soth, ‡ is soth.
2115	Working out:
2116	1 45
2117	DIVIDE 2 by 47. (PI. B.)
2118	15 + Is is soth, * is rirst, to is ztoth.
2119	Working out:
2120	47
2121	Found 30 1$ + ts
2122	-3 TFT
2123	<10 FTT to
2124	DIvIDE 2 by 49. (PI. B.)
2125	1+is asth,. fis rhath.
2126	Working out :
2127	49
2128	Found it
2129	TJa
2130	DIvIDE 2 by 51. (PI.B.)
2131	1f is sath, (b) is Toznd
2132	Working out:
2133	1 51
2134	34
2135	102
2136	DIvIDE 2 by 53. (PI. B.)
2137	13 + to is soth, f is siath, t's is Tosth.
2138	Working out :
2139	1 53
2140	Found 130 13 + To
2141	Remainder ts
2142	1 53
2143	-10 530
2144	265
2145	à dot (= multiplier 1) after "found" which should not be there. Cf. 2÷ 29
2146	and 2÷31.
2147	G 2
p. 48
2148	44 RHIND MATHE
2149	Dlvision DIVIDE 2 by 55. (PI. B.)
2150	¿ is ssoth.
2151	Working out :
2152	1
2153	Found
2154	-6 350
2155	DIvIDE 2 by 57. (PI. C.)
2156	1y is s'sti, ¿is Tiath.
2157	Working out:
2158	1
2159	DIvIDE 2 by 59. (PI. C.)
2160	1t + ta + ta is sath, ·is
2161	Working out :
2162	1
2163	Found 156
2164	53T
2165	DIvIDE 2 by 61. (PI. C.)
2166	1t +to is zoth, fis siath,
2167	Working out:
2168	Found
2169	788
2170	<10 610
2171	DIvIDE 2 by 63. (PI. C.)
2172	1f is zond, İ is m3rth
2173	Working out:
2174	. 1
2175	DIvIDE 2by 65. (PI. C.)
2176	1% is ssth, * is risth.
2177	Working out:
2178	1
2179	Found so
2180	T35
2181	DIVIDE 2 by 67. (PI. C.)
2182	5is 3a
2183	Working out:
2184	1
2185	Found
2186	335
2187	536
2188	IATICAL PAPYRUS
2189	55
2190	13+1
2191	38
2192	114
2193	Jeth, ț is słrst
2194	59
2195	81+31+3I
2196	$ is z3gth, to is stoth.
2197	61
2198	15+70
2199	To
2200	63
2201	65
2202	13
2203	3ath, * is sboth.
2204	67
2205	25 + 8 + 31
p. 49
2206	DIVIDE
2207	DIVIDE
2208	DIVIDE
2209	DIVIDE
2210	DIVIDE
2211	DIVIDE
2212	DIVIDE
2213	RHIND MATHEM
2214	2 by 69. (PI. C.)
2215	1! is ‡sth, 1is ragth.
2216	Working out:
2217	1
2218	138
2219	2 by 71. (Pl. C.)
2220	13 + # + Iu is foth, $is s6
2221	Working out :
2222	Found
2223	368
2224	Tto
2225	2 by 73. (Pl. C.)
2226	} is atwth,
2227	Working out :
2228	1
2229	Found
2230	385
2231	2 by 75. (Pl. C.)
2232	15 is soth, 3 is rhoth
2233	Working out :
2234	1
2235	50
2236	T50
2237	2 by 77. (Pl. C.)
2238	1, + 1 is zxth, k is sogth.
2239	Working out :
2240	Found 4141 1
2241	4 31081 -
2242	2 by 79. (PI. C.)
2243	1f + 15 is 6oth, * is 237th,
2244	Working out:
2245	1
2246	Found 160
2247	• 3 253
2248	• 10 750
2249	In the working read zlr and 3te
2250	2 by 81. (Pl. C.)
2251	15 is 3tth, } is rland.
2252	Working out:
2253	1
2254	5'4
2255	12 TEZ
2256	ATICAL PAPYRUS 45
2257	Divisior
2258	of 2
2259	69
2260	sth, to is rtoth.
2261	71
2262	11+÷+ 70
2263	‡ is z3zud, 5is ststh.
2264	73
2265	18+20
2266	75
2267	77
2268	13+·
2269	*is5i:th, to is rboth.
2270	79
2271	14+t5
2272	To
2273	81
2274	11
p. 50
2275	46 RHIND MATHEMATICAL PAPYRUS
2276	Division DIvIDE 2 by 83. (Pl. C.)
2277	1g+ do is Laoth], #is slund, }is #lsth, 1 is 498th.
2278	Working out:
2279	1 83
2280	Found 15+3t
2281	33=
2282	758
2283	DIvIDE 2by 85. (PI. C.)
2284	1% is srst, } is =h„th.
2285	Wurking out :
2286	1 85
2287	Found 13
2288	-3 755
2289	DIVIDE 2by 87. (PI D.)
2290	1f is 38th, E is itath.
2291	Working out:
2292	1 87
2293	[1]6
2294	TtE
2295	DIvIDE 2 by 89. (PI. D.)
2296	[13] + 1o + zo is zoth, E] is atoth, ois (53ath, Lro] is sboth.
2297	Working out:
2298	1 89
2299	Hound 1%+ to + to
2300	356
2301	-6 337
2302	-10 550
2303	DIvIDE 2 by 91. (PI. D.)
2304	1% + to is toth, 3+ so is roth.
2305	Working out:
2306	Found 7o 15 + 1o
2307	Found T5o ·t30
2308	The working is little more than a restatement of the answer.
2309	DIVIDE 2 by 93. (PI. D.)
2310	‡ is T8sth.
2311	Working out:
2312	1 93
2313	Found
2314	DIvIDe 2 by 95. (PI. D.)
2315	12 + ta is foth, Et] is ssoth, [a is stoth.]'
2316	Workiug out :
2317	1 95
2318	Found 1+1s
2319	570
2320	1 The 5 perbaps on a B.M. fragment. See p. 48.
p. 51
2321	RHIND MATHEMATICAL PAPYRUS 47
2322	DIVIDE 2 by 97. (PI. D.) Divisior
2323	1ettixtzsis stth f is styth, [#l is rrigith. bf 2
2324	Working out :
2325	1 97
2326	Found 1++ *+28
2327	DIVIDE 2 by 99. (PL. D.)
2328	1s tsth, } is risth.
2329	Working out :
2330	1 99
2331	Found
2332	2 TUE
2333	[DIvIDE 2 by 101.] (PI. D.)
2334	1 is [rorst], 1 is znd Lal is zogrd, E is a0gth.]
2335	Working out:
2336	L1 101]
2337	[202]
2338	303
2339	606
2340	It has always heen supposed of 2 in this papyrus ran from 3 to 99.
2341	The New York fragments makeit clear that in the top register in the gap in the London
2342	papyrus (see below, p. 48) stood the division of 2 by 101. Mathematically the result is
2343	surprising and disappointing, for one of the fractions in the resolution is clearly тỏт, ie. half
2344	the fraction to be resolved ; in other words a type of resolution is here adopted which has been
2345	purposely avoided throughout the long table. It may be surmised from this feeble ending that
2346	the mathematician was here at the extreme range of his ability, and that the resolution-tables
2347	of this period hardly went heyond this figure.
2348	THE GAP BETWEEN THE TWO PAPYRI. Plate E.
2349	On Plate VII of the B.M. Facs. we find a vertical gap down the centre of the page. Gap
2350	On the right we have the left-hand edge of the recto of Papyrus 10058, and on the left the between
2351	the right-hand edge of the recto of Papyrus 10057. It has been generally assumed that these papyri.
2352	two papyri were originally one, andthat they were cut apart in antiquity. This assumption
2353	was based mainly on the fact that both are clearly by the same hand, and that on either
2354	side of the gap the ruled horizontal registers correspond almost exactly in depth.
2355	York fragments, as has been pointed out in the Introduction, place this beyond doubt, for
2356	we now see that in the top register above the first of the division of bread sums stood the
2357	division of 2 by 101. As the last problem on the recto of 10058 was the division of 2 by
2358	99, it would be idle to deny that the two papyri were originally one. Fortunately the New
2359	York fragments enable us almost completely to fill up the lacuna. Before placing them,
2360	however, we must rectify some errors on the edges of the papyrus in the British Museum
2361	facsimile.
2362	The edge of 10057 has been carelessly patched, an three alterations are needed in
2363	he facsimile. On the edge of the second horizontal register is a misplaced fragment whicl
2364	1 This step was perhaps omitted.
p. 52
2365	48 RHIND MATHEMATICAL PAPYRUS
2366	must be transferred to the edge of the sixth (bottom) register. Similarly the strip on the
2367	between edge of the fourth register fits, when inverted, on to the edge of part of the fifth and sixth
2368	the main edge of registers 1-5 and protrudes
2369	at the top should be swung a little to the left round its lowest point, so as to take the place
2370	of the blank strip which follows it in the facsimile. See Plate E.
2371	the edge of Papyrus 10058, there are two
2372	In the fourth register from the top there is a fragment with the
2373	fractional figures for 600 and 70 in red, which must be placed one register lower, in the
2374	division of 97, where it forms part of the number it». On the edge of the third register is
2375	a fragment with a sign in red for a fractional number of hundreds, though what number is
2376	not quite clear. Griffith read this as 100, and would place it on the edge of the sixth register
2377	as part of 15r. This is now impossible, for New York fragments 24 and 25 must be placed here.
2378	There are only two positions left for a fragment giving fractional hundreds, one is in the top
2379	register, »3+, and the other in the fourth, s70. Our fragment is too deep for the top register,
2380	and should therefore come from the fourth.! Unfortunately it is now partly concealed beneath
2381	the frame and the mounting, and it is impossible to be certain as to its reading.
2382	Having rectified the edges on each side of the gap, we may proceed to place the New
2383	York fragments, beginning with the left-hand side, i.e. Papyrus 10057. The large fragment
2384	1 fits on exactly in registers 4, 5 and 6, and fragments 16 and 17 are obvious additions to
2385	it. Fragment 4 can be completed by means of fragments 8 and 10-14° It is clear that it
2386	comes from the top and second registers, and if we look at line 1 of the second we see that
2387	between the verb psš of the fragment and the numeral 1 of 10057 we need only the word
2388	t: for loaves, which in the other sums takes up on the average 15 mm. Fragment 4 with
2389	its additions can thus be placed within a millimetre or two.
2390	Fragment 3 with the sign • occurring three times is needed nowhere but in register 2
2391	and top line of 3; on to it fit fragments 2, 7, and 15. For fragment 9, which gives 2-times
2392	and 4-times 3, room can be found only in registers 2 or 3. In register 2 it would be out of
2393	place, for the 8-times line, which is preserved, gives not 13+3+1 but 3+1+30. It
2394	must therefore come from register 3, and it proves that the sum which stood there was the
2395	division not, as Eisenlohr supposed, of 3 loaves but of 2.
2396	The top part of fragment 6 with its rubric sšmt can only come from the top register,
2397	and is thus part of the division of 2 by 101. Its position can be fixed by an examination
2398	of the bottom line of the register. Here we see on fragment 6 the multiplier 2 (two dots);
2399	on the right edge of fragment 4 is the sign for }, and between this and the 2 we have to
2400	place only the numeral 202, which would occupy anything from 12 to 20 mm., let us say
2401	15 as an average. Thus we can place this large fragment within about 5 mm.
2402	Turning to Pap. 10058 and the other side of the gap we can at once place in the
2403	two bottom registers fragments 20-25. Fragment 5 with its red sio can only come from the
2404	top register, and fragment 19 joins it. All we now have to do to estimate the whole width
2405	of the gap is to decide the spaces between this fragment 19 and the edge of 10058, and
2406	between fragments 5 and 6 on the left, for we have already bound up fragment 6 with 10057
2407	with a possible error of only a few millimetres. Dealing first with 19 we notice that it bears
2408	in red the 4 of the fraction s*4: to the right of this must have come the fraction & in
2409	black, corresponding to the :1. in red on Pap. 10058. It is difficult to estimate the space
2410	taken up by the missing numbers, for the spacing in the top lines of these sums is very variable.
2411	It could hardly be less than 30mm. and perhapsnotmorethan 35. Turning to the left
2412	side we have to supply to the left of fragment 5 the rest of the sign for 90 and the black tü
2413	which must have followed. As the scribe was inclined to crowd here (witness the proximity
2414	1 Where it is placed, with a query, in Plate E.
2415	2 Fragment 13 might conceivably have come from one of two other places, top line of division of 2 by 101
2416	(f is «lrth) or register 4 of 10058, f is rlath (division of 2 by 95).
p. 53
2417	RHIND MATHEMATICAL PAPYRUS 49
2418	of the red 4 and the black f on fragment 19) the space occupied was perhaps as little as Gap
2419	15 mm. between
2420	under it is likely to have extended farther to the left, we may say that this point marks the papyri
2421	farthest extension to the left of the page of division of 2 sums. The only question now remaining
2422	is how much space the scribe left before heginning the rubric of the division of 2 by 101 and
2423	the table of tenths below it. To this it may be replied that our scribe never wastes an inch,
2424	and always begins a fresh "page" at the point where the longest line of the preceding page
2425	ends. Indeed, he sometimes anticipates this, and lets a long line from the last page cut into
2426	a new one. Here, however, his longest line of the last page was the top line, and we shall there-
2427	fore be fairly safe in assuming that the rubric of the division of 2 by 101 followed immediately
2428	on the io which ends the division of 2 by 89. The result of this supposition is shown in
2429	the reconstruction of Plate E. This brings the nearest gummings on 10057 and 10058 to
2430	exactly 390 mm. apart, and as this is a fair size for a page on the recto of the papyrus (see
2431	above, p. 3) it corroborates the arrangement of the fragments here proposed.
2432	The placing of some of the minute fragments dealt with above may seem somewhat
2433	arbitrary to the general reader. It must be remembered, however, that in a mathematical
2434	papyrus there is much less choice of position for a particular sign than in a literary document,
2435	and except in cases where doubt is expressed it may be regarded as morally certain that the
2436	fragments are correctly placed, with the obvious reservation that some of the more dis-
2437	connected may be a few millimetres out of position, e.g. 20-25.
2438	The fragments that remain are for the most part uninspiring (Pl. E, right). Some of them
2439	other scholars will doubtless place, if anyone cares to take the trouble, but a few will remain
2440	strays, unless the examination of the fibres in the originals gives some help. The chief interest
2441	lies in fragments 31-34 which clearly contain portions of something entirely different from the
2442	division of 2 and the bread sums. On 'two of these, 32 and 34, we see the 3rd Singular
2443	Masculine ending f, ie. "he" or "it," and 3l is to be restored snn pw n "it is a copy of,"
2444	which reminds us of the introduction to the whole papyrus. These fragments might possibly
2445	come from the verso. If they must be placed on the recto it may be that they are to be
2446	connected with the mysterious mitt which occurs in a baffling position in front of the division
2447	of 8 loaves.
p. 54
2448	50 RHIND MATHEMATICAL PAPYRUS
2449	BOOK I. ARITHMETIC.
2450	Nos. 1-6. DIVISION OF LOAVES. Plate F.
2451	Nos. 1-6. LEAVINg the long table of the division of 2 we pass on to a series of purely arithmetical problems,
2452	to each of which Eisenlohr has given a number which for reference purposes it is clearly
2453	advisable to preserve. The first six of these are concerned with the division of certain numbers
2454	of loaves of bread among 10 men, each man's share to be expressed in aliquot parts. Thus
2455	in the division of 7 loaves among 10 men each man receives } + } of a loaf.
2456	Expressed abstractly and in modern terms these examples are the resolution of the
2457	various proper fractions whose denominators are 10 into the sum of two or more aliquot
2458	parts. From New York fragment 6 (Pl. F, top left) it is clear that they were preceded by
2459	a table which ran as follows:—
2460	to *+[5o]
2461	3+ 10 + 50
2462	5 + to 3+ \| + 30
2463	}+ts
2464	*+to
2465	This is nothing more than the expression in aliquot parts of the tenths from ro to io, and
2466	the results, with the exception of 1o, To and io, are actually proved in the problems which follow.
2467	These comprise the division between 10 men of 1, 2, 6, 7, 8 and 9 loaves respectively.
2468	It is easy to understand the omission from the series of 4 loaves, for i is equivalent to 3,
2469	by the preceding tables can be resolved at once into*+1. But why omit the
2470	division of 3 loaves and include that of 1, the answer to which, 1o, is already in the required form
2471	of an aliquot part? Here we are face to face with one of those anomalies which from time
2472	to time baffle all our attempts to see things with the mind of the Egyptian mathematician.
2473	Obviousiy 1 loaf is not treated merely for the sake of completeness, for 3 and 4 are omitted,
2474	and it is hard to conceive a mind which, while ready to accept as obvious the reduction of
2475	To to 3 and thence to ț + T's, felt a necessity to prove by four lines of arithmetic that if ten
2476	men divided a loaf each would receive one-tenth of it. Careless omissions by the scribe may
2477	be the simple explanation.
2478	The examples are in the main a continued application of the resolutions of 2-fractions
2479	or division of 2 with which the papyrus has been hitherto occupied. This will be clearly seen
2480	if any one of the sums be written out in full. Thus in No. 3, the division of 6 loaves among
2481	10 men, we are first given the answer * + to, and the sum is proved by multiplying this by
2482	10 and showing that the result is 6, thus:-
2483	* + To
2484	Iwice (* + 10) =1 + 3
2485	Four times (t+1)= 2+3= 2 + (}+1s)'
2486	_ Eight times (3 + ju) =4+3+3 =4++(+30)
2487	Total, ten times (1 + 1u) = 6
2488	Here it will be seen that the multipliers are invariably 2, and that in order to multiply ! + Tu
2489	by 10 it is multiplied by 2, and by 2 again, and by 2 yet again, thus giving 4-times and
2490	8-times. The 2-times and the 8-times are then added together, giving the required 10-times.
2491	In this process the only multiplier used is 2, and, as a consequence, in the working the only
2492	fractions excepting 1-fractions (aliquot parts) met with will be 2-fractions, which by the help
2493	of the preceding tables can be at once resolved into their 1-fractions.
2494	' Using the table for 3. * Using the table for is.
p. 55
2495	RHIND MATHEMATICAL PAPYRUS 51
2496	It is thus easy to see why these bread calculations follow directly on the tables for the Nos.1-6.
2497	of 2-fractions. But this is not quite all. In the example given above the quantities
2498	13 + 1o + 3o and 1 + } are added together and stated to give a total of 6. In other words,
2499	a process of addition of fractions is employed. And yet no working is given, the result being
2500	simply taken for granted. Presumably the same process was used as in Nos. 7-23. In No. 4,
2501	the division of 7 loaves, we are still further mystified, for there we find:—
2502	3+ 30
2503	Twice is 1f + 1s
2504	4-times is 23+ to+ 3o (by table)
2505	8-times is 5} + 1o
2506	How is the 8-times obtained from the 4-times? To our minds the simplest process would be
2507	to double each term, and the answer would be 4 + 1} + * + 15. It is true that this is equi-
2508	valent to 5 + ½ + 1o, but the equivalence costs the modern mathematician a moment's thought,
2509	and was certainly not done out of hand by the Egyptian. Notice, too, that the form in which
2510	the result, namely 5 + ½ + Tu, is far less suitable than the obvious form 5} + } + 7s
2511	for the necessary addition to the 2-times, 1+ 15. How, moreover, does the reckoner add
2512	together 51 + 1o and 1} + 1s to form 7, a process for which most of us would have to
2513	employ a common denominator, even if only mental? Beyond all doubt he was working
2514	from tables of the addition of fractions, of which none have survived to us in the papyrus,
2515	but the existence of which might easily have been inferred from Nos. 21-23 below.
2516	substitution of 5} + to for 5} + 1 + 1s is a more difficult point. Here again we surmise
2517	that tables of equivalents were used, and in the choice of the less suitable equivalent for
2518	addition purposes we may perhaps trace what one constantly suspects in the Egyptian, an
2519	implicit belief in results obtained by trial, and a certain intellectual dishonesty in constructing
2520	a proof of them.
2521	No. 1. (PI. F.) No. 1.
2522	"Example of dividing 1 [loaf]' among 10 men.
2523	[You] are to mult[iply] To by 10.
2524	The doing [as it occurs :]
2525	2
2526	4 3 + 15]
2527	8 }+ 1o + 30
2528	1. This is the number in question."
2529	he method of all these sums is the same. In the second line the answer is given
2530	id the working consists of multiplying this by 10 and showing that it gives the number (
2531	loaves to be divided.
2532	For the phrase irt mi lypr, see above, pp. 23-24.
2533	No. 2. (PI. F.) No.2.
2534	* To divide [2] loaves [among 10 men].
2535	You are to multiply [} by 10].
2536	The doing as it occurs:
2537	3 + 15
2538	3 + to + 3o
2539	115+*+15
2540	['Total 2.] This is [the number in question].
2541	' Square brackets [ ] indicate lacunae in the text. They are never used in this volume as mathematical
2542	symbols.
2543	H 2
p. 56
2544	52 RHIND MATHEMATICAL PAPYRUS
2545	No. 2. Eisenlohr has restored this sum as the division of 3 loaves, the answer to which would
2546	be, according to the table of fractions which precedes these sums, ! + to• His proof is as
2547	_2×(+0= =+1
2548	4 =1;
2549	= 2} + 15
2550	Total 10 = 3
2551	Now when the Egyptian multiplies + to by 2 he should get not 4ti but !+!+ t».
2552	it is clear from that the equivalence of 2(, +1.) and \| t was known
2553	The sum therefore might quite conceivably deal with 3 loaves. It certainly does
2554	not deal with 4 loaves, ie. }+ 1s to each man, for we should expect »o as last term of the
2555	8-times line, whereas the text preserves is. The division of 5 loaves would not account for
2556	this Ts, which must be our guide, and we are therefore forced back on to the belief that the
2557	number divided was 3 or 2, either of which, as seen from the reconstructions, would account
2558	for the is. The latter is shown to be correct by the New York fragment 9, which reads:-
2559	*tts
2560	3t tu+ 30
2561	with a horizontal line marking the division of registers immediately below. The fragment
2562	would thus be in the right position for the 2-times and 4-times lines, and there can be little
2563	doubt that this is its place.? The figures show that the division is that of 2 loaves, not 3.
2564	The divisions of 3, 4 and 5 loaves were thus not given, despite the occurrence of their
2565	answers in the table of fractions preceding the problems. We might have expected the
2566	omission of 5 and 2, which give the answers and respectively, though indeed neither is
2567	quite so obvious as the division of 1 loaf, which is given. The omission of 3 loaves is hard
2568	to explain, and may be an error on the part of the scribe.
2569	No. 3. No. 3. (Pl. F.)
2570	"To divide 6 loaves among [10] men.
2571	You are to multiply [bl + ro by 10.
2572	The doing as it [occurs.]
2573	[1 H]+ 1o
2574	L2 1]%
2575	[4 2]1 + 15
2576	43 + rol+ 30
2577	Total 6. This is [the number in question]."
2578	Observe that in the multiplication by four ; is broken up into y t is by table, and in
2579	the next line 7, is resolved into to + 30.
2580	No. 4. No. 4. (PI. F.)
2581	"To divide 7 loaves among 10 men.
2582	You are to multiply 3 +3o by 10: result [7]
2583	" + 30]
2584	L2 15+15
2585	23+To+30
2586	28
2587	Total 7 loaves. This is it."
2588	however, some irregular multiplication in Nos. 4-6; see above.
2589	" There is indeed no other place for it: it is, however, a trifle lower (judging by the register line) than we
2590	should have expected from the multiplier 2 on fragment 2.
p. 57
2591	RHIND MATHEMATICAL PAPYRUS 53
2592	Note here in the multiplication by 8 that twice } + t + " is taken to be la + ro, a r
2593	result used also in Nos. 5 and 6, but never proved. It was probably one of those pieces of
2594	information regarding the addition of fractions which the scribe knew by heart or by table.
2595	nt pw. In this sum and in No. 6 the phrase mitt pu "That is the same" with which the
2596	proof concludes, ie. that is the number of loaves which was to be divided, is replaced by
2597	nt pw. The meaning must be the same, and nt must be a word meaning "it" or "that"
2598	or "the like" or something of that kind. The phrase occurs again in Sinuhe, B 115 and
2599	possibly B 126, and also in Ebers, 99, 5. It is difficult to catch the exact sense in the
2600	Sinuhe passages, but it is certain that their solution must be approached from the equivalence
2601	of nt with mitt afforded by Rhind. The Ebers passage translates quite straightforwardly: "For
2602	its (ie. the heart's) ducts (lead) to each of his limbs; this it is (i.e. it is the heart) that
2603	speaks through the ducts of every limb."
2604	No. 5. (PI. F.)
2605	"To divide 8 loaves among 10 men.
2606	Touaretomultiply3+oby1 results.
2607	1 + 1o + 30
2608	- 2 1[ + 1o]
2609	31]
2610	- 8 65+Ts
2611	Total This is the number in question."
2612	See note on No. 4 for line 2 of the multiplication.
2613	The word mitt standing in front of line 3 (see Plate E) is a complete puzzle. It certainly
2614	has nothing to do with this sum. See notes on the unplaced New York fragments, above,
2615	p. 49.
2616	No. 6. (PI. F.)
2617	"To divide 9 loaves among 10 men.
2618	You are to multiply 3 + 1 + 3o by 10.
2619	The doing as it occurs:
2620	1 3+3+30
2621	-2 13+1+3
2622	4 3}+ra
2623	-8
2624	Total 8 loaves. This is it."
2625	Note that the words "the doing as it occurs" have slipped out of position, standing
2626	in the text before "you are to multiply." This is doubtless due to the scribe's efforts to
2627	compress into three lines (the top register being very shallow) a sum which in his original
2628	had an entirely different arrangement.
2629	The second line of multiplication is got from the first by the resolution of 3 into * + I5
2630	and of ts into to +35.
2631	Nos. 7-20. FIRST GROUP OF COMPLETIONS (śkm). Plate G.
2632	The completion-calculations fall into two distinet groups of type so different that it is r
2633	surprising to find the same verb śkm used for both. In the first group, Nos. 7-20, it is
2634	important to notice that no problem is set. The reckoner begins with some fractional quantity
2635	and adds to it two of its fractional parts, either its half and its quarter, or its third and
2636	1 See GARDINEr, Notes un the Story of Sinuhe, 46 and 158.
2637	OS.
2638	7-20.
p. 58
2639	54 RHIND MATHEMATICAL PAPYRUS
2640	its two-thirds, and gives the result. Thus in No. 13 we find a calculation which in modern
2641	7-20. times we should set down as follows:-
2642	« = 1i +Tz
2643	zu = ]= tv1+
2644	1a = 7# +775
2645	Total a(1++)=$
2646	This is in effect equivalent to multiplying 1s + riz by 1? + 1. What the Egyptian
2647	actually did was to experiment with various quantities, adding either their half and their
2648	quarter, or their third and their two-thirds, and recording the result as valuable when it
2649	happened to be an aliquot part. In fact, we are here face to face with some of those actual
2650	experimental methods on which so much of Egyptian mathematies is based. We are not solving
2651	set problems but discovering by trial useful facts for future use.
2652	The process employed in the additions is virtually that of a common denominator, and
2653	has been fully discussed in the Introduction, pp. 17-19.
2654	The best translation for śkm would appear to be the perfectly literal "completion." Both
2655	the simple verb km and its causative śkm appear in the papyrus. The former, although it is
2656	in Egyptian used both transitively' and intransitively, here occurs only in the latter sense
2657	"to be complete": thus in No. 22 we find hr km } } 1s ts r 1,"Therefore } + } + T» + T5
2658	is complete up to unity," ie. adds up to unity.
2659	No. 7. No. 7. (Pl. G.)
2660	"Example of completion.
2661	1 # + 98
2662	8 + 3a
2663	32
2664	Is + TTz
2665	13+·
2666	Total $"
2667	Here *+1s is experimented with by the addition of its half and its quarter. The
2668	Egyptian sets down the given quantity, ‡ + 3s, preceding it by a 1 (a mere dot) as is
2669	usual in the case of a quantity which is to be operated on by multiplication or division. He
2670	then takes ; of this quantity, which is f + 3t, and then f of it, which is to + rig. He then
2671	adds together the original quantity or unit, its half and its quarter, and finds the result to be 3.
2672	The addition is done by means of reduction of all the separate 1-fractions to the common
2673	denominator 28. Under each fraction is placed in red (here shown by italics)" its value in terms
2674	of this common denominator, and in view of what has been said above concerning this process
2675	it will surprise no one to find such fractional values as and 1$ + *. • Finally the values, written
2676	in red, are added up and found to amount to 14. The sum ofthe fractions isthus #f or2.
2677	No. 7b. No. 7B. (Pl. G.)
2678	* + 28
2679	} +. =6
2680	ie t ,lg
2681	16+1 ·
2682	Total 3
2683	This is identical with No. 7: some of the red values are omitted.
2684	1 E.y. Shipwrecked Sailor, 127, and Pap. Petrograd 1116 A, recto, 101.
2685	2 Only in the case of these additions by common denominator has the attempt been made to indicate red ink
2686	in the translations.
p. 59
2687	RHIND MATHEMATICAL PAPYRUS 55
2688	No. 8. (PI. G.) No. 8.
2689	Total 3
2690	Here the mathematician experiments with the fraction ž, which he first sets down,
2691	preceded by the figure 1, to show that it is the unit which he is about to multiply or divide.
2692	He then takes two-thirds of it, which his tables tell him to be , and then one-third, reached
2693	as always by halving two-thirds. He now adds the original fraction & and the two obtained
2694	from it, using not the expected common denominator 12, but 18. Under each fraction is
2695	written in red its value in terms of the common denominator 18, which in two cases is not
2696	a whole number. The three values add up to .9, and since nine-eighteenths is ‡, the sum of
2697	the three fractions is #!
2698	No. 9. (PI. G.) No. 9.
2699	1 4+ to
2700	* + 7
2701	#+ 50
2702	Total 1
2703	The quantity here experimented on is }+ Io. Its half and its quarter are added to it,
2704	and the result is stated to be 1. This is clearly incorrect. If the original fraction had been
2705	} + Tz the answer would, however, have been right: the second fraction in the second line
2706	would have been ds and that in the third line st. Perhaps some inkling of the truth was in
2707	when he wrote so instead of 7o. That he was not happy about the
2708	calculation is indicated by his failure to prove his result by the common denominator method
2709	the red numerators under each fraction never having been inserted.
2710	No. 10. (Pl. G.)
2711	1 ++ =s No. 10.
2712	9
2713	Total
2714	In halving & + is use has been made of the fact, known from the table of resolution
2715	of 2-fractions, that this quantity is equivalent to 7. In the third line there is a gross error in
2716	the halving of 4, the 2 and the 7 having been added instead of multiplied, giving 9 instead of fa.
2717	The total is nevertheless correct, and the error is clearly due to a stupid copyist.
2718	No. 11. (Pl. G.) No. 11.
2719	1
2720	Here we have once more the error 7 × 2 = 9, but it has been noticed and the correct
2721	Ti is placed after it in much lighter ink. The error, however, persists into the next line,
2722	where we find ts for 2s. Despite faulty working the total is given correctly. Cf. No. 10.
2723	1 The ignoring of the fact that to add to a number its third and its two-thirds is equivalent to doubling it is
2724	a further testimony to the experimental nature of these calculations.
p. 60
2725	56 RHIND MATHEMATICAL PAPYRUS
2726	No. 12. No. 12. (Pl. G.)
2727	TE
2728	Total #
2729	Here the greatest confusion seems to reign between multiples of 7 and those of 9.
2730	In order to give the correct sum, . the series should be tt, 2s, 3t.
2731	lighter ink: in the second line the original reading was i's, but this was altered to șs in
2732	No. 13. No. 13. (PI. G.)
2733	1 Tis + Tỉ=
2734	13+4
2735	$++%
2736	#E +748
2737	ktấtt tố
2738	Total $
2739	ure added uanttyt teomesis • Is operated on, and it is found that when its half and its quarte The result is proved by reducing all the fractions to the
2740	common denominator 28, instead of to the L.C.M. 448. The values of the fractions so reduced
2741	are inserted in red, and their sum is seen on addition to be 34. Although these values are all
2742	fractional they involve no fractions save 1 and its powers.
2743	No. 14. No. 14. (PI. G.)
2744	1 T's
2745	1
2746	36
2747	72
2748	Total tó
2749	are added to it the result is 1e- The fraction t% is here operated on, and it is stated that when its half and its quarte Chis is manifestly false. The common denominator used i
2750	the addition is 18.
2751	No. 15. No. 15. (Pl. G.)
2752	1 32 + zin
2753	3+1+
2754	#* + +3i
2755	#tati tơ
2756	т2a + "7=
2757	ốtta t ss ss
2758	Total T* WRONG.
2759	added to it the result is ir. This is untrue, but the answer would have been correct had The sum states that if the quantity to + ahs be taken and its half and its quarter
2760	the original quantity read # + zlz. Either the scribe or a reviser of his work was aware
2761	of the error, for against the total is placed an abbreviated form of the verb thi "to be wrong,"
p. 61
2762	RHIND MATHEMATICAL PAPYRUS 57
2763	No. 16. (Pl. G.) No. 16.
2764	Total
2765	The quantity here operated on is }, and it is shown that if two-thirds of it and one-
2766	third of it be added to it the result is unity. The example is so simple that the addition of
2767	the three tractions is achieved without the use of a common denominator.
2768	again, as in No. 8, ignored the fact that to add to a quantity two-thirds of it and one-third
2769	of it is simply to add three-thirds of it, or, in other words, to double it. Note that, as always,
2770	one-third of the quantity is reached not by direct division by 3, but by halving two-thirds.
2771	No. 17. (PI. G.) No. 17.
2772	1
2773	Total
2774	Here it is shown that if the quantity } be taken and two-thirds of it and then one-
2775	third of it added to it the result is }. Again, the Egyptian has failed to see that to add to
2776	any quantity its third and its two-thirds is equivalent to doubling it. (Cf. No. 16.)
2777	The working shows points of interest. Thus the second line is arrived at by means of
2778	a table of multiplication of fractions, and indeed two-thirds of t is actually given in the table
2779	in No. 61 as f + 1s. The third line is obtained as usual by halving the second, but in doing
2780	this use has been made of the fact that f + 1's is 3 (see the table of the division of 2). The
2781	fractions are added without the common denominator. Probably the
2782	scribe used the fact that f + Is is 3, then added the t and saw that 3 was }.
2783	No. 18. (PI. G.) No. 18.
2784	Total 1
2785	To f are added its two-thirds and its one-third and the result is found to be }. The
