User:Guillermind81

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History of Science.

Interested in ancient and early modern science, philosophy.

Observation, Experiment, and Ancient Physics

The following are some musings regarding the place of empirical evidence (mainly in the form of observations and experiments) in the context of ancient Greek science (by which I mean those inquiries directed at natural phenomena). For the most part, the treatment takes after Knorr (1989) and Lloyd (2000), with an emphasis on those studying physical phenomena primarily or exclusively from a mathematical framework rather than from the life sciences. By extension, I would not be dealing with those personalities whose inquires belong more to philosophy proper, except in those cases where doing so helps illustrate a point.

At the outset, it is important to comment on the often-widespread assumption that the ancient Greeks disregarded the "practical" arts and did not seriously consider the empirical aspects of their science. As a rough approximation, there is some truth to this; nonetheless, such a simplification conveniently ignores other threads that did exist and which, alongside the often heavy intellectualism, form part of Greek science all the same. Islamic and Renaissance scholars and their early modern successors would eventually pick up some of these threads in the course of developing their own scientific enterprise, leading to the delicate interplay between the empirical and the theoretical that marks the physical sciences of our day.

Classical Greece

The origins of Greek science, if one could point to any series of events that gave it “birth,” are shrouded in mystery in the distant past. It is commonly assumed that ancient Greeks borrowed quite a bit of technical knowledge from neighboring civilizations—the Greeks themselves said as much. Herodotus, for instance, mentions a lot of Egyptian travel among the earliest philosophers, and later figures such as Democritus were said to have ventured into distant lands to acquire knowledge. Yet, although ancient Greeks acknowledged these borrowings, they just as quickly took credit for originating their own ideas about nature and the underlying principles of the universe.

The evidence for Greek science only becomes reliable enough the closer we get to the age of Classical Greece, and even there we are still largely dependent on fragments and on commentaries by later authors. Be as it may, we have hitherto better testimony of individuals active in this period who sought explanations not only theoretically but also engaged in careful observations of physical phenomena, two of whom are discussed below.

Hippasus (c. 530–450 BC)

Pythagoreans took an interest in studying music by way of mathematical ratios, which became part of the science of harmonics.

A Pythagorean, Hippasus is one of the earliest personalities to whom the labels natural philosopher, mathematician, and music theorist can be assign with some confidence (Huffman, 2019). Pythagoreans are said to have paid attention to mathematics and had a sometimes mystical fascination with the relationships between numbers and things. Philolaus, for example, is quoted as saying:

And indeed all things that are known have number. For it is not possible that anything whatsoever be understood or known without this.

What is known about Hippasus fits the description not only of Pythagoreans but of the other early pre-Socratic philosophers. He held that fire was the principal element of all things, and that the universe was limited and perpetually moving. He may have contributed to mathematics, although it is difficult to be certain about his exact results. Hippasus was also interested in music (specially resonance), a trait he shared with some of his fellow Pythagoreans.

Indeed, the correspondence between the central musical concords of the octave, fifth, and fourth and the whole number ratios 2:1, 3:2, and 4:3, which were already known to instrument makers, appears to have been particularly important to the Pythagoreans: a handful of reports mention some of them were busy establishing the ranking order of these three fundamental concords (Barker, 2020). Hippasus, however, went a step further by demonstrating that these correspondences hold in general following a scientifically sound procedure. According to Aristoxenus, Hippasus had four bronze disks made of equal diameters but with varying thicknesses in the above ratios, then hanged these disks and struck them to produce sounds. The results would have shown that a disk half as thick as another will be an octave apart from the sound produced by the other disk.

If true—and there is little reason to doubt Aristoxenus—this might be the first recorded instance, in all of Greek history, of a person devising an experiment to show that a physical law can be expressed mathematically (Zhmud, 2012). For such a feat to happen, three things are required: (1) isolating those parameters that are essential and amenable for testing from those which are not, (2) devising instruments or procedures specific for that purpose, and (3) finding a way to evaluate the observed results with respect to the theory. Although we do not know what motivated Hippasus to devise such a crude (by modern standards) confirmatory experiment, it nonetheless fits the criteria delineated earlier and represents an incipient but noteworthy attempt in the history of science.

Archytas (c. 435–360 BC)

Commonly referred to as the last of the Pythagoreans, Archytas was a contemporary of Plato and a remarkable man by any account, distinguished simultaneously “as a philosopher, a mathematician, an inventor of ingenious gadgets, a statesman and a military commander” in addition to being admired “for his personal qualities, his kindness, resourcefulness, self-control and affection for children” (Barker, 2007).

By ancient testimony, Archytas was considered one of the three most prominent mathematicians of Plato’s generation, alongside Leodamas and Theaetetus. Archytas’ mathematical prowess was indeed legendary: Eutocius tell us that he was the first to solve the duplication of the cube, namely, to find two mean proportionals between two line segments, through an ingenious geometrical argument. His method uses a semicircle rotating in three-dimensional space and the curve formed by it cutting another three-dimensional surface. He also provided a proof that if n and n + 1 are any two consecutive whole numbers, then there is no rational number b such that n:b = b:(n + 1), a mathematical result that is of importance in music theory.

Archytas appears to be among the first to refer to the four liberal arts of arithmetic [logistic], geometry, astronomy, and music [harmonics] as kin (later to become the medieval quadrivium). In physics, he may have contributed to the development of the science of optics—we are told he studied the visual ray and argued for an emission theory of vision—and, more uncertainly, laid some mathematical foundations for the science of mechanics. An analysis of Archytas’ extant work in fact shows that he had a sophisticated conception of science and did not simply gave formal interpretations of the facts but was actively seeking the mathematical structures underlying them. For example, his research methodology is preserved thus:

To know what was heretofore unknown, one has either to learn it from another, or to discover oneself. What one has learnt, he has learnt from another and with another’s assistance; what one has found, he has found himself and by his own means. Discovery without research is difficult and rare; by research easy and practicable, but without knowing (how) to research it is impossible to do research.

While Ptolemy, in his Harmonics, had this to say about Archytas’ work on music:

Archytas of Tarentum, of all the Pythagoreans the most dedicated to music, attempted to preserve that which is in accordance with reason, not only in connection with the concords but also in the divisions of the tetrachords, on the grounds that commensurability between the differences is intrinsic to the nature of melodic intervals.

That is, the ratios Archytas assigned to the intervals of an attunement were not chosen arbitrarily, merely those which fit the musical data best, but were required to conform to a ‘rational’ principle, that is, one grounded in mathematics. At the same time, Archytas presumably wanted to represent the way contemporary musicians attuned their instruments rather than devising entirely artificial attunements for theoretical purposes (something his followers were bound to do). Other testimonia do attribute to Archytas an analysis of three harmonic systems in use at the time, namely, the enharmonic, diatonic, and chromatic.

Archytas argued elsewhere that sound is produced by movement and that the pitch of a sound depends upon the speed of its movement. He sought to substantiate his thesis by referring to various observations, including the trajectories of projectiles, voice propagation, the sounds produced by several instruments, and in one instance, precise measurements with a reed pipe (kalamos). Although the second part of Archytas’ thesis turned out to be wrong, it shows that he was not content with mere speculation but consistently attempted to strengthen his arguments with empirical support.

