I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenæum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain claims about the longevity of mathematical concepts. This particular volume is attractive to me as I look for better ways to present geometry this fall.
Here are some definitions regarding the word “line” and related terms:
“36. A straight line is a line such that any part of it, however placed on any other part, will lie wholly in the part if its extremities lie in that part, as AB. [visual example provided]
37. A curved line is a line no part of which is straight, as CD. [visual example provided]
38. A broken line is made up of different straight lines, as EF. [visual example provided]
Note: A straight line is often called simply a line.
45. Postulate. A straight line can be produced indefinitely.
49. Axiom. A straight line is the shortest line that can be drawn from one point to another.
50. DEF. The distance between two points is the length of the straight line that joins them.
51. A straight line determined by two points may be considered as prolonged indefinitely.
52. If only the part of the line between two fixed points is considered, this part is called a segment of the line.
53. For brevity, we say ‘the line AB,’ to designate a segment of a line limited by the points A and B.”
(7-8; all emphasis from the original)
Compare this to the descriptions* in Pearson’s Geometry Common Core:
“A line is represented by a straight path that extends in two opposite directions without end and has no thickness. A line contains infinitely many points.
A segment is part of a line that consists of two endpoints and all points between them.” (11-12)
One obvious thing that struck me was the way in which Wentworth uses “line” and “line segment” interchangeably, as well as his use of “line” in the way that we would generally use “curve”. Our modern use of “curve”, in turn, leads to confusion in some mathematics classes: Why is a line a curve? Isn’t a curve always curved?
But the key words in Wentworth’s definitions reveal a nuanced difference of perspective that reflects why he can jauntily switch from “straight line” to “line segment” with philosophical impunity, while we can’t, and the most key of those is “indefinite” (vs. “infinite”).
If we see a line as being infinite in length (as implied but not explicitly stated by Pearson**), then lines and line segments cannot be the same thing. However, in his axioms, Hilbert writes, “A line contains at least two points.” If a line contains infinitely many points, then why does Hilbert point this out?
The key word is “indefinite” rather than “infinite”. The contextual relevance of the object “line” (Wentworth’s “straight line”) is that it connects two points. There is no need for infinities or infinitesimals here: A straight line is the shortest distance between two points, possibly extending off in one or both directions. We don’t care how far it extends; it’s indefinitely long.
Wentworth’s definition doesn’t even require an infinitely long segment: It says, in more modern parlance, that a line AB is straight if, when you connect any two points on AB, you get a segment entirely on AB.
Wentworth’s definitions are far from perfect, and reveal why there must be some objects that are undefined: He defines straightness in terms of distance, then defines distance in terms of straightness. His definition of a straight line arguably makes any circular arc a line, since any subarc will have the same curvature as the containing arc.
But he correctly stresses that what’s relevant about a line in synthetic geometry is that it’s indefinitely long; if we care about a specific distance, we’ll call it a segment. We can consider the properties of lines without them needing to be infinite; if we need “more line” for a given construction, we can simply extend it out.
In the analytic definition of a line, it is indeed infinitely long: A line is the set of all solutions of a linear equation. As the name “linear equation” suggests, this is bit of question-begging, since the equation is called “linear” because it describes a line when placed on the coordinate plane.
My greater concern is this: I have increasingly felt like High School Geometry is the red-headed stepchild of the modern mathematics curriculum. University mathematicians concern themselves far more with differential and projective geometry, while algebra continues its march to dominance in high schools. As a result, the high school course “Geometry” is an unfocused and unsorted mosh of synthetic geometry, analytic geometry, and proofs. Hilbert (and Euclid, and Wentworth for his part) does not require that a line be infinitely long, and “infinite” is a concept that students struggle with… so why insist that lines be seen as “infinitely long” (from a synthetic perspective)? “Indefinitely long” will do just as well for the properties of lines; “infinitely long” is a subordination to analytic geometry.
* We can’t say “definition” because, as every modern Geometry textbook I’ve seen insists, “line” cannot be defined.
** The second sentence of the description is in fact meaningless: “A line contains infinitely many points.” A line segment also contains infinitely many points. By “infinitely many points” in the context, I believe the Pearson authors are meaning to imply that it’s infinitely long. I’m not sure if the phrasing is sloppiness or deliberate plausible deniability: After all, they’re not saying that a line is infinitely long.