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John Wallis was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus.

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The infinite work of John Wallis

Infinity, for the tense religious climate of England in the seventeenth century, often referred to the religious pursuit of eternal life. Yet the religiously driven militaristic leadership of Oliver Cromwell’s Protectorate in the 1650s and the English Restoration of the monarchy which began in 1660 didn’t resolve the constant religious tensions. Yet the meaning of infinity during this time gave way to a different interpretation—through mathematics.

For John Wallis, a doctor of divinity, born four centuries ago today, the study of maths served as an enjoyable hobby rather than as a serious endeavour. Yet his degrees from both Cambridge and Oxford allowed him to rapidly acquire knowledge in the mathematical field. His interest in the works of Galileo’s pupils led him to introduce the infinity symbol to the academic community in 1656. He elaborated upon the ideas of the indivisible method, and created the equation 1/∞ for infinitesimal height. His work would later inspire Isaac Newton; infinitude took on a new meaning, one that carries into the infinite future.

Image: Lens by Skitterphoto. Public domain via Pixabay.

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Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than ∞ as well as less than 0. In his Arithmetica infinitorum (Arithmetic of Infinitesimals, 1655) he argued that, since the ration a/0, when a is positive, is infinite, then when the denominator is changed to a negative number, as in a/b with b negative, the ratio should be greater than a/o since the denominator is smaller. Thus the ratio must be greater than ∞.

Morris Kline, Mathematics: The Loss of Certainty (1980).

(every little sophistry you daydreamed about in calculus was once the opinion of the world’s greatest mathematicians.)

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Discovery of classic pi formula a ‘cunning piece of magic’

While most people associate the mathematical constant π (pi) with arcs and circles, mathematicians are accustomed to seeing it in a variety of fields. But two University scientists were still surprised to find it lurking in a quantum mechanics formula for the energy states of the hydrogen atom.

“We didn’t just find pi,” said Tamar Friedmann, a visiting assistant professor of mathematics and a…

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The Discovery of Titan: Huygens’ Cipher and Wallis’s Trick

(Christiaan Huygens – image source)

To mark the birthday of Christiaan Huygens (1629-1695) – the Dutch mathematician, astronomer, physicist, inventor, and all-around superstar of 17th-century science – let’s recall one of the more bizarre moments of his impressive career, when he almost gave credit for a major astronomical discovery to someone who didn’t deserve it. In 1655, Huygens was encouraged by a friend to begin a correspondence with Oxford’s John Wallis, who was just beginning to make a name for himself in mathematics. Wallis was about to publish on his new techniques for quadrature (i.e,. determining the area bound by a curve), which seemed to hold promise for solving the ancient mathematical problem of squaring the circle. Huygens was eager to explain his views on quadrature to Wallis, but also took the opportunity to send a cipher whose solution contained Huygens’ recent astronomical discovery: he was the first person to find a satellite orbiting the planet Saturn. Today we know it as Titan.

The cipher technique was nothing new. Throughout the seventeenth century scholars, especially astronomers, sent ciphers or anagrams to colleagues as a means of establishing a timeline for their work. If the astronomer was fairly certain he had made an important discovery but felt that he needed more evidence, he could record a cipher in a letter or even a published text and reveal the solution when he felt more confident in his discovery. If, in the meantime, someone else announced that he had made the same discovery, the astronomer could immediately reveal the solution to his cipher and show that he had discovered it first. Decades before Huygens’ discovery of Titan, Galileo had used the same technique to record his observations of Saturn – which seemed to have “handles” protruding from its sides – and Venus – which exhibited phases just like the moon.

Like many of his colleagues in the seventeenth century, Huygens was deeply concerned about priority and credit. The prestige for a discovery in natural philosophy and astronomy, or for the invention of a novel instrument, depended on who got there first. Accordingly, natural philosophers, mathematicians and inventors often got caught up in conflicts – which historians call “priority disputes” – about who deserved credit as the initial discoverer or inventor of something that the wider intellectual community recognized as important. The most famous priority dispute is the one between Newton and Leibniz over the invention of calculus, which became particularly heated in the early eighteenth century, but this was just one case that followed the pattern set by numerous others throughout the seventeenth century. Priority disputes tended to bring out personal, professional, and even national rivalries, including when English and French writers argued interminably during the 1660s over which nation’s physicians had performed the first blood transfusion. (Both sides were passionate about claiming priority even though the results of blood transfusions were disastrous in the seventeenth century, when physicians and natural philosophers had no concept of blood types and nearly every patient died.) Galileo was deeply concerned that his priority be acknowledged, which is why he employed the cipher technique. This didn’t prevent him from engaging in priority disputes throughout his career, including one against Simon Marius over Galileo’s most famous discovery, the four “Galilean moons” orbiting Jupiter.