2786	second line is clearly taken from a table of multiplication of fractions, though this particular
2787	case does not occur in the short table in No. 61. In adding use was probably made of the
2788	fact that 6 + 1s = 4.
2789	No. 19. (PI. G.) No. 19.
2790	1
2791	1
2792	Total
2793	It is here found that if ,½ be taken and its two-thirds and its one-third added to it it is
2794	"completed" to f. Two-thirds is reached by the use of a multiplication of fractions table,
2795	one-third by halving two-thirds. In adding the three fractions the common denominator 18 is
2796	used instead of the L.C.M. 36. The values of the fractions so reduced are inserted below them
2797	in red and add up to 3, giving three-eighteenths or one-sixth.
2798	I
p. 62
2799	58 RHIND MATHEMATICAL PAPYRUS
2800	No. 20. No. 20. (PI. G.)
2801	1
2802	$+·
2803	Total t'=
2804	Similar to No. 19. In adding the fractions the common denominator used is again 18.
2805	Nos. 21-23. SECOND GROUP OF COMPLETIONS (śkm). Plate H.
2806	N0S 23. the preceding fifteen, they are in reality entirely different. Thus in No. 21 we are asked, Although these three sums come under the same heading of <km or "completion" as
2807	"What completes } + I's to 1?" and the answer is } + T's, or as we should say fs. The
2808	problemthenis simply "Subtract } + I's from 1, expressing the answer in aliquot parts."
2809	In other words, the problem dealt with is subtraction of fractions.
2810	These examples differ in one other important respect from the preceding group. In
2811	Nos. 21-23 we are set a definite problem to solve, while in Nos. 7-20 we started out with no
2812	problem, but merely operated on certain fractional quantities and recorded the results.
2813	No. 21. No. 21. (PI. H.)
2814	"It is said to you "What completes }+Ts into 1?'
2815	Total 11 : remainder 4.
2816	Reckon with 15 to find 4.
2817	1 15
2818	3
2819	T's
2820	Total 4 : then ) + T› is what must be added to it.
2821	Therefore 3+* + ts + ts is complete up to 1."
2822	The problem is to subtract }+ Ts from 1, expressing the result in aliquot parts. First
2823	3 + 1's are reduced to the common denominator 15, their values 10 and 1 being written below
2824	them. The result is eleven(-fifteenths) and the remainder to make up unity is four(-fifteenths).
2825	All that is now needed is to express this in aliquot parts. This is done by trial by writing
2826	down various fractional parts of 15. It is observed that 1 of 15 is 3, and that 1's is 1.
2827	Since 3 and 1 = 4, therefore 4 must be \| + Y's of 15, or, in other words, i; = ! + 1s, which
2828	is the answer required.
2829	The proof consists of adding the four fractions by reducing them to the common denomi-
2830	The values are filled in beneath their respective fractions, but the writer does not
2831	to state in writing the fact that they add up to 15.
2832	The words in the top left-hand corner (in B.M. Facs.) of the section, vis.:
2833	tp n sity ky & tu m wil "Proof. Another. ! + To is the amount to be added,"
2834	cannot be fitted into this problem. It is just possible to take in tp n sity, as Griffith suggests,
2835	before hr km etc., and the position of the words favours this. But they are unnecessary, and,
2836	what is more, if as is probable (see p. 22) they mean "proof," they are unsuitable, for the
2837	following words are a statement of result rather than of proof, the proof lying in the small
p. 63
2838	RHIND MATHEMATICAL PAPYRUS 59
2839	integers crowded into the last line: the inclusion of the words here would quite spoil the force No. 21.
2840	of the following hr.
2841	Eisenlohr saw that the words * + 1o m w:l could not belong to this problem, but might
2842	have come from the next. Here he is perhaps right, but the confusion is even deeper seated
2843	than this. The phrase tp n sity only occurs in six other problems of the papyrus, viz. the
2844	group 32-38, with the exception of 36, and it is surely more than a coincidence that on one
2845	occasion on which they occur in this group (viz. in No. 35) they are immediately followed by
2846	the words ‡ + io as in the present example. We may therefore conjecture that the words do
2847	not belong to our problem at all, but are borrowed partly from No. 35. The m wih has
2848	probably been added in order to try to make sense of them. We thus have a confusion
2849	between three problems. The 5 + 1o perhaps first came in from No. 22, and afterwards
2850	occasioned the further confusion with No. 35, and hence the introduction of tp n sity.
2851	The word ky still remains a difficulty. Can it be a gloss, meaning that the words which
2852	it marks have intruded from another problem? If this is the case the error goes back to the
2853	prototype from which our scribe copied, for here, judging from its position, the word has been
2854	understood as part of the problem itself.
2855	No. 22. "What completes } + 30 (PI. H.) into 1? No. 22.
2856	20
2857	Total of its excess is 9.
2858	Reckon with 30 to find 9.
2859	1 30
2860	3
2861	6
2862	Total 9
2863	Therefore + + 1o is what must be added to it.
2864	Thus 3+*+ to + st is complete up to 1"
2865	The problem is to subtract } + 3o from 1, or, in Egyptian words, to complete } + 36
2866	The two fractions are first added together by the use of a common denominator 30, and
2867	found to come to 20+1 or This is subtracted from 38 or 1, and the result 3o, called
2868	"its excess," is resolved into its aliquot parts &+ io. The result is proved by writing down
2869	the four fractions, using the common denominator 30, and showing that they add upto 38 or 1.
2870	For ge ~ "excess" or "difference," Griffith compares Pap. Kahun, PI. VIII, no. 4, where
2871	it is written
2872	No. 23. (PI. H.)
2873	No. 23.
2874	+ To + 30 + 75 Complete into 3.
2875	11$ 55+
2876	Therefore + + 7o is what must be added to it, making }.
2877	· + + * + to + so + #o+#s+t
2878	55+3
2879	making 1.
2880	The problem is to subtract (1 + + tu + 30 + z,) from }.
2881	common denominator 45 and the values or Most of the working is omitted. first of all the given fractions are reduced to the "numerators"(11%, 55+,etc.) inserted in
p. 64
2882	60 RHIND MATHEMATICAL PAPYRUS
2883	red, each beneath the fraction to which it belongs. The sum of the fractions is ts, and the
2884	amount needed to make this to up 3 (ie. 43) is t iet, ie.*+io. The whole of
2885	this step is, however, omitted, and the answer is simply stated.
2886	The proof is unusual. In order to show that the sum of the fractions is }, } is added
2887	to them and the result shown to be unity. done in the usual way by
2888	a common denominator, viz. 45, but again the sum of the numerators, which is
2889	of course 45, is never given.
2890	Nos. 24-38. SOLUTION BY TRIAL OF EQUATIONS OF THE FIRST DEGREE. Plates H-M.
2891	These problems have acquired some notoriety in mathematical circles," for they formed
2892	at one time the centre of a controversy which to us appears so stupid and futile that we shall
2893	do little more than indicate its nature. After the appearance of Eisenlohr's commentary in
2894	377, Rodet wrote in the Journal Asratque, 1881, 184-232 and 390-459, an article, Les pretend
2895	roblemes d'algebre du manuel du calculateur egyptien, in which he accused Eisenlohr of havir
2896	wrongly attributed a knowledge of algebra to the Egyptians. Cantor" at once replied and was
2897	followed by Eisenlohr himself* and by Revillout."
2898	The fact is that Eisenlohr and Cantor had lent themselves to Rodet's attack by a
2899	most unguarded use of algebraical symbols in their treatment of these problems. The Egyptians
2900	used no such symbols, they solved the problem by the trial-method of a "false supposition"
2901	followed by a proportion, not, however, definitely formulated by them as such (see below), and
2902	it is as foolish to ask whether this is algebra as it is to ask the same question with regard
2903	to many of our processes of solving modern arithmetical problems. The matter is not one of
2904	essence but of form, a fact well known to every schoolboy when he is warned to keep clear
2905	of writing x in his arithmetic examination.
2906	The, problems are divisible into three groups. The first group includes Nos. 24-29, where
2907	the solution is obtained by taking a trial number, treating it in the manner prescribed in the
2908	problem and finding the correct number by proportion. Thus in No. 24 we are asked to find
2909	a quantity which when its seventh part is added to it becomes 19. We take a trial number 7,
2910	which for obvious reasons is convenient. We add its seventh part and the result is 8. We
2911	have now to work out the proportion
2912	8: 7 : 19 : 2
2913	or, ås the Egyptian would say, we have to divide 19 by 8 and multiply the result by 7, for
2914	he never formulates a proportion as such. No. 28 differs from the rest in the form of its
2915	working out, but it is clear that this is due to an error or an omission of some kind.
2916	The second group includes Nos. 30 to 34, and differs from the first not in the nature of
2917	its problems, though these are a little more complicated, but in the method by which they are
2918	Here instead of the trial method a direct method by division is adopted. Thus in
2919	No. 31 we are required to find a quantity such that when } and 1 and of it are added to
2920	itit becomes 33. The solution is reached by directly multiplying 13 +} + ‡ to find 33.
2921	essence this method is much the same as the trial method, the trial figure here being unity,
2922	but the arrangement of the whole gives it a very different aspect.
2923	The third group, Nos. 35-38, is of a similar nature except that the statement of the problem
2924	is clothed in concrete language. Instead of dealing with quantity in the abstract we deal with
2925	actual amounts of corn. the method of proof in these cases the reader is referred to
2926	the discussions of the separate problems.
2927	1 11f erroneously in black.
2928	* They are generally referred to in mathematical treatises as the hau-calculations ("tí w).
2929	3 Zeitschrift für Math. und Physil, 27, 117.
2930	* Journal Asiatique, 1882, 515-8. » Revue Égyptoloyique, II, 287-303.
p. 65
2931	RHIND MATHEMATICAL PAPYRUS 61
2932	No. 24. (PI. H.) No. 24.
2933	"A quantity whose seventh part is added to it' becomes 19.
2934	>1 7
2935	1
2936	8
2937	16
2938	4
2939	2
2940	1
2941	2+*+$
2942	-2
2943	,4
2944	The doing as it oceurs:—The quantity is 167 + %
2945	one seventh 1s 2z + 8
2946	Total 19 "
2947	Abstractly expressed in modern terms the problem is: "If x + fx = 19, find x" A
2948	trial number is first selected, and it is precisely that which we should choose ourselves, namely 7,
2949	for the simple reason that its seventh part is an integer and thus easily obtained.
2950	7 plus one-seventh of 7 amounts to 8, and all we have now to do is, as we should say, to
2951	solve the proportion
2952	8 : 7 :: 19
2953	as the Egyptian says, to divide 19 by 8 and multiply by 7.
2954	The Egyptian working is arranged as follows:—
2955	Step 1: The trial number 7 is set down and one-seventh of it is added to it, giving 8.
2956	Step 2: This 8 is operated on in the usual fashion to produce 19, or, as we put it, 19 is divided
2957	by 8. The result as shown by the ticks is 2+*+฿.
2958	Step 3: This last quantity is multiplied by 7, giving 16} + ฿.
2959	Proof: One seventh of this quantity is taken and added to it, the result being the required 19.
2960	The working of this proof is omitted.
2961	The term 'lí w, literally, as the word-sign shows, "a heap," seems to be used here as
2962	a mathematical technical term equivalent to our "quantity." It is a good example of the
2963	•concrete nature of Egyptian mathematics.
2964	No. 25. (PI. H.) No.25.
2965	"A quantity whose half is added to it becomes 16.
2966	2
2967	1
2968	3
2969	2 6
2970	12
2971	2
2972	1
2973	1
2974	10%
2975	1 The antecedent "a quantity" being undefined, the following clause, whatever its syntactical form, may
2976	qualify it relatively. In this case the fof lprf picks up the pre-placedsubject w.
p. 66
2977	62 RHIND MATHEMATICAL PAPYRUS
2978	No. 25. The doing as it occurs:—The quantity is 103
2979	a half is
2980	Total 16 "
2981	The equation here solved is x + ]x = 16. The method is similar to that of No. 24.
2982	The trial number taken is 2. This, when its half is added to it, becomes 3.
2983	first step. In the second step 16 is divided by this 3, giving 5} and in the third step
2984	this last is multiplied by 2. Expressed as a proportion the sum is:—
2985	: 2 ::16: x.
2986	No. 26. (PI. H.)
2987	"A quantity whose fourth part is added to it becomes 15.
2988	Reckon with 4: you are to make their quarter, namely 1.
2989	Reckon with 5 to find 15.
2990	5
2991	10
2992	The result is 3.
2993	Multiply 3 by 4.
2994	1 3
2995	2 6
2996	12
2997	The result is12.
2998	1 12
2999	3
3000	Total 15
3001	The quantity is 12
3002	its quarter is 3
3003	Total 15 "
3004	The equation here solved is x + *x = 15. The method is that of proportion used ir
3005	the two preceding examples with two small differences in the statement. In the first place
3006	the first step is described in full instead of being merely stated in figures; and in the second
3007	place, the proof is stated twice over, first symbolically in figures and then in words: in neither
3008	case is it called irt mi hpr (cf. pp. 23-4 above).
3009	The trial number is 4, which when increased by its quarter gives 5. The proportion
3010	to be solved is thus
3011	and x is found by dividing 15 by 5 and multiplying the result by 4.
3012	Note the treatment of the 4 as a plural ("You are to make their quarter"), and com-
3013	pare SETHE, V.Z.Z., 44-51.
3014	No. 27. No. 27. (PI. J.)
3015	"A quantity whose fifth part is added to it becomes 21.
3016	5
3017	1
3018	Total 6
p. 67
3019	RHIND MATHEMATICAL PAPYRUS 63
3020	No. 27.
3021	12
3022	3
3023	Total 21
3024	2
3025	15 (sic)
3026	The quantity is 17%
3027	its fifth is
3028	Total 21 "
3029	The equation solved is x + țx = 21. The trial number chosen is 5, which, when its
3030	fifth is added, becomes 6. The proportion to be worked out is:-
3031	6 : 5 : 21 : x
3032	In the second step 21 is divided by 6, and in the third the resulting 3} is multiplied by 5.
3033	In the last line of this step 15 has been erroneously written for 14.
3034	No. 28. (PI. J.) No. 28.
3035	"Two-thirds added and one-third taken away: 10 remains.
3036	Make one-tenth of this ten': the result is 1: remainder 9.
3037	Two thirds of it, namely 6, are added to it; total 15. A third of it is 5.
3038	It was 5 that was taken away: remainder 10.
3039	The doing as it occurs: "
3040	This problem is incomplete and elliptically worded. Written in full it would run as
3041	follows:—
3042	"A number: two-thirds of it is added to it and one-third of the total is subtracted.
3043	Result 10. Find the number. Answer 9."
3044	In algebraical terms the equation to be solved is:—
3045	x+3x- 3(x + }x) = 10.
3046	The working given is most singular. Instead of taking a trial number, working it in
3047	the manner indicated, and finishing with a proportion as in the other examples, one-tenth of
3048	•the remainder 10 is taken from it and the resulting 9 is taken as the answer. Either the
3049	scribe knew the correct answer and botched up a "working" to obtain it, or he was aware
3050	that the result of the processes indicated in the setting out was equivalent to adding on
3051	one-ninth of itself to the original number.
3052	There has been a considerable error in copying at this point of the text. After the
3053	words irt mi hpr we expect the proof of the example No. 28, instead of which we find what is
3054	clearly part of the working out of an entirely different problem, No. 29. The copyist has
3055	evidently missed out both the end of 28 and the beginning of 29. It is natural to suggest
3056	that his eye wandered from the irt mi hpr of the first sum to that of the second, and so
3057	missed out all that lay between, but the problem is not so straightforward as this (see notes
3058	on No. 29). Whether one or more whole problems are omitted as well as the portions of 28
3059	and 29 we have no means of ascertaining.
3060	The two signs A and A° signify, as the sense shows, addition and subtraction
3061	respectively. But of what are they abbreviations? A doubtless stands for the verb pri,
3062	which is used later in the sum for subtraction. If the sign for addition represents a verb of
3063	1 The B.M.Facs. has, crroneously, 20. = Here made to face as in the hieratic.
p. 68
3064	64 RHIND MATHEMATICAL PAPYRUS
3065	No. 28. motion at all and is not a mere mathematical symbol it probably stands for hil, "to go
3066	down," which is frequently used in Egyptian in contrast to pri in its original meaning of
3067	"to go up."" If, however, pri gets its technical sense of "to be subtracted" from the later
3068	meaning "to go out," the addition sign may be an abbreviation of 'k, "to go in." Compare
3069	the technical use of the two verbs for "income" and "outgoing" in account papyri such as
3070	The legs-sign in the sense of "to subtract" has been discussed by Spiegelberg in his
3071	Rechnungen aus der Zeit Setis I, Text, 40, but though he alludes to the Rhind passage he
3072	does not note that the sign there used faces the opposite way to those quoted by him from the
3073	account papyri, or, in other words, that the sign used in the latter for "subtract" actually
3074	stands in Rhind for "add." The same sign stands in the Moscow Papyrus for "to square."
3075	No. 29. No. 29. (Pl. J.)
3076	1 10
3077	22
3078	To 1
3079	Total 13%
3080	9
3081	Total 22%
3082	71
3083	Total 30
3084	20
3085	10
3086	It has already been noted in dealing with No. 28 that here under No. 29 we have nothing
3087	more than the latter part of a problem. The original problem must have read as follows:—
3088	"A number: two-thirds of it are added to it, and one-third of the sum is added. A
3089	third of the total is found to be 10.
3090	In algebraical terms the equation to be solved was :—
3091	{x+7x+(x+7x)} = 10.
3092	Of the portion preserved the first four lines down to and including the total 13} in red
3093	ink are part of the working out. The rest constitute the irt mi lpr or proof.
3094	14 + to, and how was it obtained? Had the scribe conscientiously worked out No. 28 we These first four lines contain the multiplication of 10 by 1f + 1. What is this quantity о а ріоiеть
3095	should have been able to answer this question exactly; as it is, our answver must be approxi-
3096	mate. Presumably he began with a trial figure of 1. To this he added its two-thirds, obtaining
3097	He then took one-third of this and added it on, obtaining 22. Dividing this by 3 he
3098	got 31. All that remained was to solve the proportion
3099	20 : 10 :: 27 : x
3100	or, in other words, to divide 27 by 20 and multiply by 10, just as in the previous examples.
3101	The division was doubtless set out as follows:—
3102	20
3103	10
3104	5
3105	2 Result 11 + t*
3106	The multiplication of l4 + tu by 10 has survived, and the result is 13%.
3107	1 GARDINER, Admonations of an Egyptian Sage, p. 51, and the examples there quoted, where there is no
3108	implication of movement up and down, but rather in and out or to and fro.
p. 69
3109	RHIND MATHEMATICAL PAPYRUS 65
3110	therds of it, ic. 9, giuing 92%, stcondly sise addition toephi last ofits third pao 1 N 2
3111	total 30, and finally the division of 30 by three, giving the correct 10.
3112	It would have been interesting to see how the Egyptian managed the more than usually
3113	complicated multiplications and additions of fractions in the early steps of the working out.
3114	That he did not shirk the use of a common denominator is clear both from the preceding
3115	examples and from the form of his proportion, which merely conceals the use of the improper
3116	It is curious that the scribe should have preserved a portion of the working out as well
3117	as the irt mi lpr, for the simplest explanation of his error in copying is to suppose that his
3118	eye wandered from irt mi hpr in No. 28 to the same words in No. 29 and omitted all between.
3119	Yet here we have a small fragment (four lines) of what must have lain between. A possible
3120	solution of this difficultymay be found in the fact that in the remnant left the words irt mi
3121	ypr, which ought to follow the fourth line, do not occur. In other words, the blunder is
3122	lunes of the working out of No. 29 because in his prototype they were so placed that he read
3123	them as following
3124	difficult to find in our own copy.
3125	It might possibly be suggested that the whole of what is left is the working out, and
3126	that the problem read : "A number: its 1 and its in are added to it, two-thirds of this
3127	sum are added, and the total, when divided by 3, is 10. Find the number." This is practically
3128	untenable, firstly because it would give a problem of too complicated a nature, and secondly
3129	because it would not account for the use of red ink in "Total 13)" It must also be more
3130	than a coincidence that the first four lines are exactly what we need in the last step of the
3131	working out.
3132	No. 30. (PI. J.) No. 30.
3133	"If a scribe says to thee
3134	' 10 has become } + Ti of what?'
3135	•Let him hear:—
3136	You are to multiply }+11 to find 10:
3137	* + тT
3138	4 3T:
3139	8 67o+:
3140	sn is multiplied 23 times to find 3+ i Total
3141	Total, this quantity that says it, 13gg.
3142	83t**+138
3143	1t1t2
3144	Total 10 "
3145	The problem here solved is (3 + 11)x = 10, and the answer is 13gy. The method of
3146	solution is to make trial multiplications of } + i, in the usual way. The multipliers 1, 4
3147	and 8, totalling 13, bring us to 93i, which is within st of our goal. The working out of the
3148	addition of the fractions is not shown, and the scribe has further omitted to mark the 3u
3149	as dit or "remainder" as he should have done. All that now remains is to find what fraction
3150	of 3t iu amounts to su and since 3+, is fi, a step which is completely omitted, the
3151	reply must be eb. Or, in other words, su must be multiplied 23 times to find } + th-
p. 70
3152	66 RHIND MATHEMATICAL PAPYRUS
3153	No. 30. The final answer is thus 13gg. This is proved by actually taking } and to of this, adding
3154	them, and finding that they amount to 10. It is easy to see how these two fractional parts are
3155	obtained, but their addition is omitted.
3156	The translation of this problem offers some difficulties. It is inconsequently stated, for,
3157	though it begins "If a scribe say to thee," yet later we read "this quantity that says it is
3158	•.," as if the opening statement had been "If a quantity says to thee } + ín of me
3159	is 10; what am I?" The f of sdm-f ought to refer to the scribe, but in a somewhat similar
3160	case in No. 37 it can only refer to the quantity which speaks, and in view of the mixed
3161	nature of the statement here it may originally have done so. Grammatically we could also
3162	translate " 10 has become } + Yu : what obeys (this condition)?" In other words, what quantity
3163	obeys the condition that } + ît of it is 10? This may indeed be the correct translation.
3164	For the śdm-n-f form of hpr-n cf. perhaps hin in Nos. 45 and 46.
3165	No. 31. No. 31. (PJ. J.)
3166	"A quantity to which its two-thirds, its half and its seventh are added becomes 33.
3167	13+3+
3168	-2 43+· +=s
3169	+ T's (read {+)
3170	18%
3171	++*+1=
3172	Total 32%
3173	Remainder +
3174	‡+‡+™‡+zв+ž
3175	(Remainder) 3} + ‡
3176	(since) → is 21.
3177	1 42
3178	28
3179	21
3180	6
3181	Total 99 (read 97).
3182	156 +575 + Tt6
3183	/388
3184	Total 33."
3185	This problem is difficult to follow, partly because the arrangement of the working is
3186	extraordinary, and still more because a portion of it has been misplaced by the scribe in
3187	No. 38. The problem is x + }x + ]x + }x = 33. We choose for our trial value 1, and set
3188	out to divide 33 direct by 1 + } + ½ + ł, or rather to multiply this quantity to find 33.
3189	When the multipliers 2, 4, 8 and i, total 14%, have been applied the integers and the
3190	simpler fractions in the products are added up and found to give 32}, which is ½ short of
3191	the required 33. This 32), however, does not include the smaller fractions \|++*+ 2s
3192	+ 3s (which we have placed to the right of a vertical line for the sake of clearness), though
3193	the scribe when he writes "Total 32)" gives no hint of this. In Step 2, which in the papyrus
3194	is misplaced after Step 4, these smaller fractions are added, using 42 as common denominator,
3195	and come to 1%. In other words, 1 + } + 1 + 1 when multiplied by 14% comes to 32} + *7.
p. 71
3196	RHIND MATHEMATICAL PAPYRUS 67
3197	The amount still needed to make up the by which 32} falls short of 33 is clearly & - 177 No. 31.
3198	or 3+7. 42 Thus to get the exact quotient of our original division we must still divide +*
3199	by 1 + } + * + 4. This can only be done by expressing both divisor and dividend in terms
3200	of some common denominator, and as the dividend is already in forty-seconds the obvious
3201	thing is to express the divisor in terms of the same, which is done in Step 3, which by an
3202	error of the copyist had strayed into No. 38. The result is 3 and we have now only to
3203	divide 3} + * by 97, or in Egyptian terms, to multiply 1 + 2 +*+‡ to find 97. Thus
3204	in Step 4, where the division is actually performed, we have in the right-hand column these
3205	four numbers 1, 2, } and ; in the left-hand column are these same numbers each divided
3206	by 97, giving it, 37 (or as the Egyptian's tables told him 56+87s+7t6), r'a and 33s, while
3207	in the centre column are the same numbers 1, 2, } and divided by 42, which of course
3208	are the actual products of the multiplicand 13 + } + + with the four fractions of the first
3209	column respectively.
3210	We now add the ticked quantities in Steps 1 and 4. On the left we get 147 + sr+ 36
3211	+ots+7ts+ rztsts, which will be our answer; and ontheright32+(7++ T
3212	354++-twoch, as the words "Total 33," misplaced in the papyrus,
3213	to the correct 33.
3214	consists of a division which, owing to its nision which, oving toegue comrrexity, is done in svo parts, Steps yr and 1 complexity,
3215	In order to render the final addition easy the scribe of the original XIIth Dynasty document
3216	placed Step 4 under Step 1, and probably moved Steps 2 and 3 to the left. The copyist was
3217	puzzled by this arrangement, omitted or transferred to No. 38 the necessary Step 3, and
3218	in Step 2 after Step 4, adding to it the "Total 33," which should end the sum.
3219	No. 32. (Pl. K.) No. 32.
3220	"A quantity whose third and whose quarter are added to it becomes 2.
3221	1 15+= 228
3222	1T8 152
3223	76
3224	#+75 38
3225	$+T·7 19
3226	(52C)228 1
3227	(S2C) TTE 2
3228	Total lf+ io+ riz+=te is this quantity that says it.
3229	3+$+Ts+I+3·2
3230	[+15+3t33+456
3231	Insert (?)
3232	$ (sic) 144
3233	Total 228
3234	Proof:
3235	1 1+*5+T1E+238
3236	· +27 +
3237	Total 17 + ·
3238	Remainder 1
3239	K 2
p. 72
3240	68 RHIND MATHEMATICAL PAPYRUS
3241	No. 32. 1+ T17+ 22N+ 1N #etststixtt*xtatet.lz
3242	76 503 15
3243	Total 228,
3244	i.e. la quarter. 912
3245	456
3246	228 "
3247	The problem is x(1++ 4) = 2, and the answer 1, + 1e + il+ dx• The method
3248	of this sum is similar to that of the last and begins with a direct multiplication (Step 1)
3249	of 1 + * +÷ to find 2. There is, however, one slight difference in that here each product
3250	as it is obtained is expressed in terms of a common denominator 144 and the result placed
3251	in a third column in the multiplication. When the multipliers 3, 1, t y› have been tried
3252	it is noticed that the products corresponding to them in the third column add up to
3253	285, or just 3 short of 288, which in terms of 144ths would be the required 2. It therefore
3254	remains to divide this remaining zi» by 1! + *, and since this last, in terms of 144ths, is
3255	228 (obtained in the first line) this is equivalent to dividing 3 (ie. 2 +1) by 228, and the
3256	quotient is obviously rla+ rtx. Thus the answer to the problem is 1, + 1e t riz + g2x.
3257	The proof (Step 2) now begins, with no heading. It consists of adding to the number
3258	just found its third and its quarter and showing the result to be 2. We get the , as always
3259	through the 3, and the f through the 1: but instead of now proceeding to add up the whole,
3260	the third and the quarter the reckoner interrupts the proof for the insertion of Step 3, two
3261	pieces of rough working used in Step 1, riz. the reduction of 1! + 1 to îfi (for ! in the
3262	first line read 1) and the multiplication of 12 by 12, as is perfectly clear from the ticking
3263	of the multipliers 4 and 8. This last piece of work is presumably the source of the common
3264	denominator used in Step 1, but it is not easy to see why that number should be obtained
3265	as the product of 12 and 12: we have indeed no evidence of the lines on which the Egyptians
3266	chose their common denominators. The sign § placed in front of this third step probably
3267	indicates a 'stop' for the insertion of pieces of work omitted in Step 1 (see notes on No. 70).
3268	Step 4 is headed tp n śity, probably "proof." In this step the quantity found, namely
3269	1f + 72 + TI# +»2x, is added to its third and its fourth parts, which were found in Step 2,
3270	and the result is shown to be the correct 2. The method is peculiar. First the simpler
3271	quantities 1%, †and ‡ are added, total 1} + 4. It is now only necessary to show that the
3272	more complicated fractions, placed by us to the right of a vertical line, add up to the
3273	remaining . This is done by reducing them all to the common denominator 912. Under each
3274	fraction is placed as usual its numerator with respect to this denominator (shown here in
3275	italic figures). The total of these numerators is 228, which a simple sum shows to bel of 912.
3276	No. 33. (PI. K.)
3277	"A quantity whose two-thirds, halt and seventh are added to it becomes 37.
3278	1 15+2+!
3279	2 ++2x
3280	4 9% + 1+
3281	8 18h + \|
3282	-16 304 + 1 + **
3283	28 10%15
3284	1 42
3285	sức 3 28
3286	21
3287	10}
3288	Total t0; remainder 2
p. 73
3289	RHIND MATHEMATICAL PAPYRUS 69
3290	No. 33.
3291	Total 99 (sic: read 97))'
3292	*= 1
3293	3etw7o+T7B 2
3294	Total 37
3295	Proof:
3296	1 16 +
3297	8
3298	10% +r3se + FuTe + TiN# (read 11':z)
3299	143 4 15
3300	+ tnise+15re
3301	21+2x + ++753 +5732
3302	13+4+17+25 14
3303	(Total) 36 + + ‡ + **; remainder 28+N7
3304	3621} 1358 194 194 643
3305	5432
3306	36211
3307	} 2716
3308	1358
3309	.*- 194
3310	Total 51731;
3311	remainder 2583"
3312	The problemisa(1++1+4)= 37,and the answer is 16+++176 The
3313	method is as follows. In Step 1 the quantity 1 + 3 + * +} is multiplied in order to
3314	make 37. The multiplier 16 almost gives this result, producing as it does36+3++
3315	In Step 2 these fractions 3, 4, and 2s are added in terms of the common denominator 42 and
3316	found to give 42, which falls short of 1 by r. We have now only to multiply 1+3++7
3317	to find this 2s In Step 3, which must be restored to its place here from No. 38, whither it
3318	has been wrongly transferred by the scribe," the multiplicand 1 + } + ½ + \| is reduced to 23.
3319	• and to get f, it is clear (Step 4) that the multiplier must be 3t, or st t ets t 7ł6 a value
3320	found from the tables earlier in the papyrus. The answer is therefore16+s5+o7,+77.
3321	In the proof tlis quantity is taken as multiplicand and multiplied suecessively by 1, 3,
3322	2 and ț. The more complicated fractions in the product (here placed to the right of a vertical
3323	line) are reduced to common denominator 5432 and their respective values in terms of that
3324	denominator are written in in red (here in italies) under them. The whole numbers and larger
3325	fractions (30 + } + * + ]) (to left of vertical line) are first set down and are stated to fall
3326	short of the required 37 by 2n + it (no proof of this is given and the fact must have been
3327	drawn from tables or worked out elsewhere). We have now only to show that the fractions
3328	on the right of the vertical line add. up to ds + These last, when reduced to the common
3329	denominator 5432, give 194 and 643, the sum of which is 258}, which will be found identical
3330	with the sum of the italicized numerators (in terms of the same denominator 5432) on the
3331	right of the vertical line. Thus the fractions on both sides of the line add up to 311k or1,
3332	and the total product is 37, as it should be.
3333	Supplied from No. 38.
3334	in No. 31, g.0. = Perhaps this step may have been omitted here even in the original MS. owing to its having already occurred
p. 74
3335	70 RHIND MATHEMATICAL PAPYRUS
3336	No. 33. A further check is obtained by reducing the fractions on the left of the vertical line,
3337	viz. 3 1 and ew to the same common denominator 5432, and adding their numerators (Step 6).
3338	These come to 5173} which, when subtracted from 5432, again gives 258}. In this step the
3339	multiples},4 and Is ought to be marked with a tick.
3340	No. 34. (Pl. L.)
3341	"A quantity whose half and whose quarter are added to it becomes 10.
3342	1+*
3343	2
3344	7
3345	*+=s
3346	*+Iz1
3347	Total this quantity 5t +* + 1*
3348	Proof:
3349	5.1, 1++
3350	2%+1 +TE+3s
3351	lf+*! +25+36
3352	Total 9% + 3
3353	Remainder # +}
3354	₺ is 14
3355	7
3356	Total 21
3357	{++1+28+=+36"
3358	8 4 4 2
3359	The problem is x(1 + \| + ł) = 10, and the answer is 5}+ + + **. The method is as
3360	The quantity 1½+I is multiplied by trial to find 10.
3361	exact result is achieved by the use of the multipliers 1, 4, t and \| + *. Note that the
3362	multiplier # + 2's is simply the Egyptian way of writing 7 (the table earlier in the papyrus
3363	In the proof the answer 52 + + is simply halved and quartered, the results are
