Sample recommendation letter:
Dear Search Committee Chair,
I am writing this letter for Mr. John Smith who has applied for a position in your department. I should start by saying that I cannot recommend him too highly.
In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you.
His dissertation is the sort of work you don't expect to see these days. It definitely demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him to work for you.
Sincerely,
A. D. Visor (Prof.)
[In the margin of his copy of Diophantus' Arithmetica, Fermat wrote]
To
divide a cube into two other cubes, a fourth power or in general any power
whatever into two powers of the same denomination above the second is
impossible, and I have assuredly found an admirable proof of this, but
the margin is too narrow to contain it.
And perhaps, posterity will thank me for having shown it that the ancients
did not know everything.
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.
We have a habit in writing articles published in scientific journals to
make the work as finished as possible, to cover up all the tracks, to
not worry about the blind alleys or describe how you had the wrong idea
first, and so on. So there isn't any place to publish, in a dignified
manner, what you actually did in order to get to do the work.
Nobel Lecture, 1966.
The solution of problems is one of the lowest forms of mathematical research,
... yet its educational value cannot be overestimated. It is the ladder
by which the mind ascends into higher fields of original research and
investigation. Many dormant minds have been aroused into activity through
the mastery of a single problem.
The American Mathematical Monthly, no. 1.
The effort of the economist is to "see," to picture the interplay of economic
elements. The more clearly cut these elements appear in his vision, the
better; the more elements he can grasp and hold in his mind at once,
the better. The economic world is a misty region. The first explorers
used unaided vision. Mathematics is the lantern by which what before
was dimly visible now looms up in firm, bold outlines. The old phantasmagoria
disappear. We see better. We also see further.
Transactions of Conn. Academy, 1892.
Natural selection is a mechanism for generating an exceedingly high degree of improbability.
To call in the statistician after the experiment is done may be no more
than asking hm to perform a postmortem examination: he may be able to
say what the experiment died of.
Indian Statistical Congress, Sankhya, ca 1938.
Poetry is as exact a science as geometry.
Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Mathematicians are like lovers Grant a mathematician the least principle,
and he will draw from it a consequence which you must also grant him,
and from this consequence another.
Quoted in V. H. Larney Abstract Algebra: A First Course, Boston:
Prindle, Weber and Schmidt, 1975.
A work of morality, politics, criticism will be more elegant, other things
being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.
Leibniz never married; he had considered it at the age of fifty; but the
person he had in mind asked for time to reflect. This gave Leibniz time
to reflect, too, and so he never married.
Eloge de le Leibniz.
Whereas at the outset geometry is reported to have concerned herself with
the measurement of muddy land, she now handles celestial as well as terrestrial
problems: she has extended her domain to the furthest bounds of space.
Hodder and Stoughton, The Story of Euclid. 1901.
For hundreds of pages the closely-reasoned arguments unroll, axioms and
theorems interlock. And what remains with us in the end? A general sense
that the world can be expressed in closely-reasoned arguments, in interlocking
axioms and theorems.
Constructions. 1974.
To your care and recommendation am I indebted for having replaced a half-blind
mathematician with a mathematician with both eyes, which will especially
please the anatomical members of my Academy.
[To D'Alembert about Lagrange. Euler had vacated the post.]
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.
A scientist can hardly meet with anything more undesirable than to have
the foundations give way just as the work is finished. I was put in this
position by a letter from Mr. Bertrand Russell when the work was nearly
through the press.
In Scientific American, May 1984, p 77.