The seventeenth century is commonly called the scientific revolution principally for the emergence of two branches of science, although much more was actually going on. Firstly, the subject of this series, astronomy, and secondly the branch of science we now know as physics. The name physics had a significantly different meaning in the medieval Aristotelian philosophy. As we know astronomy and physics are intimately connected, in fact, intertwined with each other and this connection came into being during the seventeenth century. We have already seen in an earlier episode how the modern concepts of motion began to emerge from Aristotelian philosophy in the sixth century reaching a temporary high point in the early seventeenth century in the works of Galileo and Beeckman.

Galileo is often regarded as the initiator, founder of these developments and lauded with titles such as the father of physics, which is just so much irrelevant verbiage. In fact as we saw in the case of the laws of fall he was just following developments that had long preceded him. On a more general level the situation is no different. Kepler was apparently the first to use the concept of a physical force rather than a vitalist anima. Simon Stevin (1548–1620) resolved the forces acting on an object on an inclined plane, effectively using the parallelogram of forces to do so. In hydrostatics he also discovered the so-called hydrostatic paradox i.e. that the pressure in a liquid is independent of the shape of the vessel and the area of the base, but depends solely on its depth. Thomas Harriot (c. 1560–1621) actually developed a more advanced mechanics than Galileo but as usually didn’t publish, so his work had no impact. However, it clearly shows that Galileo was by no means the only person considering the problems. All of these early discoveries, including Galileo’s, suffered from a problem of vagueness. Nobody really knew what force was and the definitions of almost all the basic concepts–speed, velocity, acceleration etc.–were defective or simply wrong. The century saw the gradual development of a vocabulary of correctly defined terms for the emerging new physics and a series of important discoveries in different areas, mechanics, statics, hydrostatics, optics etc.

I’m not going to give a blow-by-blow history of physics in the seventeenth century, I would need a whole book for that, but I would like to sketch an aspect that in popular accounts often gets overlooked. The popular accounts tend to go Galileo–Descartes–Newton, as if they were a three-man relay team passing the baton of knowledge down the century. In reality there were a much larger community of European mathematicians and proto-physicists, who were involved, exchanging ideas, challenging discovery claims, refining definitions and contributing bits and pieces to big pictures. Each of them building on the work of others and educating the next generation. What emerged was a pan European multidimensional cooperative effort that laid the foundations of what has become known as classical or Newtonian physics, although we won’t be dealing with Newton yet. Once again I won’t be able to give all the nodes in the network but I hope I can at least evoke something of the nature of the cooperative effort involved.

I will start of with Simon Stevin, a man, who few people think of when doing a quick survey of seventeenth century physics but who had a massive influence on developments in the Netherlands and thus, through connections, in France and further afield. Basically an engineer, who also produced mathematics and physics, Stevin motivated Maurits of Nassau, Stadholder of the young Dutch Republic to establish engineering and the mathematical sciences on the new Dutch universities. Stevin’s work influenced both the Snels, Rudolph (1546–1613) and his son Willebrord (1580–1626), the latter translated Stevin’s work into French and Latin from the Dutch, making it available to a much wider audience.

Stevin set up a school for engineering at the University of Leiden with Ludolph van Ceulen as the first professor of mathematics teaching from textbooks written by Stevin. Van Ceulen (1540–1610), who was Willebrord Snel’s teacher, was succeeded by his pupil Frans van Schooten the elder (1581–1646), whose pupils included his own son, Frans van Schooten the younger (1615–1660), Jan de Witt (1625–1672), Johann Hudde (1628–1704), Hendrick van Heuraet (1633–1660?), René-François de Sluse ((1622–1685) and Christiaan Huygens (1629–1695), all of whom would continue their mathematical development under van Schooten junior and go on to make important contributions to the mathematical sciences. An outlier in the Netherlands was Isaac Beeckman (1588–1637), a largely self taught natural philosopher, who made a point of seeking out and studying Stevin’s work. This group would actively interact with the French mathematicians in the middle of the century.

