The Rhind Mathematical Papyrus (RMP), and the Egyptian Mathematical Leather Roll (EMIR), which were discovered together in the Ramesseum at Thebes, more than a century ago, have been in the British Museum since 1864. The RMP was of immediate interest to historians as soon as it was translated, but the EMLR remained unrolled for 50 years. On its eventual successful unfolding and translation, the interest and enthusiasm of the historians waned sadly, when it was found to contain only equalities of the sums of various Egyptian unit fractions, and thus since 1927, few historians have even bothered to give the EMIR any further serious consideration. Of those writers mentioned by the author, -- Scott and Hall, Glanville, Vogel, Neugebauer, and Van der Waerden, -- none of their views concurs with that expressed in this article, so that an expression of opinion from any of them, would be of interest to the student who may already have read his ``Mathematics in the time of the Pharaohs'' [London: The MIT Press (1972; Zbl 0491.01004)]. To conceive the import of this article, one should know that, ``The Recto of the RMP occupies one-third of the 18-foot papyrus, and is the most extensive of arithmetical tables to be found among ancient Egyptian papyri'' [Arch. Hist. Exact Sci. 12, 291--298 (1974; Zbl 0286.01001)]. This Recto consists of the divisions of the integer 2, by all the odd numbers from 3 to 101, the answers being expressed as the sum of no more than four unit fractions, none of which is as small as \(\frac1{1000}\). One must also understand that in their division and multiplication of all numbers and unit fractions, it was only necessary for the Egyptian scribes to multiply by 2, that is, in modern terms, to use the Twice-times Table, and this they certainly knew. If, for example, \(23\times 15\) was to be calculated, the product was treated as follows: \(23\times (1 + 2 + 4 + 8) = (23 + 46 + 92 + 184) = 345\), and no other way. Conversely, the division of 345 by 15 was done by the continuous multiplication of 15 by 2, until 345 was reached, as follows: \(15\times (1+2+4+8+16) = (15+30+60+120+240)\), no further, for 480 is greater than 345. Now what combination of these products produces 345? Clearly, \((15+30+60+240) = 345\), so the quotient is the sum of the corresponding multipliers of \(15\times (1+2+4+8+16)\), which is, \((1+2+4+16)=23\). In these examples, all the numbers are integers. But when either the multiplicand or the multiplier, the dividend or the divisor, contained unit fractions, then quite intricate difficulties could supervene. Thus suppose the multiplication of \(23\frac17\) by \(15\) was required. The first doubling cannot be \(46\frac27\), for no fraction can have a numerator other than unity for it could not be written, nor could it be \(46\frac17\frac17\) because it was never permitted to put two equal unit fractions together. No historian has yet explained, or even hazarded a guess why this restriction was made by the ancient Egyptians. The author here suggests the reason could be that a second doubling would produce \(92\frac17\frac17\frac17\frac17\) while successive doublings would produce 8, 16, 32, etc., equal unit fractions for even relatively small multipliers, which would take up impossible space on the papyrus, for the fractions like \(\frac1{29}\) or \(\frac1{101}\). For even unit fractions the problem still existed though to a lesser degree. Thus the doubling of \(\frac1{18}\), \(\frac1{44}\), \(\frac1{56}\) would produce \(\frac19\), \(\frac1{11}\), \(\frac17\) after only one, two or three doublings, so that odd unit fractions re-appear, and the difficulty again presents itself. The scribes therefore solved the problem in their own way. Nowhere is it shown in available papyri how they did it, and today modern historians are not all agreed upon what their method was. The Recto as mentioned earlier, gives the 50 scribal values in unit fractions, for the doubling of all the odd unit fractions up to \(\frac1{101}\) but does not show how their excellent answers were achieved. This we would all like to know. What A'hmosé, the scribe of the RMP does in the Recto, is to state what the selected answer is, and then proves it to be correct by multiplication. His errors are rare and few of the equalities can be improved upon, even though modern computers have produced thousands of alternative values for comparison. A present day mathematician observing that the Recto states, \(\frac17\times 2\) is equal to \(\frac14 + \frac1{28}\) might reason thus: \[ \frac27 = \frac4{14} = \frac{3+1}{14} = \frac6{21} = \frac{3+2+1}{21} = \frac8{28}= \frac{7+1}{28} = \frac14 + \frac1{28} \] and say, ``That's how it was done''. It would only be necessary to try the method for \(\frac2{53}\), \(\frac2{89}\) or \(\frac2{97}\) to decide that there must be another way of doing it without using computers. Now the EMLR contains 26 equalities of the addition of unit fractions, in duplicate like, \[ \frac19 + \frac1{18}= \frac16, \quad \frac17+\frac1{14}+\frac1{28}= \frac14, \quad \frac1{15}+\frac1{25}+\frac1{75}+\frac1{200}=\frac18, \] relations which would be most useful in the calculations of the Recto of the RMP. In conformity with the Recto, the scribe of the EMIR does not show any steps of his calculations, nor indeed does he prove the equalities are correct. The author in this article shows that the EMIR tables properly arranged and suitably extended, would be of the same value to the Egyptian scribes, as a standard book of mathematical tables would today, be of use to a modern calculator. Over several pages of comparison and discussion of the various unit fraction equalities of the EMIR and the Recto of the RMP, he establishes relations between them hitherto not remarked or observed, and thence offers suggestions why the two papyri were found together. He concludes the article with Figure B, which is his own conception of another EMLR in hieratic characters, as it might have been prepared by a competent, dedicated, Egyptian scribe, who would have been thereby the author of the world's earliest text of Mathematical tables. If such papyri were ever prepared over two millenia, none happens to have come down to us. We have only the poorly bound EMLR. Autorreferat.