Euclid’s “Geometric Algebra”Mariusz WodzickiEuclid considers plane figures (sq¨mata) which are certain subsets ofthe plane. This includes all rectilinear figures. Below, I focus my attentiononly on the latter ones.Let us consider the following three equivalence relations on setSofall rectilinear figures. Figures σ and σ0are said to:• be congruent (“fitted” to each other, to use a literal translation ofEuclid’s terminology) if one can be “placed” over the other withperfect match. Such a placement can be always accomplished as aresult of rigid motion of the whole plane ( notation: σ ∼cσ0, thecongruence class of σ will be denoted¯σ);• be scissors-equivalent if both figures can be dissected into a finitenumber of mutually congruent figures (notation: σ ∼sσ0, the equiv-alence class of σ will be denoted [σ]);• be equivalent in the sense of Euclid if our figures are obtained byremoving a finite number of congruent figures from two scissors-equivalent figures (notation: σ ∼eσ0; the equivalence class of σ willbe denoted |σ| ). Subsequently, I will often refer to such figures as“equal”.Each subsequent relation is a priori weaker than its predecessor:σ ∼cσ0⇒ σ ∼sσ0⇒ σ ∼eσ0. (1)Each satisfies the conditions spelled out in the Common Notions section ofBook I. Only the last equivalence relation, however, satisfies an additionalrequirement of Euclidean Geometry, first implicitly referred to in the proofof fundamental Proposition i.35:1If one removes congruent pieces from two“equal” figures then one again obtains“equal” figures.(2)1(...) the triangleEABwill be equal to triangleDZG. Let (triangle)DGEhave been takenaway from both. Thus the remaining trapeziumABHDis “equal” to the remaining trapezium .1Equivalence classes of congruent figures are naturally represented by“free” figures, i.e., figures literally “lifted into the air” from their actuallocation on the plane.Addition of segments. Given two straight line segments (“segments”,in short)ABandGD, we can attach one to the other in a straight line. Theresulting segments depend on whether we attachGDtoAB, or vice-versa,and whether we place pointGatBor atA. The resulting segments arehowever congruent to each other.Moreover, if we replace segmentsABandGD, by segmentsABandGDcongruent toABandGD, respectively, then the resultsing segmentsare still congruent. We obtain thus a well defined operation of additionon the setEof congruence classes of segements which can be identifiedwith the set of “free” segements:+ :E×E−→E. (3)Equipped with binary operation (3) the set of free segments becomes acommutative semigroup.One can define addition of angles exactly in the same manner.Addition of figures. If we try to do the same with arbitrary recti-linear figures, the resulting figures will not be congruent to each otherin most cases, they will be scissors-equivalent instead. This behavior isonly seemingly different from the behavior of the operation of attach-ment for segments: for segments, the relations of being congruent andscissors-equivalent coincide!IfSdenotes the set of all (finite) rectilinear figures, letS'˜S/∼s(4)denote the set of scissors-equivalence classes of rectilinear figures. At-tachment of figures yields thus a well defined binary operation on setS':+ :S'×S'−→S'(5)which makes it a commutative semigroup in complete similarity to theset of free segmentsE.Euclid’s equivalence relation ∼eis compatible with this operation ofaddition: given two “equal” figures, σ ∼eσ0, attaching either of them to a2figure τ produces “equal” figures. In particular, oparation (5) induces anoperation of addition also on the set of Euclid’s equivalence classes:S''˜S/∼e, (6)so that we have the following commutative diagramS'×S'S'S''×S''S''w+uuuuw+(7)where the vertical maps correspond to the canonical mapS'−→S''whichsends any equivalence class, [σ], of relation ∼sto the correspondingequivalence class, |σ|, of relation ∼e.Multiplication of segments. Given two segments λ and µ, one canconstruct a rectangle having λ and µ as its sides. Its congruence classdepends only on the congruence classes of λ and µ. We will denote itλ × µ.We thus obtain a pairing:×:E×E−→S. (8)If we fix one of the segments, say λ, then the induced mapE−→S, µ 7→ λ ×µ, (9)is injective: rectangles having pairwise equal sides are congruent.Euclid’s crowning achievement in Book I of the Elements is the follow-ing theorem.Proposition i.45 Given a segment λ and a figure σ, there exists a segmentµ such that |σ| = |λ × µ|. Segment µ is unique up to congruence.The same in modern mathematical idiom:for any segment λ, the mapE−→S'', µ 7→ |λ ×µ|,defines an isomorphism between the semigroup,E,of free segments and the semigroup,S'', of Euclid’sequivalence classes of figures.(10)3Let us fix some segment η ∈E. The inverse map to (10)mη:S''−→E, |σ| 7→ the unique µ ∈Esuch that |σ| = |η × µ|, (11)whose existence is guaranteed by the above theorem of Euclid, can beused to define multiplication onE:E×E−→E, µµ0˜ mη(µ ×µ0). (12)It is essential to remember that this multiplication depends on the choiceof segment η even though our notation does not reflect it.Exercises.(a) What ηµ is equal to?(b) Show that multiplication (12) is distributive with respect to addi-tion of segments.(c) What is the meaning of µµ0?Associativity and commutativity of multiplication (12) is a questioninvolving a priori theory of volume for 3-dimensional figures. There ishowever an approach that allows one to stay within the framework ofplane geometry. It is based on the theory of similar figures which is devel-oped in Book VI, and on the celebrated Theorem of Pappus, another greatmathematician from Alexandria, who lived there some 6 centuries later.He was the last significant mathematician of Antiquity.With all this additional nontrivial work it is possible to demonstratethat the set of free segments,E, equipped with operations of addition, (5),and multiplication, (12) becomes a semifield.2Having chosen the unit segment η, one can identify Pythagorean Arith-metic as the arithmetic of segments of special kind, namely of positiveintegral multiples of η (Euclid would have said: of segments that are mea-sured by η).To me this seems to be the key to understanding the methods of BookVII of the Elements. On the other hand, theory of