Huygens himself became embroiled in priority disputes later in his career, including a nasty one with Robert Hooke over who had invented a spring-powered pocket watch. So his concern about priority for discovering Titan was justified. But the technique he used to protect his priority in this and many other cases, the cipher, faced increasing criticism over the course of the seventeenth century since it did not seem to be absolutely secure. Couldn’t a talented codebreaker find the solution to the cipher and steal credit for the discovery? Indeed, Huygens evidently didn’t realize that when he sent his ciphered discovery to Wallis, he was sending it to a professional codebreaker who was not above a little opportunism to increase the prestige of English astronomers. Even if he had known about Wallis’s cryptanalytical prowess, though, Huygens could scarcely have predicted how exactly Wallis would demonstrate the vulnerability of the cipher technique.

(John Wallis – image source)

In fact, Huygens made his cipher particularly vulnerable by hinting at the nature of its solution. His letter to Wallis contained this cryptic Latin message: “ADMOVERE OCULIS DISTANTIA SIDERA NOSTRIS, VVVVVVVCCCRR-HNBQX”, the intelligible part of which means roughly “to direct our eyes to distant stars.” Huygens planned to reveal later that this code contained his discovery of Saturn’s moon in the form of an anagram. When the letters are rearranged it reads, “ SATURNO LUNA SUA CIRCUNDUCITUR DIEBUS SEXDECIM HORIS QUATUOR”, or “Saturnʼs moon is led around it in sixteen days and four hours.” The announcement would reflect the message in the cipher itself: Huygens had made a new discovery by directing his eyes to the distant stars. It seems that Huygens wanted not only to protect his priority, but also to add some dramatic flair when he revealed that the solution to his cipher had been under people’s noses the whole time.

Wallis, no stranger either to astronomical observation or to ciphered announcements, replied to Huygens with a cipher of his own. Apparently lacking Huygens’ sense of drama, though, Wallis didn’t hint at the solution with a clever arrangement of the letters. Instead he simply gave Huygens a list of letters in alphabetical order, which would be arranged to form the solution when Wallis was prepared to announce the discovery:

(image from Wallis Correspondence vol. 1, cited in full below.)

Then Wallis chose to bide his time, saving the solution to his ciphers until Huygens revealed his. In March of the following year, Huygens sent his colleagues in Oxford a copy of his newly published De Saturni luna observatio nova in which he announced his discovery of Saturn’s moon and explained that he had recorded this discovery in a cipher he had sent to friends the year before.

Wallis picked this moment to explain what his own cipher had concealed. He wrote to that two astronomers in Oxford, Paul Neale and Christopher Wren, had discovered Saturn’s moon several years before Huygens, which he could prove with the solution to his cipher. When properly arranged, Wallis explained, the letters he had sent produce the following phrase: “Saturni Comes quasi lunando vehitur. Diebus sexdecim circuitu rotatur. Novas nuper Saturni formas Telescopo vidimus primitus. Plura speramus.” This translates to, “A companion of Saturn is carried in a curve. It is turned by a revolution in sixteen days. We have recently observed new shapes of Saturn with a telescope. We expect more.” Wallis’s cipher established that the Oxford astronomers had observed the moon around 1649, when astronomers noticed that Saturn had apparently begun to change its shape. (When observed through a telescope, the Saturn takes on different shapes depending on the angle of its rings relative to the observer.) Huygens was disappointed that his discovery had been anticipated, but he conceded to the Oxford astronomers. They had established their priority according to Huygens’ own rules.

