# What is Newman's “infinite number of curves”?

In John Henry Newman's The Idea of a University (c. 1850), he writes that, in mathematical science, we are told of

...the existence of an infinite number of curves, which are able to divide a space, into which no straight line, though it be length without breadth, can even enter.

He uses this as an example of an idea which, though it might run counter to intuition, need not be immediately rejected. But, in modern terms, what mathematical concept is Newman referring to?

• Unfortunately many philosophers use mathematical metaphors without understanding what they are talking about. – Alexandre Eremenko May 30 '18 at 22:26
• Even after reading it in context, The Idea of a University, p.464, I still can not parse the sentence. Can not enter what? The space? The "infinite number of curves" dividing it? A straight line can not "enter" any foliation of space by non-straight lines, I guess, but that is hardly counter to intuition. Is it just that the space can be foliated by infinitely many "lengths without breadth"? But that can be done already with straight lines. – Conifold May 31 '18 at 20:59
• I'm guessing from the timing (c. 1850) that he might have referred to some construction in the Bolyai–Lobachevskian/hyperbolic geometry or in projective geometry. – Bence Mélykúti Jun 10 '18 at 15:02
• He might be talking about families of curves from differential equations, such as the family of all circles centered at the origin being the solutions to the differential equation $yy' + x = 0,$ and the family of all circles tangent to the $x$-axis being the solutions to the differential equation $[1 + (y')^2]^3 = [1 + (y')^2 + yy'']^2.$ Also, the topic of envelopes and involutes was very common in calculus texts in the 1800s. – Dave L Renfro Jun 10 '18 at 23:25
• It also occurs to me that "straight line" might include what we would now call a line segment, so the issue is roughly with being densely filled (or entirely filled; the distinction was probably not one many non-math people made then). I'd have to look at some of the 1800s books I have in calculus and analytic geometry (over 20 in hardback print form; easily over 1000 as .pdf files) to get a feel for how line segments were referred, something I don't have time to do now. Incidentally, the reason for "straight" in this phrase is that "line" then typically meant what we would now call "curve". – Dave L Renfro Jun 11 '18 at 8:44

As others have pointed it out, it's difficult to know what he meant. Remember that he was reporting on something he had been told which he probably hadn't fully comprehended at the time, or of which he had only an inaccurate recollection at the time of writing.

This answer is the configuration that came to my mind as having some chance of being the one he had had in mind. (If he didn't have it in mind but had only been told of at some point, then I expect the example to be more complicated than mine.)

When Newman writes 'which are able to divide a space', I take it that the space can be the Euclidean plane; he would have written the space if he had meant the three-dimensional space. My guess then is a hyperbolic pencil of circles (the set of blue circles in the drawing is a finite sample of this): The set of blue circles, including the vertical straight line which is missing from the middle of the drawing, contains infinitely many curves. If you pick countably infinitely many (or finitely many) of them including the straight line such that the sizes of the circles on both sides are unbounded (which also means that there is one arbitrarily near the straight line on either side), then those lines partition the plane ('divide a space') into shapes such that none of those shapes contains a straight line.

Addendum. A simpler (but equivalent, as we'll see) example is an unbounded set of concentric circles and the concentric annuli defined by them, e.g. with equal spacing: $$\Big\{\{x\in\mathbb{R}^2\,|\, n<|x|<n+1\}\ \Big|\ n\in\mathbb{N}\Big\}.$$ This is projectively isomorphic to my first example because there is a Möbius transformation which maps one configuration to the other. You can map

• the two focuses (limiting points) of the pencil of circles, and the point where the vertical line intersects the segment between the two focuses

to

• the centre of the concentric circles, to $\infty$, and to an arbitrary point of any circle of the set of concentric circles, respectively.

Compare with this:

A family of concentric circles centered at a single focus C forms a special case of a hyperbolic pencil, in which the other focus is the point at infinity of the complex projective line. The corresponding elliptic pencil consists of the family of straight lines through C; these should be interpreted as circles that all pass through the point at infinity. (Pencils of circles, Wikipedia)

• I am a bit confused by your last paragraph. Are you saying that, by picking only one blue circle ("finitely many"), we partition the plane into two regions, neither of which contains a straight line? – Doubt Jun 10 '18 at 19:20
• You're right, I updated it. It doesn't work with finitely many curves: then you could insert a straight (e.g. vertical) line on each side far from the blue circles. Additionally, you could draw another one on each side close to the vertical line in the middle, parallel to it (between the vertical line in the middle and the circle closest to it). – Bence Mélykúti Jun 10 '18 at 20:26
• This is pure speculation on what Newman might have meant. Can we say that this idea was known in 1850? – Gerald Edgar Jun 11 '18 at 11:49
• I find it improbable that "he would have written the space if he had meant the three-dimensional space". Perhaps just "space" with no article. – Peter Taylor Jul 11 '18 at 9:25