History of Mathematics
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1 History of Mathematics Paul Yiu Department of Mathematics Florida tlantic University Spring : Pythagoras Theorem in Euclid s Elements
2 Euclid s Elements n ancient Greek mathematical classic compiled in the Third century.. The Elements consists of 13 books. I VI VII IX X XI XIII Plane geometry Number theory Theory of irrational constructible quantities Solid geometry 1
3 Euclid, ook I ook I II III IV V VI Total Definitions ommon notions 5 Postulates 5 Propositions
4 Euclid, ook I: The definitions (1) point is that which has no part. (2) line is breadthless length. (4) straight line is a line which lies evenly with the points on itself. (10) When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. (11) n obtuse angle is an angle greater than a right angle. (12) n acute angle is an angle less than a right angle. (15) circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. (16) nd the point is called the center of the circle. (23) Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. 3
5 Euclid ook I: The postulates (1) To draw a straight line from any point to any point. (2) To produce a finite straight line continuously in a straight line. (3) To describe a circle with any center and radius. (4) That all right angles equal one another. 4
6 Euclid ook I: The fifth postulates (5) That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 5
7 The common notions (1) Things which equal the same thing also equal one another. (2) If equals are added to equals, then the wholes are equal. (3) If equals are subtracted from equals, then the remainders are equal. (4) Things which coincide with one another equal one another. (5) The whole is greater than the part. 6
8 Euclid I.1 On a given finite straight line to construct an equilateral triangle. onstruct the circles (, ) and (,) to intersect at a point. Then is an equilateral triangle. D E 7
9 Euclid I.4 If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. ongruence test SSS: XY Z if = XY, = YZ, = ZX. X Z Y 8
10 Isosceles triangles I.5 n isosceles triangle [is] that which has two of its sides alone equal, and a scalene triangle [is] that which has its three sides unequal. 1 Euclid I.5. In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Proof. Extend and tod ande respectively. hoose an arbitrary point F on D, and (by I.3) constructgone such that G = F. The triangles F and G are congruent by I.4. It follows that F = G and F = F = G = G. F G gain, by I.4, the triangles F and G are congruent. D E From this, F = G, and =. 1 Euclid seems to take isosceles and scalene in the exclusive sense. ut it is more convenient to take these in the inclusive sense. n isosceles triangle is one with two equal sides, so that an equilateral triangle is also isosceles. 9
11 Euclid I.6: converse of I.5 (6) If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. 10
12 Euclid I.4,8,26 (congruence tests) Euclid did not use the term congruence of triangles. When he says two triangles are equal, he means they are equal in area. Euclid I.4, SS: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles equal the remaining angles respectively, namely those which the equal sides subtend. Euclid I.8, SSS: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will have the angles equal which are contained by the equal straight lines. Euclid I.26, S or S: If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle. The RHS test is not in the Elements. 11
13 ongruence tests (1) SSS: XYZ if = XY, = YZ, = ZX. X Z Y (2) SS: XYZ if = XY, = XYZ, = YZ. X Z Y 12
14 ongruence tests (3) S: XYZ if = YXZ, = XY, = XYZ. X Z Y (4) S. We have noted that this is the same as S: XYZ if = YXZ, = XYZ, = YZ,. X Z Y 13
15 ongruence tests (5) RHS. The SS is not a valid test of congruence. Here is an example. The two triangles andxyz are not congruent even though = YXZ, = XY, = YZ. Y X However, if the equal angles are right angles, then the third pair of sides are equal: Z Y 2 = 2 2 = YZ 2 XY 2 = XZ 2, and = XZ. The two triangles are congruent by the SSS test. Without repeating these details, we shall simply refer to this as the RHS test. XYZ if Z = YXZ = 90, = YZ, = XY. X 14
16 Parallel lines Euclid I.27. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another. Euclid I.28. If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. 15
17 Parallel lines Euclid I.29. straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles. Euclid I.30. Straight lines parallel to the same straight line are also parallel to one another. 16
18 Parallelograms Euclid I.34. In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas. D Euclid I.35 Parallelograms which are on the same base and in the same parallels are equal to one another. D E F 17
19 Some constructions with parallel lines Euclid I.31. Through a given point to draw a straight line parallel to a given straight line. I.42. To construct, in a given rectlineal angle, a parallelogram equal to a given triangle. I.44. To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle. I.45. To construct, in a given rectlineal angle, a parallelogram equal to a given rectlineal figure. 18
20 Euclid I I.36. Parallelograms which are on equal bases and in the same parallels are equal to one another. I.37. Triangles which are on the same base and in the same parallels are equal to one another. I.38. Triangles which are on equal bases and in the same parallels are equal to one another. I.39. Equal triangles which are on the same base and on the same side are also in the same parallels. I.40. Equal triangles which are on equal bases and on the same side are also in the same parallels. I.41. If a parallelogram have the same base with a triangle and be in the same parallels, then the parallelogram is double of the triangle. 19
21 Euclid I.46: onstruction of a square Euclid I.46. To describe a square on a given straight line. (4) (5) (7) D (8) (3) (2) (6) (9) (1) 20
22 Euclid VI.3: ngle bisector theorem If an angle of a triangle is bisected and the straight line cutting the angle cuts the base also, the segments of the base will have the same ratio of the remaining sides of the triangle; and if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of the section will bisect the angle of the triangle. D E 21
23 External angle bisector theorem This is also true for the bisector of the external angle: IfE is a point on the extension of the side, E bisects an external angle of if and only if E : E = :. D Figure 1: E orollary. Let and be given points. The locus of a point P whose distances from and are in a constant ratio (not equal to 1) is a circle. 22
24 Euclid I.47 In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. There are many proofs of this fundamental theorem in geometry. The proof given by Euclid is important because it gives a way of converting a rectangle into a square. 23
25 Euclid I.42,43 Euclid I.42. To construct, in a given rectilineal angle, a parallelogram equal to a given triangle. F G D E Euclid I.43. In any parallelogram the complements of the parallelograms about the diameter are equal to one another. H D E K F G 24
26 Euclid I.44,45 Euclid I.44. To a given straight line to apply, 2 in a given rectilineal angle, a parallelogram equal to a given triangle. F E K D G M H L Euclid I.45. To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure. F G L D E K H M Figure 2: Euclid I.45 2 The term application is explained in Proclus commentary: application starts with one side given and constructs the area along it, neither falling short of the length of the line nor exceeding it, but using it as one of the sides enclosing the area. 25
27 Proof of I.47 K K G G H H F F M M D L E D L E SquareFG = 2 F = 2 D = rectangledlm. Similarly, SquareHK = rectangleelm. Therefore, SquareFG + squarehk = squarede. 26
28 Euclid I.48: converse of I.47 If in a triangle the square on one of the sides be equal to the squares on the remaining sides of the triangle, the angle contained by the remaining sides of the triangle is right. Y a c a b Z b X Proof. Let be a triangle for which = 2. onsider a right triangle XYZ with YZ =, ZX = and XZY = a right angle. y Euclid I.47, XY 2 = YZ 2 + ZX 2 = = 2 ; XY =. Therefore, the triangles and XY Z are congruent by Euclid I.8, and = XZY = a right angle. 27
29 useful corollary of Euclid I.47 Let the perpendicular from the right angle vertex of a right triangle intersect the hypotenuse at X. Then 2 = X, 2 = X. X 28
30 Euclid II.14 To construct a square equal to a given rectilineal figure. H G E F D Proof without words: H H G E F G E F D D 29
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