3364	added to it and the whole is shown to amount to 10. For convenience, however, the whole
3365	numbers and simpler fractions, the left of a vertical line, are added first and
3366	seen to amount to The quantity still needed to make up 10 is thus *+ d, and it
3367	only remains to show that the fractions to the right of the line are equivalent to this f + %
3368	This is done by reducing all the fractions to the common denominator 56.
3369	denominator f is 14 and j is 7, total 21. In the last line the fractions 7+1+r+ 2u+ 2s+36
3370	are set down, and under each is placed in red (here in italies) its numerator in terms of the
3371	common denominator 56. These clearly add up to 21, and the proof is complete.
3372	No. 35. (PI. L.)
3373	"I go three times into the hekat-measure; my third part is then added to me and I
3374	return fully satisfied.
3375	What is it that says this?
3376	The doing as it occurs:
3377	,2 2
3378	Total 3%
p. 75
3379	RHIND MATHEMATICAL PAPYRUS 71
3380	You are to divide 1 by 3}: No. 35.
3381	súC Tố
3382	sic
3383	Total
3384	Proof :
3385	*+Tr
3386	sic }
3387	Total
3388	Proof:
3389	320
3390	To 32
3391	64
3392	Total 96
3393	amounting in corn to
3394	1 96 (* + 3b + 87) hekat and 1 ro
3395	2 (+ s + 3z) hekat and 2 ro
3396	(=+32) hekat and 2 ro
3397	Total Total 1 hekat"
3398	The problem is on the same lines as Nos. 31-34, except that the amount to be made
3399	up is a concrete unit, the hekat of capacity. The equation to be solved is x (3 + }) = 1 hekat,
3400	and the answer in its final formis(4+3+z7) hekat plus 1 ro.
3401	In the first step,curious as it may seem to us, unity is solemnly multipliedby three
3402	and its third part is added to it, the result being 3}. Next 1 is divided by 3}, or, in other
3403	words, 3} is multiplied by various trial fractions to find 1. The fractions To and } (the latter
3404	from the former by doubling) are seen to give the required result, and the answer is therefore
3405	* + To of a hekat.
3406	This answer is now proved in terms of three separate units :—
3407	(1) In terms of pure fractions of the hekat (3, It, etc.).
3408	(2) In terms of ro, the ro being złu of a hekat (see p. 25).
3409	(3) In terms of the 4, , 3, ete. of a hekat, expressed in the customary Horus-
3410	eye notation (see page 25).
3411	It is very much as if we were to solve a problem in Avoirdupois Weight, and then to test our
3412	result firstly inpure fractions of a ton, secondly in ounces, and thirdly in hundredweights,
3413	quarters, pounds, and ounces.
3414	The first proof, marked tp n sity, needs little comment: it consists in multiplying&+ to
3415	and showing it to give1. In the multiplication by 2 we expect to be resolved by
3416	the table into 3 + Ts, yet this is not the case, some more direct table being used which
3417	gave the result of multiplying } + tu by 2 in the useful form 2 + Tố. A tick is omitted
3418	The second and third proofs are arranged in parallel lines the heading tp n sity
3419	doubtless refers to both. First of all the hekat is written down as 320 ro and iu+* of it
3420	This number when multiplied by 3} gives 320 ro, or 1 hekat.
3421	parallel column is headed "amounting in corn to," doubtless because it was in this notation
3422	that grain was actually measured in Egypt and not in various fractions of a hekat or in ro.
3423	The amounts given in this column are certainly obtained indirectly from the numbers of ro in
3424	the parallel column, by means of the table for converting the }, ‡, etc. of the hekat into ro
p. 76
3425	72 RHIND MATHEMATICAL PAPYRUS
3426	Note that the first line of the column gives for the first time the answer to
3427	the problem in the form in which an Egyptian would need it for practical purposes.
3428	not be forgotten that to him the b, t, d, re, de, and t of a hekat were specific measures,
3429	each with its own name and notation, just as much as our quarter, pound, ounce,
3430	above, p. 25.
3431	dieuetedouiln the sentuof ton phobprm ie mt the clamei tel, team translation has been much If we translate these
3432	words "I am filled " or "I am full" we make nonsense of the problem, for we then represent
3433	the unknown quantity as filling itself 3} times and then saying "I am filled."
3434	suggested avoiding this by taking mli•kwi as example of the rare survival of the old active-
3435	transitive use of the pseudo-participle, "If I go down 3 times into the hekat measure I fill it."
3436	But such a use, besides being a little improbable in a Middle Kingdom text (though the Rhind
3437	does contain archaisms), would require the addition of the Old Independent Pronoun & or &i
3438	Schack-Schackenburg" seeks to avoid the difficulty by reading hskwi in a metaphorical
3439	sense and translating " Ich bin dreimal genommen um die Masseinheit zu erreichen, ein Drittel von
3440	mir zu mir hinzu, dann bin ich zur Einheit complettiert." He supports this metaphiorical trans-
3441	lation of h:•kwi by the statement that "to go down into the hekat measure " in the literal
3442	sense would require the preposition m, r not being used for "into" after h:i in its litera]
3443	concrete meaning, at least in the Pyramid Texts. (This is untrue: see below.)
3444	As a matter of fact both Griffth and Schack-Schackenburg are attempting to force on
3445	the text a degree of logic which it does not possess. The scribe who stated the problem in its
3446	present form seems neither to have had before him a perfectly clear concrete picture of a
3447	measure being dipped into another so many times, nor on the other hand to be speaking
3448	entirely in the abstract terms of mathematics. He may have aimed at the latter, but an
3449	Egyptian rarely succeeded in completely dissociating himself from the concrete.
3450	Thus we need not fear to translate hikwi r lleit, "I go into" the hekat-measure." This
3451	rendering is perfectly literal, but at the same time preserves just the ambiguity between abstract
3452	and concrete which exists in the Egyptian as it stands. Coming now to iwi mlkwi, it is
3453	clear that iw-i is not a mere auxiliary, but the counterpart to h: kwi and means "I return,"
3454	which again favours a literal sense for h:•kwi. The pseudo-participle mlı•kwi involves that use
3455	of mlı in the sense of "to pay in full" or "to satisfy" (in the matter of payment) which
3456	has been illustrated by Gardiner,' and the words iw•i mli-kwi are normal Egyptian for "I
3457	return fully satisfied," scilicet, with my full hekat. For the rare geminated form h•kwi see
3458	SETHE, Verbum, II, § 116.
3459	No. 86. No. 36. (PI. L.)
3460	"I go 3 times, and my third and my fith are then added to me. I return fully
3461	satisfied. What is the quantity that says this?
3462	1 1
3463	1 1
3464	1 1
3465	1 P.S.B.A., XVI, 234. 3 Ä.Z., 41, 79-80.
3466	3 Examples of hii in the literal sense followed by r are not uncommon in the Midille Egyptian, e.g. Pap.
3467	Westear, 3, 2; Eloquent Peasant, R. 7; Cairo stela 20,007 ; Shipwrecked Sailor, 25.
3468	4 A.Z., 43, 34.
p. 77
3469	RHIND MATHEMATICAL PAPYRUS 73
3470	106 No. 36.
3471	53
3472	1
3473	2
3474	Total 1
3475	* + =5 + Tốo + =1=
3476	= + 30 +318+ 755 +35+ Tốa
3477	To+r5s+sts+ 536
3478	từ tato tsšot Tuou
3479	is + rue+21z
3480	20 10 5 35
3481	utsts+755+35+106
3482	35฿ 70
3483	tatibststotett
3484	88g 35 1% 100
3485	zu tets+stu + Tutu
3486	53 2 80 (read 60)
3487	265
3488	530
3489	265
3490	265
3491	Total 1060 "
3492	This problem is exactly similar to the last. The equation to be solved is x(3 +}+$)
3493	= 1 hekat, and the answer is (f + 3* + T7s + z1z) hekat.
3494	We begin with the (to us unnecessary) step of multiplying 1by 3f+*. The product
3495	is then added by means of the common denominator 30 (we expect 15) and found to come to
3496	yin, but the working of this is suppressed. We have now to divide 300 into 1, or, in Egyptian
3497	fashion, to operate on 106 to find 30. The multipliers & row st and złz give the required
3498	product, but the scribe, instead of giving the total of the products as 30, gives it as l. This
3499	must not be regarded as a mere error; although working in terms of the denominator 30
3500	the reckoner has not lost sight of the logical goal of his step, namely the multiplication of
3501	Lo6 to find 1, and the unexpected substitution throws an interesting light on the psychology of
3502	the Egyptian treatment of fractions. The answer is thus+ + Tur+ztz hekat.
3503	The proof has no heading. The answer is multiplied in the usual way by 1, by 2,
3504	by } and by } (total 3} + }) and the produets are added in terms of the common denominator
3505	1060. But in setting out the step the scribe forgot to leave spaces for the entry in red of
3506	the numerators (in terms of this denominator) under each fraction. He therefore sets out the
3507	products afresh with more space, omitting, however, the two simplest fractions and }. He
3508	now enters his numerators in red and adds them for each line separately. The four totals
3509	add up to 265, which, as the first two lines of the short calculation below demonstrate, is
3510	7 of 1060. The total of the fractions is thus ‡, which, with the neglected } and t, adds up
3511	to 1. In order to apply an additional check the reckoner now uses the first two lines of the
3512	final step above referred to to reduce } and to the common denominator 1060. To these
3513	he adds another ‡, represented by 265, and finds that the three numerators amount to 1060,
3514	thus doubly proving the result.
3515	L
p. 78
3516	RHIND MATHEMATICAL PAPYRUS
3517	No. 36. In the main multiplication of fractions, which constitutes the first part of the proof,
3518	note in the second line the resolution by table of In the third
3519	and fourth lines we need not assume that \| and ! are obtained otherwise than in the orthodox
3520	manner through } and tu respectively.
3521	No reduction to ro and to the Horus-eye parts of the hekat is given. Perhaps the
3522	reckoner's heart failed him before the complicated nature of the reduction in this case: perhaps
3523	responsible for the omission.
3524	No. 37. No. 37. (PI. M.)
3525	Let it hear:
3526	1 1
3527	2 2
3528	Total 3} + ts
3529	Divide 1 by 3} + 1:
3530	3}+ T5
3531	1+*+35
3532	to +32 +64 +376
3533	Total 1
3534	Addition?: ++*++5+16+32+87+576 Total
3535	72
3536	Proof:
3537	# + 32
3538	}+ to
3539	3 of it 12 + t
3540	} of its 1 30 +=ds
3541	ș of it 3ut=30
3542	Totall
3543	Addition': $ sE dE te dE 3E zãs sü r$s Total·
3544	18 24 3
3545	Total 320
3546	160
3547	80
3548	40
3549	20
3550	10
3551	Total 90
3552	1 Cf. No. 30 and notes thereto. 2 Literally "completion."
p. 79
3553	RHIND MATHEMATICAL PAPYRUS 75
3554	Proof : amounting in corn to No. 37.
3555	90 i+35hekat
3556	180 +ts hekat
3557	sic 3 30 TE+ s5 hekat
3558	sic s of 3 10 3of$ 35 hekat
3559	s ot it 10 tof it 32 hekat
3560	Total 320 Total +++hekat"
3561	The problem is of the same type as the two preceding. The equation for solution is
3562	x{3++(x})+#= 1 hekat, and the answer is (k + 3z) hekat.
3563	The method is clear. First it is shown that the various operations to be performed on
3564	the unknown measure amount to multiplying it 3} + t* times. To obtain the measure we must
3565	therefore divide 1 hekat by 3} + †*. This is done by the usual trials, and the multipliers
3566	I and s's are ticked off as giving the correct amount. The addition of the two products
3567	corresponding to these multipliers is called km. The translation "addition" for this is perhaps
3568	a little bold, for what is actually done is to add only the last five fractions, omitting the first
3569	three, which are ‡, ‡ and #. The common denominator 576 is used and the numerators of the
3570	five fractions when reduced to this denominator are placed beneath them and found to add up
3571	to 72, which is ‡ of 576. This, together with the ‡, and neglected completes 1. Thus
3572	what is actually done is to show that the five fractions "complete" ț, } and ‡ to make unity,
3573	and in view of this km ought strictly speaking to be translated "completion" rather than
3574	"addition" (see Nos. 21-23 and pp. 12-13).
3575	Thus f + ss of a hekat is the required amount. First comes a proof in ordinary fractions,
3576	involving a "completion" similar to that above. The answer is now reduced to ro, giving 90,
3577	and a proof in parallel columns follows, on the left in ro and on the right in terms of the t.
3578	*, 8, etc. of the hekat in their Horus-eye notation.
3579	No. 38. (PI. M.)
3580	go three times into the hekat; a seventh of me is added to me and I return fully
3581	satisfied.
3582	Total 34
3583	Divide 1 by 34:
3584	37
3585	4, since 4 is multiplied 22 (times) to make 37
3586	+7E +TE
3587	Total 1
3588	Proof :
3589	1 51+27
3590	+it+st tu
3591	I's, since as is multiplied 7 times to find the top group of fractions.
3592	L 2
p. 80
3593	76 RHIND MATHEMATICAL PAPYRUS
3594	No. 38. 1 320
3595	213
3596	1063
3597	291r
3598	149120
3599	5++AB
3600	Total 1013 + ir+ dyt th
3601	amounting in corn
3602	(*+1e) hekat+(15+11+de+75) ro
3603	-2 (t 1) hekat + (33 + ir + sla + is) ro
3604	Total 319} + i, +n t d2 + 72 + lo t ản t 2o hekat + (41 + 1r) ro
3605	1
3606	Total 22
3607	1"
3608	The problem is x (34) = 1 hekat, and the answer is (# + ir) hekat plus (13 + xr
3609	+txt66) ro.
3610	Next 1 is divided
3611	{+ 6t). The answer is first proved in ordinary fractions by multiplying 6 + ir + ¿s + de by
3612	37 and showing that the result is 1 (hekat). The answer is then reduced to ro and the usual
3613	double proof follows, firstly in ro and their fractions, and secondly in terms of the Horus-eye
3614	parts of the hekat. As there are already fractional parts of the ro in the first of these proofs
3615	theynaturally persistintothe second. In the final addition of fractions in the latter
3616	proof the common denominator 66 is used, the fraction } neglected, and the sum of the rest
3617	shown to be }3 or }, thus making up the 320 ro or 1 hekat.
3618	Two points in this sum need special notice. l'hese are the phrases irt pw - (spw) 22
3619	ant 3, and irt po ds spo 7r gmt ti lit hrt. The first of these is placed in black inks after the
3620	step 12 7, and is clearly an explanation of how that result is obtained. In these multiplica-
3621	tions and divisions we are accustomed to meet only simple multipliers like 2, } and 1. Here,
3622	however, we are suddenly confronted by ss as a multiplier, and therefore some word of explana-
3623	tion is felt to be necessary.! But though the meaning is fairly obvious the syntax is difficult.
3624	Since the second word is pu the first should normally be a noun-equivalent and it can therefore
3625	only be an Infinitive or the Neuter Perfect Participle. The first is the more likely, and the
3626	from the parallel phrase) to find 3}." This gives good sense grammatically, though it is not
3627	quite the turn of phrase which we expecthere, something like "since \| must be multiplied
3628	22 times to find 31" being what seems to be needed. Nevertheless it is the only rendering,
3629	for it is impossible to get a grammatical translation on the supposition that irt is a neuter
3630	participle either active or passive."
3631	The parallel phrase undoubtedly refers to the step da in the multiplication of
3632	f + Ti + z + on by 3}, though owing to the exigencies of arrangement in narrow horizontal
3633	registers it has been separated from it by the words "Total I." It is clear from the context
3634	1 It is seldom given, however.
3635	= irt followed by spu is used frequently in the papyrus for "multiply."
3636	* If passive, we should expect iryt not irt.
p. 81
3637	RHIND MATHEMATICAL PAPYRUS
3638	that the translation must be "It is the multiplying of ¿s 7 times to find this above ist," No. 38.
3639	where ist must be a word standing for the quantity +t++* "sum of fractions," or even more generally "a quantity." To combine t; and and meaning
3640	iit into one word and to identify the result, as Eisenlohr does, with the tiit of No. 61 is
3641	impossible.
3642	Nos. 39 & 40. DIVISION OF LOAVES IN UNEQUAL PROPORTIONS.
3643	No. 39. (PI. M.) No. 39.
3644	and fifty to 4. "Method of finding the difference of share. A hundred loaves to 10 men, fifty to 6 What is the difference of share?
3645	1 4
3646	12
3647	24
3648	48
3649	2
3650	123
3651	12}
3652	123
3653	12%
3654	Difference of share 4%"
3655	This example brings us to a fresh type of problem, the division of loaves between men
3656	in unequal proportions. We are now introduced to a new technical term, twnw, whose meaning
3657	must be guessed from the context. Here 100 loaves are divided into two fifties, one of which
3658	is further divided among 4 men at the rate of 12} each, and the other among 6 men at the
3659	rate of 8% each. The twnw is stated to be 4%, and it can therefore hardly be other than the
3660	difference between the share which any one of the 4 men receives and that which any one
3661	of the 6 men receives.
3662	mathematics for this quantity (see, however, below, on No. 40).
3663	The working explains itself. First 4 is multiplied to find 50, giving the larger share as
3664	12}. Then 6 is multiplied to find 50, and the smaller share is seen to be 8}. The four larger
3665	shares and the six smaller shares are then set out in full, and the twnw stated to be 4%.
3666	No working of this step is shown, but the subtraction of } from } was one with which the
3667	mathematician was probably perfectly well acquainted.
3668	The word twnw is unknown outside this papyrus. A verb twn, determined in the same
3669	way, occurs in Ebers Medical Pap., 101, 12-13, and in Pap. Harris Mag., 8, 6, but I am unable to
3670	seize its meaning in either passage." The ox persists even in the plant-name twn, Hearst
3671	Medical Pap., 8, 4, and BruGscH, Würterbuch, Suppl., 1315. Doubtless the same root is involved .
3672	in the town-name Mtwn of Medum, Pl. XIX (determined by a lassoed " ox), and the noun
3673	mtwn of Pap. Millingen, 1, 10.
3674	11, § 142), 1 It is just conceivable that irt I1, 332). t ecurs again in No. 62. though there are no passive examples known, and we should in any case expect the geminated form pu is a passive form of the śdm-f pu of the Ebers Papyrus (SETHE, Verbum,
3675	* For further examples see 1.Z., 57, 38.
3676	* Sue, however. #.Z., 43, 74-6.
p. 82
3677	78 RHIND MATHEMATICAL PAPYRUS
3678	No. 40. No. 40. (PI. M.)
3679	"A hundred loaves to 5 men, one-seventh of the three first men to the two last.
3680	What is the difference of share?
3681	The doing as it occurs supposing the difference of share to be 5$:
3682	23
3683	17)
3684	12
3685	1
3686	Total 60
3687	60
3688	40
3689	(Total 100)
3690	You are to count with 1}
3691	23 times, it becomes 38g
3692	17% 29-
3693	12 20
3694	103+1
3695	1 Total 100
3696	The problem consists in dividing 100 loaves among 5 men in such a way that the shares
3697	are in arithmetical progression and that the sum of the two smallest shares is one-seventh the
3698	The first of these two essential conditions is not mentioned in the
3699	problem as set by the Egyptian, but it is possible that it was implied by the mention of trnw.
3700	In dealing with No. 39 we found it difficult to believe that a special technical term should
3701	have been invented for the " difference of share" in the very simple sense in which it occurs
3702	in that problem, and it is possible that the real technical meaning of twnw is that in which it
3703	is used here, namely the "common difference" in an arithmetical series.
3704	The method is as follows. A hypothetical series in arithmetical progression, namely
3705	1, 6}, 12, 17}, 23, is taken, the common difference or twnw of which is 5%, while its sum is seen
3706	to be 60. This series has the further property that the sum of its two lowest terms is one-
3707	seventh the sum of its three highest. Thus the trial numbers chosen are not really arbitrary,"
3708	but are chosen because they were already known to be suitable for the purpose. The fact
3709	probably is that it had been noticed that in this set of numbers in arithmetical progression with
3710	a common difference of 5% the sum of the last two was just one-seventh of that of the three
3711	first, while the total of all five was 60. This suggested the setting of problems involving a
3712	division on the lines demanded in our example, but in which the total of loaves is not 60 but
3713	some other number, thus involving a small sum in proportion in addition to the knowledge of
3714	the 60-series. In other words, the sum was, like those in many modern examination papers,
3715	set from the answer. Otherwise it is impossible that the Egyptian could have obtained the
3716	60-series and its twnw of 5%, which involve the preparation and solution of two simultaneous
3717	equations, which we know to have been beyond his reach.
3718	the working. Having copied down his 60-series the scribe notes that the
3719	total he needs is not 60 but 100, and as 100 is 60 + (60x }) he must multiply all the shares
3720	1 Or, "one seventh of the three superiors two inferiors."
3721	= The papyrus here has "Total 60," erroncously repeated from above.
3722	3 Obviously when two trial numbers are chosen (in this case the first or last term of the series and the common
3723	diflerence) they cannot be arbitrary, for they bear a numerical relation the one to the other.
p. 83
3724	RHIND MATHEMATICAL PAPYRUS 79
3725	in the 60-series by 1}. This gives him the required series whose total is 100, and whose twnw, No. 40.
3726	which, oddly enough, he never works out, is 9%.
3727	It is worth while to notice the knowledge of proportion displayed in this example. The
3728	Egyptian had realized that if the total of a number of proportionate shares was increased that
3729	of each separate share would be increased in the same proportion. This is indeed the method
3730	by which we now solve problems in arithmetic where there is only one unknown and where we
3731	are anxious to avoid the open use of an algebraic symbol. He does not reveal to us whether
3732	he had also realized the fact that the twnw would also be increased in the same proportion
3733	and could therefore be obtained direct from 5½ by multiplying by 13. From the fact of his
3734	having given us the actual shares and not the twnw it is not fair to argue that he did not, for
3735	despite the fact that the problem asks for the twnw the important point in practice-and these
3736	problems are almost always practical-is to find the actual share of each person.
p. 84
3737	80 RHIND MATHEMATICAL PAPYRUS
3738	BOOK II. MENSURATION.
3739	PART I. VOLUMES AND CUBIC CONTENT. Nos. 41-47. (Plates M-O.)
3740	NoS.-47. WIrH these problems begins the second great section or book of our papyrus, that which
3741	treats of mensuration. Problems 41-47 deal with the volume, or more strietly, as might be
3742	expected from the concrete Egyptian mind, the content in corn of certain containers or spaces
3743	of varying shape. The word used for container is ší, which, as Griffith notes, might mean
3744	not a bin or granary but rather a three dimensional space or figure in the mathematical sense.
3745	The solids represented by it in these problems seem to be
3746	regular figures varying in shape at the base. In Nos. 41-43 the s: is circular at the base,
3747	ie. it is a cylinder: in No. 44 it is ifd, square, and as the height is equal to the sides of
3748	the base the whole is a cube. In No. 46 the s* has no epithet, and is seen to be either square
3749	or rectangular in base (in No. 45 it is square), while in No. 47 it is distinguished from the
3750	noun dbn, which can be nothing but a round container (cylinder), and it must therefore stand
3751	for a figure on a rectangular or square base. Thus the word when used alone would seem to
3752	carry the idea of rectangularity, though when qualified by the adjective dbn (41-43) it indicates
3753	a cylinder.
3754	No. 41. No. 41. (PI. M.)
3755	"Example of working out a circular container of diameter 9 and height 10.
3756	You are to subtract a ninth of 9, namely 1; remainder 8.
3757	Multiply 8 eight times, result 64.
3758	You are to multiply 64 ten times; it becomes 640.
3759	Its half is now added to it; it becomes 960.
3760	(This is)its content in khar. You are to take a twentieth of 960, namely 48. This is the
3761	amount which will go into it in quadruple-hekat, namely 48 hundreds of quadruple-hekat of corn.
3762	Form of its working:
3763	1 8
3764	2 16
3765	4 32
3766	64
3767	1 64
3768	<10 640
3769	320
3770	Total 960
3771	to 96
3772	48"
3773	In this problem the base of the s; is circular and 9 cubits in diameter, the figure being
3774	in fact a cylinder. The diameter is decreased by its ninth part and the result squared, which
3775	gives roughly the area of the base in square cubits. This is next multiplied by the height
3776	1 An obscure noun š3°t with the house-determinative occurs LACAU, Iextes Religieux, 86, 85-88. Cf.
3777	Itl I in a list of buildings in the Golenischefi Glossary, 5, 15 (Gardiner).
3778	2 pw omitted by the scribe.
3779	3 See No. 48. That the dimensions are in cubits is clear from No. 43.
p. 85
3780	RHIND MATHEMATICAL PAPYRUS 81
3781	of 10cubits, and the resulting 640 stated to be the volume of the whole in cubic cubits. To No.41.
3782	turn this into a measure of capacity we add its half, and the result is the content in khar,
3783	from which we may see at once that lf khar is the capacity of a cubic cubit. We now divide
3784	the number of khar by 20, and the resulting 48 is the amount of corn which will go into the
3785	space, reckoned in hundreds of quadruple-hekat.
3786	For the khar and its history see Introduction, p. 26.
3787	The expression here arrived at for the volume of a cylinder is a very reasonable
3788	approximation. The correct value is given by
3789	V = Trh
3790	where r is the radius and h the height. The Egyptian uses
3791	V = (9) h
3792	giving as the value of w 314. which is not very far from the correct value 3•14159265......
3793	(ahout 34).
3794	The uses of the noun rit in this papyrus bear out the belief that its literal meaning is
3795	simply "number" (nombre, not numiro) and not "list" or "specification," as it is so often
3796	translated, perhaps in consequence of its obvious derivation from rh, "to know." It is true
3797	that the word often serves to head a list, but it will be found that, in the Middle Kingdom
3798	at least, the items in these lists are accompanied by numbers or quantities. See, for example,
3799	GRIFFITH, K.P., VIII, 44, Ä.Z., 57, 58 (from Pap. Bulaq 18), Urk., IV, 664 and 893, and À.Z., 37,
3800	92. In the Rhind Papyrus it occurs eleven times. In Nos. 63, 74 and 82B it can mean nothing
3801	but "number" or "amount," while in 86 this meaning suits quite as well as "list." Else-
3802	where we have a slightly extended use of the meaning "number," for rht is used in 46 for
3803	"dimensions," in 44 for the content in khar of a certain space, in 50, 52 and 53 for the "area"
3804	of fields, and in 69 for the "flour-content" of a loaf, an idea expressed in No. 70 and else-
3805	where by hrt, "the share" of a bushel of flour to be found in each of a batch of loaves fixed
3806	many to the bushel. In No. 62 the phrase rht-f pw, whether we take it to refer simply
3807	to the words "Total 84" or to the separate values in shaty of the three metals, can hardly
3808	mean anything but "amount" or "value."
3809	No. 42. (PI. N.) No. 42.
3810	"A circular container of 10 by 10."
3811	You are to subtract a ninth of 10, namely 14; remainder 83 + } + 1s.
3812	Tou are to multiply 83 +2+1 by 83+1+15¡result 791l +32z
3813	You are to multiply 79yux + iba by 10; it becomes 790y* + 27 + 54.
3814	Its half is added to it; it becomes 1185.
3815	Multiply 1185 by zu, giving 59%. This is the amount that will go into it in
3816	quadruple-hekat, namely 594 hundreds of quadruple-hekat of corn.
3817	Form of its working :
3818	1 87+*+78
3819	2 17,+
3820	4 35= +1*
3821	71%
3822	53+ +7+ ÷t
3823	23+++36 + 34
3824	1+12+4++is
3825	+$+++#
3826	Total 79T + sł
3827	1 ie. of diameter 10 and height 10.
p. 86
3828	82 RHIND MATHEMATICAL PAPYRUS
3829	No. 42. 1 79T1x +=3+
3830	10 7901+1+
3831	395g+ + ** + 13s
3832	Total 1185
3833	Tu 118}
3834	59}"
3835	This example is precisely similar to the last except that the numbers involved are
3836	more complicated. There is a slight error in the working, 79,ux t sl# (which is in effect
3837	79gł) when multiplied by • 10 yielding not 790,% + 2 +s* (which 790.) but 79011, a
3838	difference of #T.
3839	No. 43. No. 43. (PI. N.)
3840	"A circular container of 9 eubits in its height and 6 in its breadth. What is the amount
3841	that will go into it in corn?
3842	The doing as it occurs :
3843	You are to 1 from 9; remainder 8.
3844	Operate on 8; you are to add its third part to it; it becomes 10%
3845	Multiply 10} by 10}; it becomes 1133 + +.
3846	Multiply 113} + * by 4, this being two-thirds of the 6 cubits which are the breadth.
3847	It becomes 455). This is its content in khar.
3848	You are to find one-twentieth of its content in khar; it becomes 22%+1+ 1s (sic).
3849	This is the amount that will go into it int quadruple-hekat, viz.:
3850	corn, hundreds of quadruple-hekat (22] + 1)
3851	+ quadruple-hekat 6+ +.)
3852	+ quadruple-ro (2} + } + 1.)
3853	Form of working :
3854	8 1 10%
3855	-10 106
3856	7,
3857	Total 10% Total 1133 +1
3858	1 1133+ 1 4551
3859	2 227} + I* 45y + 30
3860	455%, 120 22} + * + » (read T") "
3861	This is one of the most difficult problems in the papyrus. It professes to give a method of
3862	finding the content of a regular figure in khar without first working out the volume in cubic
3863	cubits and multiplying it by 1} as in Nos. 41 and 42. The container is again a round one
3864	(š; dbn), of which only two dimensions are given, a height of 9 cubits and a breadth of
3865	6 cubits. Now the only suitable regular solid figure which can be determined by two diniensions
3866	alone is the cylinder, and the si of this problem is therefore again a cylinder, just as
3867	in the two previous examples. is the diameter of its base, to which, it
3868	will be remembered, no name was actually given in the other two problems.
3869	later that the statement of the problem is incorrect, the 9 cubits being really the diameter
3870	and the 6 cubits the height of the cylinder.
3871	The difficulty of the question lies in the fact that, though the container is precisely
3872	similar to that in Nos. 41 and 42, the result obtained for its content in khar is different trom
3873	that which would be reached by the method of the other two examples. Clearly, in the
3874	attempt to find the result directly in khar without first finding the volume in cubic cubits
3875	1 The words m i* het which follow are clearly a scribe's error and to be omitted.
p. 87
3876	RHIND MATHEMATICAL PAPYRUS 83
3877	adding, its half to it, some error has been introduced, and we have to ask where No.43.
3878	Eisenlohr treats the question at great length, but his reasoning is vitiated by his having
3879	khar incorrectly throughout and therefore having failed to grasp the real meaning
3880	of the problem. There is no need to try to solve the question by referring as he does to
3881	granaries of various irregular forms. The solution of the difficulty is due to the ingenuity of
3882	Schack-Schackenburg. To follow his reasoning we must go back to a somewhat difficult problem
3883	in the Kahun Papyri, PI. VIII. It is set out as follows:—
3884	12
3885	8 (13654)
3886	8 .-1 16 -1 256
3887	-10 160 2 512
3888	80 1024
3889	Total 256 853
3890	Total 13653
3891	There is no doubt as to what is being done here. First 1) times 12 is taken and found
3892	The first line of the step, 12, is omitted because the 12 over the
3893	figure of the cirele, used to indicate its diameter, also does duty for this first line. Next 16
3894	is squared, giving 256, and finally 256 is multiplied by 5} which is } of 8, the other given
3895	dimension of the circular figure.
3896	orlind getin K , Noee 19) surmiend chat the spbilem an to aod chnc lo nte vo dt. g
3897	with this hypothesis.
3898	Borchardt, writing in A.Z., 35, 150-52, attempted to explain it as the finding of the
3899	cubic content of a hemisphere 8 cubits in diameter. In order to do this it was necessary to
3900	suppose that the 12 over the figure of the circle belonged to the working and not to the
3901	78-9, Schack-Schackenburg proposed an alternative reading which avoids this difficulty and is
3902	certainly the correct one. According to him the problem is to determine the volume of a
3903	cylinder of diameter 12 and height 8 cubits. The diameter 12 is first multiplied by 13,
3904	producing 16. This is squared, giving 256, and this number is then multiplied by 5f which
3905	is two-thirds of the height. The result is 1365}, and this result is in lchar, though the
3906	papyrus, which is very terse in expression, does not make this point clear. It is in fact
3907	exactly the number of lihar which would be given by the method of Nos. 41 and 42 in the
3908	Rhind, as will be clear from the following working:—
3909	12 - 5.12 = 10%
3910	10% x 10% = 1133
3911	113 × 8 = 910฿
3912	910* x 11 = 13653
3913	In other words, the method of the Kahun Papyrus is a simplified method of finding
3914	the content of a cylinder direct in khar without working out its volume in cubic cubits and
3915	multiplying by lt to turn it into khar.
3916	Now this is precisely what No. 43 in the Rhind seems to be aiming at, yet it fails to
3917	get the right answer. This failure is due simply to the fact that the scribe, attempting to
3918	use the short metlod, mixed it up in his mind with the longer and more logical method, and
3919	put into the new method one of the steps of the old which is not needed in the new.
p. 88
3920	84 RHIND MATHEMATICAL PAPYRUS
3921	No. 43. This is the first step, where from nine he takes away one, which is a ninth of it. from
3922	this error it results that his result is only () of what it should be, namely 455* instead of
