March 12, 2015

Finest Hour 159, Summer 2013

Page 19

James Markovitch explains the Byss and the Abyss

“I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor’s Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go! The practical point is that if this aged, weary-souled Civil Service Commissioner had not asked this particular question about these Cosines or Tangents in their squared or even cubed condition, which I happened to have learned scarcely a week before, not one of the subsequent chapters of this book would ever have been written. I might have gone into the Church and preached orthodox sermons in a spirit of audacious contradiction to the age. I might have gone into the City and made a fortune.”
—WSC, My Early Life, 1930

The “tergiversation” that Churchill refers to in his mathematics quote may very well derive from the expression 1/x. If x approaches zero from one, 1/x approaches a limit of positive infinity. But if x approaches zero from negative one, it approaches a limit of negative infinity. One might therefore infer that as x goes from positive one to negative one, the expression 1/x goes from positive infinity to negative infinity as zero is crossed—a “tergiversation” that was controversial when negative numbers were first introduced. So Churchill perhaps understood more mathematics than he let on!

I read the quote in his autobiography decades ago, but it was only when reading An Imaginary Tale, by Paul Nahin (, which discussed the controversy surrounding the introduction of negative numbers, that I made the connection.

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Leave it to Sir Winston Churchill to remember a point of controversy about the foundations of mathematics.

Churchill continues: “When I look back upon those care-laden months, their prominent features rise from the abyss of memory….We were arrived in an ‘Alice-in-Wonderland’ world, at the portals of which stood ‘A Quadratic Equation.’ This with a strange grimace pointed the way to the Theory of Indices, which again handed on the intruder to the full rigours of the Binomial Theorem. Further dim chambers lighted by sullen, sulphurous fires were reputed to contain a dragon called the ‘Differential Calculus.’ But this monster was beyond the bounds appointed by the Civil Service Commissioners who regulated this stage of Pilgrim’s heavy journey. We turned aside, not indeed to the uplands of the Delectable Mountains, but into a strange corridor of things like anagrams and acrostics called Sines, Cosines and Tangents. Apparently they were very important, especially when multiplied by each other, or by themselves! They had also this merit—you could learn many of their evolutions off by heart. There was a question in my third and last Examination about these Cosines and Tangents in a highly square-rooted condition which must have been decisive upon the whole of my after life. It was a problem. But luckily I had seen its ugly face only a few days before and recognised it at first sight. “I have never met any of these creatures since….they passed away like the phantasmagoria of a fevered dream. I am assured that they are most helpful in engineering, astronomy and things like that. It is very important to build bridges and canals and to comprehend all the stresses and potentialities of matter, to say nothing of counting all the stars and even universes….I am very glad there are quite a number of people born with a gift and a liking for all of this; like great chess-players who play sixteen games at once blindfold and die quite soon of epilepsy. Serve them right! I hope the Mathematicians, however, are well rewarded. I promise never to blackleg their profession nor take the bread out of their mouths.”

Mr. Markovitch may be reached by email at [email protected]

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