Despite the fragmentary evidence, it is important to take stock on what Archytas’ extant work amounted to. On the more mathematical side, he highlighted the importance of epimoric (or superparticular) ratios (where (n +1) / n = 1 + 1/n for a positive integer n) and their use in the construction of harmonic divisions; he classified and defined three means (arithmetic, geometric, and harmonic), and then use these means in the analyses of attunements of instruments. On the more physical side, Archytas showed a scientific interest in the behavior of sound and pitch for their own sake, taken the sounds we hear as essential data–interests that today belong to physical acoustics (Barker, 2007). Thus, both mathematical ideas and empirical observations seem to be at work behind Archytas' interest in music, even if opinions differ about the weight he gave to one or the other. Ultimately, in Archytas' hands, the science of harmonics seemingly headed in a new direction: to explain the facts which empirical data records in a mathematically meaningful way.

Plato and Aristotle on science

The Platonic Academy attracted many Greek intellectuals eager to discuss philosophy and various other topics.

Pythagoreans were not the only ones interested in finding first principles behind physical phenomena. In a sense there were part of a much larger zeitgeist of extraordinary creativity and innovation that extended to architecture, fine arts, literature, and philosophy. It was a time when men (almost always men) sought to attain honor and recognition by all kinds of performative displays, such as reciting a poem, writing a play, winning a case, proving a theorem, drawing a map, or explaining the causes of disease. Among the philosophers who competed in this culture, the two most important ones were Plato and Aristotle. As it happens, both thought carefully about mathematics and physical phenomena, and their views, though in many ways diametrically different, were tremendously influential in the long run.

Plato’s Academy, rather than a full-fledged institution, was a congregation of individuals meeting in the first half of the 4^th century BC on the Akademia site just outside of Athens. Plato's reputation attracted several major intellectuals from across the Greek world, including his most famous student, Aristotle, who spent close to 20 years in his company. All sort of discussions took place at the Academy, including politics, aesthetics, performing arts, mathematics, and the study of nature (Mueller, 2006). Naturally, Plato had something to say about all these things, but it is the latter two that interest us here.

These topics appear predominantly in the Republic and the Timaeus, two Platonic dialogues which largely deal with, respectively, the ideal state and cosmology/metaphysics. In the Republic, Plato devises a curriculum for a cast of rulers of a just state, consisting of ten years of mathematics, five years of dialectic, fifteen years of practical experience, and then for a select few, alternating periods of philosophizing and ruling (Mueller, 2006). It is important to highlight that mathematics serves as an introduction to philosophy and not the other way around. The role of the "mixed" mathematical sciences is even more fraught. The Republic only mentions two: astronomy and harmonics. However, in each case Plato is not interested in the physical phenomena per se (e.g., the motion of planets, the sounds we hear) but something else entirely. As Baker (2007) noted:

[For Plato], when the mathematical disciplines are treated as they should be, without reference to the objects of sense-perception (which are semi-real at best), they habituate the mind to the study of genuine realities, those that are changeless and eternal, and whose existence and nature can be grasped only in thought.

In other words, mathematics and the mathematical sciences exist solely to train the soul in turning away from the world of the sensible and towards the Forms, of which the Good is the highest. The Timaeus goes one step further when it conceives the entire sensible world as an expression of the goodness of the Demiurge (divine craftsman), who shapes such less-than-perfect world after the image of the perfect Forms (White, 2006). Importantly, the tools the craftsman uses to shape the world are the mathematical disciplines already mentioned in the Republic: arithmetic, geometry, stereometry, astronomy, and harmonics, the latter two appearing as purely metaphysically motivated and devoid of empirical considerations (Barker, 2007).

After Plato’s death, his former student Aristotle left the Academy and travel abroad for a while, including a stint as a tutor of Alexander the Great. He returned to Athens around 335 BC as a resident alien and establish his own school next to the Lyceum, a public gymnasium used, among other things, for training Athenian youth.

As we would expect from someone who did much research into animals and plants, and who may have been familiar with the medical tradition, Aristotle had a much higher appreciation for empirical data than did his teacher Plato. He saw no separation between matter and Forms but conceived of these as being mind-independent features of the world. He insisted that scientific knowledge (epistêmê), which included mathematical as well as empirical knowledge, moves beyond reporting facts about the world to an explanation of them in terms of their priority relations. That is, scientific knowledge explains what is less well known by what is better known and more fundamental, or what is explanatorily faint by what is explanatorily evident.

To arrive at these fundamental explanations, as Shields (2020) argues, Aristotle foresaw a process by which:

…knowers move from perception to memory, and from memory to experience (empeiria)—which is a fairly technical term in this connection, reflecting the point at which a single universal comes to take root in the mind—and finally from experience to a grasp of first principles. This final intellectual state Aristotle characterizes as a kind of unmediated intellectual apprehension (nous) of first principles.

Accordingly, Aristotle divided the sciences into three tiers, in descending order of importance: theoretical, practical, and productive. Among the theoretical sciences, metaphysics (what he called "first philosophy") had a place of honor above all the other sciences, including mathematics and physics. In the end, although Aristotle was more mathematically competent than his mentor Plato and recognized the “mixed” mathematical sciences of astronomy, harmonics, optics, and mechanics as legitimate (but limited) approaches to the study of nature, he did not engage in a mathematical study of physical phenomena of the kind found in Hippasus or Archytas (Mendell, 2004).

In the philosophies of Plato and Aristotle, then, we encounter two attitudes toward scientific knowledge. The first heavily downplays empirical considerations and instead seeks out abstract principles beyond the phenomena to orient the soul towards the Good, where mathematics serves as a propaedeutic to dialectic. The second seeks out these first principles by reasoning about empirical data, but here mathematics is confined to a narrow corner of a fundamentally qualitative philosophical undertaking. In either case, we end up short of that unique combination of the theoretical with the empirical that has been the most successful when dealing with physical phenomena.

Hellenistic Age

The Hellenistic Age covers the time between the death of Alexander the Great (c. 323 BC) and the traditional start of the Roman Empire (c. 31 BC). This was a time of great expansion as Greeks settled everywhere beyond the Mediterranean and into Africa and Asia, covering a territory from Gibraltar to the Punjab and down as far as Aswan. In the process, the Greeks encountered first-hand the accumulated knowledge of civilizations more ancient than theirs, including Egyptian, Babylonian, Persian, and Indian.

Alongside the knowledge from ancient civilizations, the Greeks brought with them their own culture and language, which they put to use in the administration of vast amounts of land, peoples, and religions. In turn, this led to an increase in literacy and specialization in the arts, science, and philosophy. Many literary, scientific, and other works that resulted were soon amassed by the newly formed kingdoms of the Diadochi, and it was not uncommon to find leading scholars moving around the courts of Hellenistic rulers (Berrey, 2017). Two figures who were active in this period and who exemplify work in the mathematical sciences are dealt with below.

Archimedes (c. 287–212 BC)

This much is certain about Archimedes’ life: he was born in the prosperous and well-protected kingdom (formerly polis) of Syracuse in Sicily, at that time a major regional player in the western side of the Greek world. His father was an astronomer, and he was on familiar terms with the ruler of Syracuse, King Hiero II, for most of his life. There is no reason to think that Archimedes traveled anywhere—the court of Hiero II and the resources of Syracuse would have been enough to sustain his work. He communicated widely with other intellectuals, mainly those working in Alexandria. He died at an old age defending his city from Roman invasion.