3923	There still remains a difficulty about Schack-Schackenburg's solution. In the statement
3924	of the problem the height is distinctly said to be 9 cubits and the diameter 6.
3925	cylinder actually worked out is one which has these two figures interchanged, 6 being the height
3926	and 9 the diameter. How are we to account for this? The most probable explanation seems
3927	to be that in the original papyrus the statement of the problem was correct, but the scribe
3928	a mistake in the first line of working, as we have seen. A later scribe, seeing in the
3929	first line of the working the subtraction of a ninth of 9 from 9, just as in Nos. 41 and 42,
3930	naturally concluded that this 9 must be the diameter and not the height, and so he
3931	transposed the two dimensions in the statement of the problem, his mathematical knowledge
3932	not taking him far enough to test the result with the statement in its new form.
3933	There are several points to notice in the method. In the first place, after the statement
3934	of the problem comes the phrase irt mi lpr. This here refers to the full detailed working out,
3935	while the śšmt is kept as a heading for the rough working. In the.
3936	second place, there is an unfortunate mistake in both the simt and the irt mi lpr, t's being
3937	written instead of Tho in the division of 455% by 20. Curiously enough this mistake disappears
3938	when the fraction is reduced to the Horus-eye notation of the quadruple-hekat, the amount
3939	taken being clearly T&o and not źs of a hundred quadruple-hekat. The actual mistake is made,
3940	as will be observed, in the sšmt, where the caleulator divides 90 by 2 instead of multiplying
3941	it. The fact that the answer is nevertheless correctly given would suggest that the answer
3942	was copied from the prototype, while the working was actually filled in by the copyist. Possibly
3943	the full sšmt (rough working in this case) was not in the prototype in all cases, though it is
3944	curious that here the same error appears in the irt mi lpr.
3945	The statement of the answer is interesting. It is 22% + * + rło hundreds of quadruple-
3946	hekat. Of this the 221 + I are left in units and ordinary fractions, as is correct in dealing with
3947	the hundred-of-hekat (whether simple, double or quadruple), the 22 preceding the .' and the
3948	% and following it. The other fraction 18u is then reduced to the Horus-eye divisions (half,
3949	quarter, etc.) of one quadruple-hekat.
3950	No. 44. (PI. N.)
3951	"Example of reckoning out a square container of 10 in its length, 10 in its breadth
3952	and 10 in its height. What is the amount that will go into it in corn?
3953	Multiply 10 by 10, it becomes 100.
3954	Multiply 100 by 10, it becomes 1000.
3955	Take a half of 1000, that is 500, it becomes 1500: this is its content in kihar.
3956	You are to take a twentieth of 1500, it becomes 75. This is the amount that will
3957	go into it in quadruple-hekat, viz. 75 hundreds of quadruple-hekat of corn.
3958	Working out:
3959	10
3960	1 10 1 100 10 10
3961	10 100 10 1000
3962	1000
3963	500
3964	1 1500
3965	To 150
3966	75
p. 89
3967	RHIND MATHEMATICAL PAPYRUS 85
3968	1 75 No. 44.
3969	10 750
3970	-20 1500
3971	1,th 150
3972	Io of 1,th 15
3973	;of in of iuth of it 10 "
3974	We have now come to the determination of the volume of a rectangular parallelopiped,
3975	which in this particular case happens to be a cube. It is described as a ši ifd. The word ifd is
3976	generally translated "square" or "rectangular," but it is quite possible that we ought to extend
3977	its use to three dimensions to cover parallelopipedal. It is clearly a derivative of fdw, the numeral
3978	4, and its original meaning must therefore have been four-sided, perhaps with the tacit addition
3979	of rectangular. In the present instance the meaning parallelopipedal' is conveyed by implica-
3980	it need only describe its base. Thus are: don is a cylinder, and a si al is e paraleponiyed cion if not directly, for a ši i
3981	or in special cases a cube.
3982	This cube has a side of 10 cubits, and its volume is correctly calculated as 1000. This
3983	is then multiplied by 1f to obtain the number of khar contained, and the division by 20
3984	reduces this last to 75 hundreds of quadruple-hekat. The three last lines constitute a proof
3985	by continued division, if indeed they have not merely strayed hither from No. 45.
3986	There is nothing to note in the text except a slight confusion in the first line. The
3987	word 10 occurs four times, whereas we only need it three times. This is due to the fact that
3988	š; n mh 10 m iw-f mh 10 m slwf mh 10 m lcsl-f
3989	2 e no m which dlo i, ot ele iced etin theren tho aecond
3990	No. 45. (PI. N.) No. 45.
3991	"A container into which corn has gone to the amount of 75 (hundreds of) quadruple-
3992	hekat. How much is it by how much?
3993	Multiply 75 twenty times; it becomes 1500.
3994	Operate on 1500: you are to take one-tenth of it, namely 150 ;
3995	to of To of it, namely 15 ;
3996	}of Tu of tu of it, namely 10.
3997	Therefore it is 10 by 10 by 10.
3998	1 75
3999	10 750
4000	20 1500: behold this is its content.
4001	1500
4002	Toth 150
4003	Thi of foth of it 15
4004	3 of to of rith of it 10 "
4005	This problem is the reverse of the preceding. Here we are given the cubic content
4006	of a container in hundreds of quadruple-hekat, and we are asked to find its dimensions.
4007	assumed that the container is parallelopipedal, and even, as the answer shows, that it is a
4008	cube.
4009	' In L. D., II, 134A ifd appears to be used of a parallelopipedal block of stone.
p. 90
4010	86 RHIND NATHEMATICAL PAPYRUS
4011	No. 45.
4012	its dimensions." Griffith (K.P., Text, 57) read do → *r, and took the words to be a further
4013	description of the container "having side equal to side." He gave instances of a word šdao
4014	meaning a "side" or "fence," and of the use of the preposition r in the sense in which he
4015	here required it. The difficulty is that the hieratic signs for dog and I both occur elsewhere
4016	in the papyrus and are not made as they are Nor is Griffith's earlier reading II
4017	(P.S.B.A., XVI, p. 235) any more satisfactory palaeographically. There is, however, a reading
4018	which seems to me free from every objection, namely The group does not occur else-
4019	where in the papyrus, so that no internal comparison is possible, but forms not unlike this
4020	occur in Ebers and are quoted by Möller (Hieratische Paliiographie, I, p. 19, note 1). If this
4021	reading be correct, what we have here is simply a late Middle Kingdom use of the Late Egyptian
4022	ur in the sense of "how much" or "how many" (Coptic orHp), and the correct translation
4023	will be "How much is it by how much?" For the use of ns in giving the dimensions of
4024	an object see below in this same problem and also Shipwrecked Sailor, 1. 62, and for the
4025	preposition r in the sense of "by" in measurements see No. 46 and also below in the present
4026	problem. This translation has the further advantages that it supplies the statement of the
4027	problem to be solved, which would otherwise be wanting, and that the same reading of the
4028	group gives good sense in No. 73, where again we need a statement of the problem.
4029	The method is as follows. The container is assumed to have a base 10 cubits square,
4030	and indeed if the solution is to be unique it is obvious that two of the dimensions must be
4031	thus assumed. The 75 hundreds of quadruple-hekat are first multiplied by 20 to bring them
4032	to khar. This ought now to be reduced to cubic cubits by multiplying by }, for it is clear
4033	from the preceding examples that the khar is a capacity of 17 cubie cubits. This would have
4034	given 1000 cubic cubits, and by dividing this successively by 10 cubits and 10 cubits, the
4035	two assumed dimensions, the third dimension, also 10 cubits, would have been obtained. The
4036	reckoner curiously enough chooses an equally effective but less logical method. Instead of
4037	multiplying by } to reduce khar to cubic cubits he divides lihar at once by 10 cubits and
4038	again by 10 cubits, getting as his result l5, but of what unit it would not be easy to say.
4039	Only now does he multiply by his } and obtain the correct answer, 10 cubits. The method
4040	is illogical because the units in which the working is done are confused, and we suspect that
4041	the reckoner is working not by the light of reason but by a mere practical rule.
4042	There are two important points in the text. In the opening sentence hi-n is the Relative
4043	to the extent of 75 hundreds of quadruple-hekat." This seems to tell against the proposal śdm-nf-form and therefore must have perfect force, то ротвлм оо "A container into which corn has gone down.
4044	to take hil in a purely metaphorical sense both here and in No. 35, for, if it had the
4045	meaning "to be contained in,"
4046	which has either imperfect or timeless meaning, in preference to the Relative śdm-nf-form
4047	which, being perfect, demands a concrete translation.
4048	The group which I have here tentatively transcribed %has previously been read
4049	in the papyrus (except in the extraneous No. 87) for &. Moreover, the meaning of the lo Palaeographically, this is just poosible, though the ligature in question is not used
4050	word, which can be no other than "content," and the parallel with the supposed synonym rlit have
4051	noun derived from it, is almost unthinkable in Egyptian.
4052	For these reasons another reading must be sought and shoti would seem to be the most
4053	probable. The word oceurs twice in No. t6 and once in No. 60, where it is an obvious error
4054	for śkil, "batter." For the final " in an abstract noun compare śity "proof."
p. 91
4055	RHIND MATHEMATICAL PAPYRUS 8T
4056	No. 46. (PI. N.) No. 46.
4057	"A container into which corn has gone to the extent of 25 hundreds of quadruple-
4058	hekat. What are its dimensions?
4059	You are to multiply 25 twenty times; it becomes 500. This is its content.
4060	Take tw of it, namely 50
4061	2i of it, namely 25
4062	To of tu of it, namely 5
4063	} of Ii of tu of it, namely 3j.
4064	This container is 10 by 10 by 33.
4065	Its working out:
4066	1 25
4067	10 250
4068	20 500
4069	This is its content.
4070	1 500
4071	50
4072	Th of In of it 5
4073	# of 7ü of to of it 3)
4074	This container proves to be of 10 cubits by 10 by 3.
4075	That is the same."
4076	This problem is precisely similar to the last,
4077	so, it we are asked to find the rlit in place of finding is natural to translate rht not by content but by numbers, i.e. dimensions, and this PEAEE
4078	is supported by the fact that the content (in khar), which is mentioned twice in the working,
4079	is called not rlt but śtuti, for which see under No. 45.
4080	assumption is made that two of the required dimensions are 10 cubits and 10 cubits. Again
4081	too the multiplication by }, which ought to have been introduced to reduce lchar to cubic
4082	cubits, is left over until the last step, where it has no logical meaning.
4083	The working out is little more than a restatement of the method, with the sole difference
4084	that the steps in the multiplication of 25 by 20 are shown. The concluding words, mitt pr
4085	"This is equivalent (to the result obtained above)," are somewhat devoid of point, for they
4086	have only a real meaning when placed at the end of a rigorous proof.
4087	No. 47. (Pl. O.) No. 47.
4088	"If the scribe says to you, Let me know (what) 1',th becomes, in a rectangular container
4089	or a circular container:
4090	i becomes 10 quadruple-hekat of corn
4091	5
4092	(31 + 16 + «t) hekat + 1} ro
4093	2 hekat
4094	2 hekat
4095	(l1 + * +:2) hekat + 3} ro
4096	70
4097	1f hekat
4098	(ie + 32 + 87) hekat + ($ + t's) ro
4099	TỎI 1 hekat."
p. 92
4100	88 RHIND MATHEMATICAL PAPYRUS
4101	No. 47. This example is nothing more than a statement of the values of the fractions ro, zu, so,
4102	etc. of a hundred quadruple-hekat, and the words "in a rectangular or a circular container"
4103	are unnecessary, having been perhaps added by a scribe who sought to connect this problem
4104	more closely with those which precede it.
4105	The usual translation is
4106	space." To this it may be objected that lpr-f could hardly take the place of the English
4107	verb "to be" in such a case as this, for lypr quite definitely means "to come into being,"
4108	"to become" or "to happen," and in fact Egyptian would need no verb at all here, but
4109	would merely say mš:dbnr pw.
4110	s have dropped out by haplography of the bf, and that the original reading was,
4111	"Let me know what to becomes, in a rectangular or a circular container."
4112	This is perhaps borne out by the sign which follows the fraction ro in the next line
4113	If this line merely meant " Toth is 10 quadruple-hekat of corn," we should expect not the
4114	preposition → (supposing the sign to be an ) but S, and so on throughout. The
4115	sign in question is perhaps that which is familiar to us from papyri of all periods as standing
4116	for "ditto," and the words for which it here stands are lprf m: this is corroborated by the
4117	position of the sign. A similar sign is used in Nos. 72 and 73 (see, however, notes to No.72)
4118	to separate the pfów of loaves from a following numeral.
4119	The notation of the quadruple-hekat here used is perfectly regular (see Introduction,
4120	p. 26). Ten quadruple-hekat are expressed by a tall stroke after the quadruple-hekat sign,
4121	standing, as Griffith has shown,' for a vertical Five are expressed by a
4122	ligature possibly combining 5 dots, and two and one by 2 dots and I dot respectively.
4123	words hekat and ro, which it is necessary to fill in in order to produce an intelligible trans-
4124	lation, refer in every case to quadruple-hekat and to quadruple-ro, ie. to a ro which is the
4125	320th part of the quadruple-hekat.
4126	PART II. CALCULATION OF AREAS. Nos. 48-55.
4127	No. 48. No. 48. (PI. O.)
4128	" "
4129	This problem, which has no wording and consists merely of a figure with working out,
4130	is clearly the comparison of the area of a square of side 9 khet with that of a circle of
4131	The area of the cirele (on the left) is obtained as usual (cf. No. 41) by
4132	squaring i of the diameter,* viz. 8. The area of the square (on the right) is got of course
4133	by squaring the side 9. The natureofthe measurements has been cleared up by Griffith.It
4134	is plain that the units placed under the sign = are each one-tenth of the units which stand
4135	free in front of it. Moreover,since the square of 9 khet is written 8 plus ,it is further
4136	clear that the free standing units each represent ten setat or square khet, while the units under
4137	the — are each one square khet. Now the khet measures 100 cubits, and therefore a square
4138	Ichet (10,000 sq. cubits) may be regarded as the sum of 100 strips of land each one cubit
4139	broad and 100 long. Each of these strips was called a cubit-of-land, because it measured a
4140	cubit along one side of the square khet, and a thousand of these, known technically as
4141	Il-II "a thousand-of-land," would make ten square lhet, which is precisely what is
4142	1 P.S.B.A., XIV, 425. 2 Of. No. 69.
4143	3 The text has 60 setat. Cf. No. 50. * For the value of - thus obtained see p. 81.
p. 93
4144	RHIND MATHEMATICAL PAPYRUS 89
4145	represented by the units in our example. Each of these then represents a No. 48.
4146	thousand-of-land, ie. a thousand of the narrow strips each 100 cuhits by one cubit, and would
4147	be represented in early hieroglyphs by the sign for thousand.
4148	Lhe example seems curously out of pee point in this reckoning which has hardly pereived the attention it deser ves
4149	As a general rule the Egyptian does not trouble in his working out to insert the dimensions
4150	of the figures he uses. Thus he does not always openly distinguish between pure numbers and
4151	concrete lengths, weights, etc. So in finding the volume of a cylinder he will square §ths of
4152	the diameter of the base and proceed to multiply this by the height without stopping to point
4153	out whether the figures used are cubits, square cubits, or cubic cubits until the conclusion of
4154	the reckoning.
4155	o o co ea e ae etin eie ieo d al oid io a 8 setat," the unit is stated as setat. It
4156	To our modern feeling measure, viz. khet, and it is not until we multiply it by another unit of long measure, viz. tere an the is nromtl The 8 in question is, strictly speaking, in units of long
4157	8 khet, that it can logically be expressed in square units. Similarly in the second half of the
4158	sum, the finding of the area of a field 9 khet square, the logical process is to multiply
4159	9 khet by 9 khet; but the working actually written looks like the multiplication of 9 setat by
4160	thepure number 9. The explanation of this is that, if dimensions are to be stated at all in
4161	the working, the peculiar form of the Egyptian system of multiplication demands that the final
4162	dimension should be stated immediately in the first line. Thus it is obviously impossible to
4163	write 9 khet in the first line and 18, 36, and 72 setat in the following, for the first and last
4164	lines have next to be added together, which is impossible if one is in long and the other in
4165	In other words, the difficulty lies in the very nature of the Egyptian multiplication
4166	It is strictly speaking not a multiplication but an addition or counting, and in
4167	reality it is impossible to get an area by adding together lengths. The Egyptian solvedthe
4168	logical difficulty in one of two ways. He either put in no dimensions until the working was
4169	complete, or he put in the final dimensions straight away. The latter method might be
4170	justified by considering the first line of the multiplication as giving the product of say 9 khet
4171	by 1 khet, ie. 9 square khet or setat, and not as a mere statement that the multiplicand is
4172	There is a somewhat similar example in No. 53. Here, despite the obscurity of detail
4173	and meaning, we are clearly dealing with the calculation of certain areas. In the wsh tp
4174	process the products (on the left in the original) are given in square measure from the first
4175	line to the last. The multipliers (on the right) ought, of course, to be in the pure arithmetical
4176	notation, and indeed the integers 1 and 2 are: but when we come to the fractional multi-
4177	plier 1 we are surprised to find it expressed in the form peculiar to square measure, the arm
4178	(half-setat) standing instead of the pure number $.
4179	No. 49. (Pl. O.) No. 49.
4180	"Example of calculating (?) land. If it is said to thee, A rectangle of land of 10 khet
4181	by 2 khet. What is its acreage?
4182	The doing as it occurs:
4183	1 1000
4184	10 10,000
4185	100 100,000
4186	oth of 100,000 is 10,000
4187	1„th of tu of it is 1000
4188	This is its content in land."
4189	N
p. 94
4190	90 RHIND MATHEMATICAL PAPYRUS
4191	No. 49. It is not easy to deal with the text of this and the following examples,
4192	with scribe's errors of the worst description. The problem is to determine the area ofa
4193	rectangle whose sides are 10 and 2 khet. This is clear both from the setting out and from the
4194	figure. Yet the working is not consistent with this. The absence of the 2 from the working
4195	means either that the scribe was totally ignorant of the wholemethodor that there is an
4196	error in the setting and in the figure.! If we read 1 khet instead of 2 for the shorter side
4197	the working will be correct, though even then it is clumsy. The answer, as usual in these
4198	sums, is in cubits-of-land, ie. in narrow strips each 100 cubits by 1 cubit, and the correct
4199	answer is 1000 of these. This is the answer given, but it should have been obtained directly
4200	from the two dimensions of the rectangle 10 and 1khet. The multiplication of these gives
4201	10 setat or square khet, and as each square khet contains 100 cubits-of-land the answer is
4202	1000 cubits-of-land, or one thousand-of-land. The scribe, however, reduces the 10 khet to
4203	1000 cubits and multiplies it by the 1 khet, which is 100 cubits.? This gives 100,000 square
4204	cubits, which he then reduces to cubits-of-land quite correctly by dividing by 100.
4205	In the opening phrase tp n iśt, iśt can hardly be a noun—" Example of an ist of land"
4206	-for three reasons. In the first place this phrasewould probably have been rendered by tp
4207	n ist nt (or m) :lt (cf. Nos. and 52), in the second place the sense would require a concrete
4208	meaning for ist, which is not borne out by its abstract determinative, and in the third place
4209	tp n is always followed in this papyrus by an infinitive. Thus, if the text is correct, ist ought
4210	to be the infinitive of a verb isi, which the sense would require to have a transitive meaning :
4211	unfortunately no such Is it possible that the correct reading is tp n nís,
4212	"example of calling up," ie. of reckoning out, as in Nos. 44 and 56?
4213	ntf might be used here in its normal pronominal sense, "This is it
4214	(sc. the ifd) reckoned in land-area," but is perhaps better taken in the pregnant sense first
4215	demonstrated by Erman," " belonging to it," or even nominally "its property," ie. "its content."
4216	No. 50. No. 50. (Pl. O.)
4217	"Method of reckoning a circular piece of land of diameter 9 khet. What is its area
4218	You are to subtract one-ninth ot it, namely 1; remainder 8.
4219	You are to multiply 8 eight times; it becomes 64. This is its area in land,
4220	6 thousands-of-land and 4 setat.
4221	The doing as it occurs:
4222	5 of it
4223	subtract from it; remainder 8
4224	1 8
4225	2 16
4226	4 32
4227	64
4228	Its area in land is 6 thousands-of-land, 4 setat."
4229	There is little to notice here. The sum is worked out correctly in square khet by the
4230	usual rule for calculating the area of a cirele. The result is 64 square khet, which is reduced
4231	to 6 thousands-of-land (written like 60*) and 4 setat or square khet.
4232	1 Perhaps there is confusion with a right-angled triangle of base 2 khet and height 10.
4233	3 A.Z.. 34, 50 ff.; cf. d.Z., 41, 135-G
4234	4 The sign is here as in No. 48 defnitely 60, not the cursive form of 6 used above in No. 14 and in Eloquent
4235	Peasant, B 2, 136.
p. 95
4236	RHIND MATHEMATICAL PAPYRUS 91
4237	No. 51. (PI. O.) No. 51.
4238	"Example of reckoning a triangle of land.' If it is said to thee, A triangle of 10 khet
4239	in its height and 4 khet in its base. What is its acreage?
4240	The doing as it occurs:
4241	You are to take half of 4, namely 2, in order to give its rectangle. You are
4242	to multiply 10 by 2. This is its acreage.
4243	400 1 1000
4244	200 2 2000
4245	Its acreage is 23"
4246	Here we meet for the first time the triangle (śpdt, "the pointed") and its area. Is this
4247	correctly determined? Eisenlohr thought not, and mathematicians have for the most part
4248	accepted his dictum. Yet the matter is hardly so simple as he supposed. There are in reality
4249	two interdependent problems to be solved. Firstly does the solution given apply to all triangles
4250	or only to those of a special type, and secondly is it correct? The evidence at our disposal
4251	consists of the following:—
4252	(1) The names of the triangle and its parts.
4253	(2) The position of the numbers marked in the figure.
4254	(3) The shape of the figure.
4255	(4) The striking phrase "This is its rectangle."
4256	(1) The name śpdt here given to the triangle gives no clue as to the shape of the
4257	figure. It is true that "the pointed figure" calls up most readily the idea of
4258	with short base and sharp vertex, ie. with one angle definitely less than the others; but this
4259	hardly amounts to evidence, and even if this was the original meaning it is extremely likely
4260	that as a mathematical technical term it applied to the triangle in general.
4261	The word tp r is a compound in which, as in so many Egyptian compounds, the tp
4262	adds little in meaning to the simple noun, and it is to be translated simply "mouth," Coptic
4263	тăпpо. It is beyond all doubt that it stands for the base of the triangle, and the very use
4264	of this word does lend some colour to the belief that the triangle dealt with is one with a
4265	sharp apex, the narrow base of which can be reasonably described as the "mouth," lying
4266	between the two long sides envisaged as jaws.
4267	The word mryt is the crux problem. Its literal meaning is the "bank" or
4268	"edge" of a river or sea, more particularly a "harbour" or "quay." This being the case,
4269	the obvious rendering of the word as a mathematical term would be the "edge," ie. the
4270	side of the triangle, a meaning which seems doubly suitable in the case of a triangle with
4271	narrow base, but which—and here lies the crux—presupposes that the two long sides are
4272	equal, for otherwise there would obviously be two different solutions for the area of a trianale.
4273	namely half the base multiplied by the two sides respectively. The only other possibility
4274	would seem to be to reject the tempting analogy of "side" or "edge" and to regard myrt
4275	as the vertical height, i.e. the length of the perpendicular from the apex to the base. This
4276	is just possibly the correct solution.
4277	(2) With regard to the position of the numbers marked in the figure it is clear that
4278	expected that the 10 khet written along the middle of the upper long side referred to the
4279	length of that side. It may be so, but it is far from certain. In Nos. 56-58, for instance,
4280	a measurement is written outside the left side of the pyramid figure which beyond all possible
4281	doubt gives the vertical height and not the slant height. This may, it is true, be partly
4282	due to the exigencies of arrangement in narrow horizontal bands, but it does at least show
4283	that the line on which a marked measurement is taken need not actually be drawn. But
4284	' Or, " of finding the area of a triangle in land." = sc. thousands-of-land.
4285	N 2
p. 96
4286	92 RHIND MATHEMATICAL PAPYRUS
4287	No. 51. we may go farther than this. In No. 53, a difficult problem, but not totally devoid of senue,
4288	we have a picture of a triangle divided into three by lines presumably parallel to its base.
4289	The base of the apex portion is marked 24 in its area is filled in as 7}+*+} in
4290	red, and along the lower side almost at the apex is a 7. In the working half the base, viz.
4291	1f khet, is multiplied by this 7 khet and gives the area as marker in the figure. Clearly the
4292	7 is the mryt, and yet it is here written not along the side, but exactly in the position
4293	occupied by the vertical height figure in the illustrations of the pyramids in Nos. 56-58
4294	This shows that nothing must be argued from the position of the 10 khet in No. 51.
4295	(3) It would be unwise to attribute too much importance to the shape of the triangles
4296	actually drawn in the papyrus. At the same time we should keep in mind the fact that the
4297	triangles illustrating these three problems, 51, 52 and 53, are as a matter of fact firstly isosceles
4298	and secondly erected on comparatively narrow bases.!
4299	(4) The words "This is its rectangle" make it probable that the solution was obtained
4300	graphically. It is of course just conceivable that the Egyptian, as Eisenlohr supposes, erected
4301	a rectangle on one of the long sides with half the base as the other dimension; but if this
4302	is the case, we must suppose that the solution only applied to tall isosceles triangles, isosceles.
4303	because any other would yield two possible rectangles, tall because in the case of short
4304	triangles the solution is a manifest absurdity. It would, however, seem more logical to explain
4305	the words "This is its rectangle"by means of some such graphic solution as
4306	Figs. 3 and 4. If the triangle is isosceles and the mryt is its vertical height, we actually see in
4307	the drawing the rectangle contained by half the base and the height, Fig. 3.
4308	the triangle are unequal "its rectangle" is not actually shown, but a rectangle is seen (Fig. 4)
4309	Fig. 3. Fig. 4. Fig. 5.
4310	mere symmetry of triangles is clearly double the triangle in area. The same reasoning
4311	applies to No. 52, where the area a truncated triangle is obtained by taking half the sum
4312	of the two parallel sides "in order to get its rectangle" and multiplying it by the mryt.
4313	Here it is possible to conceive the solution as having been obtained graphically as shown
4314	Did mryt here mean the slant side (with the involved assumption that the figure
4315	is regular), extreme cases would surely have shown the absurdity of the solution except in
4316	the case of very tall triangles.
4317	to the e itemnal indiotiong thus mal e it very ditficut te da any co ain condlegirn us draw any certain conclusion as
4318	We have now to ask ourselves whether the triangle treated in this example is a general
4319	one with special properties. The possibilities are three: it may be scalene, v.e.
4320	general, isosceles, or right-angled.
4321	(1) If it is scalene, the mryt must be the vertical height, for otherwise there would be
4322	two mryt and two possible solutions. In other words, if the triangle is scalene, then the
4323	Egyptian solved the problem correctly.
4324	The same is true of a similar but much damaged probiem in the Moscow papyru A gap in the papyrus
4325	akes it impossible to say where the measurement of the mrul was written in the figur
p. 97
4326	RHIND MATHEMATICAL PAPYRUS 93
4327	(2) If it be isosceles, the mryt might be either the vertical height or the length of one No. 51.
4328	of the equal sides. If it be the vertical height, it follows that the Egyptian had correctly
4329	determined the area of an isosceles triangle (probably graphically); but we cannot assume that
4330	he had also correctly solved the scalene triangle.
4331	If the wryt is the length of the long sides, as Eisenlohr supposes, we can only say that
4332	the Egyptian had formed an approximation to the area of an isosceles triangle which was
4333	fairly accurate when the base was very small and became increasingly inaccurate as the base
4334	increased.
4335	(3) If it be right-angled. This supposition, unlikely in the face of the definitely non-
4336	right-angled figures, which, however, are not final in themselves, must be ruled out in view
4337	of the fact that in the Moscow papyrus the right-angled triangle is clearly perfectly well
4338	understood and treated as half a rectangle: the two sides enclosing the right-angle are
4339	• actually called the "length" and "breadth" respectively, terms manifestly taken from the
4340	terminology of a rectangle.
4341	The matter may be summed up as follows. The mryt is either the vertical height or
4342	slant height of a triangle. In the first case the evidence is insufficient to show whether the
4343	solution may have been obtained. In the second, the triangle must obviously be isosceles, and,
4344	have a very small base.
4345	It should be noted that whatever decision we here make must have its corollary
4346	No. 52. If the mryt be the vertical height here it must be the same there also, and the solution
4347	will be correct whether the truncated triangle be isosceles or not. If on the other hand the
4348	mryt be the slant height, then the truncated triangle must be isosceles to avoid ambiguity and
4349	A very similar difficulty also arises in the case of the problem of the truncated pyramid
4350	in the Moscow Papyrus published by Turaiev.' Here we have a frustrum of a pyramid.
4351	top surface is 2 cubits (a) square and the bottom 4 cubits (b) square. The śti (h) is 6 cubits,
4352	and the volume is calculated by the formula
4353	V = "} (2" + ab + 5°)
4354	If the śti is the vertical height of the frustrum the solution is correct, and in this case the
4355	Egyptian has here made a very notable achievement. Turaiev gives him this credit; but in
4356	at the element (a" + ab + 6°) correctly should have made so gross an error as to multiply
4357	it by a third of the slant instead of the vertical height. This however hardly amounts to
4358	argument.
4359	So far we have judged the question of the solution of the triangle on purely internal
4360	There is one piece of external evidence which to Eisenlohr seemed so cogent that
4361	it led him to pronounce in favour of the mryt being the slant and not the vertical height of
4362	the triangle. In the great dedicatory inscription of the temple of Edfu, built by Ptolemy XI,
4363	mention is made of a large number of fields.? In each case four linear dimensions are given,
4364	which we may call a, b, c and d, and the area is determined by the formula
4365	Area = ("+°) (°+")
4366	• Ancient Egypt, 1917, 100-102. * BRUGSCH, T'hesaurus, 531 #.
p. 98
4367	94 RHIND MATHEMATICAL PAPYRUS
4368	No. 51. where, presumably, a and c, b and d are pairs of opposite sides.! When the field is triangular
4369	the solution is obtained by making d equal to 0, i.e. regarding the triangle as a special case
4370	of a quadrilateral with one side zero.
4371	This method of measuring land is by no means unique; there is good evidence for
4372	believing that in the Ptolemaic, and Coptic periods it was the means by which land
4373	was measured in Egypt for purposes of taxation Now there are cases in these documents
4374	in which not only is one side zero, but the other pair of opposite sides is equal. The figure
4375	then becomes an isosceles triangle and its area is determined by the formula as ("+) (+°)
4376	or șab, i.e. half the base into one of the equal sides. Eisenlohr, who believes that the triangle
4377	in No. 51 is isosceles and that the mryt is one of the equal sides, maintains that the use of
4378	an identical formula in Egypt in later times bears out his claim. This is however hardly
