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SECOND PUBLIC EXAMINATION
Final Honour School of Mathematics Part B: Paper BO1.1
Honour School of Mathematics and Computer Science Part B: Paper BO1.1
Honour School of Mathematics and Philosophy Part B: Paper BO1.1
HISTORY OF MATHEMATICS
Trinity Term 2016
FRIDAY, 10 JUNE 2016, 2.30pm to 4.30pm
You should submit answers to two questions from Section A
and one question from Section B.
Sections A and B carry equal weight.
In Section A you are expected to comment on the
context, content, and significance of the given passage.
Section B offers a choice of essay titles.
Do not turn this page until you are told that you may do so
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Section A: Extracts
Comment on the context, content, and significance of two of the following
extracts (translations into English are by J. Stedall unless stated otherwise).
1. Moreover the way of discovery by zetetics is by its own art, exercising its reasoning not now in
numbers, which was the shortcoming of the ancient analysts; but by a newly introduced reasoning with symbols, much more fruitful and powerful than the numerical kind for comparing
magnitudes, first by the proposed law of homogeneity, and then by setting up, as it were, of an
ordered series or scale of magnitudes, ascending or descending proportionally from one kind to
another by power, from which the degree of each and comparisons between them are denoted
and distinguished.
[François Viète, Isagoge, 1591]
2. Proposition I.
Plane figures considered according to the method of indivisibles
I suppose at the start (according to the Geometria indivisibilium of Bonaventura Cavalieri)
that any plane whatever consists, as it were, of an infinite number of parallel lines. Or rather
(which I prefer) of an infinite number of parallelograms of equal altitude; of which indeed the
1
altitude of a single one is ∞
of the whole altitude, or an infinitely small divisor; (for let ∞
denote an infinite number); and therefore the altitude of all of them at once is equal to the
altitude of the figure.
[John Wallis, De sectionibus conicis, 1656]
3. What mortal I ask, could ascertain the number of diseases, counting all possible cases [. . .] and
say how much more likely one disease is to be fatal than another [. . .]? Or who could enumerate
the countless changes that the atmosphere undergoes every day and from that predict today
what the weather will be a month or even a year from now?
[Jakob Bernoulli, Ars conjectandi, 1713, translation by J. Newman]
4. We have assumed until now that the function for which one seeks the expansion as a series
of sines of multiple arcs, may be expanded as an ordered series, in powers of the variable x,
and that there enter into this last series only odd powers. One may extend the same results
to any functions whatever, even to those which are discontinuous and entirely arbitrary. To
establish clearly the truth of this proposition, it is necessary [. . .] to examine the nature of the
coefficients which multiply sin x, sin 2x, sin 3x, sin 4x. [. . .] One arrives in this way at a very
remarkable result expressed by the following equation:
1
πϕx = sin x
2
Z
Z
(sin x ϕx dx) + sin 2x (sin 2x ϕx dx)
Z
Z
+ sin 3x (sin 3x ϕx dx) + · · · + sin ix (sin ix ϕx dx) + etc.;
the right hand side will always give the sought expansion of the function ϕx, if one carries out
the integrations from x = 0 to x = π.
[Joseph Fourier, Théorie analytique de la chaleur, 1822]
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5. Substitutions are the passage from one permutation to another. [. . .]
As the concern is always with questions where the original disposition of the letters has no
influence, in the groups that we will consider one must have the same substitutions whichever
permutation it is from which one starts. Therefore, if in such a group one has substitutions S
and T , one is sure to have the substitution ST .
[Évariste Galois, late addition to ‘Premier Mémoire’, probably written 29 May 1832, published
1846, translation by Peter M. Neumann]
6. By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought.
These objects are called the “elements” of M.
[Georg Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 1883, translation by
P. E. B. Jourdain]
Section B: Essay
Choose one of the following topics.
7. Describe and assess the contributions of René Descartes to the mathematics of the seventeenth
century.
8. Discuss, with reference to specific examples, the changing standards of rigour in mathematics
from the seventeenth to the nineteenth centuries.
9. Modern mathematicians are often divided into ‘pure’ and ‘applied’ mathematicians. To what
extent can we apply this distinction prior to the twentieth century? You should illustrate your
answer with specific examples.
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