MI314 History of Mathematics: Episodes in Non-Euclidean Geometry

Transcription

1 MI314 History of Mathematics: Episodes in Non-Euclidean Geometry Giovanni Saccheri, Euclides ab omni naevo vindicatus In 1733, Saccheri published Euclides ab omni naevo vindicatus (Euclid vindicated om all faults), in which he attempted to prove Euclid s 5 th postulate. He began by introducing the geometric object now known as a Saccheri quadrilateral. C B Saccheri quadrilateral has two right angles at the base, = =B R 90. EV rop. 1: Saccheri then showed that =C = ðñ C B. He then proceeds by setting out three possible hypotheses. Hypothesis of the Right ngle (HR): =C = R 90, Hypothesis of the Obtuse ngle (HO): =C = ą R 90, Hypothesis of the cute ngle (H): =C = ă R 90. 1

2 EV rop. 3: He then shows that HR ùñ B C, HO ùñ B ą C, H ùñ B ă C, which can be used to show that HR ùñ sum of angles of a 2R 180, HO ùñ sum of angles of a ą 2R 180, H ùñ sum of angles of a ă 2R 180. The rest of this part of the text is structured as an extended reductio ad absurdum. art 1 He uses HR to prove Euclid s 5 th postulate. art 2 He uses HO to show that the geometry so created is inconsistent with Elem. I 16 & ⒘ [nd, hence, Saccheri believes HO is false.] art 3 He uses H to argue that it leads to consequences that are repugnant to the nature of the straight line. [nd, hence, Saccheri believes H is false.] If his arguments were sound, this would lead to a proof of the 5 th postulate, however, there are problems with the conclusions of arts 2 and ⒊ We will examine some of the arguments om these two parts to see if we can uncover where Saccheri has gone wrong. EV rops. 11 & 12 N 1 N 2 N 3 N 4 N 5 M 1 M 2 M3 M 4 M 5 rops. 11 & 12 (HR, HO). If a line falls on two given lines such that it is perpendicular to one and makes an acute angle with the other, then the two given lines will meet in the direction of the said acute angle. 2

3 roof. et line fall on and, such that K and = is an acute angle. To show that will intersect. We cut of segments M 1 M 1 M 2 M 2 M 3, and so on; and draw M 1 N 1 M 2 N 2 M 3 N 3, and so on. Then N 1 ď N 1 N 2 ď N 2 N 3 ď N 3 N 4 ď... If this process is continued indefinitely, some N n will fall beyond. Then if we draw N n M n, M n will fall beyond and M n will meet. Key. We construct a sequence of segments om towards which increases at least arithmetically, and hence some segment will fall beyond. oes this argument work on the sphere? re there constructions in the argument that are not permissible on the sphere? Note that the theorem may still be true, even if the argument does not work. EV rop. 13 (HR, HO) ' rop. 13. If a line falls on two given lines and makes the internal angles in the same direction less than two right angles, then the two given lines will meet. Moreover, this implies that in a triangle, two angles can be equal to two right angles. roof. et line fall on given lines and, such that = ` = ă 2R. I say that will meet. We drop K, so that, by EV rops. 11 & 12, will meet. But in HO, = ` = ą R, since the angles of a are greater than 2R. Therefore, we can set = ` = ` = 2R. 3

4 But since meets at some point, say 1, then we can say = 1 ` = ` = 2R. But = 1 ` = = 1 (see the figure), so = 1 ` = 1 2R. So in 1, two angles are equal to 2R. But in Elements I 17, it is shown that two angles of a triangle are less than two right angles. Key. We use the properties of HO to construct a triangle that has two angles equal to two right angles. EV rop. 14 rop. 14. The hypothesis of the obtuse angle is absolutely false, since it destroys itself. In fact, we have shown that HO contradicts Elements I ⒘ Hence, we should look closely at Elements I 17 and its use of Elements I 16, to see what this contradiction really means. EV rops. 32 & 33 (H) C H Z N M B K K K rop. 32. We show that if H is true, then there will exist a line,, with the following properties: is a limit to the set of lines which meet B and also to a set of lines which have two distinct perpendiculars, like Z. 4

5 It meets B at one point, infinitely distant. It always approaches closer and closer to B. It is a straight line. rop. 33. The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of a straight line. What is the real basis of this objection? What do we know about the nature of a straight line, and how do we know it? 5