Wallis let Huygens think he had been scooped for over two years. In 1659 Huygens wrote to Wallis to tell him that he was ready to publish his Systema Saturnium. This book would explain why Saturn’s appearance changed over time: by supposing that Saturn is surrounded by a ring one could explain all the various shapes it appeared to take. (In fact, this was another discovery he had recorded in a cipher. He recorded the encoded version in print in De Saturni luna.) Responding to Huygens, Wallis again chose a crucial moment to reveal a secret: his cipher had been a fake. Wallis wrote that the Oxford group had, in fact, observed and recorded Saturn’s moon but they had mistaken it for a regular fixed star. When Huygens included a cipher in first letter to Wallis, the experienced cryptanalyst had created a clever pseudo-anagram with several possible solutions. Together with Neale and Wren, he had made a few guesses about Huygens’ discovery, then wrote out a cipher whose letters could be arranged to correspond to whichever discovery Huygens announced. Wallis’s anagram was a practical joke, and he thought Huygens should know the truth before he acknowledged the English astronomers’ fake discovery in print.

(image of the cipher from Huygens’ Systema Saturnium, 1659.)

It’s not entirely clear why Wallis and his friends deceived Huygens, but it seems that in part they wanted to convince him that the cipher system was insufficient to protect his priority. In general, though, Wallis rarely missed an opportunity to promote English natural philosophy and mathematics, and he may have seen this as a chance to humiliate a foreign rival. In the end, though, Wallis’s trick had little affect on either Huygens’ behaviour or his reputation. He continued to prefer ciphers as a means to record his achievements, and after he was elected a fellow of the Royal Society in 1663 he sometimes sent ciphers to the Society’s secretary, Henry Oldenburg. And although Wallis had embarrassed him in the matter of Saturn’s moon, Huygens still earned great prestige for solving the puzzle of Saturn’s changing shape. In addition, Huygens continues to be recognized for his accomplishments in physics and optics, his name having been associated for centuries with the wave theory of light. If anything, the Titan affair made Huygens wary of Wallis himself, although he appears to have regarded the Oxford professor’s trickery with an air of amusement. He wrote to a French colleague after Wallis had revealed the truth, “indeed [he] has a quick mind and it is enjoyable to see how he tries at all costs to maintain the honour of his nation.”

For historians of science, though, Wallis’s trick should not be remembered as merely amusing. His pseudo-cipher reveals much about the shaky ground on which credit and priority were founded in seventeenth-century mathematics and natural philosophy. This episode suggests that one reason for the profusion of priority disputes in the seventeenth century is the lack of a secure means to establish one’s findings in public. It could take months or years to publish scientific texts, which were complicated and expensive to print. Furthermore, Wallis’s trick was believable in the first place because certain areas of study were particularly crowded. Huygens had a sense that if he did not make and announce discoveries in a timely manner then someone else would beat him to it, and he may well have been right. These problem were compounded by the fact that no one agreed about what constituted a discovery. Did Huygens deserve credit for recognizing Saturn’s moon, or did Neale and Wren deserve credit for observing the satellite but thinking it was a star? In cases like these there was plenty of latitude for different commentators to give credit to whomever they preferred. Between Huygens and Wallis, the former was more skilled in matters of astronomy, but the latter was more skilled in appreciating the sociology of his intellectual community and taking advantage of its ambiguities.

Sources

Philip Beeley and Christoph Scriba, eds., The Correspondence of John Wallis vol. 1 (Oxford: Oxford University Press, 2013).

A. Rupert Hall and Marie Boas Hall, “The First Human Blood Transfusion: Priority Disputes,” Medical History 24 (1980): 461-465

Rob Iliffe, “ʻIn the Warehouseʼ: Privacy, Property and Priority in the Early Royal Society,” History of Science 30 (1992): 29-68.

W. T. Lynn, “The Discovery of Titan,” The Observatory 148 (1889): 181-182.

E. W. Maunder, “The Discovery of Titan,” The Observatory 147 (1889): 147-150.

Paolo Palmieri, “Galileo and the Discovery of the Phases of Venus,” Journal for the History of Astronomy 32 (2001): 109-129.

Albert van Helden, “ʻAnnulo Cingiturʼ: The Solution to the Problem of Saturn,” Journal for the History of Astronomy 5 (1974): 155-174;

Albert van Helden, “Saturn and His Anses,” Journal for the History of Astronomy 5 (1974): 105-121.

Richard S. Westfall, “Science and Patronage; Galileo and the Telescope,” Isis 76 (1985): 11-30.

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