{it From simple to complex and intricate}{it From simple to complex and intricate}{it From simple to complex and intricate}{it Rules of the ``game"!}{it Rules of the ``game"!}{it Rules of the ``game"!}{it The notion of quadrature}{it The notion of quadrature}{it The notion of quadrature}{it Quadrature of the rectangle}{it Quadrature of the rectangle}{it Quadrature of the rectangle}{it Quadrature of the rectangle}{it Quadrature of the rectangle}{it Quadrature of the rectangle}{it Proof of claim}{it Proof of claim}{it Proof of claim}{it Proof of claim}{it Proof of claim}{it Proof of claim}{it Proof of claim}{it Quadrature of the triangle}{it Quadrature of the triangle}{it Quadrature of the triangle}{it Quadrature of the triangle}{it Quadrature of the triangle}{it Quadrature of the polygon}{it Quadrature of the polygon}{it Quadrature of the polygon}{it Quadrature of the polygon}{it Quadrature of the polygon}{it Quadrature of the polygon (cont.ed)}{it Quadrature of the polygon (cont.ed)}{it Rectilinear vs curvilinear figures}{it Rectilinear vs curvilinear figures}{it Rectilinear vs curvilinear figures}{it Rectilinear vs curvilinear figures}{it Three famous problems from antiquity}{it Three famous problems from antiquity}{it Three famous problems from antiquity}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates of Chios {m (470-410 BC)}}{it Hippocrates' Elements}{it Hippocrates' Elements}{it Hippocrates' Elements}{it Hippocrates' lune}{it Hippocrates' lune}{it Hippocrates' lune}{it Hippocrates' lune}{it What is a lune?}{it What is a lune?}{it What is a lune?}{it Comments (1)}{it Comments (1)}{it Comments (1)}{it Comments (1)}{it Comments (1)}{it Comments (2)}{it Comments (2)}{it Comments (2)}{it Hippocrates' Theorem}{it Hippocrates' Theorem}{it Hippocrates' Theorem}{it Hippocrates' Theorem}{it Strategy of proof}{it Strategy of proof}{it Strategy of proof}{it The proof}{it The proof}{it The proof}{it The proof}{it The proof (conclusion)}{it The proof (conclusion)}{it The proof (conclusion)}{it The proof (conclusion)}{it Epilogue}{it Epilogue}{it Epilogue}{it False argument}{it False argument}{it False argument}{it False argument (cont.ed)}{it False argument (cont.ed)}{it False argument (cont.ed)}{it False argument (end)}{it False argument (end)}{it False argument (end)}{it Ferdinand Lindemann and more}{it Ferdinand Lindemann and more}{it Ferdinand Lindemann and more}{it Ferdinand Lindemann and more}{it Possible ideas for the final project}{it Possible ideas for the final project}{it Possible ideas for the final project}{it Possible ideas for the final project}Hippocrates’ Quadrature of the Lune(ca. 440 BC)presentation byAlberto CorsoMA 330 — History of Mathematics– p. 1/27From simple to complex and intricate•Ancient Greeks were enthralled by the symmetries, thevisual beauty, and the logical structure of geometry.•Particularly intriguing was the manner in which thesimple and elementary could serve as foundation forthe complex and intricate.•This enchantment with building the complex from thesimple was also evident in the Greeks’ geometricconstructions.– p. 2/27Rules of the “game”!•The rules of the game required that all constructions bedone only with compass and (unmarked) straightedge•These two fairly unsophisticated tools—allowing thegeometer to produce the most perfect, uniformone-dimensional figure (the straight line) and the mostperfect, uniform two-dimensional figure (the circle)—must have appealed to the Greek sensibilities for order,simplicity and beauty.•Moreover, these constructions were within reach of thetechnology of the day.– p. 3/27The notion of quadrature•These seemingly unsophisticated tools can produce arich set of constructions (from the bisection of lines andangles, to the drawing of parallels and perpendiculars,to the creation of regular polygons of great beauty).•A considerably more challenging problem in ancientGreece was that of the quadrature of a plane figure.•The quadrature (or squaring) of a plane figure is theconstruction—using only compass and straightedge—ofa square having area equal to that of the original figure.If this is the case, the figure is said to be quadrable (orsquarable).– p. 4/27Quadrature of the rectangleLet ABCD an arbitrary rectangle.We must construct, with compassand straightedge only, a squarehaving area equal to that of ABCD•OABDCE FGHExtend line AD to the right, and use the compass to markoff segment DE with length equal to that of CD.Bisect AE at O, and with center O and radiusAO = EO,describe a semicircle as shown.At D, construct line DH perpendicular to AE, where H is thepoint of intersection of the perpendicular and the semicircle.Construct the “desired” square ... DFGH.– p. 5/27Proof of claimWhy does the square DFGHhave the same area asthe rectangle ABCD?•OABDCE FGHabcSet a, b, c to be the lengths of segments OH, OD and DH.Pythagoras theorem gives us thata2− b2= c2.Observe that:DE = CD = a − b and AD = a + b.It follows that:Area (rectangle ABCD) =AD × CD=(a + b)(a − b) = a2− b2= c2= Area (square DFGH)– p. 6/27Quadrature of the triangleGiven a triangle ABC, constructa perpendicular from C meetingAB at point H.•HA BDECMWe know that the area of the triangle ABC is12AB × CH.If we bisect CH at M and construct a rectangle ABDE withDB = EA = MH, we obtain a rectangle with the same areaas the triangle ABC.But we already have seen how to square a rectangle.– p. 7/27Quadrature of the polygonGiven a general polygon we cansubdivide it into a collection ofntriangles, by drawing diagonals (eg n=3).A1A2A3A BCDENow, triangles are quadrable.We can construct squares with sidesa1, a2, a3and areas A1, A2, A3.Construct a right triangle with legsa1, a2and hypotenuse d1. Next construct a trianglea1a2a3d1d2with legs d1, a3and hypotenuse d2. We have:d22= d21+ a23= (a21+ a22) + a23= A1+ A2+ A3.– p. 8/27Quadrature of the polygon (cont.ed)•Obviously, this procedure can be adapted to thesituation in which the polygon is divided into anynumber of triangles.•By analogous techniques, we could likewise square afigure whose area is the differencebetween twosquarable areas.– p. 9/27Rectilinear vs curvilinear figures•With the previous techniques, the Greeks of the 5thcentury BC could square wildly irregular