Ornaments and icons, symbols of complexity or evil, aesthetically appealing and endlessly useful in everyday ways, knots are also the object of mathematical theory, used to unravel ideas about the topological nature of space. In recent years knot theory has been brought to bear on the study of equations describing weather systems, mathematical models used in physics, and even, with the realization that DNA sometimes is knotted, molecular biology.

This book, written by a mathematician known for his own work on knot theory, is a clear, concise, and engaging introduction to this complicated subject. A guide to the basic ideas and applications of knot theory, Knots takes us from Lord Kelvin’s early—and mistaken—idea of using the knot to model the atom, almost a century and a half ago, to the central problem confronting knot theorists today: distinguishing among various knots, classifying them, and finding a straightforward and general way of determining whether two knots—treated as mathematical objects—are equal.

Communicating the excitement of recent ferment in the field, as well as the joys and frustrations of his own work, Alexei Sossinsky reveals how analogy, speculation, coincidence, mistakes, hard work, aesthetics, and intuition figure far more than plain logic or magical inspiration in the process of discovery. His spirited, timely, and lavishly illustrated work shows us the pleasure of mathematics for its own sake as well as the surprising usefulness of its connections to real-world problems in the sciences. It will instruct and delight the expert, the amateur, and the curious alike.

This edition of Isaac Newton’s Principia is the first edition that enables the reader to see at a glance the stages of evolution of the work from the completion of the manuscript draft of the first edition in 1685 to the publication of the third edition, authorized by Newton, in 1726.

A photographic reprint of this final version, the present edition exhibits on the same page the variant readings from the seven other texts. This design allows the reader to see all the changes that Newton introduced and to determine exactly how the last and definitive edition, published a few months before Newton’s death, grew from earlier versions.

A series of appendices provides additional material on the development of the Principia; the contributions of Roger Cotes and of Henry Pemberton; drafts of Newton’s preface to the third edition; a bibliography of the Principia, describing in detail the three substantive editions and all the known subsequent editions; an index of names mentioned in the third edition; and a complete table of contents of the third edition.

Why is the sky dark at night?

The answer to this ancient and celebrated riddle, says Edward Harrison, seems relatively simple: the sun has set and is now shining on the other side of the earth. But suppose we were space travelers and far from any star. Out in the depths of space the heavens would be dark, even darker than the sky seen from the earth on cloudless and moonless nights. For more than four centuries, astronomers and other investigators have pondered the enigma of a dark sky and proposed many provocative but incorrect answers. Darkness at Night eloquently describes the misleading trails of inquiry and strange ideas that have abounded in the quest for a solution.

In tracing this story of discovery—one of the most intriguing in the history of science—astronomer and physicist Harrison explores the concept of infinite space, the structure and age of the universe, the nature of light, and other subjects that once were so perplexing. He introduces a range of stellar intellects, from Democritus in the ancient world to Digges in the reign of Queen Elizabeth, followed by Kepler, Newton, Halley, Chéseaux, Olbers, Poe, Kelvin, and Bondi.

Harrison’s style is engaging, incisive yet poetic, and his strong grasp of history—from the Greeks to the twentieth century—adds perspective, depth, and scope to the narrative. Richly illustrated and annotated, this book will delight and enlighten both the casual reader and the serious inquirer.

Between 1650 and 1750, four Catholic churches were the best solar observatories in the world. Built to fix an unquestionable date for Easter, they also housed instruments that threw light on the disputed geometry of the solar system, and so, within sight of the altar, subverted Church doctrine about the order of the universe.

A tale of politically canny astronomers and cardinals with a taste for mathematics, The Sun in the Church tells how these observatories came to be, how they worked, and what they accomplished. It describes Galileo's political overreaching, his subsequent trial for heresy, and his slow and steady rehabilitation in the eyes of the Catholic Church. And it offers an enlightening perspective on astronomy, Church history, and religious architecture, as well as an analysis of measurements testing the limits of attainable accuracy, undertaken with rudimentary means and extraordinary zeal. Above all, the book illuminates the niches protected and financed by the Catholic Church in which science and mathematics thrived.

Superbly written, The Sun in the Church provides a magnificent corrective to long-standing oversimplified accounts of the hostility between science and religion.

A Physics Today Best Book of the Year

The first biography of a pioneering scientist who made significant contributions to our understanding of dark matter and championed the advancement of women in science.

One of the great lingering mysteries of the universe is dark matter. Scientists are not sure what it is, but most believe it’s out there, and in abundance. The astronomer who finally convinced many of them was Vera Rubin. When Rubin died in 2016, she was regarded as one of the most influential astronomers of her era. Her research on the rotation of spiral galaxies was groundbreaking, and her observations contributed significantly to the confirmation of dark matter, a most notable achievement.

In Vera Rubin: A Life, prolific science writers Jacqueline Mitton and Simon Mitton provide a detailed, accessible overview of Rubin’s work, showing how she leveraged immense curiosity, profound intelligence, and novel technologies to help transform our understanding of the cosmos. But Rubin’s impact was not limited to her contributions to scientific knowledge. She also helped to transform scientific practice by promoting the careers of women researchers. Not content to be an inspiration, Rubin was a mentor and a champion. She advocated for hiring women faculty, inviting women speakers to major conferences, and honoring women with awards that were historically the exclusive province of men.

