habituating
him to precision. It is one of the peculiar excellences of mathematical
discipline that the mathematician is never satisfied with an _a peu
pres_. He requires the _exact_ truth.... The practice of mathematical
reasoning gives wariness of the mind; it accustoms us to demand a sure
footing."[12] Mill, however, is no guide except for exceptionally gifted
youth. He began to learn Greek when he was three years old, and by the
time he had reached the age of twelve had read a good part of Latin and
Greek literature and knew elementary geometry and algebra thoroughly.

The three English historians who have most influenced thought from 1776
to 1900 are those whom John Morley called "great born men of
letters"[13]--Gibbon, Macaulay, and Carlyle; and two of these despised
mathematics. "As soon as I understood the principles," wrote Gibbon in
his "Autobiography," "I relinquished forever the pursuit of the
Mathematics; nor can I lament that I desisted before my mind was
hardened by the habit of rigid demonstration, so destructive of the
finer feelings of moral evidence, which must however determine the
actions and opinions of our lives."[14] Macaulay, while a student at
Cambridge, wrote to his mother: "Oh, for words to express my abomination
of mathematics ... 'Discipline' of the mind! Say rather starvation,
confinement, torture, annihilation!... I feel myself becoming a
personification of Algebra, a living trigonometrical canon, a walking
table of logarithms. All my perceptions of elegance and beauty gone, or
at least going.... Farewell then Homer and Sophocles and Cicero."[15] I
must in fairness state that in after life Macaulay regretted his lack of
knowledge of mathematics and physics, but his career and Gibbon's
demonstrate that mathematics need have no place on the list of the
historian's studies. Carlyle, however, showed mathematical ability which
attracted the attention of Legendre and deemed himself sufficiently
qualified to apply, when he was thirty-nine years old, for the
professorship of Astronomy at the University of Edinburgh. He did not
succeed in obtaining the post but, had he done so, he "would have made,"
so Froude his biographer thinks, "the school of Astronomy at Edinburgh
famous throughout Europe."[16] When fifty-two, Carlyle said that "the
man who had mastered the first forty-seven propositions of Euclid stood
nearer to God than he had done before."[17] I may cap this with some
words of Emerson, wh