4379	decisive. This method of field measuring was admittedly no more than an approximation for
4380	taxation purposes, and fractions less than zt of a square khet, sometimes even 3'½ of a square
4381	khet, were omitted. Such a method necessitated only the measuring by rope of such lines as
4382	were already present, namely the four sides, whereas a correct determination would have
4383	involved measuring a diagonal and two perpendiculars dropped on to it, a very much more
4384	complicated process. In all probability large numbers of fields were approximately rect-
4385	angular, in which case the error was small; and finally be it noted that the error was always
4386	in the tenant's favour, for the area of a quadrilateral field whose sides are a, b, c and d is
4387	==(ab. sinab), where the angle ab is that contained by the two sides a and b. This is
4388	obviously a maximum when the sines of ab, bc, cd, and da are all unity, ie. when the figure is
4389	a rectangle. Thus the formula Area = ("+°) (°+*), ie. † Eab, always gives a result smaller
4390	than the truth except in the case of a rectangle, when it is correct, or, in other words, the
4391	tenant neverlostby this rough system of measurement.
4392	In short we are hardly justified in arguing that, because the tax-gatherers of later Egypt
4393	reckoned quadrilaterals in a manner which disregarded the correct solution for the area of a
4394	triangle, the Egyptian mathematician of the Middle Kingdom was not acquainted with this
4395	correct solution. The question must be decided on other grounds than these.
4396	No. 52. No. 52. (PI. P.)
4397	"Example of reckoning a truncated triangle of land. If it is said to thee, A truncated
4398	triangle of land of 20 khet in its height, 6 khet in its base and 4 khet in the cut side. What
4399	is its acreage?
4400	You are to combine its base with the cut side: result 10. You are to take a half
4401	of 10, namely 5, in order to give its rectangle. You are to multiply 20 five times, result
4402	10 (sic). This is its area.
4403	The doing as it occurs:
4404	1000 2000
4405	500 2 4000
4406	-4 8000
4407	Total 10000, making in land 20 (read 10)
4408	This is its area in land."
4409	1 Simon in his Geschichte der Mathematik in Altertum, 47-8, has rightly. pointed out that in the later portion
4410	of this inscription, which was still unpublished when Eisenlohr wrote, this formula is in some cases not adhered
4411	to. He takes this to indicate that the Egyptians were themselves aware of its approximate nature, and in certain
4412	cases corrected the result. It would seem more likely that the variations from the formula are merely due to the
4413	inaccurate copying of a seribe or of a seulptor.
4414	2 See KENYON, Catalogue of Greek Pupyri in the Brit. Mus., II, 129 ff. (Pap. CCLXVII); GRENFELL, HuNT and
4415	Tebtunis Pupyri, Pt. I, 385 ff.; CRum, Coptic Ostraca, 42 ff., Ostracon D.12; HALL, Coptic and Greek Texts
4416	of the Christian Period in the Brit. Mus., 128, Coptic Ostracon 29750.
p. 99
4417	RHIND MATHEMATICAL PAPYRUS 95
4418	Theword!:k occurs in the Pyramid Texts, 673 c,' where it quite obviously means No. 52.
4419	"to cut off the tail," whence the determinative in our passage. The cutting line is clearly
4420	assumed to be parallel to the base.
4421	The word mryt is here rendered by the ambiguous "height." In the light of the dis-
4422	cussion on No. 51 it will be seen that it refers either to the vertical height of the figure, in
4423	which case the solution was obtained graphically, see Fig. 5, p. 92, and is correct, even if
4424	the triangle be not isosceles, or else to the slant height, in which case the triangle was clearly
4425	envisaged as isosceles, and the solution is wrong.
4426	Two sets of working are given, the first without heading and the second under the
4427	title of irt mi lpr. In the first the unit is the khet. The two parallel sides are added and
4428	give 10 khet. This is now halved (5), and the result multiplied by the mryt in khet, namely
4429	20. The result should be 100 square khet, but it is mentally divided by 10 in order to reduce
4430	it to thousands-of-land, which is the form in which the answer is expected in these sums.
4431	In the second working the units are badly muddled. First 10 khel, the sum of the two
4432	parallel sides, is reduced to 1000 cubits and halved, giving 500 cubits. Then the 20 khet of
4433	the mryt are reduced to 2000 cubits and multiplied, not, as one would expect, by the 500
4434	cubits, but by these turned back into 5 khet. This has the advantage of giving the result
4435	direct in cubits-of-land, ie. in hundredths of square khet, and a division by 1000 reduces these
4436	to 10 thousands-of-land (wrongly written 20). The confusion of units here is doubtless entirely
4437	due the desire using square khet, the Egyptian land-measurers preferring for
4438	practical use the cubit-of-land and the thousand-of-land.
4439	No. 53. (PI. P.)
4440	No. 58.
4441	4y setat
4442	-2 9 setat
4443	2} setat
4444	ly setat
4445	Total (5)+1) setat
4446	To of it is (1) + 1) setat + 10 cubits-of-landº
4447	Tu of it subtracted, then this (?) is the area.
4448	1 7 setat
4449	1 thousand-of-land and 4 setat
4450	3 setat
4451	15+)setat
4452	Total 1 thousand-of-land and (5} + }) setat
4453	(7} + 7 + 1) setat.
4454	It is hardly worth while to spend much time on a problem which is clearly incomplete
4455	and incorrect. All that is to be made out is that the second calculation is the finding of the
4456	area of the small triangle whose base and height* are marked in the figure as 2½ and 7 re-
4457	spectively. In this problem the multiplier 1 is actually written as if it were setat (see p. 89),
4458	while there is continual confusion between thousands-of-land, which should be shown in free-
4459	standing units, and setat, which should be placed beneath the rectangle sign.
4460	The first calculation is hopeless, and appears neither to have a meaning in itself nor
4461	to bear any relation to the figure. It begins by multiplying 4} by If, correctly, despite
4462	inaccurate ticking and the introduction of an unnecessary step. The next lines are totally
4463	unconnected with this, and clearly come from some other reckoning. Possibly something has
4464	" I owe the reference to
4465	≥ The sign, which exactly resembles the hieratie for 30, clearly means 10 cubits-of-land from No. 54. It is
4466	perhaps very cursively made in both cases.
4467	3 For the ambiguity in this term see Nos. 51 and 52.
p. 100
4468	96 RHIND MATHEMATICAL PAPYRUS
4469	been omitted by haplography. There may even have been confusion with No. 54, where 7
4470	setat and the number 10 also occur.
4471	For the dimidiated parts of the setat here used see Introduction, pp. 24-5.
4472	No. 54. No. 54./ (PI. P.)
4473	"To divide (7 setat) of land into 10 fields.
4474	10
4475	(1 + #) setat + 7} cubits-of-land
4476	-2 (1# + #) setat + 2, cubits-of-land
4477	(2%+ [*]) setat + 5 cubits-of-land
4478	5} setat + 10 cubits-of-land."
4479	The problem, as is clear from the working, is to divide 7 setat of land (unfortunately
4480	omitted by the scribe in line 1) into 10 fields, for "fields" must be the meaning of the second
4481	3ht. The method is to divide 7 by 10 (see, however, notes on No. 55), result } + *.
4482	quantity is then expressed in setat and cubits-of-land, multiplied by 10, and shown (though
4483	the actual addition is missing) to yield 7 setat.
4484	Note that in the first step, division of 7 by 10, no units are mentioned: contrast No. 55,
4485	and see commentary there.
4486	The notation of the setat here used is perfectly regular, see pp. 24-5. The special hieratic
4487	signs are used for the t, the f and the } of the setat, which, with i, and as, are the only
4488	fractions permitted, odd amounts being entered in cubits-of-land, expressed by placing the
4489	number under the arm or cubit-sign. Ten cubits-of-land has, however, a special sign, precisely
4490	like the hieratic for 30 (cf. No. 53), but perhaps in reality a cursive writing of a 10 under an
4491	arm: half a cubit-of-land is shown in hieratic by a sign equivalent to that used for the pure
4492	No. 55. No. 55. (PI. P.).
4493	" To divide 3 setat of land into 5 fields. You are to operate on 5 setat (sic) to find
4494	three setat of land.
4495	1 5
4496	sic 1
4497	siC TT
4498	*t Tü results.
4499	You are to multiply } + to five times:
4500	} setat + 10 cubits-of-land
4501	1% setat + 7} cubits-of-land
4502	24 + # setat + 2j cubits-of-land
4503	Thus you find the acreage to be 3 setat."
4504	The problem is exactly similar to the last. In the setting out the scribe has written
4505	"1 setat" instead of the simple numeral 5, which it remotely resembles. There is considerable
4506	confusion of units and dimensions. A. modern worker would divide 3 setat by 5 and get
4507	his answer direct in setat. The Egyptian here quite illogically divides 3 setat by 5 setat
4508	and gets his answer as a pure fraction. Yet there is a reason for this. The very nature of
4509	Egyptian division makes it impossible to obtain the quotient otherwise than in the form of
4510	pure number, for it is obtained by adding together certain of the trial multipliers on the left,
4511	which can only be pure numbers, not weights or lengths. Thus if we wish to divide 3 square
p. 101
4512	RHIND MATHEMATICAL PAPYRUS 97
4513	miles by 5 we can do so directly and get the answer in acres, square poles, square yards, No. 55.
4514	square feet, etc. An Egyptian has no direct process for doing this; he can only divide 3
4515	by 5, giving } + 1a and turn this afterwards into acres, square poles, square yards, etc.
4516	Thus in the present case an Egyptian could not mark t ili as setat, because the setat-
4517	notation does not recognize such a quantity; it must be expressed as b setat plus 10 cubits-of-
4518	land. Similarly when in No. 54 % + . obtained as a pure number,
4519	into setat it must be turned into the
4520	it would have to be shown in the Horus-eye notation. The 2 is allowed to stand, being one
4521	of the fractions for which a sign exists,
4522	3 setat, which is 121 cubits-of-land, plus 71, cubits-of-land. Egyptian cannot write } setat any
4523	more than it can write hekat.
4524	The use of the verb libi here and in No. 54 is unusual. This verb followed by the
4525	preposition lent generally means "to take "subtract from" (Nos. 43, 50 and 64).
4526	Yet there seems no way of avoiding the conclusion that it here refers to division.
4527	PART III. ANGLE OF SLOPE OF PYRAMIDS, &c. Nos. 56-60.
4528	(Pl. Q.) No. 56.
4529	"Example of reckoning out a pyramid 360 in length of side and 250 in its vertical
4530	height. Let me know its batter.
4531	You are to take half of 360: it becomes 180.
4532	You are to reckon with 250 to find 180.
4533	Result } + 1 + sn of a cubit.
4534	A cubit being 7 palms, you are to multiply by 7:
4535	7
4536	13+t5
4537	in+ zs
4538	Its batter is 5g's palms."
4539	The meaning of this and the following examples depends entirely on the interpretation
4540	given to the three terms whi-tot, pr-m-ws and śkd. It is fairly obvious from the figure that
4541	the wh:-tbt is a ground measurement, and if so it can hardly be other than the diagonal or
4542	side of the base. Similarly pr-m-uś is clearly a measurement of
4543	height, not necessarily vertical, and as such it might be either the
4544	, taity es rlatie ai tueopamest ao tul i
4545	wle:-tbt. See Fig. 6.
4546	Eisenlohr took the wh:-tbt to be the diagonal of the base,
4547	PQ, and the pr-m-ws to be the slant height from a corner to G
4548	the apex, DP; the śkd would then be the cosine of the angle
4549	DPQ made by the edge with the diagonal of the base. In support Fig. 6.
4550	of this interpretation he argued that it gives a batter which agrees
4551	well with that of certain existing pyramids and secondly that it is unlikely that wh;-tbt and
4552	pr-m-vś should mean the same as snti and l:in lıno of No. 60, which he held to be beyond
4553	all doubt the length of the side of the base and the vertical height.
4554	certainly erroneous. In the first place the monument dealt with in
4555	No. 60 is not a pyramid but an iwn, and there is no reason at all why similar measure-
4556	ments in this and a pyramid should not have had totally different technical names.
4557	1 It was first attacked by Borchardt in Ä.Z., 31, 9 f.
p. 102
4558	98 RHIND MATHEMATICAL PAPYRUS
4559	place Eisenlohr's interpretation ives for the skd a measurement tor which i
4560	is impossible to see any practical use. he Egyptians always thought concretely, even whe
4561	mathematics, the clue to the understanding of this problem certainly lies in
4562	the true significance of the reduction of the answer to palms per cubit.
4563	provide the mason with a simple practical rule for dressing at the required angle the
4564	the outer facing of the pyramid. All he has to do in this case, when confronted
4565	with a solid block of stone, parallelopipedal in shape, is to measure one cubit upwards
4566	on the outer edge, and then 5, palms inwards at right-angles. The line joining the point
4567	to the point from which he started gives the correct angle of dressing. Of the
4568	outer blocks of a pyramid only a very small minority lie on the slant edges, while the vast
4569	lie in the sloping sides. The figure needed by a mason would thus be not the
4570	angle of the slant edges given by Eisenlohr's interpretation, but that of the sides.
4571	more, the slant edges of a pyramid are in actual building merely determined secondarily as
4572	the intersections of the sides.
4573	We can make the śkd correspond with this ratio if we take the pr-m-wś to be the
4574	vertical height DG, and the whi-tbt to be a side of the base, for the angle of slope (DÑG)
4575	of a side QDR is one whose cotangent is half a side of the base divided by the vertical
4576	GF
4577	height, ie. GD
4578	No one who has grasped the essentially practical nature of all Egyptian mathematics
4579	will doubt the accuracy of the above solution. Unfortunately no corroboration can be obtained
4580	from the names of the measurements themselves. The word wl:-tbt, a compound formed of
4581	the verb why; "to seek" and the noun țbt "a sandal," should refer to a ground measure-
4582	ment, but that is as far as we can go. The compound pr-m-ws, perhaps the origin of mupauís,
4583	has by some been taken to mean "that which comes forth from the saw," ie. a measurement
4584	which can only be seen in section. If this were the case the word wś could hardly have been
4585	written without the saw or knife-determinative. The house-sign after wś seems to be needed
4586	as a determinative to ws itself and can hardly apply to the whole compound pr-m-ws.
4587	therefore should be some kind of building, but I can find no examples of its use.
4588	The word used for batter is equally obscure. If the s is causative it may be formed
4589	from kd "to build" or "form," and it would be one of those cases where the addition of
4590	the causative prefix barely alters the meaning. "that which forms,"
4591	i.e. the measurement which builds up the pyramid, as indeed it is, for, the base once marked
4592	the laying of outer blocks accurately cut to the slid determines the structure.
4593	this case we should rather expect with śkd the sign of the man building and not merely the
4594	abstract determinative.
4595	The method of this problem needs little comment. Note that the dimensions 360 and 250
4596	are given in no particular unit. The Egyptian was doubtless aware that the measurement he
4597	proposed to find, being a ratio and not a length, was independent of the unit of the original
4598	dimensions.
4599	The 360 is halved, result 180, and this last is then divided by 250. The result is then
4600	illogically stated to be 1+1+ of one cubit, instead ofthe pure number+3
4601	Here the Egyptian really means to say that 180 is to 250 in the same proportion that 1 + }
4602	+ su of a cubit is to a cubit, or in other words that the angle determined by the base 180
4603	and perpendicular 250 can also be determined by base (2 + } + 'v) cubit and perpendicular
4604	He has introduced the 1 cubit simply in order to reduce his result to a practical
4605	form for the use of the mason.' The sum is made still more practical by the reduction of the
4606	1 + 3 + su cubit to palms, namely 53,.
4607	1 For a most interesting practical application of this śled-value in the building of a mastaba see PETRIE,
4608	Medum, PI. VIII, and text thereto. There is some admirable material relevant to these problems in BORCHARDT,
4609	Gegen die Zahlenmystik an der grossen Pyramide bei Gise; Berlin, 1922.
p. 103
4610	RHIND MATHEMATICAL PAPYRUS 99
4611	No. 57. (Pl. Q.) No. 57.
4612	"A pyramid 140 in length of side, and 5 palms and a finger in its batter. What is
4613	the vertical height thereof?
4614	You are to divide one cubit by the batter doubled, which amounts to 10g' You are
4615	to reckon with 10% to find 7, for this is one cubit.
4616	Reckon with 10%: two-thirds of 10g is 7.
4617	You are now to reckon with 140, for this is the length of the side:
4618	Make two-thirds of 140, namely 93%. This is the vertical height thereof."
4619	In this example we are given the length of the side and the amount of the batter in
4620	palms and fingers (a finger being one-fourth palm) per vertical cubit. We are asked to
4621	find the vertical height.
4622	The working, though correct, is slightly obscured to our modern way of thinking by
4623	the fact that, instead of finding EG (see Fig. 6, p. 97) from the datum EF by halving, and
4624	then determining DG by means of the batter, the batter is doubled (twice 57 palms = 10g)
4625	and the proportion used is:—
4626	DG : EF : 7 : 10]
4627	There is an admirable parallel to this illogical style of working in No. 45.
4628	58. (Pl. Q.)
4629	"A pyramid whose vertical height is 93}. Let me know its batter, 140 being the length
4630	of its side.
4631	You are to take half of 140, namely 70. You are now to reckon with 93% to find 70.
4632	Reckon with 93}: its half is 463,
4633	its quarter is 23g.
4634	You are to make ‡ + i of a cubit. Reckon with 7; its half is 3}; its quarter is 1$ + 4;
4635	total 5 palms 1 finger. This is its batter.
4636	Working out:
4637	1 931,
4638	463
4639	23)
4640	You are to make } + ‡ of a cubit.
4641	Now a cubit is seven palms.
4642	1$(read 1t + +)
4643	Total 5 palms 1 finger.
4644	This is the batter."
4645	This example is concerned with the same numbers as the last, but here we are given
4646	the length of the side and the height and are asked to find the batter. The method is logical
4647	and consists simply in halving the side and dividing the resulting 70 by the height 93}. The
4648	numbers are suitably chosen, for 70 is precisely } or (} + ł) of 93J. It then only remains to
4649	reduce 1 + 7 of a cubit to palms and fingers, which is done by multiplying 7 by it. Result
4650	5} palms or 5 palms 1 finger.
4651	ino ir ml 1 šsp 4 pw. The position of iw in front of ir is interesting syntactically. Cf.
4652	ive ir didit lir nb dbn in No. 62. Gunn quotes also Urk., IV, 366, 13.
4653	The traces after the numeral do not suit (for its use in the Nominal Sentence with
4654	pw cf. Nos. 64, 70 and 71 and notes), nor yet read by Borchardt.
4655	• Literally "10} results," ie. from the doubling, nut the division. The translation given avoids the ambiguity.
4656	02
p. 104
4657	100 RHIND MATHEMATICAL PAPYRUS
4658	No. 59. No. 59. (PI. Q.)
4659	"A pyramid the vertical height(sic) whereof is 12 and the side(sic) 8. (Find its batter.)
4660	You are to reckon with 8 to find 6, for this is half the height.
4661	1 8
4662	4
4663	2
4664	You are to take a half and a quarter of 7, for this is 1 cubit.
4665	7
4666	It comes to 5 palms 1 finger. Behold this is its batter.
4667	What.....?"
4668	No. 59b. No. 59B. (Pl. Q.)
4669	"You are to reckon a pyramid of 12 (sic), whose batter is 5 palms 1 finger. Let ‹me) know
4670	the height thereof.
4671	to 10%. Two-thirds of it is 7.
4672	Nos. 59 and 59B are two quite distinct problems, the second of which is the reverse
4673	of the first. Unfortunately the scribe did not realize this, and has made No. 59B appear as
4674	part of the working of No. 59 by introducing it by the word ir-lr•k and failing to use red
4675	ink for the opening word or words. The verbal form śdm.lyr-k is never used in this papyrus
4676	to introduce a problem. In the original from which our scribe copied the problem was probably,
4677	like its fellows, introduced directly by mr "a pyramid." It is possible that this was in black
4678	instead of being, as it should be, in red. We know enough of our scribe's intelligence to assert
4679	that this would be quite sufficient to conceal from him the fact that a fresh problem had
4680	In No. 59 there is an unfortunate error in the setting out, forthe side and the height
4681	have been transposed. In order to give a batter of 5 palms 1 finger it is the height which
4682	must be 8 and the side 12. Otherwise the batter would be only 2} spans.
4683	phneo which mh tịn phoen o ngo g. t,
4684	In No. 59B we are given the side and batter and asked to find the height. After the
4685	words "a pyramid of 12" we expect m wh;-tbt.f "in its side"; but it is quite possible that
4686	the Eoyptian is sufficient as it stands. and that "a pyramid of 12" was the technical
4687	expression for "a pyramid built on a base 12 square."
4688	No. 60. No. 60. (PI. R.)
4689	"A cone (?) of 15 cubits in its base and 30 in its height. Let me know its batter.
4690	Reckon with 15: its half is 71.
4691	Multiply 7, four times (sic) to find 30:
4692	The result is 4. This is the batter thereof.
4693	Working :
4694	1 15
4695	7%
4696	15
4697	30 "
p. 105
4698	RHIND MATHEMATICAL PAPYRUS 101
4699	Here we are once more asked to find the batter of a certain structure, but it is no longer No. 60.
4700	an ordinary pyramid. It is described as an l' iwn. The sign i means and doubtless
4701	originally depicted a column or pillar, but the house-determinative which here accompanies
4702	it makes it clear that here a structure, or at any rate a solid figure and not a column, is meant.
4703	It is difficult to sayhowmuch shouldbe placed on the illustration, even supposing
4704	that our scribe has copied it accurately from his original. It is a plain triangle, and as such
4705	forms a contrast to the figures of pyramids above, all of which have the low rectangular base
4706	typical of the pyramid word-sign in the hieroglyphs. This figure suggests several possibilities,
4707	in addition to that of a triangle, which we may obviously discard. They are a cone, a prism
4708	isosceles in section and lying on its unequal face, and a pyramidal structure other and smaller
4709	than a royal tomb,' for this last alternative is not to be neglected.
4710	Do the other technical terms used help us to decide between these alternatives? śntt
4711	means literally "the ground-plan" or "base," and it is clear that the base of our figure
4712	be determined by a single measurement. This is eminently true of the cone, but it is
4713	also true of a pyramid, and even of a prism laid on its unequal face, the batter of the two
4714	equal faces depending only on the form of its triangular section, which is of course independent
4715	of the length of the prism as it lies. The other dimension k:l n lrw means quite literally
4716	"height of top," which can surely be nothing but the vertical height. This measurement
4717	again might apply equally well to any of the three alternatives.
4718	Nor do the uses of the word iwn help us much. It occurs several times with the pyramid
4719	determinative for "heaps" of slain foes." This use is doubtless connected with that which
4720	we have here," but it would not be easy to say whether this favours the cone, the pyramid,
4721	or the prism; perhaps the picture is less well suited to the last than to the two first. In
4722	view of this fact, and remembering the literal applicability of the base determined by a single
4723	dimension to the cone and pyramid, I am inclined to think one of the two latter more probable
4724	than the prism. As between the two the fact that pyramids have already been dealt with
4725	tells slightly but not quite decisively in favour of the cone.
4726	The working out begins in a similar manner to that of the pyramid problems. We halve
4727	the śntt and get 7%. We ought now to divide this by the height, 30, from which we should
4728	get the answer 1 cubit or 17 palms. But instead of this the scribe divides the 30 by the
4729	There is clearly something wrong here and
4730	a close examination of the text shows that a serious confusion has taken place. The words
4731	The first part can only mean "Reckon 7½ four times to find 30"; then if we omit the signs
4732	(P" * the rest of the sentence will mean "It becomes 4. This is its batter" Now
4733	it is clear in the first place that the words śpw 4 are not needed, for it is only after
4734	we have operated on 7; to find 30 that we find the required multiplier to be 4. This
4735	however is a small error. The real question to be decided is the meaning of \|"*.
4736	Von Calice,* reading śtrly for śtwty, see No. 46, takes st as the 3rd Person Singular Neuter
4737	ending to lypr and translates rly as "that vertical height in which the slanting side diverges
4738	by one cubit from the vertical." Both Borchardt" and Schack-Schackenburg® have seen the
4739	The latter points out that not the Neuter lypr-s but the Masculine lpr•f is
4740	used in mathematics to express result, and that śt must therefore be taken with rly; in fact
4741	we have here nothing but the technical term śtrhy (śtwty) used in Nos. 45 and 46 for "content."
4742	He therefore proposes to delete the word śtrly•f, which he thinks has been wrongly introduced
4743	" That this is what ix intended by m in Nos. 56-59 is clear from the size of the dimensions.
4744	* Pap. Harris I, 77, 3; DümıcHex, Historische Inschriften, 1, 18.
4745	3 See, however, Sethe in BorcharDr, Grabdenkmal des Sahure, Band II, Text, 81, note 4.
4746	+ .I.Z., 10, 117. * t.Z., 31, 13. * I.Z., 41, 77-8.
p. 106
4747	102 RHIND MATHEMATICAL PAPYRUS
4748	No. 60. from one of the examples in which it occurs. There can be no doubt that this view is correct.
4749	delete the ko from the text, for though the form lpr
4750	m X "it becomes X" the Berlin Papyrus 6619 and is grammatically possible,
4751	hpr being a śdm-f-form without ending, the form lpr.f m X is usual in our papyrus.
4752	further argument against the retaining of the text as it stands is the fact that if rly or even
4753	vertical height in which the sloping side diverges from the vertical by a
4754	cubit, it would be nonsense to add, as the papyrus does, "This is its śkd," for the śkd is the
4755	reverse of this measurement, namely the divergence from the vertical in a vertical height of
4756	useless to attempt to defend the text as it stands.
4757	no place here and must have come from some other problem, and the amount of contamination
4758	cannot be determined.? It was at least sufficient to mislead the scribe into dividing 30 by 7}
4759	to prevent his attempting to express his result in palms, as should
4760	be done in the case of a batter. The problem affords no case for the belief that the batter
4761	was in some cases measured by the tangent instead of the cotangent of the base angle.
4762	1 And perhaps below in No. 62, see p. 14, note 5.
4763	somewhat sie ma y be cenduton bet bei thoe ihe tof caicd ty and in, thichocor y nlem cien, guph, 1 ah, "content," and the
p. 107
4764	RHIND MATHEMATICAL PAPYRUS 103
4765	BOOK III. MISCELLANEOUS PROBLEMS.
4766	No. G1. (PI. R.) No. 61.
4767	Line 1 3ot3istt1
4768	2 * of f is " + Ix
4769	3 * of \| is ¡ +I*
4770	4 3 of 1 is 1e + 3a
4771	5 3ofsis 1
4772	1 of ! is 1
4773	7 1 of } is 12
4774	8 t'= of ! is dt
4775	9 1 of 3 is t's+s* 1 3 of it is tx[+ **]
4776	10 [1?
4777	11 [1?
4778	12 [t. 3 of it is tu +3u]?
4779	13 [s, → of it is ta]?
4780	14 Lt= of it is 1o]?
4781	15 [b] # of it is zu
4782	16 t, } (of it) is t* + Fiz B of it is ar]?
4783	17 %, + of it is te Et of it is aa]?
4784	18 Tr, 3 of it is de tit } of it is #5
4785	19 Tr, 1 of it is 3s #of it is *t
4786	No. 61 brings us over on to the verso of Pap. 10058. It is a table of multiplication
4787	of fractions. That it is not part of the original plan of the treatise is evident from the fact
4788	thai it lies between Book II, Mensuration, and Book III, Miscellaneous Problems, that it lies
4789	outside the double vertical ruling, in a margin obviously intended to be left blank, and that
4790	it is very carelessly written. It has in fact been placed here by the scribe in order that it
4791	might be in an accessible spot for reference when needed.
4792	Its main interest lies in the fact that, though a single table, it contains two different
4793	forms of statement. Thus in lines 1-4 we find the form ÷ of ; is \| + , while in lines 15-19
4794	we find the form 1, I of it is if. In line 9 the statement is made twice, once in the second
4795	form and once in the margin in the first form, while in lines 5-8 the second form seems
4796	originally to have stood but to have been altered afterwards to the first.
4797	These variations have an interesting significance which has not been insisted on by the
4798	commentators. An Egyptian cannot take one-ninth of }: he can take one-third of any quantity
4799	by simply taking two-thirds and halving it, but he cannot obtain one-ninth direct from one-third,
4800	for he cannot divide by 3, only by 2. The consequence is that to speak of taking one-ninth
4801	is technically incorrect, and the Egyptian should avoid the use of the phrase even in a table
4802	of results. Thus in line 9 we find in the column the correct form of statement: "One-ninth,
4803	} of it is i" + »" The less correct " One-ninth of } is î's +»" being added in the margin
4804	owing to an error explained below. It would seem that the succeeding lines of the table
4805	were written in the correct form, but without the marginal addition.
4806	In lines 5-8 the position is very curious. The scribe would seem to have written
4807	originally }, ½ of it is }, and so on, though the other form of statement, that used in lines
4808	1-4,would havebeen quiteunexceptionable, theonlymultipliers involved being 3 and , both
4809	of which are legitimate. Perceiving this he seems to have altered these four lines by deleting,
4810	veryperfunctorily, the zo's and adding mn's. In the excess of his zeal he has altered
p. 108
4811	104 RHIND MATHEMATICAL PAPYRUS
4812	ine 9 in a similar manner, writing it, however, afresh in the margin, though this line should
4813	really have been left as it was. The point may seem a small one, but it has a real signiticance
4814	for the study of Egyptian mathematies.
4815	The gap in the middle of the table probably contained five lines. Of these the first two or
4816	three doubtless gave fractions of one-ninth and the remainder hegan the fractions of one-fifth.
4817	With this table should he compared the very similar but more complete table of fractions
4818	of Byzantine date published by Thompson.! The tablet which remains gives in fractional form
4819	the fifteenth part of all whole numbers from 1 to 15, and the sixteenth part of all whole
4820	numbers from 1 to 16.
4821	No. 61b. No. 61B. (PI. R.)
4822	"To make two-thirds of an aliquot part. If it is said to thee, What is two-thirds of b,
4823	you are to make its double and its six times: that is two-thirds of it.
4824	likewise in the case of any aliquot part which may occur."
4825	The translation of tit gbt or tiit gbt is fixed by the working. To find two-thirds of x
4826	we take twice x and six times x and presumably add them. Clearly this is a short statement
4827	of a practical rule for multiplication by }, for since } = } + ¿ all we have to do is to multiply
4828	5, the denominator of our fraction, by 2 and then by 6, invert the results and add. Thus
4829	A lit got is thus an aliquot part.? The word tit means a "sign " or "figure" of something.
4830	gbi means "to be weak," and the compound must be "a weak sign." Why this
4831	should be the technical term for an aliquot part it is not easy to see, unless a weak sign is
4832	one which is placed beneath the fractional sign, or "inverted" as we now say. The word
4833	tist also occurs in Pap. Kahun, Plate VIII, l. 50, in a mathematical technical sense."
4834	ti;t is subtracted, remainder 11." Here the number from which it is subtracted is apparently
4835	though in the obscurity of the passage this is not quite certain, and in this case it is
4836	hard to see why the writer did not merely say "Subtract 1." According to Maspero's inter-
4837	pretation * of the problem the 12 are the 12 months of a year, in which case tiit would actually
4838	stand for a month! In fact, so uncertain is the meaning of the passage that it is impossible
4839	to draw from it any conclusion as to the meaning of tiit.
4840	The existence of this rule in the papyrus is not without interest. We have seen that
4841	the Egyptian regarded * as an aliquot part and was apparently able to take two-thirds of
4842	an integral number by a single process. Moreover, his method of finding } and f was by
4843	halving and re-halving 3. Even in the treatment of fractional quantities the same was the
4844	case, } always being found as a step towards } and 6, as for instance in the table of this
4845	example (No. 61). It is for this very reason that in the table of the division of 2, 3 was
4846	never treated, though it might at once have been resolved into \| + %. It is therefore
4847	interesting to find that when a fraction had to be dealt with the equation =+ was
4848	brought into use.
4849	Note that this is the sole instance in our papyrus of a general rule, except perhaps No. 66.
4850	No. 62. No. 62. (PI. R.)
4851	A bag ins ampl af eekon alva lang cond. thiv hag hpsecien motgls for 3t it nas: to at i,
