Below I have made a comparison of some different definitions of some of the first terms in Geometry. The first and most basic are point, line, and plane. These are known as undefined terms, not because they are not defined, but because they are the basic building blocks of all other geometrical figures. There are no simpler shapes or building blocks that can be used to define them. This year is the first year that I have had students debate many of the definitions of geometry. Much of the basic vocabulary they have had exposure to for many years, but have not learned precise definitions. Below, I have listed the three "undefined" terms of geometry and definitions from various sources.
NOTES ON USAGE: Some words are used a little differently in the older texts compared to more modern versions. Euclid, Playfair use the word line to mean any curved segment; for example a circle is a type of line. Interestingly, Eclid's definition of a line is very similar to a definition that you might hear in an art class: a line being any continuous stroke made by the artist. However, to avoid confusion when talking about objects and in making my comparison chart, I tried to maintain modern word usage as much as possible so that the comparisons would be clear.
Euclid | Playfair | Holt Publishing | The Plebian |
Point: that which has no part | Point: place, without reference to extent; incapable of division and not occupying space | Point: an undefined term in geometry, it names a location and has no size | Point: an object that has no dimension of measurement and can only be described as a location |
Straight Line: a breadthless length that lies evenly between its extreme points | Straight Line: a length without reference to breadth or Thickness, extended so that its extremities are the farthest possible distance apart as the length of the line will allow such that all its points are in the same direction | Line: an undefined term in geometry, a straight path that has no thickness and extends forever | Line: the shortest path between two points, having no thickness, extended indefinitely in both directions |
Plane: a surface (that which has length and breadth only) that lies evenly with the straight lines on itself | Plane: a surface that can be derived by moving a straight line along two straight lines that make an angle | Plane: an undefined term in geometry, a flat surface that has no thickness and extends forever | Plane: a flat smooth surface without thickness that extends indefinitely in 4 directions |
You will notice that the undefined terms have some interesting properties. A point for example has zero dimension, a line has zero thickness, and so on. Sometimes these attributes make their way into science fiction or can take on a mystical aura because of these odd properties. They sometimes find their life in little intellectual games we play with ourselves. Once in class I made a remark about the some of the properties I aforementioned. I don’t remember the context, but perhaps, like many others, I may have tried to add a little imagination. In that particular case, one of my students quickly began questioning why we would waste time talking about something that couldn't exist in the real world. Since then, I have learned my lesson and tried to stress the practical application to "real" world problems, as opposed to philosophical toys. For example, instead of talking about infinity, I might say, as you see below in my definition of a line, that it extends indefinitely. What I mean is that it goes on as far as you would like. For Euclid a line has a beginning and an end, but he goes on to say it can be extended. In modern forms of plane geometry, Euclid's definition of a straight line, is really a line segment or part of a line. The modern preference being to say that a line segment is part of a line, versus Euclid saying that a straight line can be extended. Notice that we don't really even really differentiate between line and straight line. In modern terms all lines are straight, other continuous one dimensional objects we generally refer to as curves.
The reference to dimensions, measurements, or parts in the definitions is in its application simply a matter of interest. For instance, something takes on the quality of a point in calculations when only a reference point or location is wanted. A line or a curve is a good approximation when only distance or length is involved in a calculation, where a second dimension of width is relatively uniform or irrelevant. So, I like Playfair's tendency to say, "with out reference to..." instead of saying that a line has no thickness. I think in the future, I will adopt similar language.
Of course, there is language that I dislike as well. I find the description of what it means to be straight by Euclid and Playfair to be awkward. The phrases "lie evenly" and "same direction" are the primary violators of clarity and precision in my mind. Of course, Holt, the publishing company that sold my employer math texts, makes no attempt to explain or describe what it means to be straight. That is why I choose to describe lines as the shortest difference between two points extended indefinitely. Playfair makes the observation about lines that the shortest distance between two points is a straight line [segment]. In making this observation, I think he best summarizes the idea of straightness being direct. This observation illustrates that there is a fine line between axioms, basic assumptions, and definitions. Some of the axioms actually describe properties of things formerly defined, and therefore deepen our understanding of terms without the necessity of being proven. As such, they illuminate the meaning of those things formerly defined.
NOTES ON USAGE: Some words are used a little differently in the older texts compared to more modern versions. Euclid, Playfair use the word line to mean any curved segment; for example a circle is a type of line. Interestingly, Eclid's definition of a line is very similar to a definition that you might hear in an art class: a line being any continuous stroke made by the artist. However, to avoid confusion when talking about objects and in making my comparison chart, I tried to maintain modern word usage as much as possible so that the comparisons would be clear.
