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[HM] Egyptian Algebra and the Method of False Position
Posted:
Apr 26, 2002 3:33 PM
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This semester I'm teaching a course to prospective middle school
teachers (grades 5 - 8 in the US), and I've been drawing upon some
history of elementary mathematics to design lessons that will enrich the
students' understanding of the topics being studied and that they will
one day teach.
In connection with introductory algebra, I thought I would draw upon
some early developments in algebra, so one topic I looked at was
Egyptian algebra. As I was investigating this more closely to teach it
to my students, however, I was led to question the standard way of
viewing what's going on in the Rhind Mathematical Papyrus (RMP) with
respect to algebra. My questions were prompted by something van der
Waerden says in his Geometry and Algebra in Ancient Civilizations on
page 161. I'll first summarize the standard interpretation, then give
my response based on vdW's remarks.
Problem 26 in the RMP says: a quantity and its fourth added together
become 15. What is the quantity?
The solution goes as follows (calculations omitted):
Calculate with 4: 4 + 1 = 5.
Calculate with 5 to get 15: 15 divided by 5 is 3.
Multiply 3 by 4: the answer is 12.
In explaining this process, most say it is one of false position. You
assume the (possibly) false value of 4 and try it out. It only gives 5,
not 15, so you determine by division what you need to scale this amount
by in order for it to be correct. The scale factor is 3, so multiply
your original guess (4) by this amount, giving 12 as the answer.
van der Waerden notes that Tropfke in a 1980 history of elementary
mathematics (in German) gives another explanation, which seems borne out
by the process itself. Rather than taking 4 as an initial guess, 4 is
taken as the part-size (since fractional fourths are involved). Then
the sum 4 + 1 = 5 tells you how many fourth-parts you have in total.
Since these 5 parts must give 15 in all, each part must be of size 15
divided by 5, or 3. The original quantity was 4 fourth-parts, so it
must be of size 4 x 3 = 12.
No false position occurs here at all; it's just quite straightforward
calculation with fractional parts and part-size determination. In
support of this being the method of part-size (my term) rather than the
method of false position, vdW notes that the scribe takes four threes to
get 12, not three fours (which would be the natural product to take if
you're scaling the original guess of 4 by a factor of 3).
Looking at the resources I have at hand, I'm convinced by what
vdW/Tropfke says on this, but then I'm no expert on Egyptian history of
mathematics. As I noted above, most historians of mathematics talk
about this in terms of false position (e.g., Gillings, Boyer, and Katz
all do). So I wonder whether others on this list have thought about
this and what they might have to say. Treating this as the method of
part-size makes the process more elementary and more transparent to
students who are studying it, it seems to me.
Calvin Jongsma
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