By JE Littlewood
MATHEMATICS WITH minimal uncooked MATERIAL
FROM THE MATHEMATICAL TRIPOS
CROSS-PURPOSES, subconscious ASSUMPTIONS, HOWLERS, MISPRINTS, ETC.
THE obstacle OF chance THEORY
FROM FERMAT'S final THEOREM TO THE ABOLITION OF CAPITAL PUNISHMENT
A MATHEMATICAL EDUCATION
REVIEW OF RAMANUJAN'S accrued PAPERS
NEWTON AND THE allure OF A SPHERE
THE DISCOVERY OF NEPTUNE
THE ADAMS-AIRY AFFAIR
'LION AND guy'
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Additional info for A Mathematician's Miscellany
Pictorial arguments, while not so purely conventional, ' can be quite legitimate. Take the theorem 1 f(x)=o(I) as 3-* oo and jT=0(l) imply /'=o(l) '. If f'^o(l) the graph y=f'(x) will for some arbitrarily large x have peaks (above or below y=0) enclosing triangles like PQR, the height of P not small, the slopes of PQ, PR not large, and so the area PQR not small. Thenf(Q)f(R) is not small, contradicting /=o(l). This is rigorous (and printable), in the sense that : ' c R Q FIG. 8 symbols no step occurs that is not For myself I think like this both unequivocal and trivial.
I lately came across (the lowest depth, from a very naive writer) functions never taking the values nought or one '. I myself favour using figures practically always (and am A acting to the principle in the book). up linguist would be shocked to learn that mean that it is open, or does not mean the same thing closed this does not is dense in in itself E ' if a set is not again that E as E is dense ' ' '. 1 I have often thought that a good literary competition would be to compose a piece in which all the normal misuses of words and constructions were at first sight committed, but on consideration not.
In the particular problem referred to the limiting value is c=|. g. | for time of here ; is ' 1 * * ' ' > ; ; ' flight given < To '. ) The average height in any ordinary sense being $h, this is a mild subtlety (the Office would not believe it until they had made a numerical experiment, after which principles requires rather ' ' *"* they believed any result guaranteed by theory). It is as intuitive as anything can be that, whatever the how does property ', c cannot be outside the range (0, 1) c : the reader react to the possibility of c^O or 1 ?
A Mathematician's Miscellany by JE Littlewood