Mathematics

Einstein’s latest anniversary marks the birth of modern cosmology

Andromeda galaxy
Edwin Hubble’s observations of stars in the Andromeda galaxy (shown) demonstrated that the universe was vastly bigger than Albert Einstein realized. Nevertheless, Einstein’s paper applying his general theory of relativity, published a century ago, became the foundation for the modern science of cosmology.

First of two parts

Sometimes it seems like every year offers an occasion to celebrate some sort of Einstein anniversary.

In 2015, everybody lauded the 100th anniversary of his general theory of relativity. Last year, scientists celebrated the centennial of his prediction of gravitational waves — by reporting the discovery of gravitational waves. And this year marks the centennial of Einstein’s paper establishing the birth of modern cosmology.

Before Einstein, cosmology was not very modern at all. Most scientists shunned it. It was regarded as a matter for philosophers or possibly theologians. You could do cosmology without even knowing any math.

But Einstein showed how the math of general relativity could be applied to the task of describing the cosmos. His theory offered a way to study cosmology precisely, with a firm physical and mathematical basis. Einstein provided the recipe for transforming cosmology from speculation to a field of scientific study.

“There is little doubt that Einstein’s 1917 paper … set the foundations of modern theoretical cosmology,” Irish physicist Cormac O’Raifeartaigh and colleagues write in a new analysis of that paper.

Einstein had pondered the implications of his new theory for cosmology even before he had finished it. General relativity was, after all, a theory of space and time — all of it. Einstein’s showed that gravity — the driving force sculpting the cosmic architecture — was simply the distortion of spacetime geometry generated by the presence of mass and energy. (He constructed an equation to show how spacetime geometry, on the left side of the equation, was determined by the density of mass-energy, the right side.) Since spacetime and mass-energy account for basically everything, the entire cosmos ought to behave as general relativity’s equation required.

Newton’s law of gravity had posed problems in that regard. If every mass attracted every other mass, as Newton had proclaimed, then all the matter in the universe ought to have just collapsed itself into one big blob. Newton suggested that the universe was infinite, filled with matter, so that attraction inward was balanced by the attraction of matter farther out. Nobody really bought that explanation, though. For one thing, it required a really precise arrangement: One star out of place, and the balance of attractions disappears and the universe collapses. It also required an infinity of stars, making it impossible to explain why it’s dark at night. (There would be a star out there along every line of sight at all times.)

Einstein hoped his theory of gravity would resolve the cosmic paradoxes of Newtonian gravity. So in early 1917, less than a year after his complete paper on the general theory was published, he delivered a short paper to the Prussian Academy of Sciences outlining the implications of his theory for cosmology.

In that…

15 Female Mathematicians Whose Accomplishments Add Up

In many periods of history, women have been discouraged from applying their minds to mathematics—but a few persevered. The world-altering contributions of these 15 notable female mathematicians include making hospitals safer, laying the groundwork for the computer, and advancing space flight.

Hypatia (c.355–415) was the first woman known to have taught mathematics. Her father Theon was a famous mathematician in Alexandria who wrote commentaries on Euclid’s Elements and works by Ptolemy. Theon taught his daughter math and astronomy, then sent her to Athens to study the teachings of Plato and Aristotle. Father and daughter collaborated on several commentaries, but Hypatia also wrote commentaries of her own and lectured on math, astronomy, and philosophy. Sadly, she died at the hands of a mob of Christian zealots.

Maurice Quentin de La Tour via Wikipedia // Public Domain

Emilie Du Chatelet (1706–1749) was born in Paris in a home that entertained several scientists and mathematicians. Although her mother thought her interest in math was unladylike, her father was supportive. Chatalet initially employed her math skills to gamble, which financed the purchase of math books and lab equipment.

In 1725 she married an army officer, the Marquis Florent-Claude du Chatalet, and the couple eventually had three children. Her husband traveled frequently, an arrangement that provided ample time for her to study mathematics and write scientific articles (it also apparently gave her time to have an affair with Voltaire). From 1745 until her death, Chatalet worked on a translation of Isaac Newton’s Principia. She added her own commentaries, including valuable clarification of the principles in the original work.

