In January 1801, an elated astronomer, Joseph Piazzi announced that he had discovered a piece of stone, possibly a new planet in the night sky. He had tracked its location for 41 days after which he said it disappeared from the night sky; perhaps it had moved along its orbit to the other side of the sun so that it is now in the day sky, not to be seen.
The response from astronomers, none of whom other than Piazzi had seen this planet, was an admixture of excitement, skepticism and anticipation. From the observations for 41 days by Piazzi, which included only 21 points in the sky, it could be concluded that the planet lives in between Mars and Jupiter and that it may visit our night sky again in about an year. But where exactly in the night sky and when exactly after an year? When and where must an astronomer point his telescope in order to see this planet again? These questions had no answer. Astronomers feared that the piece of stone may disappear into darkness for eternity.
Intense attempts to calculate the path of the planet followed. The fundamental problem was to find a path, a curve of some sort, like an ellipse that passes through the 21 points. The difficulty, however, was that those known points covered a mere 3 degree arc and were subject to significant errors. No one knew how to solve this problem.
The excitement around this problem infected onto a 24 year old mathematician. This young man set aside most other commitments to solve this problem. Almost an year later, on December 7, 1801, an astronomer pointing his telescope where the young man told him to, spotted a faint object. It was later confirmed that this object is indeed the planet they were looking for. They had found the piece of stone in the darkness.
This piece of stone is now known as Ceres, the largest among the asteroids, not a planet. The young man was Carl Friedrich Gauss.
Along with the stone, two other bright objects were discovered. One is the now-famous Gaussian distribution, which Gauss developed to model the errors in the observed data points. The other is the least-squares curve fitting technique, which Gauss derived from the Gaussian distribution of error. One curious question here is, is it an accident that the endeavor of finding a stone in the darkness led to revelation of such universal mathematical concepts? Perhaps the difficulty in finding Ceres in the sky was indicative of a deficiency of the day’s mathematics. The problem was only superficially about finding a stone in the darkness; lurking underneath, was a fundamental mathematical problem that we didn’t know how to solve. More generally, lurking beneath every natural event that we can’t predict, there is either a fundamental physical law that we don’t know or a mathematical problem that we can’t solve.
There is a second interesting feature of this event. What was the value of this endeavor from the perspective of Gauss and the astronomers? A success in this endeavor would mean finding the stone in the darkness; a failure would mean losing the stone. What does any of this mean? What do we gain if we find the stone and what do we lose if we don’t? From the perspective of Piazzi and many other astronomers, it was of profound value to make new friends in the sky, who can help them navigate the sky in the future. In this case, the end product of this endeavor was to rediscover Ceres in the sky and track her location, so that they can keep in touch with her. From the perspective of Johann Bode, another astronomer, the hopeful end product of this endeavor was to show that God indeed didn’t leave a large gap between Mars and Jupiter; he placed Ceres appropriately in between, in accordance with Bode’s law. However, as Gauss mentions in his book, from his perspective, the value of this endeavor lies in establishing the importance of the mathematical problem that lies beneath. The end product was to solve this mathematical problem. Ceres was a piece of stone to some; she was a friend in the sky to others and to Gauss, she was a messenger of a mathematical treasure.
Interestingly enough, Gauss had considered the problem of predicting the orbit of a planet starting with very few points known, as a mathematical curiosity even before he became aware of the case of Ceres! It was usual for astronomers to find orbits of comets with very few points known. However, they could make a variety of intuitive assumptions about the orbit, making the problem simple. Gauss, as any mathematician would, naturally sought to solve the most general version of this problem, with no assumptions about the orbit. However, he anticipated that astronomers would overlook this problem considering it to be a mathematician’s fantasy with no practical value. The case of Ceres, as Gauss himself notes, was exciting to him because it inflated the value of this problem among astronomers.
Of the discoveries made during this episode, Ceres is perhaps the least remembered today. However, she assumed the primal position in 1801, during the event. The endeavor was powered by the value attached to finding Ceres. Natural sciences are about knowing our home, the universe, better. I believe that mathematics, in an unusual way, is also a part of the physical universe. The more we explore the latter, the more we will learn about the former.
Before concluding, I want to remark on the role of risk and trust in availing the value of exploration. Scientific research is an exploration. Like all explorations, it involves, at-least, the inevitable risk of not finding anything. By its nature, the outcome of a scientific exploration can’t be evaluated by someone who is not an expert. In other words, the question of whether a researcher has been honest in doing what they should do can only be answered by another researcher sharing the same expertise, and perhaps the same source of income ;). Therefore, such explorations can be carried out only when the explorers are trusted. Scientists are trustees. The story of the discovery of Ceres, however, was remarkable. With no knowledge of its details, Gauss’ methods were validated by the astronomers. In fact both the fields of astronomy and mathematics were mutually validated by each other.
Thought Flow 