Beyond that, the person of Archimedes confronts us with a puzzle—one could say that there is no one Archimedes but two. One is the engineer, wonder worker, and technological genius that is represented in the numerous anecdotes that began to accumulate shortly after his death. The other is the mathematician, who had a penchant for rigorous demonstrations, and who taunted his colleagues in the search for ever more intricate proofs. That this is one and the same person we owe it to the historical record alone.

How are we to reconcile these two sides of Archimedes? If the anecdotes about his life offer any indication, we should expect him to have engaged in empirical research, perhaps reporting on measuring objects on balances, or bodies in a liquid, alongside some diagrams with calculations or a list of values. However, if we look at most of his extant works, we would be disappointed to find that there is nothing of the sort. What we get instead are highly abstract, idiosyncratic monographs about two- and three-dimensional curvilinear objects and stability conditions for some of these objects including, inter alia, proofs for the surface area and volume of a sphere inside a cylinder, the area of an ellipse, the area under a parabola, the volume of a segment of both a paraboloid of revolution and hyperboloid of revolution, and the area of a spiral. Archimedes the engineer appears all but vanished! Nonetheless, I believe there are enough clues scattered in a handful of places to at the very least qualify the otherwise abstract nature of Archimedes’ work.

Measurement and estimation

Archimedes was renown as a mathematician, as a mechanical writer, and as the inventor of ingenious devices.

Measurement and estimation are important aspects of science and ones which Archimedes was familiar with, including knowing the difference between exact and approximate results. Already in Measurement of the Circle, which is likely one of his earliest works, Archimedes approximates the ratio of the circumference of a circle to its diameter (the constant we know as π) by using inscribed and circumscribed 96-side polygons to get to the estimates 223/71 < π < 22/7, which are accurate to two digits. Archimedes evidently knew that the exact ratio would be nested within these bounds, and the use of this technique—now part of interval analysis—would ensure to account for all errors that could arise during calculations (Miel, 1983).

The use of upper and lower bounds, however, was not limited to geometry. In the Sand-Reckoner, Archimedes sets out to determine the Sun’s apparent diameter in what is ostensibly the first extant account of an astronomical measurement, as Shapiro (1975) notes:

While Archimedes’ account is rather terse and omits features which we would expect today…it is, nonetheless, a sophisticated piece of work: the procedure followed and apparatus used are described; correction factors are applied; and, most significantly, the result is given with an experimental error in the form of an upper and lower bound, rather than as a unique value.

Thus, the use and reporting of interval values in this work show that Archimedes was aware of the problem of exactness in physical measurements, for he states that “it is not easy to measure this angle with precision, for neither the eye, nor the hands, nor the instruments for measuring it are reliable enough for determining it exactly,” adding that this situation was known to others (Simms, 2005). Indeed, Archimedes' range of 0.450° to 0.549° for the Sun's apparent diameter does include the actual range (0.527° to 0.545°).

Later in the same work, Archimedes conceives a way to use poppy seeds to create a comparative magnitude for grains of sand. Archimedes assumed that a myriad grains of sand would not be larger in volume than a poppy seed, then found that 25 seeds, placed side-by-side, filled more than a finger-breadth. He then almost doubled the estimate to 40 seeds, so that 1 poppy seed = 1 myriad grains of sand = 1/40^th of a finger-breadth in length—an estimate he can now use to calculate the number of grains of sands needed when the diameter of the universe is known (Berrey, 2017). Archimedes also shows in On Floating Bodies that he had a clear conception of what we call specific gravity, which presupposes knowing how to measure the densities of objects when submerged in a fluid.

Natural philosophy

Every scientist, past or present, operates under certain assumptions regarding nature, whether explicitly or not. Working out these assumptions was once the preserve of natural philosophy. While Archimedes is largely silent about philosophy, there are a couple of places where he betrays familiarity with philosophical arguments on the study of nature. In the Sand-Reckoner, Archimedes presents a cosmological claim similar to Anaximander’s, where the Sun is placed closer to the outermost edge of the kosmos—a concession necessary in his search for the largest possible universe. In Prop. 2 of On Floating Bodies I, Archimedes states that “any fluid which is so located that it remains motionless will have the form of a sphere which has the same center as the Earth,” alluding to themes discussed in Aristotle’s On the Heavens and Physics, where the natural motion of heavy objects such as water is towards the center of the universe (Meli, 2006). There is also a fragment attributed to Archimedes by Hippolytus of Rome that preserves an explanation of the distances and order of the planets by way of numerical estimates embedded into Pythagorean cosmology, which is based on musical scales.

Elsewhere, Knorr (1982) has argued convincingly that one of the basis for the Arabic treatise Kitab Ii'l-Qarastiin, extant in the work of Thabit ibn Qurra, is a lost work on the lever by Archimedes. Despite the similar topic, Archimedes follows a rather different approach to that found in On the Equilibrium of Planes I and Quadrature of the Parabola: he begins with a standard Aristotelian notion of force (i.e., that bigger circles overpower smaller circles), then derives a justification of the equilibrium of weightless beams implicit in that tradition, and finally provides a geometrical exposition of the specific causes of the weighted beam using procedures found in later Archimedean writings. If the attribution to Archimedes is true, it represents an early attempt to combine dynamic with static mechanical concepts, suggesting some acquaintance with ideas about heaviness and lightness found in prior natural philosophical works, particularly the Aristotelian Mechanical Problems (Knorr, 1982).

Referents to empirical phenomena

This brings us to our last point, which is referents to empirical phenomena found in Archimedes’ work, including physical treatises no longer extant. Let us recall what the latter included: by Archimedes’ own account, there are the Equilibria and the Mechanics (which are referred to by other names); to these, we should add Of Balances, On Supports, Catoptrics, and On Sphere-Making. Setting authenticity issues aside, this list should give us hope that Archimedes did not completely neglect his engineering side.

But there are other signs closer to home. In Quadrature of the Parabola, Archimedes presents two proofs that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. The second proof is purely geometrical, so it does not concern us here, but the first proof is mechanical and based on the principle of the lever. Archimedes first invites the reader to imagine “a plane perpendicular to the horizon,” then describes a balance or scale-beam constructed in that vertical plane whose fulcrum is the midpoint (Prop. 6). He proceeds to hang rectangular weights on one side of the balance and a triangle on the other side so that they are in equilibrium. Because weights are inversely proportional to their distances from the fulcrum, the ratio between the two distances is equivalent to the ratio of the two figures. Thereafter, in Prop. 6-13, Archimedes divides a parabolic segment and an enclosing triangle into parallel components and weighs them on the balance, leading to inequalities of the inscribed and circumscribed polygons. Finally, the proof that the area of a parabolic segment is 4/3 the area of its inscribed triangle is shown in Prop. 16-17, without reference to the balance, using the previously proved theorems and a double reductio ad absurdum (Berrey, 2017).

As can be glanced from the above summary, Archimedes’ mechanical proof in Quadrature of the Parabola is unintelligible without familiarity with empirical phenomena related to levers: He makes constant allusions to metaphorical “balances,” “suspension,” “weights,” and “equilibrium,” without bothering to provide definitions for any of these terms. Knorr (1982) further notes that the lost treatise on the lever we discussed earlier employed similar terminology regarding balances, and that terminology also appears in Hero's account of Archimedes' work on mechanics. Moreover, there are hints that Archimedes knew of an empirical procedure to determine the center of weight of plane figures in both Hero and Proposition 6 of Quadrature of the Parabola (Valente, 2020).