On the French side with have a much more mixed bunch coming from varying backgrounds although Descartes and Mersenne were both educated by the Jesuits at the College of La Flèche. Nicolas-Claude Fabri de Peiresc (1580–1637), the already mentioned René Descartes (1596–1650) and Marin Mersenne (1588–1648), Pierre de Fermat (1607–1665), Pierre Gassendi (1592–1655), Ismaël Boulliau (1605–1694) and Blaise Pascal (1623–1662) are just some of the more prominent members of the seventeenth century French mathematical community.

René Descartes made several journeys to the Netherlands, the first as a soldier in 1618 when he studied the military engineering of Simon Stevin. He also got to know Isaac Beeckman, with whom he studied a wide range of areas in physics and from who he got both the all important law of inertia and the mechanical philosophy, borrowings that he would later deny, claiming that he had discovered them independently. Descarte and Beeckman believed firmly on the necessity to combine mathematics and physics. Beeckman also met and corresponded with both Gassendi and Mersenne stimulating their own thoughts on both mathematics and physics.

On a later journey to the Netherlands Descartes met with Frans van Schooten the younger, who read the then still unpublished *La Géometrié*. This led van Schooten to travel to Paris where he studied the new mathematics of both living, Pierre Fermat, and dead, François Viète (1540–1603), French mathematicians before returning to the Netherlands to take over his father’s professorship and his group of star pupils. Descartes was also a close friend of Constantijn Huygens (1596–1687), Christiaan’s father.

Peiresc and Mersenne were both scholars now referred to as post offices. They both corresponded extensively with mathematicians, astronomers and physicists all over Europe passing on the information they got from one scholar to the others in their networks; basically they fulfilled the function now serviced by academic journals. Both had contacts to Galileo in Italy and Mersenne in particular expended a lot of effort trying to persuade people to read Galileo’s works on mechanics, even translating them into Latin from Galileo’s Tuscan to make them available to others. Mersenne’s endeavours would suggest that Galileo’s work was not as widely known or appreciated as is often claimed.

Galileo was, of course, by no means the only mathematician/physicist active in seventeenth century Italy. The main activist can be roughly divided in two groups the disciples of Galileo and the Jesuits, whereby the Jesuits, as we will see, by no means rejected Galileo’s physics. The disciples of Galileo include Bonaventura Francesco Cavalieri (1598–1647) a pupil of Benedetto Castelli (1578–1643) a direct pupil of Galileo, Evangelista Torricelli (1608–1647) another direct pupil of Galileo and Giovanni Alfonso Borelli (1608-1679) like Cavalieri a pupil of Castelli.

On the Jesuit side we have Giuseppe Biancani (1565–1624) his pupil Giovanni Battista Riccioli (1598–1671) and his one time pupil and later partner Francesco Maria Grimaldi (1618–1663) and their star pupil Giovanni Domenico Cassini (1625–1712), who as we have already seen was one of the most important telescopic astronomers in the seventeenth century. Also of interest is Athanasius Kircher (1602–1680), professor at the Jesuit University, the Collegio Romano, who like Peiresc and Mersenne was an intellectual post office, collecting scientific communications from Jesuit researchers all over the world and redistributing them to scholars throughout Europe.

Looking first at the Jesuits, Riccioli carried out extensive empirical research into falling bodies and pendulums. He confirmed Galileo’s laws of fall, actually using falling balls rather than inclined planes, and determined an accurate figure for the acceleration due to gravity; Galileo’s figure had been way off. He was also the first to develop a second pendulum, a device that would later prove essential in determining variation in the Earth’s gravity and thus the globes shape.

Grimaldi was the first to investigate diffraction in optics even giving the phenomenon its name. Many of the people I have listed also did significant work in optics beginning with Kepler and the discovery of more and more mathematical laws in optics helped to convince the researchers that the search for mathematical laws of nature was the right route to take.

As we saw earlier Borelli followed Kepler’s lead in trying to determine the forces governing the planetary orbits but he also created the field of biomechanics, applying the newly developed approaches to studies of muscles and bones.

Torricelli is, of course, famous for having invented the barometer, a device for measuring air pressure, of which more in a moment, he was trying to answer the question why a simple air pump cannot pump water to more than a height of approximately ten metres. However, most importantly his experiments suggested that in the space above the mercury column in his barometer there existed a vacuum. This was a major contradiction to traditional Aristotelian physics, which claimed that a vacuum could not exist.

