THE IMAGINATION AND SQUARE ROOTS

1. Picture this.

Picture Rodin's Thinker, crouched in mental effort. He has his supporting right elbow propped not on his right thigh, as you or I would have placed our right elbow, but rather on his left thigh, which bolts him into an awkward striving, his muscles tense with thought. But does he, can we, really feel our imaginative faculty at work, striving toward, and then finally achieving, an act of the imagination?

Consider the range of our imaginative experiences. Consider, for example, how immediate is the experience of imagining what we read. Elaine Scarry has remarked that there is no "felt experience" corresponding to this imaginative act. We experience, of course, the effect of what we are reading. Scarry claims that if we read a phrase like

the yellow of the tulip

we form, perhaps, the image of it in our mind's eye and experience whatever emotional effect that image produces within us. But, says Scarry, we have no felt experience of coming to form that image. We will return to this idea later.

Perhaps one should contrast reading with trying to think something up for ourselves. Rainer Maria Rilke's comment on the working of our imagination,

We are the bees of the invisible

paints our imaginative quests as not entirely unfelt experiences (following Scarry), but not contortions (following Rodin) either. Our gathering of the honey of the imaginative world is not immediate; it takes work. But though it requires traveling some distance, merging with something not of our species, communicating by dance to our fellow creatures what we've done and where we've been, and, finally, bringing back that single glistening drop, it is an activity we do without contortion. It is who we bees are.

Thinking about the imagination imagining is made difficult by the general swiftness and efficacy of that faculty. The imagination is a fleet genie at your service. You want an elephant? Why, there it is:

[ILLUSTRATION OMITTED]

You read "the yellow of the tulip."

And, again, there it is: a calligraphed swath of yellow on your mental movie screen.

More telling, though, are the other moments of thought, when our genie is not so surefooted. Moments composed half of bewilderment and half of expectation; moments, for example, when some new image, or viewpoint, is about to reveal itself to us. But it resists emerging. We are forced to angle for it.

At those times, it is as if the waters of the imagination are roiling; you have cast your fishing line from a somewhat shaky boat, and you feel a tug on that line, but have no clear sense what you have hooked onto. Bluefish, old boot, or some underwater species never before seen? But you definitely feel the tug.

I want to think about the inner articulations of our imaginative life by "re"-experiencing a particular example of such a tug. The example I propose to consider occurs in the history of mathematics. It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining. Moment, though, is not the right word here, for the period, rather, stretches over three centuries. And anticipation carries too progressivist and perhaps too personal a tone, for this "act" doesn't take place fully in any single mind. There are many "bees of the invisible" in the original story.

If we are successful, we will be reenacting, for ourselves, the imagining of a concept that, for the original thinkers, had never been seen or thought before, and that seemed to lie athwart things seen or thought before. Of course, thinking about things never thought before is the daily activity of thought, certainly in art or science. The cellist Yo-Yo Ma has suggested that the job of the artist is to go to the edge and report back. Here is how Rilke expressed a similar sentiment: "Works of art are indeed always products of having-been-in-danger, or having-gone-to-the-very-end in an experience, to where one can go no further."

In contrast to the instantly imaginable "yellow of the tulip," the square root of negative quantities was a concept in common use in mathematics for over three hundred years before a satisfactory geometric understanding of it was discovered. If you deal exclusively with positive quantities, you have less of a challenge in coming to grips with square roots: the square root of a positive number is just a quantity whose square is that number.

Any positive number has only one (positive) square root. The square root of 4, for example, is 2. What is the square root of 2? We know, at the very least, that its square is 2. Using the equation that asserts this,

([square root of 2])2 = [square root of 2]·[square root of 2] = 2

try your hand at estimating [square root of 2]. Is it smaller than 3/2? Do you see why [square root of 3] · [square root of 5] = [square root of 15]?

Square roots are often encountered geometrically, as lengths of lines. We will see shortly, for example, that [square root of 2] is the length of the diagonal of a square whose sides have length 1.

Also, if we have a square figure whose area is known to be A square feet, then the length of each of its sides, as in the diagram below, is [square root of A] feet.

The square root as "side"

Suppose that each box in this diagram has an area equal to I square foot. There are a hundred boxes, so A = 100, and the dimensions of the large square are [square root of A] by [square root of A] — that is, 10 by 10.

In Plato's Meno, Socrates asks Meno's young slave to construct a square whose area is twice the area of a given square. Here is the diagram that Socrates finally draws to help his interlocutor answer the question:

[ILLUSTRATION OMITTED]

The profile of this diagram is a 2 × 2 square (whose area is therefore 4) built out of four 1 × 1 squares (each of area 1). But in its midst, we can pick out a catercorner square (standing, as it seems, on one of its corners). By rearranging the triangular pieces that make up the diagram, can you see, as Socrates' young friend in the Meno did, that the catercorner square has area 2, and therefore each of its sides has length [square root of 2]?

The sides of the catercorner square play a double role: they are also the diagonals of the small (1 × 1) squares. So, as promised a few paragraphs earlier, we see [square root of 2] as the length of the diagonal drawn in a square whose sides are of length 1.

The...