Postulates! Are Euclid-ing Me? Quiz

Answer:

The approach to geometry taken in 'Elements' relies heavily on propositional logic, starting with the most fundamental aspects (definitions and postulates often left standing as "self-evident" statements) and, from those fundamentals, slowly building up ever more elaborate propositions. The dependence upon propositional logic reflects the fact that geometry was studied as one of the liberal arts in antiquity and - alongside arithmetic, music and astronomy - formed part of the quadrivium in medieval curricula. In contrast, modern geometry requires a great deal of algebraic manipulation.

Throughout history, four of the five postulates from Book I of 'Elements' have not been considered controversial. The first postulate, which in plain terms says that a straight line can be drawn between two points, builds upon two definitions from earlier in the book, namely that "a point is that which has no part" and "a line is breadthless length". The English language wording for these definitions, and all other terms used here, are taken from Sir Thomas Heath's detailed work titled 'The Thirteen Books of Euclid's Elements'.

The second postulate states that a finite straight line (a line with a defined start point and a defined end point) can be extended. The third postulate outlines that a circle can be described if there is a known centre (a point) and a known distance of a straight line with one end coincident with the centre (this line is the radius). This is taught in schools these days alongside the equation for a circle: r^2 = (x-a)^2 + (y-b)^2 where "r" is the radius, "x" and "y" are variable coordinates and "a" and "b" are the coordinates of the centre point (x = a and y = b). Note the similarity between the equation for a circle and the Pythagorean theorem!

Whilst the fourth postulate has utility in that it brings the concept of angles into the mix, it is probably fine to be somewhat shallow here and say that if something is ninety degrees, it is ninety degrees! It is very much up there with statements such as 1 = 1. The fifth postulate - and its converse - is more complex and over time it has been shown that it doesn't hold in all circumstances. In short, the postulate deals with circumstances where two straight lines meet or, as the case may be, don't meet. Euclidean geometry requires the postulate to hold, whereas a different branch called non-Euclidean geometry arises to deal with systems, such as hyperbolic geometries, where the postulate fails.

Adding text to an original publication was not uncommon in the tradition of Ancient Greece. In this case two apocryphal - or otherwise referred to in the literature as "spurious" - books have been appended to the original thirteen instalments, though not all editions of 'Elements' include them. The apocryphal Book XIV presents a number of propositions that culminate in a statement on the ratio of the respective surface areas of a dodecahedron and a twenty-sided icosahedron given the constraints of propositions that are provided before it. The spurious Book XV presents material on the five Platonic solids (regular forms of the tetrahedron, cube, octahedron, dodecahedron and icosahedron).