Thus, Archimedes’ working knowledge of levers and centers of weight is attested not only in the anecdotes about his engineering feats but, more importantly, in his extant and lost works. But whence the insight regarding buoyancy? In Prop. 1 of On Floating Bodies II, for example, Archimedes restates that an object’s apparent weight would be lighter than its true weight by the weight of a displaced fluid yet says nothing on how this came about. We are tempted to think that Archimedes was inspired by weighing objects in water (Graf, 2004) or, more intriguingly, by work on pneumatics such as that of his contemporary, Ctesibius, on the water organ (Berryman, 2009). We simply do not know.

Summary

The evidence reviewed thus far is enough, in my view, to justify the claim that Archimedes did pay attention to more than just mathematics (pace Plutarch). To wit, in some instances, Archimedes paid attention to measurement and estimation, to natural philosophy, and to empirical referents of idealized physical objects. It was this Archimedes, not Plutarch’s Platonist one, that provided an inspiration for later mechanicians such as Hero, Anthemius, and their Renaissance and Early Modern successors (Bernard, 2018).

Unfortunately, the fate of those few treatises where Archimedes might have engaged more directly with empirical facts is that they did not survive. Part of the blame lies with the priorities of scribes and commentators centuries after him, as we shall see. But part of the blame also goes back to Archimedes himself. In the authorial choice between the engineer and the mathematician, Archimedes appears to have sided more with the latter, presumably because in geometry, unlike in engineering, his results would remain irrefutable (Netz, 2000)—and judging by the sometimes unfair modern reputation of more empirical-minded people like Ptolemy, he may have been correct.

Hipparchus (c. 190–120 BC)

Hipparchus was one of the greatest and most important observational Greek astronomers in antiquity.

A native of Nicaea (now in Turkey) but working mostly from the island of Rhodes, Hipparchus was without a doubt the most important astronomer prior to Ptolemy and a fundamental player in the integration of Babylonian astronomical knowledge into Greek mathematical astronomy.

As a mathematician, Hipparchus developed trigonometrical methods and may have been the first to introduce a table of chords. He used numerical methods extensively in addition to geometry to study astronomical phenomena. It is also likely that Hipparchus, in the course of rebutting Stoic logic, did original work in combinatorics (Acerbi, 2003), and presumably wrote a book on pre-modern algebra and another one on arithmetic [logistic]. He was probably the first to investigate stereographic projection, which was essential for the design of later astronomical instruments such as the plane astrolabe (Neugebauer, 1949). He is also known to have designed a four-cubit rod dioptra.

Sadly, except for his short commentary on the Phainomena of Aratus and Eudoxus, almost all of Hipparchus’ works are not extant or survive only in fragments. Among these lost works one should note On Things Carried down by Their Weight, where he apparently explained the phenomenon of acceleration in a manner very different from Aristotle. Another work, Against the Geography of Eratosthenes, dealt mostly with the nascent subject of mathematical cartography and Eratosthenes' contributions to it. Arguably, Hipparchus also wrote on optics, though it is no longer possible to assign a title to this work.

Yet, it was in astronomy where Hipparchus’ fame mostly and duly rested, as evidenced by the titles of his other works now lost: On the Movements of the Solsticial and Equinoctial Points, On the Length of the Year, On Intercalary Months and Days, On the Risings of the Twelve Zodiacal Signs, Treatise on Simultaneous Risings, On Sizes and Distances, and On the Moon’s Monthly Motion in Latitude. To better appreciate the accomplishments of Hipparchus in this area, however, it is necessary to go over the state of astronomy right before the time he was active.

Summary

The overall picture is of an author contributing to a full range of the mathematical sciences, and the most obvious model for another figure we shall discuss shortly.

Scholasticism and experimental method in Alexandria

In the Hellenistic age, the new kingdoms founded by Alexander the Great’s successors… Although not the only such place, the Mouseion of Alexandria (presumably connected to the Great Library) was by far the most well-known and consequential. A brief survey of people who either worked in Alexandria or that had some connection to its Mouseion and Library and who were engaged in physical investigations should be telling.

First, Eratosthenes of Cyrene, Ctsebius of Alexandria,… Philo of Byzantium…

All told, the writings attributed to these authors demonstrate respect for empirical data, without neglecting entirely the theoretical part despite underdeveloped. However, there was another trend among Alexandrian scholars that suggests a preoccupation with formal structure and rigor…  idea of concealing empirical data may have stemmed from the same concern of concealing heuristics from mathematical works.

Scholars commonly regard these strands as developing independently and separately from natural philosophy, with little to no interaction. Although this may seem true overall due to intellectual demarcations, it certainly was not true all the time (Feke, 2021). Even the most abstract mixed mathematical works would not dispense with physical properties derived from experience, though the level of idealization and reliance varies from case to case. Additionally, one cannot rule out the possibility that such apparent demarcations are partly the result of the paucity of our evidence. At any rate, we do have proof to the contrary in the work of two figures who were active not long after the close of the Hellenistic Age, and whom I deal with next.

Imperial Rome

Hero (c. 10-70 AD)

The work of Hero—as we shall see—is notably less abstract but shows a bigger preoccupation in connecting physical theories with mechanical or mathematical principles. Strato. Mechanical advantage.

Ptolemy (c. 100-170 AD)

Across all of antiquity, Ptolemy is arguably the person that best captures our modern idea of what a scientist is. Simplicius quote.

Legacy

The Antikythera mechanism, an example of the scientific achievement of the ancient Greeks.

All major figures of the generation preceding Newton—Torricelli, Roberval, Gregory, and Huygens, to name a few—were intimately familiar with the ancient tradition. Galileo arguably wanted nothing more than being remembered as the new Archimedes, though his talents evidently laid elsewhere. Before him, Viète moved away from pre-modern algebra into modern symbolic algebra not solely on his reading of Pappus and Diophantus, but more so by his dealings with the geometry and arithmetical calculations underlying Ptolemaic astronomy. Even the empiricist tradition, which took on a decisively (though not exclusively) British flavor, traces some of its roots back through the work of Robert Grosseteste and Roger Bacon (via Alhazen) to Aristotle himself.

Harmonics and physical acoustics, ancient and modern optics, Archimedes’ mechanics, Hero's theory of matter, Hipparchus' and Ptolemy's astronomy. Predicting phenomena in Euclid, Ptolemy.

Level of commitment to the phenomena varied by author (and even within author) but referents to empirical data never completely absent.

It is simply worth noting the materialist point: again and again, and from Euclid onwards, we find authors referring their geometrical lines and angles back to tangible entities—whether the substance of mirrors, the refracted forms of objects, the burned flax set ablaze by a “burning mirror”… or an obliquely perceived chariot-wheel.

As we saw earlier, Archimedes makes use of the principle of the lever in Quadrature of the Parabola in a way that presupposes familiarity with balances. This principle also appears in Prop.7-8 of On the Equilibrium of Planes I, but there are two important differences. First, Archimedes doesn't just state what the law of the lever is, he produces a mathematical proof of the physical law, setting a powerful precedent for others to follow. Second, there is no mention of balances or scale beams, or of suspending weights on a beam perpendicular to the horizon. Instead, the scale beam is a weightless line, weights are applied to single points on the line, and even inclination disappears after Prop. 8 (Lindberg, 2007). A similar approach occurs in On Floating Bodies I, where the physical phenomena enter early and then quickly recede from view, and it seems clear as the study progresses that Archimedes embarked on this work for intrinsic reasons rather than any immediate utility in mind.