Rubin’s papers and correspondence yield vivid insights into her life and work, as she faced down gender discrimination and met the demands of family and research throughout a long and influential career. Deftly written, with both scientific experts and general readers in mind, Vera Rubin is a portrait of a woman with insatiable curiosity about the universe who never stopped asking questions and encouraging other women to do the same.

Mark Peterson makes an extraordinary claim in this fascinating book focused around the life and thought of Galileo: it was the mathematics of Renaissance arts, not Renaissance sciences, that became modern science. Galileo's Muse argues that painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day, steeped as they were in a medieval cosmos and its underlying philosophy.

According to Peterson, the recovery of classical science owes much to the Renaissance artists who first turned to Greek sources for inspiration and instruction. Chapters devoted to their insights into mathematics, ranging from perspective in painting to tuning in music, are interspersed with chapters about Galileo's own life and work. Himself an artist turned scientist and an avid student of Hellenistic culture, Galileo pulled together the many threads of his artistic and classical education in designing unprecedented experiments to unlock the secrets of nature.

In the last chapter, Peterson draws our attention to the Oratio de Mathematicae laudibus of 1627, delivered by one of Galileo's students. This document, Peterson argues, was penned in part by Galileo himself, as an expression of his understanding of the universality of mathematics in art and nature. It is "entirely Galilean in so many details that even if it is derivative, it must represent his thought," Peterson writes. An intellectual adventure, Galileo’s Muse offers surprising ideas that will capture the imagination of anyone—scientist, mathematician, history buff, lover of literature, or artist—who cares about the humanistic roots of modern science.

A New Scientist Book of the Year
A Physics Today Book of the Year
A Science News Book of the Year

The history of science is replete with women getting little notice for their groundbreaking discoveries. Cecilia Payne-Gaposchkin, a tireless innovator who correctly theorized the substance of stars, was one of them.

It was not easy being a woman of ambition in early twentieth-century England, much less one who wished to be a scientist. Cecilia Payne-Gaposchkin overcame prodigious obstacles to become a woman of many firsts: the first to receive a PhD in astronomy from Radcliffe College, the first promoted to full professor at Harvard, the first to head a department there. And, in what has been called “the most brilliant PhD thesis ever written in astronomy,” she was the first to describe what stars are made of.

Payne-Gaposchkin lived in a society that did not know what to make of a determined schoolgirl who wanted to know everything. She was derided in college and refused a degree. As a graduate student, she faced formidable skepticism. Revolutionary ideas rarely enjoy instantaneous acceptance, but the learned men of the astronomical community found hers especially hard to take seriously. Though welcomed at the Harvard College Observatory, she worked for years without recognition or status. Still, she accomplished what every scientist yearns for: discovery. She revealed the atomic composition of stars—only to be told that her conclusions were wrong by the very man who would later show her to be correct.

In What Stars Are Made Of, Donovan Moore brings this remarkable woman to life through extensive archival research, family interviews, and photographs. Moore retraces Payne-Gaposchkin’s steps with visits to cramped observatories and nighttime bicycle rides through the streets of Cambridge, England. The result is a story of devotion and tenacity that speaks powerfully to our own time.

Sir John Herschel, one of the founders of Southern Hemisphere astronomy, was a man of extraordinarily wide interests. He made contributions to botany, geology, and ornithology, as well as to astronomy, chemistry, and mathematics. Throughout his scientific career he kept a diary, recording his public and private life. The diaries from 1834 to 1838, years he spent making astronomical observations at the Cape of Good Hope, are reproduced in this book and prove to be much more than an ordinary scientist’s logbook. They present personal and social history, literary commentaries, the results of close observations of nature and numerous scientific experiments, the excitement of travel, political intrigues, gossip, and philosophical reflections—all interpreted through an alert and versatile mind. In the present transcription, the material has been enriched with selected correspondence of Sir John and his wife Lady Herschel (née Margaret Brodie Stewart).

Sir John devoted his working time at the Cape primarily to a systematic observation of the southern sky, complementing his earlier “sweeping” of the northern sky at Slough, England. He later became one of the founders of photography, but at the Cape he used a simple optical device, the camera lucida, in the production of numerous landscape drawings. Many of these, along with reproductions of sketches contained in the diaries and botanical drawings made by Sir John and Lady Herschel, are used to illustrate this book. Sir John was also a leading figure in the foundation of the educational system of the Cape and a supporter of exploratory expeditions into the interior.

As the son of Sir William Herschel, in his day the most famous British astronomer and the discoverer of the planet Uranus, Sir John was already celebrated when he arrived from England. Every individual of note, resident at the Cape or visiting, went to see him. He was supported in his work by his wife, who ran an enormous establishment and bore a huge family, but who nevertheless found time to travel in the country around the western Cape with him and to assist in his observations.

The diaries and letters are supplemented by especially valuable editorial notes that provide much needed and highly interesting information concerning persons and events mentioned and described by Sir John. All the original manuscript material used in this volume is archived at the Harry Ransom Center at the University of Texas at Austin. Sir John’s camera lucida drawings are from the South African Public Library in Cape Town.