The integral sign reminds me of the salt water taffy machines in the  sweet shops of the Oregon coastal towns we  visited during the summers of my girlhood. I remember pressing my nose against the glass partition to watch  the machine pulled the strands of candy in a mesmerizing pattern over and over,  stretching them into almost living things – supple and touchable.  As I look back, those summers appear pleasantly long and unfocused like the desultory walks along the beaches, trading taffy flavors and seashells with my sisters.   The fact that a mathematical sign can invoke such nostalgia made me reach out for an old textbook tonight.  Full of equations yellowed out with highlighters by my former self, it patiently repeats to  me that the Volterra series has the ability to capture the effects of memory.  Differing from the Taylor series that approximates the response of a non-linear system such that the output is solely dependent on the inputs at a particular time, the Volterra series calculates the output using the inputs at all other times.  Specifically, it is a series of infinite sums of multidimensional convolutional integrals and can be used to calculate the intermodulation effect of audio signals.

The integral of rational functions can also be used to calculate the drag force effect on the free fall of a sky diver.  But this fact was of no use to John Gillespie Magee whose parachute failed to open one summer day in 1941.  But how he had lived!

High Flight

by John Gillespie Magee, Jr.

Oh, I have slipped the surly bonds of earth,

And danced the skies on laughter-silvered wings;

Sunward I’ve climbed, and joined the tumbling mirth

Of sun-split clouds…and done a hundred things

You have not dreamed of…wheeled and soared and swung

High in the sunlit silence. Hov’ring there,

I’ve chased the shouting wind along, and flung

My eager craft through footless halls of air.

Up, up, the long, delirious burning blue

I’ve topped the windswept heights with easy grace

Where never lark, nor even eagle flew.

And while with silent, lifting mind I’ve trod

The high untrespassed sanctity of space…

…put out my hand, and touched the face of God.

John was nineteen when he died.  It was a farmer who saw him falling from the plane. The same force must have acted upon Icarus of the Grecian myth, another boy who fell out of the sky.  W. H. Auden offers some commentaries on how  these tragedies were perceived and interpreted by arts.

Musee des Beaux Arts

About suffering they were never wrong,

The Old Masters; how well, they understood

Its human position; how it takes place

While someone else is eating or opening a window or just walking dully along;

How, when the aged are reverently, passionately waiting

For the miraculous birth, there always must be

Children who did not specially want it to happen, skating

On a pond at the edge of the wood:

They never forgot

That even the dreadful martyrdom must run its course

Anyhow in a corner, some untidy spot

Where the dogs go on with their doggy life and the torturer’s horse

Scratches its innocent behind on a tree.

In Breughel’s Icarus, for instance: how everything turns away

Quite leisurely from the disaster; the ploughman may

Have heard the splash, the forsaken cry,

But for him it was not an important failure; the sun shone

As it had to on the white legs disappearing into the green

Water; and the expensive delicate ship that must have seen

Something amazing, a boy falling out of the sky,

had somewhere to get to and sailed calmly on.

W. H. Auden

It is cold comfort to be offered the permanence in the nature of gravity.  Looking at the poems from another angle,  I see that there are worse things than to be desensitized by arts. After all, having a susceptible heart  I need  all the help I can get to be inoculated against the rampant pain radiating from too many tragedies, both real and mythical.