Defining Euclid Trivia Quiz

Answer: Point

The approach to geometry taken in the relevant books that make up 'Elements', the classical text attributed to the Ancient Greek mathematician Euclid, is based on propositional logic. Hundreds of propositions - some straightforward and others complex - are set forth in the text but they are all underpinned by definitions and postulates. A definition in this case is the same as one would expect from a dictionary; a concept or object is given a description in order for it to be used in a consistent, logical manner. A postulate is effectively a proposition whose validity is supposedly "self evident".

The first definition of Book I of 'Elements' relates to a "point" which, on its face, seems a fairly non-controversial thing to define but the description of "that which has no part" has attracted much debate both in antiquity and in the modern era. This debate alone could be enough for a quiz so we'll stick to one objection which is that it isn't a "true definition" because it is entirely negative (in other words it tells us what a point isn't but not what it actually is).

In Sir Thomas Heath's excellent analysis, 'The Thirteen Books of Euclid's Elements' (1908), he presents a statement from the philosopher of late antiquity named Simplicius who argues that a negative construction is satisfactory in this case, "[since a body,] has three dimensions, it follows that a point [arrived at after successfully eliminating all three dimensions] has none of the dimensions, and has no part".

Answer: Line

The second definition put forward in Book I of 'Elements' is that of a "line". The description that a "line is a length without breadth" was not a new conceptualisation; scholars have traced it back to Plato (or the Platonic School at the very least). Proclus, a scholar of late antiquity, provides detailed commentary about the first book of 'Elements' and he arrives at a formulation that broadly states that a point is zero-dimensional but the motion of that point creates dimension in the form of a line. In Definition 3, a little more detail is offered by Euclid where it is states that "the extremities of a line are points".

In Definition 4, Euclid introduces the "straight line" which is a concept that is used extensively throughout the rest of the work; it is defined as "a line which lies evenly with the points on itself".

Answer: Surface

In Definition 5, Euclid describes a "surface" as "that which has length and breadth only" and in so doing introduces a second dimension. The concept of a surface paves the way for the introduction of plane geometry and two-dimensional shapes but a few more definitions are required before reaching that particular destination!

Answer: Plane angle

This definition is the first to expressly mention angles. Building on the definitions set out by Euclid for a "point" (Definition 1) and a "line" (Definition 2), the "plane angle" can be expressed as being the inclination between two lines that meet at a point.

The complete definition specifies that this is the case for lines that meet on a "plane surface" which is itself described in Definition 7. Nothing in the definition states that the lines must be straight but Definition 8 introduces the term "rectilineal" to refer to an angle that is contained by straight lines.

Answer: Obtuse angle

Euclid provides a somewhat convoluted description of a "right angle" in Definition 10 of the first book of 'Elements'; this convolution is largely down to the propositional nature of geometry that was prevalent in antiquity. The classical method is to build up the definition from first principles and, in the case of a right angle (which we would now largely refer to as an angle equal to ninety degrees), relies upon descriptions of one straight line standing upon another straight line such that two adjacent angles are equal to each other.

Definition 11 is fairly straightforward in its description of an "obtuse angle" being one that is greater than a right angle but there is a quirky omission in the list of definitions - the reflex angle! A reflex angle (covering the range of 180 to 360 degrees) makes no appearance and a number of suggestions have been put forward ranging from Euclid being unaware of the construction (seems unlikely) to the geometers of antiquity wishing to maintain rigour within the constraints of the visualisation capabilities of the day. The lack of reference to a reflex angle is also evident in 'Republic' by Plato, where reference is made to the "three kinds of angles" (right, obtuse and acute).

Answer: Acute angle

The "acute angle" is now taught in schools as being an angle that is between zero and ninety degrees or π / 2 radians (not inclusive). If the angle were to be zero degrees then there would be no angle at all and the lines would coincide with each other (though the lines may separate farther away from the point of measurement if the lines aren't straight); if the angle were ninety degrees, then it would be a right angle by definition.

Answer: Boundary

Definition 13 is doing more of the preparatory work needed to begin formulating descriptions of shapes. The conceptualisation of a "boundary" as being the "extremity of something" is from the same type of reasoning as that seen in Definition 3 where a "point" is said to be "the extremities of a line". It is worth noting that the wording of this definition is different in Sir Thomas Heath's work, where a boundary is said to be "that which is an extremity of anything".

It could be argued that both "something" and "anything" can include a line and as a point is the extremity of a line (Definition 3) and a boundary is the extremity of something/anything (Definition 13), then there is nothing logically preventing a point from being considered a boundary. In which case, Definition 13 is a generalisation of Definition 3 and Definition 3 is a special case of Definition 13.

Answer: Circumference

In Definition 14, Euclid describes a "figure" as "that which is contained by any boundary or boundaries". In Definition 15, he then uses figure to define the first shape - the circle: "a plane figure contained by a single line [which is called a circumference]...". Sources differ on the exact translation from Greek into English but in some books, the text in square brackets is not included whilst it is in others.

The remaining portion of the definition goes on to express that lines from a point in the circle (Definition 16 clarifies that this point is at the "centre") to the boundary or circumference are all equal in size. Such a line is readily referred to in the modern era as the "radius".

Answer: Diameter

This fairly straightforward definition of "diameter" is well understood and is one of the first aspects of circle geometry taught in mathematics. Definition 17 references the fact that the diameter "cuts the circle in half" and in so doing implicitly gives rise to another shape, a "semicircle" (this is made explicit and formalised in Definition 18).

A diameter can also be referred to as twice the length of the radius, though Euclid did not make use of the word radius in 'Elements'.

Answer: Square

The last few definitions set out in the opening book of Euclid's 'Elements' largely relate to the provision of descriptions for a number of two-dimensional figures. In Definition 19, he defines a "quadrilateral" as a figure contained by four straight lines. After providing descriptions of shapes such as the equilateral, isosceles and right-angled triangles in Definitions 20 and 21, Euclid moves on to making the distinction between the quadrilateral figures, namely a square, oblong, rhombus, rhomboid and trapezium (Definition 22).

Euclid concludes his list of Book I definitions with Definition 23 which relates to parallel lines.