4852	assignable to each precious metal ?
4853	1 See above, p. 8.
4854	2 In No. 70 fi, by itself seems to have the same meaning. Sinee only aliquot parts could be written, the
4855	word ought to mean any written fraction.
4856	* GRIFFITH, P.K., Text, 18. 4 Op. cit., 101.
4857	• Or, with Gardiner (see below). "What is the amount of."
p. 109
4858	RHIND MATHEMATICAL PAPYRUS 105
4859	Now what is given for a deben of gold is 12 rings, for silver 6 rings, and for a deben No. 62.
4860	You are to add together that which is given for a ring (sic, read deben) of
4861	each precious metal; result 21. You are to reckon with this 21 to find 84 rings, for that is
4862	what has been bought in this bag. It comes to 4, which you assign to each metal.
4863	The doing as it actually occurs:
4864	4 is multiplied (?) twelve times; the gold turns out to be 48. This is its amount.
4865	six times ; the silver turns out tobe 24.
4866	three times; the lead turns out to be 12
4867	twenty-one times Total 84."
4868	There is little difficulty in getting the correct mathematical drift of this problem. Eisen-
4869	lohr did it in 1877. But it may be doubted whether even in 1923 it is possible to give a
4870	final and absolutely certain translation.
4871	The problem is that a bag, or
4872	ever stated in so many words) of gold, silver and lead. The total value of the bag is 8
4873	rings, and we are also given the price per deben of each metal.
4874	each metal in the bag. Answer, 4 deben.
4875	The working explains itself and is perfectly modern in type. The values in rings of a
4876	deben of each metal are added together and come to 21. Thus if the bag contained 1 deben
4877	of each it would be worth 21 rings. But in reality it is worth 84. Therefore the number of
4878	deben of each metal must be 4.
4879	The clue to correct translation. lies in realizing, as Gardiner was the first to do, that,
4880	inthe phrase "Add together that which is given for a ring of each metal," the word ring
4881	is a mistake for deben. To keep "ring" and translate " Add up that which is given in rings
4882	for each metal" involves two errors. In the first place, the Egyptian for "in rings" is
4883	not lir šty but m šty, and, in the second, the n written in hieratic without a dot over it is
4884	the Genitive Exponent "of" or "belonging to," not the preposition "for," which has the dot.
4885	It is true that the papyrus is not fully consistent in the matter (see Nos. 39 and 40), but in
4886	the present example the distinction is clearly made. Moreover, in the phrase hpr-lr m 4 didi•k
4887	n it nbt it is to be noted that didi.k is Masculine, not Neuter like didi-t above: it
4888	must therefore refer to the numeral 4 and be the Relative Form in its rather uncommon
4889	continuative sense," " The result is 4, and this is what you are to attribute to each metal."
4890	Were the meaning of this last phrase more concrete, "what you are to put in (the bag) of
4891	each metal," we should expect m 'it nbt rather than n. There is an exact parallel in the
4892	• Moscow Papyrus, where the former translation gives good sense and the latter no sense at all.
4893	Coming now to points of detail, the word krft is found again in Pap. Ebers, 53, 12-14,
4894	where it is clearly some kind of cloth bag in which some portion of the date fruit is placed
4895	in order to be soaked and boiled. The meaning "bag"seems required in our passage. The
4896	Kalenderische
4897	from the same root, need not have
4898	the same meaning.
4899	For in meaning "to buy" see the Old Kingdom sale of a house quoted below; also
4900	Gardiner's note Ä.Z., 43, 34.
4901	For the position of le in inu ir didit lir nb dbn compare No.58 and note there.
4902	In the working-out portion the papyrus is much damaged, and has been patched, in
4903	ancient times, by someone who either knew roughly what ought to stand in the gaps or who
4904	had actually the broken fragments before him. The former is the more likely alternative, for
4905	the mender has failed to supply the opening words of No. 63, and, what is more,
4906	been unable to complete the beginning of line 9 in the present example. This as it stands
4907	1 Ä.Z., 43, 46-7. = See, however, p. 14, note 5.
p. 110
4908	106 RHIND MATHEMATICAL PAPYRUS
4909	is a puzzle. We expect ir-lr-k w:l! tp m "You are to multiply 4 twelve times..." Yet
4910	this is not what stood there, for under the ex is the top of a sign which from its shape
4911	and position can only be o. The next group is a clear O. This is followed by a blank
4912	due to a patch extending vertically into the lines above: the small black trace shown here
4913	in the facsimile is non-existent in the original. In the blank space we can hardly read any-
4914	thing but e. We thus get the phrase irt pw which we have already met in No. 38, and in
4915	both cases the context is the same, irt pa X r spu Y. At the same time the explanation
4916	attempted of the phrase in No. 38 will hardly apply here, for there the words were used in
4917	excuse or justification of an unusually bold step, whereas here they introduce the irt mi lpr
4918	(here practically constituting a proof) and take the place of the more usual ir lr•k X r spw Y.
4919	Possibly the explanation of irt pu as a passive form of the well-known śdm•f pw suggested on
4920	"It (namely the result 4 obtained above) means that 4 must be
4921	multiplied by 12, 6 and 3 respectively to get the required values." This, however, is merely
4922	conjecture.
4923	The main interest of the problem lies in the evidence it gives of the existence of a
4924	system of exchange based on the value of "rings" made of various metals. The word for
4925	"rings" is here written 7, or 2e7i Its phonetic reading is almost certainly given
4926	by the writing "B, in line 3. This was decomposed by Grifith' into two separate
4927	words " and f T. He supposed that ity denoted generally the goods to be bought,
4928	or that it might be a real or imaginary substance used as a common measure for the debens
4929	of all the metals. His translation was "84 pieces of shati." This separation of the signs would
4930	leave šty without a determinative, which, as it is an uncommon word, is improbable, and
4931	occurs once again in Egyptian literature, in an Old Kingdom inscription* recording the sale
4932	of a house and some of its fittings or effects. There it is written -, and followed by a sign
4933	which is either a form of the old determinative of metal or an actual pictogram of a ring.*
4934	Sethe took this word to be st, the well-known word for loaves or cakes, but as a house would
4935	hardly be sold for 10 cakes he had to assume that 10 measures of cakes were intended, which
4936	is not probable.
4937	Whatever view we may adopt with regard to the reading of the word it is clear that
4938	the § was a unit of some kind whereby the values of various objects could be compared
4939	and exchanges made. In this case it is probable that the word-sign represents the actual
4940	unit, and if so it may originally have been not the picture of the seal-stone but a ring of " or ie lor te boni noe tho pictuo
4941	sign used in this problem for Q is in reality a mistake for e"
4942	Now a unit written QI=, determined by the weight sign, has long been known from
4943	account papyri of the New Empire. Thus Griffith gives instances from Pap. Bulag Il of
4944	"rings" of gold and of silver, in which the values of certain commodities are expressed. "Half-
4945	rings " are also used, as well as "rings" simply, which comparisons show to have been silver.®
4946	Dynasty from Kahun, now in the Berlin Museum, and shown that the
4947	• F.S.B.A., XIV, 436-9.
4948	2 Suggested by Gardiner, A.Z., 43, 47.
4949	3 SETHE, Aegyptische Inschrift auf den Verkauf eines Hauses; SorTAs, Élude critigne sur un ucte de vente
4950	immobilière; cf. Chassinat in Recueil de Travaux, 39, 79-88; BIsSINg, Ein Hauskauf im IV Jahrtausend vor
4951	Chr. (Sitzungsb. der Bayerisclen Akad. d. Wiss., Philos.-philolog.-hist. Kl., 1920).
4952	* It differs, however, from the seal-stone sign which oceurs elsewhere in the inscription.
4953	• Hieratische Paläcgraphie, I, 40, note 1.
4954	• See SPIEGELBErG, Rechnungen aus der Zeit Setis 1, 89 #.
p. 111
4955	RHIND MATHEMATICAL PAPYRUS 107
4956	these papyri was a weight equivalent to one-twelfth of a deben. Comparing this with the No. 62.
4957	Rhind example, where "what is given for a deben of gold is 12 rings," he concluded that the
4958	rings referred to in the Berlin papyri were of gold.
4959	Thus it is clear that in the New Kingdom a regular currency in "rings" of silver and
4960	of gold, more particularly the latter, had become usual in Egypt. Since, however, the "ring"
4961	was clearly a weight we must not
4962	objects of gold, which would almost have constituted a coinage even if not inscribed. The
4963	Rhind papyrus doubtless takes us a stage farther back towards the origin of this system, to
4964	a time when the "ring" was purely a weight and had not associated itself with any particular
4965	metal.
4966	No. 63. (Pl. S.)
4967	"[Example of dividing] 700 loaves among 4 men, } to one, ½ to another, [} to another,
4968	and f to another]. Let me know the share of each of them.
4969	You are to add together }, ($.) } and ‡; result 1} + $. You are to divide 1 by 1} + ‡;
4970	result } + T. You are to take }+ T of 700, namely 400. You are to take } of 400, which
4971	is 266}; } of 400, which is 200; } of 400, which is 133}; and ‡ of 400, which is 100. These
4972	are the shares of each man among them.
4973	The doing as it occurs:
4974	Number, 700
4975	* + ** is 400
4976	of 400 to one: 2663
4977	} of 400 to another : 200
4978	} of 400 to another: 133}
4979	1 of 400 to another : 100
4980	Total 700 "
4981	At the beginning of this problem and the end of the last the papyrus has been torn
4982	and a patch placed over the gap. Parts of two lines are lost, and the mender seems also to
4983	have copied on to the new piece as best he could the signs of a third line, which was still
4984	present but which he was forced to cover up in order to get an overlap for his pateh. The
4985	first line should probably be restored tp n psš: Eisenlohr's tp n irt is not long enough to fill
4986	the space. In the second line we must restore } n ki ‡ n ki, or something similar.
4987	The problem is curiously stated, it appearing at first sight that } of the 700 loaves are
4988	to go to the first man, half to the second and so on. This of course is impossible, and the
4989	The solution is on modern lines. The four fractions are added and give lf + *. The
4990	first man then receives i+ of 700, the second i#+] and so on. In Egyptian this working
4991	is expressed differently. The sum 17 + of the fractions is turned upside down, ié. in
4992	Egyptian 1 is divided by it. The result is ‡+ t7. The total 700 is then multiplied by this
4993	+ › which is the same thing as dividing it by lt+. The result is 400, and we have
4994	now only to multiply 400 successively by 3, 4, % and to get the shares.
4995	"Example of distributing differences. If it is said to thee, 10 hekat of barley' to 10
4996	men, the difference of each man over his neighbour being of a hekat of barley.
4997	The mean share is } hekat (read 1 hekat). Take 1 from 10; the remainder is 9. A
4998	hali of the common difference is taken, namely t* hekat. Multiply it 9 times, result (t+ t6)
4999	1 Plural strokes omitted in plate.
5000	r 3
p. 112
5001	108 RHIND MATHEMATICAL PAPYRUN
5002	No. 64. hekat. Add to the mean share. You are now to subtract } hekat for each man down to the
5003	last.
5004	The doing as it occurs:
5005	13+ Tr 11 +1+1 1+Te 11 +1 1T:
5006	+++1 ++ Tii 1+1+ = +TR + + +16
5007	Total 10.1 "
5008	Put into modern language the problem is to form a series of 10 terms in arithmetical
5009	progression, their sum being 10 hekat and the common difference } hekat. The Egyptian method
5010	is to find the last (i.e. highest) term. This is done by taking the mean share, ie. the share
5011	which each would get if the division were made equally. Unfortunately the scribe has here
5012	written ¿ hekat instead of 1. To this is next added half the common difference, i.e. } of g:
5013	which is ie, multiplied by the number of terms less one, ie. 9. This gives the last term.
5014	This rule was doubtless obtained empirically. If an arithmetical series be written down
5015	it will be seen at once that, supposing the number of terms to be odd, the middle term of
5016	the series is the mean share, i.e. the whole sum divided by the number of terms (n), and
5017	share above this adds on the common difference, so that the last term consists of the
5018	mean share (m) plus the common difference (d) multiplied by the number of terms on either side
5019	When the number of terms in the series is even the mean share lies midway between
5020	the two terms m -¿, m + §. But the rule still holds, for the first term above the mean
5021	share is m + §, the next m + 3d 3, the next m +5s, and the last will bem+d."-! The last
5022	and highest term having been found, it is only necessary to keep subtracting the common
5023	difference } from it to get all the previous terms.
5024	In modern arithmetic we do not use quite the same method, for we avoid the use of
5025	the mean share m. Thus if a be the first term and l the last, we have the equation
5026	1 = a + (n-1) d
5027	Now the Egyptian mean share in is clearly "+l :. a = 2m -l. Substituting for a in the
5028	equation we get 1= 2m-l+ (n-1)d or l=m + "-.d, which is the Egyptian
5029	equation.
5030	Unless we are prepared to give prw two different meanings in the same example, which
5031	is almost impossible, we must translate it throughout in the sense demanded by the phrase
5032	prw n s nb r śnwf, which can from the context only mnean "The excess" (or difference) of
5033	each man over his fellow," cf. twnw in No. 40. The opening words, " Example of dividing
5034	which we may best express in English by "distribute."
5035	prw...m it likit } pw. This form of nominal sentence seems redundant, either m or
5036	pw being unnecessary. Yet we have the same form again in Nos. 70 and 71.
5037	ślt. I can find no other examples of the figurative use of the verb in this sense. The
5038	metaphor must be either "to weave in" the men one after the other, or "to catch" them
5039	as in a net. The verb occurs with the abstract determinative in difficult passages, Prisse 0, 7
5040	and 6, 9, and B.M. 10509, 2, 10.*
5041	hry phwi. Literally "he who has the end," ie. "the last" Cf. Urh., IV, 110t, where,
5042	however, a sense of inferiority in rank is also implied.
5043	For the sign § at the end of line 1 see under No. 70.
5044	1 Strictly speaking the lkil-sign should precede if the stroke is to be read 10 and not 1.
5045	¿ For prw = " exeess," "surplus," see GARDINER, J.E.A., IX, 19, u. 5; SerHs, Einsetzung des Veziers, note 130.
5046	* Also Pap. Petrograd 1116 A, recto, 111 (Gum).
p. 113
5047	RHIND NATHEMATICAL PAPYRUS 109
5048	No. 65. (PI. S.) No. 65.
5049	"Example of reckoning vut 100 loaves for 10 men, a sailor, a foreman and a watchman
5050	with double.
5051	Its working:
5052	You are to add up the crew, result 13. Reckon with 13 to find the hundred
5053	loaves:result73+3.
5054	Then shall you say, This is the ration of the 7 men, and of the sailor, the foreman and
5055	the watchman with double.
5056	73+ 3s
5057	75+ 35
5058	75+ 30
5059	73+ 35
5060	73 + as
5061	73+33
5062	73+ 33
5063	Sailor 15g tz0t 7
5064	Foreman
5065	Watchman 15%+*+ 1
5066	Total 100."
5067	The problem is a simple one. A hundred loaves are to be divided among 10 men,
5068	three of whom are to receive double portions. Three doubled is 6, and 6 + 7 is 13. Thus
5069	all we have to do is to divide the 100 loaves into 13 portions, giving two portions to each
5070	of the favoured men and one each to the rest.
5071	Note the resolution of 3, by table into șt + 1s
5072	The reading of the second line has given some trouble. The word* is certain,
5073	despite the curious form of the &, and the word which follows it can hardly be other
5074	than Q → We thus get the compound rmtt "pr, which on the analogy of rutt iśt must mean
5075	"men of the crew or gang," or "the crew or gang" simply. Sethe has pointed out that pr
5076	is used of a group of men working on land as well as of the crew of a ship.' In this case we
5077	have to do with a ship's crew if the reading nfw, sailor, is correct, and judging by the form of
5078	the hieratic it is much more probable than the only other possibility, which is eyp, a follower.
5079	In t: t: 100 the article is feminine to agree with the feminine numeral št. See SETHE,
5080	V.Z.Z., 50, and contrast p; t; 1000 in No. 74, where the numeral l: (1000) is masculine.
5081	No. 66. (PI. S.) No. 66.
5082	"Ten hekut of fat has been issued for a year. What is the daily portion thereof ?
5083	Its working out:
5084	You are to turn the 10 hekut of fat into ro, making 3200. Now turn a year into
5085	days, result 365. You are to divide 3200 by 365. Result 83t rotatri
5086	making in ro(sic) is hekat and (33 + to t zrvu) ro. This is the daily portion.
5087	The doing as it occurs:
5088	1 365
5089	2 730
5090	1460
5091	2433
5092	Total 83 + *"+ 21'
5093	You may do similarly for any problem put to you resembling this example."
5094	' In BORCHARDI, Dus Grabdenkmal des Suhure. I1, 85, noto 6.
p. 114
5095	110 RHIND MATHEMATICAL PAPYRUS
5096	No. 66. The method needs no explanation, except that in the working 8 2920 has been
5097	omitted after 4 1460. It is worthy of notice that in the contracts of Hapzefa the daily
5098	portion is obtained from the yearly by dividing not by 365 but, as he expressly states, by
5099	360, the five epagomenal days being there neglected.
5100	In line 1 pri is clearly used in its technical sense of "to be issued or delivered" from
5101	a storehouse or government department, for which see the Siut contracts and Pap. Bulaq 18
5102	passim. The form here must be the Pseudo-participle, indicating a state, best translated by a
5103	Perfect in English, "has been issued." The problem is, as usual, concrete. An official has
5104	received a year's supply of fat, and is asking himself how much he can afford to use daily.
5105	The phrase mi tp pn is our authority for translating tp as "example" in the heading
5106	of the problems throughout the papyrus. Note here the formulation of a general rule and see
5107	on 61B, p. 104.
5108	No. 67. No. 67. (PI. T.)
5109	"Example of reckoning the produce of a herdsman. Behold now this herdsman came
5110	to the numbering of cattle' with 70 oxen: said this accountant of cattle to this herdsman,
5111	How few are the head of oxen which thou hast brought! Where then are thy numerous head
5112	of oxen? This herdsman said to him, What I have brought thee is two-thirds of one-third of
5113	the cattle which thou didst entrust to me. Count for me and thou wilt find me complete.
5114	The doing as it occurs:
5115	Multiply 70 by 43
5116	Result 315: these are what
5117	were entrusted to him
5118	}of $ 1 315
5119	Divide 1 by * + *8 210
5120	105
5121	of ș of it is 70: these
5122	are what he brought."
5123	The mathematics of the problem are simple. The question is, If two-thirds of one-third
5124	This
5125	4½, and this has only to be multiplied by 70 to give the answer 315.
5126	The translation is less easy. The situation seems to be that the accountant of cattle
5127	has entrusted a certain number of cattle to a herdsman to rear, with instructions to produce
5128	two-ninths of them on the day of cattle-numbering. That this system of letting out cattle
5129	was practised in Egypt is very clear from the accounts preserved among the Kahun papyri.?
5130	Both there and here the cattle produced at the numbering are termed b:kw, which Griffith
5131	renders "produce" and Maspero meanings being equally common in Egyptian.
5132	In the light of the present example "produce" would seem the better translation, for the
5133	herdsman is simply delivering over a certain number not of his own cattle but of some which
5134	have been entrusted (śip) to him by another.
5135	The crux of the passage lies in line 3, in the hieratic group which follows the
5136	tn. Eisenlohr failed to transliterate it. Griffith read and translated the clause "very
5137	few are the heads of oxen you are contributing: what is the whole number of your heads
5138	of oxen of various kinds?" There are four objections to this. In the first place, there is no
5139	word for " what," tr being merely an interrogative particle meaning "pray." In the second
5140	1 Cf. Beni Husan, I, PI. VIII, 1. 17, and PI. XIII; L.D., II, 31 ; Url., IV, T5, 14.
5141	* See GRIrrITH, K.P., Text, 43 and 45-47, and the interesting inscription Beni Hosan, 1, PI. VIII.
5142	* GRIFFITH, up. cit., 101.
p. 115
5143	RHIND MATHEMATICAL PAPYRUS 111
5144	place, tnw no could not mean "the whole mumber," but only "every number." In the third
5145	place, țnw "a number" has the w written out in M.K. texts, and therefore cannot be the word
5146	we have here. In the fourth place, the group after the bird cannot be read
5147	would give us a form for • which is quite foreign to this and any other papyrus of the
5148	period, and the sign below it would be very short for ‹m and lacks the final turn downwards
5149	which this sign almost always has in
5150	we have here is not tmu "number" but tni " where," and the group
5151	following the bird is nothing sign &i which frequently determines tni. The form
5152	is curious, it is true, but its relation to those of papyri of about the same period, such as
5153	Ebers and Westcar, is not hard see. Unfortunately no other instance of the sign oceurs
5154	in Rhind.
5155	is now clear: "Pray where are your many head of cattle?" or, in
5156	other words, "What has become of all the cattle I entrusted to you?" which is a perfectly
5157	natural question to follow the statement " How few cattle you have brought!" The accountant
5158	doubts that the herdsman has brought the whole two-ninths. The herdsman replies, Here are
5159	70, work it out, and you will find that it is two-ninths of 315, which, as you can verify from
5160	your roll, is what you delivered to me.
5161	{1,5 in line 3. It is difficult to see what else could be read, since the sign for
5162	$ is exactly similar to that used in No. 62, where however it may just be for O and
5163	not § (see p. 106 and note 5). It is clear, too, that the ancient scribe who patched up the torn
5164	papyrus was of the same opinion, for in mending line 2 he has written this sign almost
5165	like the hieroglyphic Q. I have followed Griffith in translating "head of cattle," though I
5166	am far from convinced that we have reached the correct solution.
5167	lịsb nỉ gm-ki wi km•kwi. Griffth divided the words lısb-nỉ gm kwi km-kwi, and translated
5168	"I have reckoned and I found that I had completed my contribution." This is improbable,
5169	since the pseudo-participle ym•kwi, unless an archaism, could only be passive in meaning in a
5170	Moreover, the verb gmi should be followed by a complete pseudo-nominal
5171	thus for "I found that I was complete" we expect gm-ni wi km-kwi!
5172	"to be complete in one's payment" see Pap. Bulag 18 passim.
5173	No. 68. (PI. T.)
5174	"If a scribe says to thee, Four gangers; they have drawn 100 great quadruple-hekat
5175	of corn. The gang of the first ganger consists of 12 men, that of the second 8, that of the
5176	. third 6, and that of the fourth 4, total 30.
5177	You are to divide 100 by 30 ; result 3}, making in corn (3f +( + *ł) hekat and
5178	1§ ro. Multiply by 12 for the first, 8 for the second, 6 for the third, and 4 for the fourth.
5179	1 (34 + Tv + 7#) hekat + 15 ro
5180	2 (6}++32) +3:,
5181	(13++ Ti+T7) +1* "
5182	28 "+3;,
5183	Total, the first, 40 hekat.
5184	(34 + 1* + .*) hekat + 15 ro
5185	+3}"
5186	4 (13*+T+T7) +15,
5187	-8 (261+!+) +3:,
5188	Total, (26) + " + 12) hekat + 3} ro, the second.
5189	1 At the same time the Pseudo-participle can in independent sentences be used in the Ist Person Singular
5190	without a preceding pronoun; also after "-» (see Shipwrecked Sailor, 109, 157, 169, 174, 177, where, however,
5191	contrast 39, 131, 155).
p. 116
5192	112 RHIND MATHEMATICAL PAPYRUS
5193	No. 68. 1 (34+ 7e + 1t) hekat + 13 ro
5194	+35,
5195	sic 4 (13} + 1 + nt) +(1)5"
5196	Total, the third, 20 hekat.
5197	1 (3* + Tr + 77) hekat + [13 ro]
5198	2 +35"
5199	-4 (131 +1r+ E) +15"
5200	Total, the fourth, (13} + T + 67) hekat [+ 13 ro]
5201	List of these :
5202	gangers: great quadruple-hekat of corn :
5203	The first 12 25 + 10 + 5 40
5204	The second 8 (26} + * + =) hekat + 3); ro 265
5205	The third 6 20 20
5206	The fourth 4 (134 + 1 + 77) hekat + 13 ro 13%
5207	Total 30 100 hekat 100 "
5208	The problem is that the 100 hekat of corn has been earned by 4 gangs together and is
5209	given to the four gangers to divide into four portions proportionate to the size of their gangs.
5210	The further division within each gang does not enter in here at all.
5211	The working has a very complicated appearance, partly from the fact that it contains
5212	a great deal of repetition, and still more from the fact that the scribe, copying probably from
5213	a tabulated original on to a papyrus divided into narrow horizontal strips, was forced to
5214	destroy the tabular arrangement, and thus several pieces of the work have got out of place.
5215	The total number of men is found to be 30. The hundred quadruple-hekat of corn are
5216	divided by this, and the result is 3} quadruple-hekat per man. To get the shares of the gangs
5217	we have to multiply this by 12, 8,6 and 4. The 3) is first turned into the Horus-eye notation,
5218	giving (34+ 1+ 67) hekat and 13 ro. The four multiplications are all worked out separately,
5219	despite the fact that since the multipliers in the first are 2, 4, and 8 all the results could
5220	have been obtained from this one piece of work. The sum ends with a table giving each gang,
5221	the number of men composing it, its share in quadruple-hekat expressed first in correct hekat
5222	notation, and second in ordinary pure integers and fractions. The last line gave the totals
5223	of each column. The ."* under the sign e for 100 in the last column should of course be
5224	omitted, since the column is not in hekat notation. This is a curious reminiscence of the
5225	converse mistake in the setting of the problem, where 100 quadruple-hekat is written
5226	instead of .5.
5227	shn, "to embrace," must be used figuratively here of "drawing" wages collectively.
5228	I can find no other instances.
5229	Nos. Nos. 69 to 78. EXCHANGE OF BREAD AND BEER. Plates U—W.
5230	69-78.
5231	These examples deal with thestrength, alelT of bread and of beer, with
5232	the exchange of loaves of various sizes, and with the exchange of bread for beer. The
5233	word pfśw (read pśw or fśw? See SETHE, Verbum, I, 216) must mean literally the "cooking"
5234	or "cooking-value."1 The pfiw of a loaf of bread is simply the number of such loaves
5235	which can be made out of a hekat of corn. Thus if the pfśw of a loaf is 12 (per hekat)
5236	the loaf must contain one-twelfth of a hekat of corn. Similarly the pfśw of a jug of beer of
5237	a certain size is the number of such jugs which can be made out of a hekat of corn."
5238	1 First explained by Dümichen, Ä.Z., 1870, 41 ff.
5239	2 It seems impossible to find an English word to cover both meanings.
p. 117
5240	RHIND MATHEMATICAL PAPYRUS 113
5241	loaves of bread and jugs of beer could be exchanged for one another, or for loaves or jugs of beer of different sizes, or The importance of the pféw clearly rested even for other commodities whose value in relation to the hekat on the fact that it formed a basis on which No69-78.
5242	of corn was known or could be found.'
5243	It should be is a rather important difference between the pfśw of
5244	bread and that of beer, due to the natural difference between a solid and a fluid. The pfsw
5245	of a loaf, giving as it does the amount of corn in the loaf, practically determines its size,
5246	apart from small variations due to cooking. The pfśw of a dé-measure of beer cannot of
5247	course alter the size of the measure, but, as it gives the amount of corn used to produce
5248	the beer, it determines the strength. In other words, the pfśw determines the size of loaves of
5249	bread and the strength of beer.
5250	In the present papyrus the pfśw are all reckoned per simple hekat, while in the calendar
5251	inscription of Medinet Habu they are reckoned per quadruple-hekat. In the Rhind the pfśw of
5252	loaves runs from 5 to 45. At Medinet Habu we read of bit-cakes of pfśw up to 100, and
5253	prśn-cakes of pfśw up to 30, in both cases per quadruple-hekat. These sacrificial cakes were
5254	clearly much smaller than the ordinary loaves and doubtless made of a finer quality of flour.
5255	The usual type of pfśw entry in the Medinet Habu inscription? is as follows:—
5256	An
5257	"Bit-bread of pfśw 30; amount of corn used hekat, yielding 15 bit-cakes.
5258	The pfśw of beer, as Griffith has shown, tended to increase in course of time,
5259	to say the beer became less strong. In the Middle Kingdom Papyrus Bulaq 18 the pfśw is
5260	invariably 2 dé per simple hekat. Here in the Rhind it is 2, 2} and 5 to the hekat.
5261	an inscription of Tuthmosis IV at Karnak beer of pfśw 4 is mentioned. Finally, in the calendar
5262	of Medinet Habu the pfśw of beer is 5 (špnt-jugs), 10 and even 20 (dś-jugs in the last two
5263	cases) per quadruple-hekat, which gives 14, 2% and 5 for the simple hekat.
5264	Quite distinet from the pfśw in the mind of the Egyptian is another term, which is in
5265	nothing but its inverse. This is the lurt," which here means the content of a loaf in
5266	Thus if the pfiw of bread per hekat is 4, then each loaf has a content of f hekat.
5267	In these examples various kinds of grain and flour are mentioned, and it will be best
5268	to diseuss at once their translation into English.
5269	of offerings are added up to give 4 khar (?) of it mhty and 1 khar (?) of it šm, and these two
5270	are combined in the words "Total, sś 5 khar (?)." In the Annals of Tuthmosis III this general
5271	meaning is also clear. Thus in one passage (Urk., IV, 694) the harvest of the land of Retenu
5272	is said to consist of šś "s; it śwt bdt, "various kinds of corn (including) it, sut and bdt" That
5273	the correct rendering, and that the šś must be a general term including the three
5274	species it, śwt and bdt, is clear from a comparison of the two passages quoted. The uses of
5275	ss in Rhind fully bear out this conclusion. Thus in Nos. 35 and 37 it is used of a quantity
5276	of grain in a case where the particular species is immaterial and the amount is all that matters.
5277	It is employed in the granary sums Nos. 42 ff. under precisely similar circumstances, and so
5278	too in No. 68. In No. 82 it seems to include both śwt and bdt (if this be the right reading),
5279	but the example is too obscure in meaning to allow of certainty.
5280	" In the tomb of Amenemhab a certain official is labelled "Overseer of the granary of the king, who reckons
5281	the pfsw of bread and beer." Thescene shows the provision of supplies for an army. Urk., IV, 912.
5282	" DümIcHen, Kalender-Inschriften, PI. I.
5283	3 Called rlt in No. 69, perhaps less correctly. Contrast No. 74, where vlt =the number of loaves. Pap.
5284	Moscow has hit throughout.
5285	Q
p. 118
5286	114 RHIND MATHEMATICAL PAPYRUS
5287	N059-78. first otth ceealst pesifically ientioeid in the eat, yrui ane turji tnm o, hlc amd duum the it, bdt and bs:! The
5288	Coptic coro. It occurs but once in Rhind, in the very unsatisfactory No. 82, where it is used
5289	to prepare bread for feeding geese.
5290	bdt, Coptic Bwte, is a kind of spelt, Triticum dicoccum, much cultivated in Europe and
5291	known in Germany as Emmer. It is mentioned three times in Rhind. In No. 79 no clue is
5292	given as to its use: in No. 82 it is made into bread for geese, and in No. 84 it forms the
5293	food of oxen.
5294	it includes two types of barley, Hordeum hexastichum and Hordeum vulgare, Coptic EIwT.
5295	The Egyptians distinguished two kinds, that of Upper and that of Lower Egypt, written in
5296	later times simply "Upper Egyptian" and "Lower Egyptian" with the word barley omitted.*
5297	The former occurs in No. 74.
5298	The fourth kind of grain, bš:, is of infrequent occurrence.
5299	only in No. 71, where it is made into beer. There is in this passage a seribe's error which
5300	led Eisenlohr to take bš: for the reading of the measure known to have been called hkit.
5301	occurs three times in the Kahun Papyri, Pls. XV, 66, XVIII, 3, and XX, 3. Griffith in his
5302	• note (K.P., Text, 44) quotes examples from the Ebers Papyrus and Pap. Bulaq 18.3 In the
5303	Kahun papyri it occurs on each occasion in lists between Upper Egyptian barley and dates,
5304	and in the pfsw problems of the Moscow papyrus it is always associated with bnr, dates, in
5305	the brewing of beer, in a puzzling connection the exact drift of which I cannot at present
5306	perceive. Stern in his Glossar zum Papyrus Ebers identifies it, wrongly, with the grain now
5307	known as dura, Sorghum vulgare, which appears to be a late importation into Egypt.
5308	nd (Nos. 69 and 70) has generally been identified with the Coptic woeIT, and in con-
5309	sequence translated "flour" The equation may be correct, but it is phonetically far from
5310	satisfactory. In Rhind as elsewhere this substance is used in the making of bread. In Urk.,
5311	IV, 688, it occurs with sś and śwt in a list of tribute from Syria. In SPIEGELBErG, Rech-
5312	nungen aus der Zeit Setis I, Pl. IVb, it appears to be obtained from bdt, despite the author's
5313	remarks on pp. 39-40 of the Text. The Rhind examples give no clue to its exact nature.
5314	The most puzzling of all these types of grain or flour is undoubtedly that called wdyt.
