View from the Pennines: What Has Mathematics Done for Us?

I am sometimes astonished at my own ignorance. I can spot a siskin amongst the birds which come up from the valley bottom to feed from the peanut dispenser and recognize the tumbling ight of the lapwings, but I only learned last year that the lungs of birds have a completely di erent arrangement and mechanism from the mammalian lung I had always imagined to be the standard evolutionary solution to breathing. A bird breathes using two leathery air sacs, in which very little gas exchange occurs, separated by a very densely packed region of blood vessels. Air is sucked into the rst sac, then forced across the blood vessels into the second sac. It is then exhaled from the second sac. Blood is pumped in the opposite direction so as to maintain an optimal oxygen concentration di erence along the ow (lowest concentration of oxygen in the blood nearest the second sac, where the air also has the lowest oxygen concentration). Apparently [8] this arrangement is more e cient than the alveoli of the mammalian lung, and is particularly useful in ight, where a great deal of energy is expended. I feel that there must have been a conspiracy amongst my friends that in over forty years no one has ever let me in on that secret! I felt the same sense of paranoia and ignorance at a recent talk in Manchester by Richard Zare, a chemist from Stanford University [11]. He mentioned in passing work he did some time ago on the growth of bubbles as they move up a liquid such as beer or champagne. I am not unfamiliar with beer (although less at home with champagne), but to the extent that I had given the matter any thought I would have said that the increase in size of the bubbles as they rise is due to the reduction in pressure they experience, the pressure being proportional to the depth of the bubble. Not so, Zare informed the audience. The pressure e ect is small compared to the in ux of further dissolved carbon dioxide from the liquid as the bubble rises. Moreover, Zare has also explained why the bubbles in Guinness appear to descend, and the same set of thoughts made me wonder about the trains of bubbles observed in less zzy beers and more zzy wines. It turns out that the