Analysis and synthesis (and other peculiarities): Euclid, Apollonius. 2 th March 2014

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Joy Davis
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1 Analysis and synthesis (and other peculiarities): Euclid, Apollonius 2 th March 2014

2 What is algebra? Algebra (today): Advanced level : Groups, rings,..., structures; Elementary level : equations. The language of elementary algebra is connected to Analytic Geometry. Classical mathematics was mainly Geometry (Geometric algebra a hidden tool?): Elementary level : Euclid; Advanced level : Apollonius, Archimedes,... Pappus

3 What is algebra? Development of algebra: Al-Khwarizmi ( ?),the great Arabian mathematicians, Fibonacci, Cardano, Tartaglia, Bombelli, Viète, Descartes Three forms: Rhetoric, syncopated, symbolic; Descartes: algebra (elementary) reaches its actual form; Is Algebra the very substance of mathematics (concealed by the Ancients)? Analysis versus synthesis

4 Analysis and synthesis Euclid s Elements are not only a classic. They also constitute the fundamental model later adopted for each scientific discipline. They have, and should have, an important role in our cultural education. The sharing of these ideas does not mean to accept uncritically their content and the form of their presentation. In particular, the drastic choice of Euclid to favour the synthesis as the unique form of presentation of his results must be critically assessed. The following examples are particularly interesting.

5 Problem (II.11 of the Elements). Given a segment AB we have to determine a point X on it so that the square construct AB XB. The problem may be propos AB : AX = AX : XB... Proportion theory (the more

6 Euclid s solution is given by the following picture

7 The proof (in geometrical terms...) [ CG AG + AH 2 = GH 2 (a + x)x + a2 ( a ) ] 2 4 = 2 + x GH 2 = HB 2 = AH 2 + AB 2. CG AG = AB 2 [ (a + x)x = a 2 ] (a + x)x = a 2 is equivalent to x 2 = a(a x),...

8 Hidden analysis (?) a : x = x : (a x) x 2 = a(a x) x 2 + ax = a 2, x a 2 x + a2 4 = a2 + a2 4, ( x + a ) 2 = a 2 + a2 2 4, x = a 2 + a2 4 a 2. It is quite easy to revert this argumentation, even in a geometrical form. Since it is not required a particular mathematical ability to recover the analysis, we could assume that Euclid had left this task to the reader.

9 Problem (IV.10): an isosceles triangle such that Analysis: AH = x, HC = y, AH + HC = a; BH bisector: AB : AH = BC : CH (VI.3), and AH = a : x = x : (a x): Euclid s construction!

10 In numerical terms

11 Hidden analysis or privilege given to the logical structure? No numbers in the Elements: 36, 72,... The theorem of the bisector is VI.3 and requires the tools of proportion theory (Book V). It cannot be used in advance. One thing is to use some arguments to produce the analysis of a problem but a different thing is to produce a proof with a minimum of instruments. The Elements give no room to the intuition (visual or logical. The following example shows that Euclid does not use the visual symmetry as a demonstrative tool.

12 Proposition III.16. A line at right angle to the diameter of a circle from its extremity is a tangent.

13 An example: a proof in Apollonius Conics: I.33.

14 By contradiction.

15 A little help given by modern algebra: T V = V P = x, T P = 2x, P Q = y. The inequality becomes 4x(x + y) > (2x + y) 2, that is 0 > y 2. This result, and many other results in advanced mathematics (of the classical period), cannot be considered in terms of analysis/synthesis if taken as isolated results. They became matter of analysis when the whole approach is changed. This result, in particular, may be come an object of analysis within the algebraic theory of conic section.

16 An example in Pappus (Collections VII, Prob. 105). The circle and the points D, E are given. It is required to determine a point B such that DB and EB intersect the circle in C, A so that AC is parallel to DE.

17 Suppose the problem solved and draw the tangent AZ at A intersecting DE in Z. ẐAB = ÂCD = ĈDE. Hence ẐAB + BDZ = π. A, B, D, Z are concyclic. Then EB EA = ED EZ. EB EA = EF 2 is given. Hence ED EZ is given. The point Z is determined. Hence A is determined (in two possible ways...)

A plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.

A plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane. Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their