Without running the risk of anachronism, one could label Archimedes’ extant physical treatises as belonging to “mathematical physics”: the physical properties of phenomena studied mainly as a license to do mathematics. Indeed, it was mostly their mathematical, not physical, character that ensured their preservation in Late Antiquity and the Middle Ages. Only later, starting in the late Renaissance, did those seeking to make natural philosophy more geometrico reappropriated Archimedes’ approach to the physical found in On Equilibrium of Planes and On Floating Bodies, their influence being up to then rather limited (Clagett, 1959).

When mathematical-minded people in Early Modern Europe began to look for new ways to do science, they did not turn to Plato or Aristotle, but to the examples of Archimedes, Pappus, etc. Much more so than was the case with Renaissance humanism earlier, the ancient heritage provided early modern scientists not only with models to emulate but problems to cut their teeth into. And in the process of reviving long-forgotten questions and debates, parts of the Greek scientific corpus—e.g., the study of conic sections, of mechanical curves, of centers of gravity, and of simple machines—jump-started the mathematization of nature or, alternatively, the added physicalization of the mixed mathematical sciences (Schuster, 2017). Ultimately, the ancients' example also exposed the limitations of applying classical methods to study physical phenomena as more and more challenging objects were introduced (Damerow & Renn, 2010). Overcoming these difficulties took many years and was one of several factors that led to the advent of the calculus.

Capecchi (2018) makes a convincing—and in broad strokes essentially correct—argument regarding the legacy of Greek science by stating the following:

1. Early Modern science originated from Hellenistic mathematics, not so much because of its rediscovery but, rather, because its “applied” components, namely mechanics, optics, harmonics, and astronomy (“old sciences”), continued to be transmitted throughout the Middle Ages without complete interruption.
2. “New sciences” such as dynamics, acoustics, hydraulics, pneumatics, etc. had exactly the same starting methodology and logic organization of “old sciences”: they were applied mathematics. The crucial difference was in the different natural phenomena examined and in the richer deductive mathematical apparatus with the use of algebra, analytical geometry, and (later) calculus.
3. “Old” sciences played a role as a whole. “New” mechanics, for instance, derived not only from “old” mechanics but also from harmonics, optics, and astronomy.
4. Most protagonists of the “new” sciences could be qualified as mathematicians.
5. The modus operandi typical of mathematics, proceeding from clear definitions and strict reasonings, was adopted by some new philosophers, which led to the experimental and/or mechanical philosophy. This would eased the appropriation by mathematicians of most fields of natural philosophy.

Why thus no revolution?

The preceding survey was highly selective, as I deliberately left out many other lesser-known figures in the mathematical tradition and excluded the evidence from the life sciences, particularly medicine, with their significant records of observations and experiments. Further, I cannot fail to mention that another and comparatively late tradition that had a place for observations and experiments is the alchemical corpus dating back to the first through sixth centuries AD. Although this tradition originated from old metallurgical and dyeing practices, during the Roman era it increasingly began to include theoretical as well as mystical aspects, which gave this discipline a very different character than the sober one we have seen in the works surveyed here (Merianos, 2017).

Indeed, when broadly defined, the full scope of ancient science is much wider but tellingly not thicker: the Encyclopedia of Ancient Natural Scientists or EANS—the most authoritative source—has 2,043 entries on ancient scientists across all disciplines, 995 of whom have dates attested, covering a span of over a thousand years, from c. 610 BC to c. 615 AD (Keyser & Irby-Massie, 2008). Following much stricter criteria, Zhmud and Kouprianov (2018) further reduced the original 2,043 entries to 407, most dealing with mathēmatikoi, an ancient term designating the kind of mathematical scientists our sample is drawn from.

Thus, the question "why no scientific revolution?" in fact has two aspects to it: (1) to evaluate the scientific achievement of the ancient Greeks across some twelve centuries, and (2) to determine the influence that achievement (or what was available of it) had on the work of scientists in Early Modern Europe. The difficulty with answer (1) satisfactorily, which is less on an issue with (2), is that the extent of the evidence at our disposal makes it very hard to draw the necessary inferences. For we need to be evenhanded.

ON THE ONE HAND, we should not underestimate how much our understanding of ancient Greek science depends on the vicissitudes of transmission, particularly what got through the filter of Late Antiquity and the subsequent translations and transliterations efforts that followed it (Acerbi, 2018). By now, in this short survey, the list of works attested but now largely or completely lost includes some that may have had a bigger connection with empirical phenomena, to wit:

1. Archytas’ works on harmonics and (possibly) optics and mechanics.
2. Archimedes’ Catoptrics and his treatises on levers and centers of weight.
3. Almost the entirety of Hipparchus’ oeuvre on astronomy and physics.
4. Eratosthenes' Geography and his work on geodesy.
5. Ctesibius’ works on pneumatics.
6. Philo’s Mechanical Compendium, from which only a few books survived.
7. Hero’s other treatises on horology and surveying.
8. Ptolemy’s work on mechanics.

We should add to this list various other extant and lost works (e.g., Diocles’ On Burning Mirrors, Menelaus’ On the Weights and Distributions of Different Bodies, Euclid’s Phenomena) which, though not covered here, nonetheless belong to the same tradition.

ON THE OTHER HAND, although the late commentators and scribes did much to skew the survival of more theoretical works, they certainly did not create that predisposition out of whole cloth (Høyrup, 1990). That intellectualism is already evident in both Plato and (for different reasons) Aristotle, in the scholasticism of the Alexandrians, and in the academic gerrymandering of the Neoplatonists. However much internally motivated, mathematical scientists in antiquity were not entirely immune to these trends, and on more than one occasion shared similar concerns, as evidenced by introductory remarks found in some of their works. Keep in mind that, for each of these cases, a push to downplay empirical considerations or question the legitimacy of using mathematics to make claims about physical phenomena was made.

• Hippasus may or may have not been exiled or even murdered for his allegedly impious ways, but even the exemplary Archytas found himself at the wrong end of Plato’s diatribe, as the latter did not look kindly to time devoted to studying the perceptible without transcending to the imperceptible. Aristotle was in principle more sympathetic to allow mathematical analysis of empirical data—but in practice, neither he nor his followers used this approach, partly because for them the qualitative aspects of phenomena were their essential features.

• During the Hellenistic Age, when more people engaged in what we consider scientific work, we can still see these efforts getting stifled. Archimedes regularly complains about the scholastic attitude of his Alexandrian colleagues, and one wonders whether he would have considered presenting his work less detached from empirical data should the situation have been otherwise. And it says something about combining observational data with mathematical modeling that Hipparchus had no attested follower in astronomy until Ptolemy came along three centuries later.

• These attitudes continued throughout the Imperial Age. Hero, as well as Ptolemy, felt the need to defend their theoretical/empirical approach to phenomena, which no scientist today has to do. Simplicius and Philoponus drew limits to what the mixed mathematical science of mechanics could do, much like how Epicurus and his followers famously questioned the usefulness of mathematical astronomy earlier. And although Iamblichus speaks of something like a research program in mathematical physics, this remained programmatic, never carried out.

