MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000
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 The content of Euclid
Book II involves the geometry of rectangles. Book II contains two
definitions and fourteen propositions. It is often viewed as geometrical algebra
because if you investigate what Euclid is actually saying, most of the
propositions can be translated into algebraic equations with which we are
familiar. This is very interesting considering that at the time Euclid wrote the
Elements, algebra had not yet been invented <<Link to Missy’s>>.
It is also interesting that many of the algebraic equations that can be used to
represent Euclid’s work in Book II can also be used to represent the work of
the Babylonians long before Euclid’s time <<Link to Chuck’s>>.
We can look at Book II in a way in which not only the geometrical approach
Euclid took makes sense, but also the algebra that lies beneath it is clearly identifiable.
Proposition 1If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.  If we take what is given we first
draw two straight lines. We’ll call them lines a and b, as seen in the diagram
at the left.
Now as you see since line DE is equal to line AF, the rectangles contained by the uncut straight line and each of the segments are rectangles ACGF and CDEG. Therefore, it is obvious that the rectangle contained by the two straight lines (rectangle ADEF) is equal to the rectangles contained by the uncut line and each of the segments (rectangle ACGF and rectangle CDEG). My interpretation of proposition 1 is given any two lines with one cut randomly any number of times, the area of the rectangle contained by the two lines is equal to the area contained by the uncut line and each of the segments. This looks a lot like our algebra. If we ignore the rectangles and just focus on the lines this is saying that a whole is equal to the sum of parts. There are no limitations in this proposition of what the whole is or how large it is. There are also no limitations on the number of parts. Thinking of the proposition in this aspect, and using the algebraic notation that we are used to dealing with we can make a line y = y1 + y2 + ….yn where n is the number of segments, and by this proposition any number x multiplied times the original line will give us xy = yx1 + yx2 + …yxn. This is what we call the distribution law for multiplication over addition.
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We next want to cut one of
them into a number of segments, so for simplicity we will cut line a into two
segments and call this point C, shown at the right.
The proposition then states
that the rectangle contained by the two straight lines is equal to the
rectangles contained by the uncut straight line and each of the segments. So let
us first look at the rectangle contained by the two straight lines which we will
call rectangle ADEF shown in the figure at the left. Since this is a rectangle,
we’ll call the sides equal to line a = X, and the sides equal to line b = Y.
Now we look at the rectangle
contained by the uncut line and each of the segments.