5315	It occurs in Nos. 72 to 78 and 82. In most of these cases it is made into bread, but in 77 it
5316	is used for beer, and in 78 for both bread and beer. In No. 74 the amount of Upper Egyptian
5317	barley contained in a certain number of loaves is found: the next words are "Then shalt
5318	thou say, this is (the amount of) wdyt," from which it would at first sight appear that wdyt
5319	is the meal obtained by grinding Upper Egyptian barler. In No. 82, however, bread for
5320	the feeding of geese is made from wdyt, and later in the sum we find a rather unintelligible
5321	calculation of the amount of śwt* and bdt which must be ground in order to produce this
5322	other instances of wayt in Egyptian.
5323	No. 69. No. 69. (PI. U.)
5324	"Three and a half hekat of flour made into 80 loaves. Let me know the content of a
5325	single loaf in flour. Let me know their strength.
5326	1 See on these cereals ScHuiz. Die Getreide der alten Aegypter, and Beiträge zur Kenntniss der Geschichte der
5327	Spelzweizen im Altertum, in Abhandlungen der Naturforschenden Gesellschaft zu Halle, N.F., Nos. 5 and 6, 1916 and
5328	1918; also his articles in Berichte der deutschen botanischen Gesellschaft, XXXIV, on the same subject; compare
5329	HrozNy, Das Getreide im alten Babylonien, Vienna, 1914.
5330	2 Sethe in A.Z., 44, 19.
5331	3 Add Pap. Leyden 349, verso 2, 8, along with nd. 4 Reading not quite certain, see p. 124.
p. 119
5332	RHIND MATHEMATICAL PAPYRUS 115
5333	You are to reckon with 3} to find 80: No. 69.
5334	1
5335	10 35
5336	<20 70
5337	7
5338	2g
5339	The strength is 223 + 4 + 3t
5340	223+ + + *1
5341	-2 453++++·=
5342	1lf++s
5343	>1 320
5344	-2 640
5345	160
5346	Total 1120 in ro
5347	You are to reckon with 80 to find 1120.
5348	The doing as it occurs:
5349	1 80
5350	-10 800
5351	2 160
5352	1 4 320
5353	Total 1120
5354	The content of a single loaf in flour is 3's hekat + 4 ro.
5355	1 3z hekat + 4 ro
5356	(I's + 67) hekat + 3 ro
5357	(* + 3b + 6ł) hekat + 1 ro
5358	(#+16+32)
5359	(7+*+3)"+
5360	(23 +1+35 + =4) hekat + 1 ro
5361	Result 3) hekat of flour."
5362	Theworking is simpler than it looks: it is in great disorder owing to the efforts of the
5363	scribe to fit it into the narrow the papyrus was divided. It consists of
5364	two parts, (1) the finding of the pfśw and its proof, and (2) the finding of the rht and its
5365	The pfśw is obtained by maltiplying 3} to get 80, or, as we put it, dividing 80 by 34.
5366	The result is 22% + } + 2, and by way of proof this number is multiplied by 3½ and shown to
5367	give 80, though the actual addition is not inserted.
5368	The reckoning of the contentbegins with the reduction of 3} hekat to ro, viz. 1120 ro.
5369	Next 80 is multiplied to find 1120, i.e. 1120 is divided by 80. This gives the content of a
5370	loaf as 14 ro, or, in the Horus-eye notation, sh hekat and 4 ro. This result is finally proved
5371	For Toe." see above, p. 114.
5372	wít nt ti, which occurs again in No. 70, is interesting grammatically. Since the word t3
5373	is masculine w't cannot be the New Egyptian indefinite article, "a loaf," which would require
5374	w ni ti, and it must therefore be the abstract noun of number, whose existence was first
p. 120
5375	116 RHIND MATHEMATICAL PAPYRUS
5376	monstrated by Sethe and which corresponds to the Coptic ore. The literal translation wou
5377	be "A unit of loaves." he use is suitable here because we are finding the content not
5378	any particular loaf but of the unit loat.
5379	No. 70. No. 70. (PI. U.)
5380	"7}+ *+ } hekat of flour made into 100 loaves. What is the content of a single
5381	loaf in flour? What is their strength?
5382	You are to reckon with 7}+*+ to find 100:
5383	1 7+1+1
5384	2
5385	31%
5386	63
5387	Total 99% + *: remainder $
5388	E'3 is Ith. For * double the fraction.
5389	*7 + T26 IS #
5390	The strength is 12%+ *2+ T2T
5391	12%+*+ Th
5392	-2 2531276
5393	503+ TE tETt TEE
5394	65+87+112
5395	38+ T8N + 504
5396	1ộ+ te t330tTO08
5397	Total 2520 (read 100).
5398	You are to reckon with 100 to find 2520:
5399	100
5400	10 1000
5401	-20 2000
5402	The content of a single loaf is (7* + 77) hekat + } ro of flour.
5403	1 (16 + "t) hekat + 3 ro
5404	(k + #+3b) hekat + 2 ro
5405	100 (7} + ÷ + $) hekat of flour."
5406	The method is exactly that of No. 69. The strength is found first by dividing 100 by
5407	7}+ ł+* answer 123 + ** + T2*• This result is then proved by multiplication.
5408	of products in this should be 100, but we find instead of it 2520.. This is the correct answer
5409	to the next step, namely the reduction of 7} + 1 + 1 hekat to ro, which is the first part of the
5410	finding of the hrt, but which the scribe has omitted. The case is one of simply haplography
5411	of →.
5412	In front of the first word of this example stands in black the sign (see Pl. S,
5413	No. 64). Griffith is probably right in suggesting that it is here used in its
5414	meaning of "stand" or "stop" (cf. p. 68), and refers to the irregularity of the page at this
5415	It is quite impossible to read it into the structure of the first line of No. 64.
5416	use, perhaps quite differently, in account papryi see SPIEGELBERG, Rechnungen aus der Leit Setis I,
5417	Text, 48 and 58, Plates VIII and XIII. Compare too the '!n of Pap. Harris A, 1, 13 and 2,4.
5418	1 A.Z., 47, T-16, and V.Z.Z., 42-44.
p. 121
5419	RHIND MATHEMATICAL PAPYRUS 117
5420	Note here again, in line 1 and later, the unusual hieratic form of the numeral 7. It is No. 70.
5421	used in No. 53 of 7 setat : here it is used of 7 hekat, as also in No. 75 and No. 84.
5422	as to the meaning of this word in No. 61B. can ia a oicte an tigero pr, tica cen m the vham sue.
5423	In the last line but three we have the curious redundant form of nominal sentence with
5424	both in and pw used in No. 64 and again in No. 71.
5425	No. 71.
5426	"One des-measure of beer, a quarter of which has been poured off.
5427	made up with water and tasted with regard to what the strength is.
5428	the 1 des into besha-grain : result half (a hekat) of besha-grain. You are to subtract a quarter
5429	of it, namely & hekat: the remainder is 4 + } hekat. You are to operate on 1+} to find 1.
5430	Result 2%. This is the strength, namely 23"
5431	With this example we come to the pfśw of beer. At the time when this papyrus was
5432	written the pfśw of a des of beer was 2, i.e. each des contained } hekat of grain, or in other
5433	words each hekat produced 2 des of beer.
5434	It is important to notice that the vertical stroke which follows the determinative of
5435	dš both in line 1 and in line 2 is the numeral 1 and not merely a stroke accompanying the
5436	determinative &, which would be an improbable writing for this period. We must therefore
5437	translate not "A des-jug" but "One des-measure." In other words, though dś may originally
5438	have been the name of a jug of a certain shape used principally or wholly for beer, it had
5439	by this time become quite definitely a measure of liquid capacity. It has not been sufficiently
5440	recognized that the Egyptians had a series of measures of liquid as well as of solid
5441	capacity, each measure being used for some particular liquid or liquids and for those alone.
5442	Thus in Pap. Kahun, PI. XXVI, 1-33, we have an account of various kinds of vessels
5443	to be made by a potter. The text is not easy, but it seems clear that the capacity of the
5444	various vessels was given; and if this is correct the dś was certainly a measure, for the tnft
5445	vessel is to be made with a capacity of 2 dst Similarly in the Siut contracts Hapzefa con-
5446	tracts for a st: of beer for every quarter-ds which is offered. Here it is impossible to deny
5447	that the ds is a definite measure, and indeed its use throughout these very formal contracts
5448	makes it obvious. Unfortunately we have no evidence for determining the relations of these
5449	various measures to one another or to the hekat. An exception exists in the case of the pg:, a
5450	honey measure, which, as is clear from Harris I, 39, 6, is equivalent to 4 hnw, i.e. } of a hekat.
5451	In the present problem we are given a des-measure of beer, one quarter of whose
5452	contents has been poured away and the jug filled up with water. Required to find the strength
5453	of the mixture now in the jug. The original beer contained \| hekat of corn, and what is left
5454	after the pouring out will contain less (4x),i.e. f+} hekat. The filling up with water
5455	does not alter the amount of corn now represented in the jug, so that the corn content of
5456	the mixture is still (1 + }) hekat. To get the pfśw we have to invert this, which gives
5457	i or 2%. This is the pfsw of the mixture, ie. it is the number of ds-measures of beer of this
5458	strength which can be made from a hekat of corn.
5459	'There is a scribe's error in the second line which led Eisenlohr to read bš; as the name
5460	of the unit of capacity which we now know to be lkit. After lpr-hr bs; the writer has first
5461	missed out the hekat sign * which is needed to tell us what the unit here is, having con-
5462	fused it by haplography with the determinative ' of bs:, and secondly he wrote the ordinary
5463	pure fraction } instead of the sign for ] hekat in the Horus-eye notation. This error led
5464	Eisenlohr to read "Result } besha" instead of "Result besha-corn } hekat." Notice that in
5465	1 Similar evidence is to be found in Pap. Bulaq 18. See for example PI. XXIV, where a mnsi-vase of beer
5466	seems to contain 3 ds and a kby-vase 2 dó. See 1.Z., 57, 56, note 13.
p. 122
5467	118 RHIND MATHEMATICAL PAPYRUS
5468	the next line the hekat sign is inserted before the } hekat but is omitted thenceforward, the
5469	unit being already clearly enough marked.
5470	đn is u ig then, al Beghe ian f of t a peuer mineie, the anly espeple Pyr. 20 D,
5471	more general is suspicious.
5472	dp has a strange determinative for which I can find no parallel.
5473	The phrase dp-ntwf r pfśw m "It has been tasted regarding strength what" is impossibly
5474	elliptical, and can hardly be right as it stands. We expect something like dp-ntwfrrh pféw•f
5475	m m "It has been tasted in order to know what is its strength."
5476	For bš: corn see above, p. 114.
5477	In the last words we remark the same redundant form of nominal sentence, pfsw m
5478	2% pw, that we saw in Nos. 64 and 70. In the plate the m has been omitted.
5479	No. 72. No. 72. (PI. V.)
5480	"Example of exchanging loaves for loaves. If it is said to thee, 100 loaves of strength 10
5481	exchanged for a number of loaves of strength 45.
5482	You are to make the excess of 45 over' 10, namely, 35. You are to reckon with 10 to
5483	find 35 ; result 34. You are to multiply 100 by 3}; result 350; add 100 to it; result 450.
5484	shalt thou say, This means that the 100 loaves of strength 10 are exchanged for
5485	450 loaves of strength 45, making in wdyt-flour 10 hekat."
5486	The method is curiously roundabout. The obvious method is to divide 45 by 10, result
5487	4%, and to multiply this by 100, result 450. For some reason the Egyptian prefers to deal
5488	with the excess of the pfśw 45 over the 10, which is 35. He then finds the number of loaves
5489	corresponding to this pfśw to be 350, to which he adds the 100 to get the answer 450. This
5490	is really an astounding procedure, for the working out of the number of loaves corresponding
5491	to the imaginary pfśw 35 involves all the knowledge necessary to work out directly the number
5492	corresponding to 45. The problem was to solve the proportion
5493	a : 6 = c : x, where a is 10, 6 45, and c 100.
5494	Instead of multiplying b by c and dividing by a the Egyptian works out the proportion
5495	a: 6-a :c: x-c
5496	Finally he adds a to the second term and e to the fourth, which of course does not affect
5497	the proportion and gives the value of x.
5498	When both the number of loaves and their pfśw are to be given the pfśw comes imme-
5499	diately atter the word for loaves and is followed by the number preceded by a sign resembling
5500	This sign degenerates into a mere dot. In Pap. Kahun, Pl. XXVIa, this sign stands
5501	before the pfśw as well as before the number of loaves, and the same is true of Pap. Bulag
5502	18 (see for example PI. XXIV of that papyrus). That it is not the preposition r "to the
5503	amount of," or similar, is proved by t; t; • 100 (No. 73, line 2), where the article is made
5504	feminine to agree with the numeral st, 100, showing that the Egyptian said "The 100 loaves"
5505	"the loaves up to 100," or in other words that he did not pronounce the •. A
5506	similar sign perhaps in No. 47, where see the notes.
5507	'iw, "excess," compare No.
5508	db: pw t; t; etc. It is possible to take this as a nominal sentence with db; as a noun, but
5509	the sense produced is not good, "This is the exchange of the 100 loaves of strength 10 for
5510	450 loaves of strength 45." Here pw should refer to the answer 450, found just above, and in
5511	this case we should not expect to find the 450 mentioned in the sentence.
5512	" Delete in the plate the reference letter a over the s.
5513	2In Rhind the pféw may be preceded by a dot (Nos. 77 and 78), but not by the •-like sign.
p. 123
5514	RHIND MATHEMATICAL PAPYRUS 119
5515	possible that we have here a passive use of the śdm:f pw formula of the Ebers Papyrus, " This No. 72.
5516	means that the 100 loaves of strength 10 are exchanged for 450 loaves of strength 45."
5517	however, quote no instances of such a construction, except possibly the irt pw of Nos.
5518	38 and 62, where, however, the form, if it is indeed passive, is śdm-twf and not śdm•wf.
5519	For wdyt see above, p. 114. The determinative: has been omitted through haplography
5520	with the following ."*, which is the sign for hle:t, the stroke which follows indicating 10 according
5521	to the regular notation. This 10 hekat is the amount of flour contained in the 100 large or
5522	450 small loaves: it is not actually used in the working.
5523	No. 73.
5524	"If it is said to thee, 100 loaves of strength 10 exchanged for strength 15, how many
5525	is that in exchange for them ?
5526	You are find the amount of the 100 loaves in wdyt-flour [namely 10 hekat]. You
5527	are to multiply 10 by 15; result 150. Then shall you say, This is their exchange.
5528	The doing as it occurs: 100 loaves of strength 10 exchanged for 150 loaves of
5529	strength 15: 10 hekat."
5530	in place of the absurd method of No. 72 we find a straightforward
5531	of the 100 loaves of pfśw 10 to the amount of flour used in making them, which is clearly
5532	10 hekat. This 10 hekat when turned into loaves of pfśw 15 will obviously yield 150.
5533	An unfortunate error of copying has obscured this simple working. After the words
5534	ir-lyr-ki lrt t: t: 100 m wdyt the original must have read .""\|, "namely 10 hekat," but the
5535	scribe's eye, misled perhaps by the similarity of :/' (especially when written without its cross-
5536	stroke) and , has wandered to the opening words of No. 76, ki t: 10, which here make
5537	nonsense. Perhaps in his prototype the first line of No. 73 ended at wdyt, and the scribe
5538	instead of dropping his eyes to the line below went straight on to the left into the opening
5539	words of No. 76.
5540	wr pw " db:-ś. Here we have again the late Egyptian wr meaning "how many" or
5541	"how much" which we met in No. 45. The sentence seems clumsy and the r could well
5542	be omitted, "How much is their exchange?" We must remember, however, that r db: is the
5543	usual Egyptian for "in exchange for" (Coptic eTB€), so that the construction is less unnatural
5544	than it appears at first sight. Note the Pronominal Suffix s, Feminine Singular to agree with the
5545	feminine numeral st 100. Cf. No. 65 and note, and contrast db; f in No. 74.
5546	at the end of the sum stands for the 10 hekat of flour involved in the
5547	calculation, as is clear from No. 72.
5548	No. 74. (PI. V.)
5549	A thousand loaves of strength 5 exchanged for (loaves of) strength 10 and
5550	20. What is their exchange?
5551	You are to reduce to corn (?) the thousand loaves of strength 5; result 200 hekat of
5552	Upper Egyptian barley. Then shall you say, This is (the amount of) wdyt-flour.
5553	You are to take half of 200 hekat, namely 100 hekat.You are to multiply 100 hekat
5554	by 10; result 1000: this is the number of strength 10. You are to multiply the 100 hekat
5555	by 20; result 2000. This is the number of strength 20.
5556	The doing as it occurs:
5557	A thousand loaves of strength 5, making in wdyt-flour 200 hekat.
5558	Exchange, 1000 of strength 10,
5559	Exchange, 2000 ot strength 20, 100 hekat."
5560	be reckoned in breadottivo dixerent strengtas aoanda0lroceeoo lozvesten 10 and 20. The 1000 loaves of strength E
p. 124
5561	120 RHIND MATHEMATICAL PAPYRUS
5562	are reduced to hekat, giving 200 hekat. This is then divided into two equal halves ot 100 hekat
5563	each, and one half is turned into 1000 loaves of strength 10 and the other into 2000 loaves
5564	of strength 20. That the exchange is to consist of equal amounts of bread of strength 10
5565	and of strength 20
5566	which the working shows to have been
5567	In the first line he writes
5568	db: m 10 • 20, which would mean "exchanged for 20 loaves of strength 10." He has
5569	•is not
5570	needed."
5571	pfś-ler-k p: t: 5.1000. The first sign of this phrase can hardly be transcribed otherwise
5572	than lf. The meaning must be "Make the pfśw-reckoning with the 1000 loaves," in other
5573	words, find the amount of corn in them. Note the Masculine p; t; 1000, since the numeral 1000
5574	(h:) is Masculine, and contrast the Feminine used with 100 (št) in Nos. 70 and 73.
5575	Masculine suffix db;-f also agrees with 1000.
5576	No. 75. No. 75. (PI. V.)
5577	"Another.
5578	. ou are to express the 155 loaves of strength 20 in wdyt-flour; that is, (7} + ł) heka
5579	Multiply by 30; result 232}
5580	The doing as it occurs:
5581	155 loaves, strength 20, making in wdyt-lour (77 + 4) hekat
5582	exchanged for 232% strength 30, (7} + ‡) hekat."
5583	o comment is here necessary, except that we again have the unusual form of tl
5584	umeral 7 that we have found in Nos. 53 and 7
5585	No. 76. •(PI. V.)
5586	" Another. A thousand loaves of strength 10 exchanged for a number of loaves of
5587	strength 20 and 30.
5588	Let him hear:
5589	1% 1
5590	Total
5591	Multiply it to get 30:
5592	1 2%
5593	-10 25
5594	- 2 5
5595	Total 12
5596	Find the content of the 1000 loaves in wdyt-flour, namely 100 hekat.
5597	Multiply by 12; the result thereof is 1200, their exchange in loaves of 20 and of 30.
5598	1000 loaves of strength 10, making in wdyt 100 hekat.
5599	1200
5600	1200 30, 40 hekat."
5601	The problem seems at first sight similar to No. 74, but from the working we perceive
5602	that there is a difference in the conditions, though it is never stated in words. In No. 74 the
5603	total amount of corn in the loaves of the two different strengths was to be the same, whereas
5604	here the number of loaves of the two strengths is to be the same.
5605	1 Unless he used it as a mere separating mark. See note on No. 72.
p. 125
5606	RHIND MATHEMATICAL PAPYRUS 121
5607	We solve the problem by means of the equation złx + 3,x = 100, where x is the number No. 76.
5608	of loaves of either kind, and the Egyptian does what is in effect the same thing. He adds
5609	zo and șu using as common denominator 30. The sum is then ! or so, which by the
5610	multiplication process is shown to be is. Then the number of hekat (100) multiplied by the
5611	12 gives the number of loaves of each kind.
5612	śdm•f is out of place here, as in No. 37. It occurs in examples (e.g. No. 30) where
5613	the question is put by a scribe, "If a scribe says to you .... lét him hear...." It is
5614	just possible that what stood here in the original was sšmt.f "its working"
5615	sprt im pu. A very unusua. way of expressing result in our papyrus. lprt is of course
5616	a Neuter Participle. Cf. GRIFFIT, P.K., Pl. VIII, Siut, Pl. 7, line 300, and Pap. Moscow.
5617	No. 77. (PI. V.)
5618	"Example of exchanging heer for bread. If it is said to you, 10 des of beer exchanged
5619	for (bread of) strength 5.
5620	You are to turn the 10 des of beer into wdyt-flour, that is 5 hekat. You are to multiply
5621	the 5 hekat by 5,result 25. Then shall you say, This is their exchange.
5622	The doing as it occurs:
5623	Ten des of beer; 5 hekat of wayt-flour
5624	exchanged for 25 loaves of strength 5: 5 hekat of wayt-flour."
5625	This needs no comment. The pfiw of the beer is, as throughout the papyrus, 2 des per
5626	hekat, and the exchange in bread is obtained by means of a reduction to hekat of wayt.
5627	No. 78. (PI. W.)
5628	"Example of exchanging bread for beer. If it is said to you, A hundred loaves of strength
5629	10 exchanged for a quantity of beer of strength 2.
5630	Multiply by 2; the result thereof is 20. s aves ohenrehath you sat, wås fo the it at hango hekat
5631	This offers nothing new except that the irt mi lpr of the previous examples, ie. the
5632	tabulation of the working, is omitted.
5633	No. 79. (PI. W.)
5634	"An inventory of a household (?). No. 79.
5635	2801 houses
5636	5602 49 cats
5637	4 11204 343 mice
5638	Total 19607 2301 (sic) spelt
5639	16807 hekat
5640	Total 19607."
5641	The meaning of this table was first explained by Rodet (Journal Asiatigue, 1881,
5642	450 ff.).* It is evidently based on a nursery problem of the following nature:—
5643	Seven houses; in each are 7 cats; each cat kills 7 mice; each mouse would have eaten
5644	7 ears (or grains) of spelt ; each ear of spelt will produce 7 hekat. What is the total of
5645	" Rodet gives a good parallel from the Liber Abaci of Leonard of Pisa, 1202 A.D., edit. Boncompagni,
5646	Rome 1857, I, 311, which deserves guoting again : vetulae vadunt Romam; quarum quaelibet habet
5647	et in quolibet burdone sunt sacculi 7; et in quolibet sacculo panes 7; et quilibet panis habet cultellos
5648	?; et quilibet cutellus habet vaginas 7. As I was going to St. Ives" is perhaps its lineal descendant. Quaeritur summa omnium praedictorum." The nursery-rhyme beginning
p. 126
5649	122 RHIND MATHEMATICAL PAPYRUS
5650	No. 79. We thus get in effect a geometrical progression whose first term is 7 and whose common
5651	The Egyptian solves this in two ways, firstly he
5652	came the number 2801. The modern expression for the sum of a geometrical series is
5653	a" =i, where a is the first term, " the common ratio and n the number of terms. Apply-
5654	case we get s (the sum) = 7.1 7- 1 1-1 = 7 x16806 = 7 x 2801. which
5655	shows exactly what are the figures used by the Egyptian.
5656	It is further to be noticed that when a = r, ie. when the series is
5657	powers of any number, ym is the last term, and the equation then becomes s=r,=1,
5658	where l is the last term, a formula which the Egyptians may have obtained empirically. If this
5659	is the case, it is possible that the only geometric series with which they dealt were
5660	nature, viz. sums of the powers of a number. In any case the solution of even this limited
5661	type of geometric series is very flattering to their mathematical intelligence.
5662	The signs forming the heading
5663	it is difficult to suggest any restoration which will fit or account for the traces.' Perhaps
5664	'HA is as likely as anything. imt-pr" means originally "the inventory of the
5665	contents of a house," and hence "a deed of conveyance" or even "a will." Here we should
5666	have the word in its literal sense.
5667	No. 80. No. 80. (PI. W.)
5668	"As for a vessel in which are corn-measures for the clerks of the slave-prison:
5669	Expressed in henu
5670	1 hekat 10
5671	5
5672	2}
5673	*+:
5674	++1s
5675	"+""
5676	a table for expressing the hekat and its Horus-eyeparts.
5677	½, $, etc., in terms of henu, of which there are 10 to a hekat. It is not easy to see why this
5678	should need the imposing title which is here given to it, and since this table is repeated at
5679	the beginning of the next example it is possible that there has been an error of some kind
5680	in the copying, and that we have lost the table which originally stood under this heading.
5681	The dbl!, as Griffith points out, must be the wooden vessel with which labourers are
5682	seen measuring out grain in the tomb representations. Determined with the grain-sign it occurs
5683	in the Protestation of Innocence in Chapter 125 of the Book of the Dead, "I have not
5684	increased or reduced the measuring-vessel." Among the gifts dedicated by Tuthmosis III to
5685	Amün are figured seven db!ı marked "Measuring-vessels of gold for measuring the divine
5686	lysi determined by the grain-sign seems to be unknown as a measure; determined by
5687	the rope it is the common word for a measuring-tape or the plumb-line of a balance.
5688	this passage it is probably the same word as !!: determined by the grain-sign which occurs in
5689	GRIFFITH, Siut, Pl. XV. line 9, where Khety, speaking of his benefits to his city, says, "I
5690	was abundant in Lower Egyptian barley .... making the city to live by the l: and by the
5691	1 The B.M. Fucs., despite its suspect appearance, is almost accurate.
5692	* See GRIFFITII, K.P., Text, 29. 3 Urk., IV, 635.
p. 127
5693	RHIND MATHEMATICAL PAPYRUS 123
5694	hekat." From this, as well as from the determinative, we may assume that the ly; was an No. 80.
5695	actual measure like the hekat, and not a vessel of size not necessarily fixed like the dbl.
5696	This does not throw very much light on the relation of the title of this example to
5697	the table which stands under it. What, moreover, is the reason for the introduction of the
5698	clerks of the slave-prison (šn'), since surely the use of the hekat and the henu was common to
5699	all transactions whether governmental or otherwise at this period.
5700	or complex of factories where all sorts of provisions etc. were made.
5701	The prisoners taken by the king are set to work in the šn of Amun. We might render
5702	No. 81. (Pl. W.) No. 81.
5703	" Another reckoning of the henu.
5704	Now } hekat is 5 (henu)
5705	", 25
5706	+
5707	+ T"
5708	5+32
5709	Now (! + +l) hekat in henu is 82 +#
5710	(= + 7) 75
5711	(*+ +35) + 3-3 67 + 1m (sic) That is 3 of a hekat
5712	(3+ $) That is ‡ (?3) " "
5713	(++*) 33+1 That is 3 (?3) " "
5714	(4+==+67) + 13 ro » (3} + #) + 3 (sic) That is + (sic) "
5715	, 25 That is 1 " "
5716	(d+16) + 4 ro " 2 That is 1 "
5717	(E+3) + 3-5 „ 1 + [}] That is [*]
5718	Now (* + TE) hekat + 4 ro is 2 henu That is } of a hekat
5719	(1a + 32) + 2 ro , 1 " That is to
5720	(1=+«4) +1 ro That is zo
5721	+3 ro That is f0 "
5722	16 + 1} ro= That is zo (? sic) of a hekat
5723	32 hekat + 1* + 1} ro (sic)* (sic)3 = = That is 50 That is to (sic) of a hekat " "
5724	.-n-+ »2} " 5 " That is 1 Tat is b " " " "
5725	(++:) (}+) hekat „ 8} (sic) hemu That is (1 + 4) That is (t+ 4+) of a hekat "
5726	„6} henu That is (t+ }) hekat
5727	(++#) "(t+ #) (sic) henu That is (#+*)
5728	(6+=+3z) + 3} ro „ 6% henu Tat is 3 hekat
5729	(4+T6t 64) " +15 ro „33 That is }
5730	* hekat ,,1* " That is
5731	T6 "(+ #)henu That is Tє „
5732	» ($ + 1i) That is 32 »
5733	„ (# + =2) That is Ft "
5734	1 The B.M. Facs. gives a vertical stroke instead of the ligature for 6.
5735	: Below this, on the bottom edge of the papyrus, the tops of the figures it hekat (?) and 4. Not shown in
5736	B.M. Facs.
5737	3 The 1 is certain, though wrong. * The scribe wrote 1f and then crossed it out. Read 1g.
5738	R 2
p. 128
5739	124 RHIND MATHEMATICAL PAPYRUS
5740	No. 81. This is simply a table for expressing the various more complicated fractions
5741	very incorrect.
5742	It begins with a repetition of the table of No. 80, on which see note above.
5743	succeeds a more elaborate table, of which the following is a typical line:—
5744	ot hekat + 3 ro is = henu: that is, in of a hekat.
5745	In the first column is the required fraction of the hekut, expressed correctly in the Horus-
5746	eye notation with smaller fractions added in ro. In the central column is the corresponding
5747	number of henu, and in the last the quantity expressed as a pure fraction of a hekat. last column is added in red, except in the first group of nine quantities, a thn ta th go ta oa oem ld., where it is placed This
5748	before in black, and is hopelessly incorrect and incomplete. It is possible that the scribe in
5749	writing in the black portions of the text forgot to leave room for this first red section in the
5750	proper place and had to crowd it in as best he could. For the relation of Column 3 to
5751	Column 1 see Introduction, p. 25.
5752	No. 82. No. 82. (PI. X.)
5753	"Estimate of the food of a poultry farm.
5754	Reckoned in bread per day: wdyt-flour.
5755	Fatted geese : that which 10 birds eat is 21 hekat.
5756	making in 10 days 25 hekat.
5757	making in 40 days 100 hekat.
5758	That which must be ground in order to produce(?) it:
5759	spelt (?) (166} + 1 + 3z) hekut and 3} ro.
5760	wheat (66} + ‡+ Tr + r4) hekat' and 1% ro.
5761	That which is to be subtracted at the rate of one-tenth,
5762	(6)+"+3L) hekat and 3} ro.
5763	Remainder to be given (93ł + 1e+ 6+) hekat and 13 ro.
5764	making in grain in hekat (93)+1 +"4) and 13 ro.
5765	making in double-hekat (47} + * + "t) hekat and 3} ro."
5766	(56) nerided general sthse be d hstprbler tiould aprp tin te te tu tind te ethelest ofe rerai
5767	many obscurities of detail.
5768	It is clear from line 5 that the bread needs 100 hekat of wdyt-flour. But in line 6 we meet
5769	with a crux. The first signs are clearly ""5e, and in view of the parallel ntt r libt
5770	below it is difficult to avoid the conclusion that these words mean " That which must be ground."
5771	Then follows a ligature," and next a sign which I cannot transliterate (it looks like a bird)
5772	and beneath it a xa. Grifhth read this last sign as e, the numeral 100, referring to the
5773	100 hekat of flour: this is unlikely, for 100 hekat is written '. The next sign Griffith reads
5774	,probably rightly, though ithas this form nowhere else in the papyrus, nor indeed in
5775	any papyrus, and though the ligature for which ought to follow it is missing: it is exactly
5776	like !. The probable meaning of the line is "That which must be ground to produce it
5777	(i.e. the 100 hekut of wdyt) is 166g hekut of spelt. The arrangement of the hieratic shows that
5778	in the next line the words "That which must be ground to produce it" must be repeated,
5779	and we thus have for line 7 "That which must be ground to produce it is 66} hekat of
5780	1 Note the unique use here of ,O to represent 33, hekut and the retention of the resulting ! hekat
5781	in defiance of the rule of correct notation.
5782	* Perhaps an incorreut determinative i to nd.
p. 129
5783	RHIND MATHEMATICAL PAPYRUS 125
5784	wheat." Clearly all is not right here. If these are alternatives, it is difficult to believe that so No. 82.
5785	much more spelt than wheat would be needed to produce the same quantity of wayt: if, on
5786	the other hand, both grains are used, are we to believe that it takes 233} hekat of grain to
5787	make 100 hekat of flour?
5788	The sum does not end here The 166} is henceforth neglected. One-tenth
5789	of the 663 is taken, viz. 6}, and subtracted from 100 hekat, giving 93}, and this is then halved
5790	to turn it into 46} double-hekat, wrongly given as (47)+* + "t) hekat + 3) ro.
5791	Now what is actually done here is to take the 100 hekat and subtract from it one-
5792	tenth of two-thirds of it, ie. In of 663, and the result is presumably the amount of grain
5793	needed to produce the 100 hekat of wdyt. This is equivalent to saying that grain increases
5794	in bulk when ground by rith of its own bulk, 93} becoming 100, which is reasonable. That
5795	this is the essence of the sum is clear from No. 82B, where the food of the geese is said to
5796	be 50 hekat (presumably of wdyt), and the amount of grain (šá) needed is, or would have been
5797	but for an error in the working, 23} double-hekat or 46} hekat. Here again grain appears to
5798	increase in bulk by with when ground.!
5799	case it would seem dle to specuate as to the origin o the intro-
5800	into the problem of the 166- hekat of speit, and we must assume that some
5801	occurred in the wording of the whole.
5802	The word is not known elsewhere, and its reading is uncertain: it is
5803	perhaps iwt, the rest of the signs being determinatives.
5804	šdi is commonly used of fattening geese for market. Cf. SETHE, Urk., IV, 754, and
5805	numerous pictures of forcible feeding in the tombs, e.g. El Bersheh, I, PI. XXII. For geese
5806	actually labelled r šd see WRESZINSKI, Atlas zur altaegyptischen Kulturgeschichte, Taf. 400.
5807	No. 82B. (PI. X.) No. 82b.
5808	"Amount of what a fatted guose eats:—
5809	ten geese, 1f hekat
5810	making in ten days, 12% hekat (read 12%)
5811	in 40 days, 50 hekat
5812	making grain. in double-hekut (23)+*+ ") hekat + (4} +1+1°) ro."
5813	As has been pointed outby Griffith, this problem must be separated from No.82,
5814	in effect another example of the same kind. that ten geese eat 50 hekat
5815	must as before refer to wdyt-flour, and it is required to turn
5816	this into grain. In order to do this a tenth of two-thirds of it is subtracted and the result
5817	halved to reduce it to double-hekut. The working is omitted and'the answer is incorrect;
5818	it should be 23} hekat, ie. (23} + 1e + it) hekat + 1fro. The number 23 in the last line is
5819	written 10 + 13.
5820	No. 83. (Pl. X.) No.83.
5821	"If the food of 4 ro-geese of those who are cooped up is 1 henu of Lower Egyptian
5822	barley, the share of 1 goose is n* hekat + 3 ro.
5823	It the food of a ro-goose which enters the pond* is Lower Egyptian barley (ro+3)
5824	hekat + 2 ro, that is 1 henu for one ro-goose,
5825	1 Wheat when ground into flour of modern fineness increases in bulk by 25 per cent.
5826	2 In M.K. hieratic this double group, which we must here read as:6 + f, is written for & simply.
5827	3 Ä.Z., 44, 19. + Cf. El Bersheh, I, PI. XX.

