Another Renaissance Mathematicus series comes to an end a little more than two years after it began with the questions Renaissance Science? Which Renaissance? and eight weeks later, is there any such thing as Renaissance science and if so, what is it? Having established that we were talking about the Humanist Renaissance, which began in Northern Italy in the fourteenth century, as a literary movement, and expanded into other areas in the fifteenth and sixteenth centuries. I took the middle of the seventeenth century as its culmination. However, already in that early episode I ended thus:

A important closing comment is that there is actually a very high level of continuity rather than disruption from the High Middle Ages through the Renaissance and one can regard the Renaissance both as a phase of the Middle Ages but also of the Early Modern Period; all historical periodisations are of course artificial and also to some extent arbitrary.

is a fresco by Raphael. The fresco was painted between 1509 and 1511 as a part of Raphael’s commission to decorate the rooms now known as theThe School of AthensStanze di Raffaello, in the Apostolic Palace in the Vatican. It depicts a congregation of philosophers, mathematicians, and scientists from Ancient Greece, including Plato, Aristotle, Pythagoras, Archimedes, and Heraclitus. The Italian artists Leonardo da Vinci and Michelangelo are also featured in the painting, shown as Plato and Heraclitus respectively.The painting notably features accurate perspective projection, a defining characteristic of the Renaissance era. Raphael learned perspective from Leonardo, whose role as Plato is central in the painting. The themes of the painting, such as the rebirth of Ancient Greek philosophy and culture in Europe (along with Raphael’s work) were inspired by Leonardo’s individual pursuits in theatre, engineering, optics, geometry, physiology, anatomy, history, architecture and art.

Description and picture both taken from Wikipedia

I then posed the all-important question, is there such a thing as Renaissance science, and if so, what is it? If I wished to write a series of episodes about it, then I should first establish what it is I’m writing about. To give a brief summary of that episode I stated that in my defined period of Renaissance science, from c. 1400 to c. 1650, a crossover took place between academic book knowledge and the empirical and practical knowledge of the artisan, areas that had previously been separated from each other. This crossover was driven by external forces drawn from political, social, cultural, and economic developments. Added to this was the literary Humanist drive to recover the knowledge of classical antiquity, which didn’t restrict itself to works of literature but revived interest in many half-forgotten scientific texts. These two developments blended together to produce a new wave of empirical, practice orientated knowledge, which when theorised in a further evolution, which began in the sixteenth century following into the seventeenth led to the so-called scientific revolution.

Having delineated an area that, I was happy to label Renaissance science, I then over the next forty-five episodes tried to expand on this theoretical definition by displaying how these changes developed in various areas of knowledge production. In the last two episodes I gave first a brief outline of the philosophical revivals that contributed to the downfall of the all-embracing Aristotelian philosophy of the medieval scholastics and then explained why I believe that the much-quoted Francis Bacon actually presented a summation of Renaissance science rather than pointing to the future and modern science as he is all to oft presented.

Having defined a general development in the sciences that took place in the historical period I defined as the scientific Renaissance i.e., a period defined by the scientific development that took place during it, we now have to apply the same caution that we applied above to the time period itself. Expressed very simply, there is no point in time were people stopped doing medieval science and started doing Renaissance science. Equally there is no point in time were people stopped doing Renaissance science and started doing modern science. These developments were both gradual and included much takeover of thoughts and practices from one era into the other. A key concept here is continuity, although we must be careful not to evoke a Whig historical concept of progress. I will now look back at a couple of the topics we have discussed over the last two years and point out the threads of continuity that they contain.

I will start with the botanical gardens, in themselves part of the much wider complex of natural history, *materia medica*, medical education, and botany. As I pointed out the concept of the specialist, medicinal herb garden had already existed in antiquity, and had also been fostered in the medieval monasteries, whose gardens also served as a role model for the university botanical gardens that emerged in the Renaissance. Here, there was a change of emphasis, as well as serving as a practical resource for medicinal herbs, to save having to scavenge them from nature, the university botanical gardens served as a centre for teaching and research. Also, botanical gardens did not cease to exist with the advent of modern science but are still going strong, to quote Wikipedia:

Worldwide, there are now about 1800 botanical gardens and arboreta in about 150 countries (mostly in temperate regions) of which about 550 are in Europe (150 of which are in Russia), 200 in North America, and an increasing number in East Asia. These gardens attract about 300 million visitors a year.