Torricelli’s invention of the barometer was put to good use in France by Blaise Pascal, who sent his brother in law, Périer, up the Puy de Dôme, a volcano in the Massif Central, carrying a primitive barometer. This experiment demonstrated that the level of the barometer’s column of mercury varied according to the altitude thus ‘proving’ that the atmosphere had weight that lessened the higher one climbed above the earth’s surface. This was the first empirical proof that air is a material substance that has weight. One person, who was upset by Torricelli’s and Pascal’s claims that the barometer demonstrates the existence of a vacuum, was René Descartes to whom we now turn.

Descartes, who is usually credited with being one of, if not the, founders of modern science and philosophy, was surprisingly Aristotelian in his approach to physics. Adopting Beeckman’s mechanical philosophy he thought that things could only move if acted upon by another object by direct contact; action at a distance was definitely not acceptable. Aristotle’s problem of projectile motion, what keeps the projectile moving when the contact with the projector breaks was solved by the principle of inertia, which reverses the problem. It is not longer the question of what keeps the projectile moving but rather what stops it moving. He also, like Aristotle, adamantly rejected the possibility of a vacuum. His solution here was to assume that all space was filled by very fine particles of matter. Where this theory of all invasive particles, usually called corpusculariansim, comes from would takes us too far afield but it became widely accepted in the second half of the seventeenth century, although not all of its adherents rejected the possibility of a vacuum.

Descartes set up laws of motion that are actually laws of collision or more formally impact. He starts with three laws of nature; the first two are basically the principle of inertia and the third is a general principle of collision. From these three laws of nature Descartes deduces seven rules of impact of perfectly elastic (i.e. solid) bodies. Imagine the rules for what happens when you play snooker or billiards. The details of Descartes rules of impact needn’t bother us here; in fact his rules were all wrong; more important is that he set up a formal physical system of motion and impact. Studying and correcting Descartes rules of impact was Newton’s introduction to mechanics.

Turning to another Frenchman, we have Ismaël Boulliau, who was a convinced Keplerian. Kepler had hypothesised that there was a force emanating from the Sun that swept the planets around their orbits, which diminished inversely with increasing distance from the Sun. Boulliau didn’t think that such a force existed but if it did then it would be an inversed square force in analogy to Kepler’s law for the propagation of light; a candidate for the first modern mathematical law of physics. The foundations of the new physics were slowly coming together.

Our last link between the Dutch and French mathematicians is Christiaan Huygens. Huygens initially took up correspondence with Mersenne around 1648; a correspondence that covered a wide range of mathematics and physics. In 1655 he visited Paris and was introduced to Boulliau and a year later began correspondence with Pierre Fermat. Frans van Schooten the younger continued to act as his mathematical mentor.

Huygens absorbed the work of all the leading European mathematician and physicists and as an avowed Cartesian became acknowledged as Europe’s leading natural philosopher. He realised that Descartes rules of impacts were wrong and corrected them. Huygens was also the first to derive and state what is now know as Newton’s second law of motion and derived the law of centripetal force, important steps on the route to a clear understanding of forces and how they operate. Huygens also created the first functioning pendulum clock in the process of which he derived the correct formula for the period of an ideal mathematical pendulum. It is very easy to underestimate Huygens contributions to the development of modern physics; he tends to get squeezed out between Descartes and Newton.

All the way through I have talked about the men, who developed the new physics as mathematicians and this is highly relevant. The so-called scientific revolution has been referred to as the mathematization of science, an accurate description of what was taking place. The seventeenth century is also known as the golden age of mathematics because the men who created the new physics were also at the same time creating the new mathematical tools needed to create that physics. An algebra based analytical mathematics came to replace the geometric synthetic mathematics inherited from the Greeks.

Algebra first entered Europe in the twelfth century with Robert of Chester’s translation of Muḥammad ibn Mūsā al-Khwārizmī’s ninth century *Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala* (*The Compendious Book on Calculation by Completion and Balancing*), the word algebra coming from the Arabic *al-ğabr*, meaning completion or setting together (in Spanish an algebraist is a bone setter). This introduction had little impact. It was reintroduced in the thirteenth century by Leonardo of Pisa, this time as commercial arithmetic, where it, especially with the introduction of double entry bookkeeping, had a major impact but still remained outside of academia.

