PhD Research SeriesObserving Mathematics in the Making: The Personal Notes of Thomas Harriot

16 June 2025

Thomas Harriot studied the motion of falling objects independently of Galileo, discovered Snell’s law of refraction before Snell, pioneered interpolation before Newton, and invented binary before Leibniz. Stefano Farinella studies the notebooks of this remarkable and still remarkably unknown figure.

By Stefano Farinella

In the autumn of 1609, Galileo Galilei pointed his new instrument, the telescope, towards the moon. He produced a set of drawings of the satellite, later made famous in his book Sidereus Nuncius. These drawings, among his other observations through the telescope, mark an important turning point in the history of Western scientific thought, contributing to what we commonly call the Scientific Revolution. Few people know that, on 5 August of that same year, an Englishman by the name of Thomas Harriot pointed his own telescope towards the moon and produced the first drawings of the satellite using that instrument. This English contemporary of Galileo also studied the motion of falling objects and parabolic trajectories independently of Galileo, discovered Snell’s law of refraction before Snell and Descartes, pioneered interpolation methods before Newton, invented binary before Leibniz, devised his own phonetic alphabet, and much more.

Why, then, is Harriot’s name largely unknown in the history of science? His discoveries went mostly unnoticed because, unlike Galileo, he published almost nothing in his lifetime. All of Harriot’s scientific work is to be found in his approximately 5,200 pages of personal notes, which were lost for more than 150 years after his death, until their rediscovery in 1784 by an Austrian astronomer. These pages, now jumbled and divided between two different British libraries, represent a daunting challenge for historians of science to this day. Studying these notes, especially from the second half of the twentieth century, has allowed scholars to recognise the importance of Harriot’s work, and finally give him due credit for his discoveries.

Today, Harriot’s notes are being studied in a different light at the CSMC. While the research on these notes is part of a bigger effort to paint a clearer picture of this extraordinary scientist, Harriot’s bundles are seen in the project ‘Knowledge Transformation and Strands of Notation – Thomas Harriot’s (1560–1621) Working Notes as a Representation of Early Modern Scientific Practice’ as an example of a particular category of written artefact: the mathematical note.

Mathematical Notes

What makes mathematical notes different from other types of manuscripts? Is there any advantage in studying personal notes, instead of focusing on published treatises? Imagine having to multiply two numbers, and then write down the result. If the numbers are small, you can perform the multiplication in your head, and then just write the result on paper. In this case, the note’s purpose is that of storing knowledge, which you obtained by performing the operation. If the numbers are sufficiently large, you will not be able to perform the multiplication in your head, but you will have to write the numbers down and use the rules of long multiplication to eventually reach the result you want. In this case, the piece of paper on which you performed the multiplication is actually a tool, an instrument that you use to help your brain perform the mathematical operation. Imagine someone has to look at the two pieces of paper, without knowing what you wrote on them. In the first case, they would only be able to see a number and would have no information about from whence it came. In the second case, assuming they know your number system, they would easily be able to trace back your steps and understand that the final number is the product of the two initial numbers.

This role of mathematical notes, being tools for performing operations and producing new knowledge, is what makes them so special. By looking at them, the historian can almost see the scientist at work, trying various methods to wrestle the truth out of his calculations. Studying a treatise or an article, on the other hand, only shows the final result. Studying mathematical notes as a category of manuscripts means asking multiple questions. Are the results obtained linked to the way one writes down mathematical passages? Would one think in a novel way if they used a different formalism to treat the same problem, such as geometry instead of algebra? What changes between different numbering systems, different languages, different cultures and different scientists when we look at mathematical notes? Is it possible to obtain information about the scientific process followed by the author by studying the materiality of the notes, such as the paper or the ink? And how do mathematical notes compare with other types of manuscripts?

Shipbuilding, Inks, Layers

We can learn more about Harriot’s way of doing science by taking the materiality of the manuscripts into account. One example can be seen in Harriot’s folios related to shipbuilding, where he tries to devise mathematical rules for calculating the ideal sizes of different parts of a ship. It is possible to see that some sections were added later, after completing the calculations on the page, by identifying two different inks used on the same folio. Looking more closely, one realises that these parts are later corrections, and all connected to a mistake Harriot committed in his choice of units of measurement, a fundamental point if one wants to actually build a ship. This reveals many important things about Harriot’s approach to mathematical problems, and how he had to adapt his methods when communicating with artisans and practitioners.

There is a lot to learn by studying Thomas Harriot’s manuscripts, seeing them as belonging to the category of mathematical notes. It will be possible to understand better the surprisingly advanced results of this still relatively unknown character by adopting the perspectives of history of science and manuscript studies. By delving deep into Harriot’s mathematical reasoning and practices of writing, and comparing it with other mathematicians and scientists, we can then gain insights into how notes written in a mathematical language shape the way we do science, even to this day.