Unsurprisingly, those wanting to study natural phenomena ended up mostly choosing from two methodologies: either work in a very abstract, mathematical manner, or else follow primarily philosophical (qualitative) conventions. The middle ground was little traversed.

What was new during the Scientific Revolution, then, was neither the use of observations and experiments nor of mathematics as a tool to uncover or express the principles behind physical phenomena. Instead, it was the growing number of people who insisted that natural philosophy (which would eventually transform into what we call science) must proceed based on both, even if no consensus existed on the exact ratio. With rare exceptions, that view never gained a majority in antiquity or any time before the late Renaissance. And yet, the mathematical approach to the study of nature and the use of observations or even experiments never completely “disappeared” prior to the 17^th century but continued, at different times and places, in some form or another (Lindberg, 2007).

From this follows that it would be a mistake to reduce the Scientific Revolution to merely more people rediscovering the Greek scientific corpus. For starters, the ancient Greeks who wrote these works were not laboring to produce a Scientific Revolution in their time, let alone in the 17^th century (arguably the same can be said, mutatis mutandis, of many sixteenth and seventeenth century savants). Many currents fed into the river we now call the Scientific Revolution—several other factors that facilitated the transition from ancient to modern science were at play and ought to be considered too (Renn, 2017). Still, re-acquaintance with the ancient Greek corpus proved decisive during the transformation of science in Early Modern Europe: it was in the back and the front of the minds of all those engaged in natural philosophy, whether as something to emulate or to surmount, and the revolution would have been almost unthinkable without it (Levitin, 2022).

I believe G.E.R. Lloyd (1973) summarized this best when he said:

The criticism is often made that the fatal shortcoming of Greek science was the failure to appreciate the importance of experimentation. But that is an oversimplification. It is true that the use of the experimental method is confined to certain problems and to certain individuals, but the same may also be said of the idea of the mathematization of physics. Here too the principle was known, and it is not hard to identify, with the benefit of hindsight, the opportunities for its application that were missed. In neither case is there a fundamental difference in kind, however great the differences in degree, between the methods of ancient and modern science. But both these shortcomings in ancient science reflect, and were aggravated by, the more basic organizational weakness to which I have alluded, the fact that the conditions needed to ensure the continuous growth of science never existed in the ancient world. The relative isolation of those who engaged in scientific investigation acted as an obstacle to the systematic application of methodological ideas and was a constant threat to the continuity of inquiry in most fields of science […]

Later, he added:

Yet the neglect that some of the most important ideas produced by the ancients suffered from in antiquity does not diminish the value of those ideas in themselves. The weakness of the social and ideological basis of ancient science becomes more obvious in the decline [of Late Antiquity]. But when scientific investigation was revived in the West, it was a genuine rebirth, not merely in that the work of the great ancient scientists was rediscovered, but also and more particularly in that there was a return to the spirit of inquiry of ancient science and to the models of method that it provided.

TL;DR

Where I make two summary points for those in a hurry:

A headless statue of the philosopher Chrysippus, enumerating his arguments.

First, it is commonly said that Greek science was neither given exact quantitative expression nor adequately supported by quantitative data. This is evidently true in some instances, such as in dynamics or element theory, but not everywhere (Lloyd, 1987). Other fields did display a commitment to the pursuit of quantitative precision, including the collection and evaluation of data, the formulation of theories from these, and the application of mathematics to explain—and in some cases predict—natural phenomena. For instance, in geography, efforts to accurately measure heights, distances, and the size of natural objects, such as the Earth's circumference by Eratosthenes (who also calculated its axial tilt), Posidonius, and others; in astronomy, Ptolemy’s planetary models and his catalogue of stars, the latter described not only in degrees but also in fractions of degrees; in optics and harmonics, the measurement of variables of interest (i.e., angles or pitch, respectively) using specialized instruments; in mechanics, the determination of parameters of stability and mechanical advantage. We also have a number of authors who produced geometric results ready-made for scientific application, provided the relevant observational data were obtained (e.g., Aristarchus’ On the Sizes and Distances of the Sun and Moon, Euclid’s Phaenomena, Autolycus’ Risings and Settings). No wonder, then, that authors such as Ptolemy went to considerable lengths to secure a quantitative dataset, and that concerns regarding the accuracy of data and the potential for error is not unheard of in the works of others (e.g., Archimedes, Hipparchus).

A lack of experimentation or critical observations, too, have often been cited as failures of Greek science. Again, this assertion is not entirely correct and leads to my second summary point. We have already seen that some early Pythagoreans were not afraid to use contrived experiments during their harmonic investigations, and that was also true of others after them. Oftentimes these experiments were carried out merely to confirm or refute a particular thesis, and once that goal was achieved, further investigation ceased (Lloyd, 2000). There were additionally many instances where ancient researchers hastily concluded that the data confirmed their theory, interpreting the experiments in light of the theory or even adjusting the results to provide such confirmation. Yet, there were also instances when experimental observations were carried out over time (e.g., Philo of Byzantium), or when they did not confirm but openly conflicted with preconceived ideas or theory. Such was the case with Hipparchus’ discovery of precession, which went against not only his own assumptions but that of all his contemporaries, and Hero’s concept of the elasticity of matter particles, an idea that had no precedents anywhere and which resulted from his evaluation of prior experimental data. Similar examples can be found in the medical tradition, which is not covered here. The important point is not so much that these instances existed but that they were reported, with the expectation that anyone could judge for themselves or improve upon them. As Lloyd (1987) puts it:

To express an allegiance to the principles of engaging in research and of securing a comprehensive and reliable data base, to the need to put theories to the test, to expose and root out unexamined assumptions, to withhold judgement where the evidence was insufficient, to acknowledge your own mistakes and uncertainty—all this was often no more than a matter of paying lip-service to high-sounding ideals. But if this was to bluff... it was a bluff that could be called, and we have also seen how, on occasion, it was called, and how the ideals were at least sometimes lived up to and the promises they implied fulfilled.

Although neither quantification nor experiment ever became the defining features of ancient Greek science—the way they are for us ever since the start of the 19th century—we should be fortunate that these features were present enough for others to elaborate over the centuries.

Conclusion

The Pythagoreans devoted themselves to mathematics. They both admired the accuracy of its reasonings, because it alone among human activities contains proofs, and they saw that general agreement is given in equal measure to theorems concerning attunement, because they are established through numbers, and to the mathematical studies that deal with vision, because they are established through diagrams. This led them to think that these things and their principles are quite generally the causes of existing things. Consequently, whoever wishes to comprehend the true nature of existing things should turn their attention to these—that is to numbers and geometrical forms of existing things and ratios—because it is by them that everything is made clear.

— Iamblichus, On the General Mathematical Science, 78.8-26

Ancient Greeks were able to generate and transmit scientific knowledge by methods scientists today would generally recognize, including deductive proof, observation, measurement, experiment, and mathematical modelling. They did so intermittently through a voluntary commitment to a system of conventions, rules, and norms governing their practice, with only sporadic institutional support and often independently from philosophers. Many other features typical associated with modern science—claims to originality, a desire for peer recognition, criticism and praise of others, priority disputes, and accusations of plagiarism—are likewise well-attested in the ancient sources (Zhmud & Kouprianov, 2018).