Sophie Germain (1776–1831) was only 13 when she developed an interest in mathematics, one that could be blamed on the French Revolution. Since the fighting raged around her home, Germain could not explore the streets of Paris—instead she explored her father’s library, teaching herself Latin and Greek and reading respected mathematical works. Germain’s family also tried to discourage her academic leanings. Not wanting her to study at night, they denied her a fire in her room, but she lit candles and read anyway, bundled in blankets.

Since women’s educational opportunities were limited, Germain studied secretly at the Ecole Polytechnique, using the name of a previously enrolled male student. That worked until the teachers noticed the dramatic improvement in the student’s math skills.

Although Germain never worked as a mathematician, she studied independently and wrote about the subject. She is best known for her work on Fermat’s Last Theorem, considered at the time to be one of the most challenging mathematical puzzles. A 17th century mathematician named Pierre de Fermat claimed he could prove that the equation x^n + y^n = z^n had no integer solution when n was greater than 2, but his proof was never written down. Germain proposed a new way of looking at the problem.

Germain also became the first woman to win a prize from the Paris Academy of Sciences, for writing about elasticity theory. Today that prize is known as the Sophie Germain Prize.

4. MARY SOMERVILLE

Thomas Phillips via Wikipedia // Public Domain

Mary Somerville (1780–1872) was born in Scotland, and was not particularly interested in academics as a child—she only attended school for a year. However, when she encountered an algebra symbol in a puzzle at age 16, she became fascinated with math and began studying it on her own. Her parents tried to discourage her, worried that her intellectual preoccupations might drive her insane. (At the time, a popular theory held that difficult study could damage a woman’s mental health.) But Somerville continued to study, teaching herself Latin so she could read earlier versions of works by Euclid.

She also corresponded with William Wallace, a professor of mathematics at Edinburgh University, and solved mathematical problems posed in contests, winning a silver prize in 1811.

Somerville’s first husband did not encourage her interests, but when he died, she remarried. Her second husband, Dr. William Somerville, an inspector of the Army Medical Board, was proud of her work in mathematics and astronomy. For her work translating a book titled Celestial Mechanics and adding commentary, she was named an honorary member of the Royal Astronomical Society.

Physicist Sir David Brewster called her “certainly the most extraordinary woman in Europe—a mathematician of the very first rank with all the gentleness of a woman.” When John Stuart Mill petitioned the British government for women’s votes, he filed his petition with Somerville’s signature first. She was proof that women were men’s intellectual equals.

Alfred Edward Chalonvia Wikipedia // Public Domain

The next time you download some electronica, you may want to remember Augusta Ada King-Noel, Countess of Lovelace (1815–1852). Lovelace was born during the brief marriage of poet George, Lord Byron and Anne Milbanke, Lady Wentworth. Her mother did not want her to be a poet like her father and encouraged her interest in mathematics and music. As a teenager, Ada began to correspond with Charles Babbage, a professor at Cambridge. At the time, Babbage was working on his ideas for a calculating machine called the Analytical Engine, now considered a precursor to the computer. Babbage was solely focused on the calculating aspects, but Lovelace supplied notes that helped envision other possibilities, including the idea of computer-generated music.

Lovelace also translated an article about the Analytic Engine by French mathematician Louis Menebrea. Her notes include an algorithm showing how to calculate a sequence of numbers, which forms the basis for the design of the modern computer. It was the first algorithm created expressly for a machine to perform.

Lovelace was a countess after her marriage, but she preferred to describe herself as an analyst and a metaphysician. Babbage called her “the enchantress of numbers”—but she might also be called the world’s first computer programmer.

Florence Nightingale (1820–1910) is best known as a nurse and social reformer, but a lesser-known contribution of hers continues to save lives. In her efforts to improve the survival rates of hospital patients, Nightingale became a statistician.

When the “lady with the lamp” returned from service during the Crimean War, she expressed sadness about how many soldiers had become sick and died while lying in the hospital. “Oh my poor men, who endured so patiently,” she wrote to a friend. “I feel I have been a bad mother to you to come home and leave you lying in your Crimean graves.”

As part of her plan to reform hospital care, Nightingale began gathering statistics. The figures she gathered indicated that a lack of sanitation…