It was first in the sixteenth century that algebra found its way into academia through the work of Scipione del Ferro (1465–1526), Niccolò Fontana known as Tartaglia (c.1499–1557)and above all Gerolamo Cardano (1501–1576), whose *Artis Magnæ, Sive de Regulis Algebraicis Liber Unus* (*Book number one about The Great Art, or The Rules of Algebra*) published by Johannes Petreius (c. 1496–1550) in Nürnberg in 1545 is regarded as the first modern algebra textbook or even the beginning of modern mathematics (which, as should be obvious to regular readers, is a view that I don’t share).

Modern readers would find it extremely strange as all of the formulas and theorems are written in words or abbreviations of words and there are no symbols in sight. The status of algebra was further established by the work of the Italian mathematician Rafael Bombelli (1526–1572),

(1572)

Another school of algebra was the German Cos school represented by the work of the

German mathematician Michael Stifel (1487–1567), *Arithmetica integra* (1544),

Simon Stevin in the Netherlands *L’arithmétique* (1585)

and Robert Recorde (c. 1512–1558) in Britain with his *The Whetstone of Witte* (1557).

Algebra took a new direction with the French mathematician François Viète (1540–1603),

who wrote an algebra text based on the work of Cardano and the late classical work of Diophantus of Alexandria (2^{nd} century CE) *In artem analyticam isagoge* (1591) replacing many of the words and abbreviations with symbols.

Algebra was very much on the advance. Of interest here is that Galileo, who is always presented as the innovator, rejected the analytical mathematics, whereas the leading Jesuit mathematician Christoph Clavius (1538–1612), the last of the staunch defenders of Ptolemaic astronomy, wrote a textbook on Viète’s algebra for the Jesuit schools and colleges. Two further important publications on symbolic algebra in the seventeenth century were the English mathematician, William Oughtred’s *Clavis Mathematicae* (1631),

which went through several editions and was read all over Europe and Thomas Harriot’s (1631), the only one of his scientific works ever published and that only posthumously.

The development of the then new analytical mathematics reach a high point with the independent invention by Pierre Fermat and René Descartes of analytical geometry, which enabled the geometrical presentation of algebraic functions or the algebraic presentation of geometrical forms; a very powerful tool in the armoury of the mathematical physicists in the seventeenth century. Fermat’s pioneering work in analytical geometry (*Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum*) was circulated in manuscript form in 1636 (based on results achieved in 1629) This manuscript was published posthumously in 1679 in *Varia opera mathematica*, as *Ad Locos Planos et Solidos Isagoge* (*Introduction to Plane and Solid Loci*).

Descartes more famous work was published as *La Géometrié*, originally as an appendix to his *Discours de la méthode* (1637). However, much more important for the dissemination of Descartes version of the analytical geometry was the expanded Latin translation produced by Frans van Schooten the younger with much additional material from van Schooten himself, published in 1649 and the second edition, with extra material from his group of special students mentioned above, in two volumes 1659 and 1661. Van Schooten was the first to introduce the nowadays, ubiquitous orthogonal Cartesian coordinates and to extend the system to three dimensions in his Exercises (1657).

The other branch of analytical mathematics that was developed in the seventeenth century was what we now know as infinitesimal calculus, the mathematics that is necessary to deal with rates of change, for example for analysing motion. There is a prehistory, particularly of integral calculus but it doesn’t need to interest us here. Kepler used a form of proto-integration to prove his second law of planetary motion and a more sophisticated version to calculate the volume of barrels in a fascinating but often neglected pamphlet. The Galilean mathematician Cavalieri developed a better system of integration, his indivisibles, which he published in his *Geometria indivisibilibus continuorum nova quadam ratione promota*, (*Geometry, developed by a new method through the indivisibles of the continua*) (1635) but actually written in 1627, demonstrated on the area of a parabola. This work was developed further by Torricelli, who extended it to other functions.

