Hail, Caesura

In which the Idiosopher appreciates the poetic value of zeroes of the first time-derivative.

Tom Hillman has written a Mythgard Academy bank-shot post, in which he draws a line from the song-duel between Finrod and Sauron in The Silmarillion, to a poem in Boëthius’s Consolation of Philosophy, to the beach in Long Island.  Tom points out an almost-caesura in J.R.R. Tolkien’s verse:

Softly in the gloom they heard the birds
Singing afar in Nargothrond,
The sighing of the Sea beyond,
Beyond the western world, on sand,
On sand of pearls in Elvenland.

Silmarillion, ch. 19

The alliteration on “s” in lines 3-5 is onomatopoetic to me.  We’re hearing waves on the beach.  The caesura effect comes from the repetition of “beyond” and “on sand”.  The forward progress of the poem glides gently to a halt, then resumes, like a wave losing energy as it climbs the beach, before it returns to the sea. To a first approximation, the distance a wave travels up the slope of the sand is a parabola. We see only the nose of the parabola, because another wave comes along and uses it as a lubricant against friction with the land.   What we see is Figure 1.

This is a good time of year to think about that. We’ve just passed the solstice, so the same sort of thing is happening with the sun. The sun has been in the sky perceptibly longer each day; now that’s come to an end like waves running out of energy on the sand.  The actual length of the day is a complicated function of the earth’s axial tilt, the latitude of the observer, the eccentricity of the earth’s orbit, the nutation of the earth’s rotation, and even a little factor due to the Moon.  The Naval Observatory keeps track of all this.  We can use a much simpler approximation, and treat everything as circles.  [geometric derivation with awesome ASCII art]  That yields an equation you can actually read.  The fraction of a day during which the sun is up is

2 acos[sin(λ) tan(τ sin(2πy))],
where λ is the latitude of the observer, τ is the earth’s axial tilt, and y is the number of days since December 21st divided by the length of the year in days.  The approximation is about 10 minutes shorter than the true amount of sunshine at my latitude, as shown in Figure 2.  Not bad, Copernicus!

So here we are, just past the noontide of the year.  The vegetable plants have stopped their manic growth phase. (Fortunately, so has the grass.)  The botanical world is in a caesura of its own for a few days.  The beanstalks made it to the tops of their poles just in time.  The squash vines have found every inch of space they can reach.  Now they’re hunkering down to making seeds and fruits.  My job protecting them from skulking vegetarians will begin soon enough, but now is a time to take a breath.