A History of Mathematics From Mesopotamia to Modernity

46 A History ofMathematics Octahedron Cube [Hexahedron] Tetrahedron Icosahedron Dodecahedron Fig. 3The five regular (‘Platonic’) ...

Greeks and‘Origins’ 47 should naturally also come across a theory of proportions, in which no arithmetical parts remain. (Hasse ...

48 A History ofMathematics how they can be so sure; for example, there is nothing to suggest this in theElements. (Fowler, contr ...

Greeks and‘Origins’ 49 challenge is to reconstruct what the two did. The main fact known about Democritus is that he was an ‘ato ...

50 A History ofMathematics This complex definition underlies the statement (see Section 3) that circles are in the same ratio as ...

Greeks and‘Origins’ 51 recourse is made to fractions of all sorts as approximations. It is hard to go further, and a concept lik ...

52 A History ofMathematics Soc. That is, from the line which extends from corner to corner of the figure of four feet? Boy. Yes. ...

Greeks and‘Origins’ 53 A BC DE Fig. 4Constructing a regular pentagon. The angles of the pentagon are 108◦, so angles like ADE ar ...

54 A History ofMathematics However, there is a simpler proof which is generally thought to be a probable reconstruction, as foll ...

Greeks and‘Origins’ 55 Mirror BD A C Fig. 6Idea of proof by reflection of ‘Thales’ theorem’. Sop= 2 r,say,andp^2 = 4 r^2. Rewrit ...

56 A History ofMathematics (3, 3) (tetrahedron), (3, 4) (octahedron), (4, 3) (cube), (3, 5) (icosahedron), and (5, 3) (dodecahed ...

3 Greeks, practical and theoretical 1. Introduction, and an example But, unlike Euclid, who attempts to prove musical propositio ...

58 A History ofMathematics H D F E C B A K Fig. 1Menaechmus’s construction (from Fauvel and Gray p. 86). A and E are given, and ...

Greeks,Practical andTheoretical 59 S k:2 a:2 TK L AEMNZ H DP G O R B S k:2 a:2 TKL AEMNZ H DP O G R B Fig. 2The ‘mesolabe’ (Show ...

60 A History ofMathematics including Archimedes, Heron of Alexandria, and Ptolemy. All were tremendously influential in the late ...

Greeks,Practical andTheoretical 61 One suspects that Plutarch had not read the works which he describes as ‘smooth and rapid’, s ...

62 A History ofMathematics Proposition 2.The area of any circle is to the square on its diameter as 11 is to 14. Proposition 3.T ...

Greeks,Practical andTheoretical 63 Fig. 3Regular hexagon circumscribed about a circle. B C uu A
A–A
Fig. 4Picture for Exercise ...

64 A History ofMathematics Fig. 5Heron’s slot machine. Heron in hisAutomataandBalancings,...or by using water to tell the time, ...

Greeks,Practical andTheoretical 65 Archimedes (from whom Heron derived several other results).^5 If this is true, it shows Archi ...