The other branch of calculus the calculating of tangents and thus derivatives was worked on by a wide range of mathematicians. Significant contributions were made by Blaise Pascal, Pierre de Fermat, René Descarte, Gregoire de Saint-Vincent, John Wallis and Isaac Barrow. Fermat’s work was the most advanced and included contributions to both integral and deferential calculus, including a general method for determining tangents that is still taught in schools. The Scottish mathematician, James Gregory (1638–1675), inspired by Fermat’s work developed the second fundamental theory of calculus, which states that the integral of a function *f* over some interval can be computed by using any one, say *F*, of its infinitely many anti-derivatives. Isaac Barrow (1630–1677) was the first to provide a full proof of the fundamental theorem of calculus, which is a theorem that links the concept of differentiating a function with the concept of integrating a function. Fermat’s work and John Wallis’ *Arithmetica Infinitorum* (1656) would be an important jumping off point for both Leibniz and Newton in the future.

By about 1670, the mathematicians of Europe, who knew of and built on each other’s work had made major advances in the development of both modern mathematics and physics laying the foundations for the next major development in the emergence of modern astronomy. However, before we reach that development there will be a couple of other factors that we have to consider first.

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I haven’t seen one of those “Under Construction” graphics since the early days of the internet. Nowadays, I just assume that

everyis “under construction” in some sense.(I imagine you’ll want to delete this comment once it’s inapplicable.)

I erected the Under Construction sign because I have now for a couple of years with a solid regularity posted once a week on Wednesday, with occasional extra posts such as the recent ‘Doctor, doctor…’ thrown in at random. This Wednesday another slice of ‘Emergence’ would have been due but the foetus is proving obstreperous and doesn’t want to leave the womb, so just in case anybody, relying on my regularity, passed by hoping to read a new post I informed them that it would be late. Resorting to a forceps birth, I hope that the new born will emerge later today or at the latest tomorrow.

Ah, OK. I wasn’t criticizing, it was just a pleasant nostalgia moment for me. Looking forward to the post.

I didn’t think you were criticising, just wanted to explain the situation for any passers by.

What a magnificent post! The wait was well worth it.

One sentence struck me:

I know people have said this, but I’m wondering, do you agree? It seems just like the sort of sentiment you’d debunk. “The first modern algebra textbook” is next door to “The father of modern algebra”, it seems to me.

As you immediately point out, “algebra” in the symbolic sense this book is not. That makes a

hugedifference. In many other ways the book looks to the past. Negative or even zero coefficients are not allowed, so that the treatment of the cubic equation is split up into I forget how many cases, taking up a large part of the book. The so-called “geometric algebra” of the ancient Greeks leaves its traces, and one notes the similarity to the Arabic works you mention.On the other hand, the first publication of the cubic and quartic formulas, the first mention of imaginary numbers… One foot in the past, one in the future–but isn’t that a common story?

As for the cubic and quartic formulas,

technicallythis is way less significant for furture development than the symbolic apparatus of Viète et.al. On the other hand, I’ve read thatculturally, sociallythis made quite an impact, supporting the Renaissance boast, “We can do better than the ancients!”Ah, I just reread your piece “Gunfight at the Cubic Corral”, and I saw the sentence

Yes indeed.

I’m glad to see the edit, about

The Great Art. Another thought, this time on terminology.Some historians get upset about ahistorical applications of words like scientist, physicist, etc. One argument is that that to a modern reader, “physicist” (for example) has a bunch of connotations inappropriate for, say, Archimedes or Newton. Another is that neither Archimedes nor Newton would ever have called themselves physicists, so we shouldn’t either.

David Wootton dismantled the second argument in

The Invention of Science, or at any rate argued forcefully for a more nuanced view. I want to make a different point.Cardano’s book does have

The Rules of Algebraright in the title, and of course the word ‘algebra’ itself comes from al-Khwārizmī’s book. (I’ve always regretted that ‘muquabala’ never made it into the mathematical lexicon.) But for a modern reader, calling what they did ‘algebra’ is profoundly misleading.What to do? Well, a common and entirely reasonable approach, employed here by Thony, is to elaborate immediately.

But “sauce for the goose”… No one reading a sentence like “In many ways, Archimedes was one of the first mathematical physicists” is likely to imagine him hunched over a laptop, running SageMath on path integrals.