The last sentence shows that botanical gardens now function as tourist attractions, the entrance fees helping to finance their upkeep. Although, the Renaissance botanical gardens also attracted a steady flow of visitors from all over Europe. The modern botanical gardens, many of which a government sponsored, are major scientific research centres and are networked worldwide to increase their effectivity, exchanging plants, seeds, and knowledge. This networking and scientific exchange was already developing during the Renaissance, albeit on a much smaller scale.

As you can see there is a strong continuity in the concept and existence of botanical gardens from some point deep in antiquity down to the present day. However, that continuity is not a smooth curve but has suffered breaks and seen changes in the people operating them and the functions that they have fulfilled.

I don’t intend to deal with all the topics that I have covered in the episodes of this series in the same way here, but I will bring one more example from a completely different area, mathematics and in particular algebra.

Although the word algebra is a comparatively modern coinage, stemming as it does from the title of a ninth-century book, *al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah* (*The Compendious Book on Calculation by Completion and Balancing*) by the Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850), the commonly used definition for elementary algebra is to quote Wikipedia:

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as “does an equation have a solution?”, “how many solutions does an equation have?”, “what can be said about the nature of the solutions?” are considered.

A definition that covers all of the algebra under discussion here.

Both the ancient Egyptians and Babylonians wrote about the solution of equations in their mathematical texts. The Babylonian’s even had a version of the general solution for quadratic equations as early as 1700 BCE. It is in a different form to that taught in schools today and of course only accepts positive solutions. Babylonian algebra grew out of the commercial arithmetic that they developed for their central, state-controlled economy.

We find our form of the general solution for quadratic equations with both positive and negative solutions in the *Brāhma-sphuṭa-siddhānta* (*Correctly Established **Doctrine of Brahma*) of the Indian mathematician and astronomer Brahmagupta (c. 598–c. 668) a text that did much to inform the work of al-Khwārizmī.

Al-Khwārizmī’ s book was first translated into Latin by Robert of Chester in 1145 but initially had little impact in Europe. Algebra first became truly establish in Europe with the publication of the *Liber Abbaci* (*The Book of Calculation*) by Leonardo Pisano (c. 1170–c. 1245) in 1201. This established algebra as commercial arithmetic, which was then taught throughout Europe in so-called Abbacus schools to apprentice traders, in order to be able to calculate interest rates on loans, exchange rates of currencies when crossing borders, and profit shares in joint trading ventures, amongst other things. This had also been the primary use of algebra in Islamicate culture from whom Leonardo had directly taken his knowledge of the discipline.

It was first in the sixteenth century, also within my defined timeframe for Renaissance science, that algebra first became recognised as a proper branch of mathematics during the disputes over the discovery of the general solutions of the cubic, and quartic equations. Some even refer to Cardano’s *Ars Magna* (Nürnberg 1543), a central text in those disputes, as the first modern mathematics book. Algebra only became truly establish as the core of analytical mathematics in the seventeenth century as part of the so-called scientific revolution. The sixteenth and seventeenth centuries also saw a gradual development from rhetorical algebra, written entirely in words, over syncopated algebra, with some symbolism, to our symbolic algebra.

Algebra, as the doctrine of the solution of equations, is of course a central element of the modern school mathematics curriculum. In German the general solution for quadratic equations is called the Mitternachtsformel (Midnight formula), because school children are expected to be able to rattle it off if woken up by their maths teacher at midnight.

As I have outlined above algebra winds its way through history from at least 2000 BCE down to the present, changing in presentation, and function over the centuries. It is by no means a continuous evolution but continuity over time in its history is just as important, if not more so, that the developments in any given, artificially defined period such as the Humanist Renaissance.

What I hope I have made clear in this blog post is that, although historically useful, the concepts, of a time period that we call the Humanist Renaissance, and the developments in the sciences within that period, that I, following others, have chosen the define as Renaissance science, are artificial constructs and when we use them, we should be very much aware of the continuity that exists with the periods and the science within them, that exists both before and after our defined period.