Hence, when Galileo said the universe was written in the language of mathematics, he was not uttering anything new. Nor was his approach of combining observation and experiment with mathematical analysis unprecedented or revolutionary on its own. Such pronouncements as one finds frequently in popular accounts are misleading, for they deny that at various times other persons engaged in similar enterprises, leading to methodologies that often resembled modern ones.

Even if the aims were not always the same as ours, or that many of the results obtained did not stand the test of time, the fact that something resembling our idea of science was practiced, however limited, in antiquity should fill one with awe but also humility. Though stronger today that at any other time in history, science is nonetheless contingent and fragile, and there is nothing written in the stars that guarantee its permanence forever.

The New Archimedes

From early on, the person of Archimedes has attracted the admiration not only of mathematicians but of laymen and prominent figures alike. As a result of said admiration, the name "Archimedes" became shorthand for "genius" (much like "Einstein" is for us), and efforts at identifying a "new Archimedes" appear sporadically in the historical record.

I originally aimed to trace back all the references to a "new Archimedes" I could find, from Late Antiquity to the Early Modern era, to identify (purely as an intellectual exercise) the person most deservedly of this title. After much effort, it became clear that this goal was overtly ambitious and in most cases what evidence there existed was not enough to do a full comparison. I decided instead to settle on the one person who, on my reading, best warrants the moniker: the Dutch polymath Christiaan Huygens.

Lives

Archimedes’ personality and scant details of his life could be gathered from his extant works. From these, we know his father was an astronomer, that from Syracuse he communicated with other mathematicians, and that he saw the value of heuristics, particularly his mechanical insights, in the search for proofs. He also appears to be jealous of guarding his priority claims and reputation. Other accounts (sans Polybius) are considerable late and less reliable. For the most part, they center around the idea of him being a mechanical and (less frequently) mathematical genius.

There is no shortage of information on Huygens since plenty correspondence and manuscripts of his survived. These sources, particularly the multi-volume Oeuvres Complètes (1888-1950) and the Catalogue of Manuscripts (2013), testify to Huygens’ unique mathematical and mechanical ability. During his lifetime, Huygens devoted himself to all branches of mathematics then conceived, including (beyond geometry) optics, mechanics, astronomy, and harmonics. He corresponded widely with intellectuals across Europe and was a prominent member of King Louis XIV's Académie des sciences for its first two decades.

	Archimedes	Huygens
Bio	287?–212 BCE (aged 75) Syracuse, Sicily	1629–1695 CE (aged 66) The Hague, Dutch Republic
Patronage	King Hiero II	King Louis XIV
Philosophy	Unknown	Cartesian (modified)
Methods	Exhaustion method Indivisibles and mechanics (heuristic) Ancient Analysis	Exhaustion method Indivisibles and infinitesimals (heuristic) Analytic Geometry
Style	Solutions to challenging mathematical problems, often with surprising results. Geometric measurement of planes, solids, and curves. Valued substance over form; individual results over systematic treatment of a subject.	Mathematization of challenging physical problems, often with surprising results. Geometrical infinitesimal analysis of curves and motion. Valued axiomatization and geometric constructions over analytical arguments, limits.

Works

Overall, Archimedes wrote more on (pure) mathematics, while Huygens wrote more on (mathematical) physics. Nonetheless, there is clear overlap in interests, and in some cases the contents of their work match. This may not be a coincidence, as the young Huygens modeled himself after Archimedes.

Items included are either extant or have some certainty as to their existence. Items for which no consensus has been reached are marked with a question mark (?).

Bold = Close match in contents

() = Publication or status of work

? = Undetermined

	Archimedes	Huygens
Mathematics	Quadratura Parabola Dimensio Circuli De Mechanicis Propositionibus ad Erastosthenem Methodus De Lineis Spiralibus De Sphaera et Cylindro I-II De Conoidibus et Sphaeroidibus De Polyedris (lost)	Theoremata de Quadratura Hyperboles, Ellipsis et Circuli (1651) De Circuli Magnitudine Inventa (1654) Epistola... 'Εξέτασις Cyclometriae Gregori à Sto. Vincentio (1651, supplement) Illustrium Quorundam Problematum Constructiones (1654, supplement) De Ratiociniis in Ludo Aleae (1657)
Physics	De Corporibus Fluitantibus I-II De Planorum Aequilibriis I-II Mechanica (lost, possibly known by other names) De Equilibria (lost) Arenarius Catoptrics (lost)	De Iis Quae Liquido Supernatant (1650, unpublished) De Motu Corporum ex Percussione (1656, published posthumously) De Saturni Luna Observation Nova (1656) Systema Saturnium (1659) De Vi Centrifuga (1659, published posthumously) Horologium Oscillatorium (1673) Traité de la Lumière (1690) Dioptrics (unfinished)
Instruments	Περί Σφαιροποιἶα ("On sphere-making", lost) Water-clock? per Arab authors Hydrostatic balance? per Menelaus Dioptra for astronomical sighting (Arenarius)	Descriptio Automati Planetarii (1681, published posthumously) Astroscopia Compendiaria Tubi Optici Molimine Liberate (1684) Memoriën Aengaende Het Slijpen van Glasen tot Verrekijckers (1685) Pendulum clock Water and air pumps Micrometer, level
Varia	Stomachion (in fragments) Problema Bovinum (in fragments) Liber Assumptorum? (extant but heavily edited) Ad Zeuxippus (lost)	Kort Onderwijs... Horologiën tot Het Vinden der Lenghten van Oost en West (1686) Discours de La Cause de La Pesanteur (1690, supplement) Lettre Touchant Le Cycle Harmonique (1691) Cosmotheoros (1698, published posthumously) Novus Cyclus Harmonicus (published posthumously)

Assessment

Although in temperament Huygens appears diplomatic where Archimedes is confrontational, Huygens best encapsulates what Elzinga (1971) calls the Archimedean paragon of science—only Maurolico comes to mind as a distant second because Torricelli died too young. I am not alone in this assessment, as several historians of 17th century science draw similar conclusions regarding Huygens and Archimedes.

Foremost of these personalities is the Dutch mathematician cum historian E. J. Dijksterhuis. His opinion is the most authoritative because (a) he wrote a widely praised and erudite introduction to the works of Archimedes that respected the Syracusan's mathematical style (the reprinted 1987 edition of Archimedes is still considered the best of its kind), (b) he assisted in the completion of Huygens' Oeuvres Complètes and provided an insightful summary of the man and his work, and (c) he wrote one of the most well-known books on the history of what would become modern science, The Mechanization of the World Picture (1950), which covered everything from just before Aristotle to right after Newton. In the first-half of the 20th century, there was probably no one more qualified to compare Archimedes and Huygens than him, being intimately familiar with the thoughts and fluent in the languages of both.

At the behest of the Royal Holland Society of Sciences and Humanities, Dijksterhuis delivered an address where he sought to "sketch.. the position occupied by Huygens in the scientific life of the 17th century." Apart from the many debts Huygens may have incurred with his immediate predecessors, Dijksterhuis highlighted this:

From the moment that his father, acting on a careful plan for the intellectual education of his sons, Constantijn and Christiaan, inserted mathematics into their curriculum, Christiaan is the one who, under the direction of Stampioen, has made surprising advances. At Leyden [sic] he immediately attracts the attention of van Schooten, the mathematician, by whom he is made acquainted with the new methods of Descartes. Soon he begins to make discoveries of his own, and his father, with pardonable pride, calls him in correspondence “mon jeune Archimède”.

