By Steve Nadis

In the 20th century, American mathematicians started to make serious advances in a box formerly ruled by means of Europeans. Harvard's arithmetic division was once on the heart of those advancements. *A background in Sum *is an inviting account of the pioneers who trailblazed a noticeably American culture of mathematics--in algebraic geometry and topology, complicated research, quantity idea, and a number of esoteric subdisciplines that experience hardly been written approximately outdoors of magazine articles or complex textbooks. The heady mathematical ideas that emerged, and the boys and girls who formed them, are defined right here in vigorous, obtainable prose.

The tale starts in 1825, while a precocious sixteen-year-old freshman, Benjamin Peirce, arrived on the collage. He could turn into the 1st American to provide unique mathematics--an ambition frowned upon in an period while professors principally constrained themselves to educating. Peirce's successors--William Fogg Osgood and Maxime Bôcher--undertook the duty of reworking the mathematics division right into a world-class study heart, attracting to the school such luminaries as George David Birkhoff. Birkhoff produced a stunning physique of labor, whereas education a new release of innovators--students like Marston Morse and Hassler Whitney, who solid novel pathways in topology and different parts. Influential figures from around the globe quickly flocked to Harvard, a few overcoming nice demanding situations to pursue their elected calling.

*A historical past in Sum* elucidates the contributions of those amazing minds and makes transparent why the background of the Harvard arithmetic division is a vital a part of the background of arithmetic in the United States and beyond.

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But let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. ”54 In some ways, Linear Associative Algebra seems to have come out of the blue, because Peirce had not done much original work in algebra 23 24 A H I S T O RY I N S U M before. In another sense, though, Peirce’s efforts were not entirely surprising, since they grew out of Sir William Rowan Hamilton’s invention of “quaternions” in 1843. Hamilton delivered his first lectures on quaternions in 1848, and Peirce was deeply impressed.

72 Benjamin Peirce and the Science of “Necessary Conclusions” Peirce’s views about mathematics were deeply colored by his fervent religious convictions. He considered mathematics one of the highest forms of human expression and, as such, a manifestation of God’s infinite wisdom. ”73 Peirce made no attempt, moreover, to conceal his religious feelings and was, instead, quite open about them. Indeed, in the introductory paragraph to his 1870 paper, he noted that the mathematical formulas contained therein had a divine origin—something that he considered to be true of mathematics in general.

Late in the eighteenth century, Laplace had suggested that Saturn had a large number of solid rings. Bond discovered a gap in the rings of Saturn during observations made with Harvard’s great refractor. Bond believed the rings must be fluid rather than solid, as Laplace and others had maintained. Peirce undertook a detailed mathematical analysis of the rings’ constitution, concluding that they were fluid. He showed, moreover, that the presence of Saturn alone would not keep the rings stable. But Saturn, along with its eight satellites, could keep a fluid ring in equilibrium.