There are four extant sources for this book, the Egyptian Mathematical Leather Roll (EMLR), the Reisner Papyri (RP), the Moscow Mathematical Papyrus (MMP) and the most important The Rhind Mathematical Papyrus (RMP) which was actually a training manual for scribes. Because it is there to teach this final papyrus document is crucial to our understanding of how the ancient Egyptians performed their calculations. This document along with the EMLR are in the British Museum in London, the RP is in the Boston Museum of Fine Arts whilst the MMP is in the Pushkin Museum of Fine Arts in Moscow, as you would expect from its name. Gillings wrote his book in 1971 and one or two errors have since been noted in the mathematical press as further studies have been made of the four sources and almost fifty years have passed since he wrote it, but these are largely technical and the book is mainly correct especially in its overview as to how ancient Egyptians calculated and is pretty comprehensive. Having said that it is definitely not a book for the layman, it is pretty solid mathematics and I would suggest that there is still a gap in the market for a simpler presentation which would introduce those with a curiosity in the subject to more easily come to some understanding as to how this worked.
Ancient Egyptian mathematics was largely overlooked and dismissed by scholars as simplistic especially when compared to that of ancient Greece but that overlooked the fact that it was more than capable of calculating the dimensions of the pyramids. For instance if you want a pyramid 139m high (the size of the Great Pyramid at Giza) just how big a base do you need to start with? It has also ensured that they still haven’t fallen down thousands of years after construction. Also the Egyptians had a large empire, so built irrigation canals, large granaries and temples, and of course had a comprehensive tax system to pay for all this so they must have been highly capable at least in the field of applied mathematics and engineering.
But let’s start at the beginning with the hieroglyphic representation of numbers, one is simply a vertical line, two is two of these and so on until you have nine lines drawn together to represent nine. Now this is easy and like all tally marks rapidly becomes unwieldy so you need symbols to indicate larger numbers and the earlier forms of these are shown below. I love the symbol for a million which appears to be the scribe throwing his hands in the air as if to say wow what a big number, what do I do with this?
In fact the papyrus scrolls were written in hieratic script which is sort of cursive hieroglyphic and is much more difficult to read and it is also important to note that they wrote right to left just as in modern Arabic so to our way of looking at it you would see the units first, then the tens, hundreds etc. There is a quick way of remembering this as animals or birds used in hieroglyphic writing always look towards the direction the writer. In the scrolls we have available to modern study addition and subtraction are regarded as so simple as to not need to show any working out which is unfortunate as this means we don’t actually know how they did it, you just get the required sum and then the answer. However everything beyond that is included and it should be understood that the ancient Egyptians managed their entire means of calculation by merely being able to multiply and divide by two and for reasons that are too complex to go into in this blog they also had the 2/3 times table (usually written down rather than memorised) and used this so extensively that when they needed to find a third of a number they would first get two thirds of it and then halve the answer.
So how did they multiply? Well the example given in the book is for multiplying 7 by 13 and this was done as follows. Start by writing two columns, the first of which has a 1 in it and the second has one of the numbers to be multiplied (this is the second example in the book as I think it is better to understand than the first). Under each number double the figure above until doing so in the column starting with one you would have a number larger than the number you are trying to multiply.
Simply adding up the values opposite the checked values in the first column gives the answer to 7 x 13 which is 91. If the number in the first column isn’t needed to sum to 13 in this case then you simply ignore the corresponding number in the second column. It’s simple really. Division is done the same way but a scribe asked to divide 184 by 8 would instead ask himself how many times do I need to multiply 8 to get to 184 so would create a similar chart to the one above.
Now at this point you hit the issue of fractions which we know that they understood as they had the 2/3 table but the way the ancient Egyptians handled them is definitely beyond me being able to explain here, I will simply say that with the sole exception of 2/3 they did not have any fractions with numerator other than 1, so to express ¾ for example they would write the equivalent of ½ + ¼. As you can imagine this becomes extremely messy very quickly. But the way they expressed a fraction, especially in hieratic is interesting as they drew a line over the number to indicate that it was a fraction and as the numerator was always 1 they didn’t need to show this, two thirds had it’s own specific character so that didn’t cause confusion. Later mathematical systems simply added a numerator above the line to indicate multiples of the denominator so this is where our way of writing fractions almost certainly originates from.
I only want to include one further example from the book and this one I chose as I particularly liked the calculation. I do recommend seeking out this book or the various online papers now available on the subject if you want to take this interesting branch of mathematics further. The calculation is how did they work out the area of a circle? Now courtesy of the ancient Greeks and their discovery of geometry (Euclid in particular) we know that the area of a circle is the square of the radius multiplied by the irrational number π which is 3.14 to two decimal places and that will do for most calculations. Archimedes worked it out to about that in 250BC but that is over 1300 years later than our poor scribe in ancient Egypt so how did he do it?
Well their calculation as given in problem 50 of the RMP is to take the diameter, work out a ninth of that figure, subtract that from the original number and then square the result. The sample problem takes a circle with a diameter of 9 khet (which makes the maths a lot easier), so if the diameter is 9 then a ninth of that is 1, subtracting that from the diameter gives 8. Multiply 8 by 8 as we know how to do above and that gives 64 setat as the area of the circle. It’s a hell of a big circle though as a khet is about 57 yards (roughly 52 metres) so a single setat, or square khet, is roughly 3250 yd² or 2720 m². But how accurate is the result of 64? Well (4½)² x π, which is our way of doing the calculation, gives 63.62 so it’s pretty good. If I ever needed to work out the area of a circle in my head (oddly something I don’t do often) then the ancient Egyptian way is definitely the way to do it, no messing around with π needed.
I was intrigued, so some basic algebra (also not something I do every day anymore) shows that the way the calculation works means that their equivalent of π is 256 / 81 which is 3.16 to two decimal places which explains the accuracy. It is also by far the simplest calculation that is remotely accurate as simply subtracting a ninth from the diameter is very easy. I spent a little more time with a calculator and found that the next easiest fraction that gives a better result is to subtract 4/35ths which is lot more difficult than dividing by 9.
There is considerably more in the book for a keen mathematician to have fun with such as calculations of volumes and of course the all so important dimensions of a pyramid and truncated pyramid (i.e. one still under construction). So once you can get past the slightly confusing way it is written it’s fun to work through, preferably with some paper and a pencil nearby to do some quick calculations of your own.