Provenance

title: Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970) — OCR (Apple Vision)
dialect: none
source: corpus/incoming/e-2/peet-rhind-1923/ — Apple Vision OCR (tools/e2_ocr.py, worker E-2, 2026-09-29) of corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf, 169 PDF pages; provenance and verification in corpus/incoming/e-2/STAGING.md
licence: public domain in Canada (author died ≤ 1971, verified; LICENCES.md §7) — US position (recorded, does not gate the class; PD-CA 2026-09-26): PD-US (published ≤1930); the scan is the 1970 Kraus reprint, "Reprinted by permission of the original publisher" (no new authorship)
share: public
attribution: Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970); public domain in Canada (published 1923; author T. Eric Peet (title page corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf pdf p. 1: "… by T. Eric Peet, Brunner Professor of Egyptology in the University of Liverpool … MCMXXIII — Kraus Reprint, Nendeln/Liechtenstein 1970"; pdf p. 2 reprint notice; preface signed Liverpool June 1923); death year UNVERIFIED — the PDF has no Author field and no in-folder authority record gives Peet's dates · note (PROPOSED-scan_copyright-E-2.csv): middle-egyptian-texts/README.md §3 states "died 1934" with no source — not counted (E-2 volume peet-rhind-1923) · Lead D authority lookup 2026-09-29 (outputs/REPORT-DEATH-YEARS-2.md, DEATH_YEARS_EVIDENCE.tsv): Peet d. 1934, BnF cb12504101w + IdRef); OCR text by the Kemetic project, Yousef Hanna 2026.
OCR: Apple Vision VNRecognizeTextRequest rev 3, accurate, language correction off, via tools/vision_ocr/vision_ocr (NC-4/NC-9); macOS Version 26.6.2 (Build 25G83); render pdftoppm 300 dpi gray; 3 text column(s) read separately (page 1; per page in the staged records); NOT D2: no CER against hand-read GT for this volume — the only figure is agreement with the scanner's own text layer (corpus/incoming/e-2/cer.tsv), not accuracy (CLAUDE.md §1.3)
file: Ancient Egyptian/Rhind Mathematical Papyrus (Peet 1923)/Peet, The Rhind Mathematical Papyrus, BM 10057 and 10058 (1923).vision-ocr.L0.txt

Cite as:

Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970) — OCR (Apple Vision). corpus/incoming/e-2/peet-rhind-1923/ — Apple Vision OCR (tools/e2_ocr.py, worker E-2, 2026-09-29) of corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf, 169 PDF pages; provenance and verification in corpus/incoming/e-2/STAGING.md. Layers as published by the Kemetic project (Yousef Hanna), 2026: Coptic · Cairene · Kemetic, /doc/ancient-egyptian/rhind-mathematical-papyrus-peet-1923/peet-the-rhind-mathematical-papyrus-bm-10057-and-10058-1923.vision-ocr/. Peet – The Rhind Mathematical Papyrus, British Museum 10057 and 10058 (Liverpool/London 1923; scan is the Kraus Reprint, Nendeln 1970); public domain in Canada (published 1923; author T. Eric Peet (title page corpus/incoming/middle-egyptian-texts/peet_rhind-mathematical-papyrus_1923.pdf pdf p. 1: "… by T. Eric Peet, Brunner Professor of Egyptology in the University of Liverpool … MCMXXIII — Kraus Reprint, Nendeln/Liechtenstein 1970"; pdf p. 2 reprint notice; preface signed Liverpool June 1923); death year UNVERIFIED — the PDF has no Author field and no in-folder authority record gives Peet's dates · note (PROPOSED-scan_copyright-E-2.csv): middle-egyptian-texts/README.md §3 states "died 1934" with no source — not counted (E-2 volume peet-rhind-1923) · Lead D authority lookup 2026-09-29 (outputs/REPORT-DEATH-YEARS-2.md, DEATH_YEARS_EVIDENCE.tsv): Peet d. 1934, BnF cb12504101w + IdRef); OCR text by the Kemetic project, Yousef Hanna 2026.

What the layers are

L0 witness
The text as its witness or edition has it, original spelling; its dialect is a description and is never standardized.

L0.5 transliteration
For Egyptian texts: the transliteration with the editors' marks turned into the symbols ° * < > _ ^ (ruled 2026-09-10). For papyri: the edition's Leiden marks as the same symbols, and the editors' readings beside the scribe's as ‹scribe→editors› (ruled PAP-3).

L1 Cairene
Standardized Greco-Bohairic spelling and grammar, generated by code from the layer above under the rulings of the project.

L2 Kemetic
The Kemetic alphabet, generated by code from L1.

English
Quoted translations where they exist, credited; otherwise model drafts, marked as drafts.

Source and licence

The text — part 1 of 2

Provenance

L0 witness (OCR)

What the layers are