Dijksterhuis then goes to add:

It is very likely that, in using this particular name, the father intends no more than a comparison with some great mathematician of antiquity; but actually this comparison provides a short and striking description of Christiaan’s specific mathematical ability. Indeed, there exists a remarkable congeniality between him and the great Syracusan who is to mould the style of his mathematical work all through his life.

He later remarks the close match of Huygens' earlier works with those of Archimedes, indicating a conscious choose of subject matter by the former. Further, Dijksterhuis detects in Huygens a certain hesitancy in communicating his results while hiding the heuristics that led to them, preferring to demonstrate everything with exhaustion methods and reductio ad absurdum arguments–a trait he evidently shares with Archimedes:

In the management of these antique modes of reasoning, Huygens was a complete master; the greatest after Archimedes. But, as the new discoveries succeeded one another more and more quickly, it meant an ever increasing self-conquest always to take pains to compose a rigorous demonstration. So he often preferred leaving his findings unpublished and restricted himself to communicating the results in his letters only or in a work of much later date. The consequence of this method was, however, that his findings repeatedly became known either in too narrow a circle or much too late, with all the unpleasantnesses resulting from this delay.

Other respected Dutch historians with similar remarks include Henk J. M. Bos, Aant Elzinga, and Fokko J. Dijksterhuis (no relation to the elder Dijksterhuis).

Arguably the most knowledgeable Huygens scholar of the last 50 years is Joella G. Yoder. Her book Unrolling Time: Christiaan Huygens and the Mathematization of Nature (1989) is an exquisite study of Huygens' work on the mathematics of pendula and the mechanical principles behind clocks that led to the Horologium Oscillatorium. She has written several articles on Huygens, and dedicated the better part of 25 years cataloguing all the manuscripts in Codices Hugeniani (and thus undoing the mess left by the editors of the Oeuvres Complètes!), which she published under the full title Catalogue of the Manuscripts of Christiaan Huygens Including a Concordance With His Oeuvres Complètes (2013).

In her article ‘Following in the footsteps of geometry’: The mathematical world of Christiaan Huygens (1996), Yoder summarized Huygens' approach to mathematics thus:

But Huygens was also well versed in the classical geometry of Euclid and, most tellingly, of Archimedes. His first two published works were extensions of the Archimedean corpus. De circuli magnitudine inventa (1654) presented an advanced algorithm for computing pi that improved upon Archimedes' approximations by means of inscribed and circumscribed polygons. In Theoremata de quadratura hyperboles, ellipsis et circuli, ex dato portionum gravitatis centro (1651), one of many seventeenth century books that attempted to reconstruct Archimedes' lost proofs, Huygens supplied theorems for computing the areas of the hyperbola and ellipse in relation to their centers of gravity that paralleled Archimedes' famous quadrature of the parabola. And later in the Horologium Oscillatorium (1673) Huygens presented similar results for the surface areas of the solid conics made by revolving the two-dimensional ones about their axes. Perhaps even more important to his development was an early unpublished treatise on floating bodies, in which he emulated Archimedes' mathematical treatment of problems in statics. No wonder that, not long after they began corresponding, Mersenne dubbed Christiaan a new Archimedes, a very apt title that the poetical Constantijn frequently reiterated.

Later, she added the following:

Huygens argued in his Cosmotheoros (1695) that the inhabitants of other planets would still develop Euclidean geometry because the same mathematical principles abide throughout the universe. In other words, mathematics is not an abstract construct of our earthly minds but exists independently in nature. Such a viewpoint permits, although it does not necessarily demands, an easy translation of the physical into the geometrical, substituting point for body or particle and line for magnitude or trajectory. It is the Archimedean method writ large… Huygens never seems to have questioned why this should be true. His success at expanding mathematics beyond Archimedean statics to the Galilean science of motion only strengthened his bias.

It is important not to push the comparison too hard and imagine Huygens as merely Archimedes in 17th-century garbs; clearly Huygens was a man of his time as much as Archimedes was of his. He was also very aware and to some extent encouraged the similarities people saw between him and Archimedes. The late Michael S. Mahoney put it more succinctly this way:

From the very beginning of Christiaan Huygens' career as a mathematician and natural philosopher his father referred to him as "my Archimedes", and friends and admirers soon followed suit. But not blindly: the Republic of Letters in the late seventeenth century chose its classical images carefully. The name of Archimedes conjured up not only the genius that established the mathematical principles of stability in the physical world but also the ingenuity that transformed those principles into machines that put nature to work. Archimedes was an engineer as well as a mathematician... Christiaan Huygens obviously had that talent, and contemporaries looked to it to yield fruits comparable to those found in Archimedes' writings and described in reports about him.

Yet, Mahoney also warns:

It would surely deepen our understanding of Huygens to know precisely what the cognomen meant to him. The repeated use of ευρηκα in his notes to signal the moment or occasion of discovery suggests that he took the comparison as more than a doting father's hyperbole. But to know how it influenced his choice of problems, his methods of investigation, and his style of presentation would require a study as yet undone. To do justice to the question, the study would have to explore also the differences that separate Huygens from Archimedes. These, I suspect, would turn out to have greater historical significance than have the similarities. In retrospect from today, Huygens looks less like the reincarnation of Archimedes than like the prototype of Coulomb, Carnot, and the practitioners of the Ecole Polytechnique's characteristic mélange de mathematique et physique.

To summarize, both in mathematical style and subject matter, Huygens is most deservedly of the title of new Archimedes for the following reasons:

1. Both men wrote monographs addressing specific sets of problems or areas of theory that were susceptible to mathematical analysis and difficult enough to demonstrate their considerable abilities, often starting with axioms and construction techniques suitable to the task at hand, but without making grand efforts towards systematization.
2. They also showed a preference for displaying unsuspected connections between different objects—indeed, objects, perhaps more so than phenomena, were the focus of their physics (e.g., Archimedes is more interested on what buoyancy does to bodies; Huygens is more interested on how acceleration by gravity affects pendula).
3. Both men paid little attention to speculative work and usually abstained from philosophical pronouncements of the kind found in, e.g., Aristotle or Descartes (Cosmotheoros being Huygens' one marked exception).
4. Each had one celebrated work motivated by an instrument or device (the lever in Archimedes' On Equilibrium of Planes, the pendulum clock in Huygens' Horologium Oscillatorium), and another one motivated by a natural phenomenon (buoyancy in Archimedes' On Floating Bodies, light propagation in Huygens' Traité de la Lumière). The latter led to two principles being named after them: Archimedes' principle and Huygens-Fresnel principle.
5. Each often went beyond what was needed for the occasion: Huygens stops in the middle of Part III of the Horologium Oscillatorium to reduce the areas of solid conics to that of plane circles by revolving the two-dimensional ones about their axes. Archimedes does something similar when he extends the law of the lever from commensurate distances to the case of incommensurate distances in On Equilibrium of Planes I. This showmanship is clearly overkill and has little to do with the physical situation and everything to do with the mathematics both were engaged with.
6. Neither men developed any radical new methods or left an Elements or a Principia intended as the last word on some subject, though Huygens seemed to have entertained the idea of writing one on more than one occasion.
7. The nature of their genius meant that they had more admirers than followers: there was no school under their names and, though influential, one is hard-pressed to speak of an Archimedean or Huygennian movement in the years surrounding their deaths.
8. Finally, and relatedly, although both communicated and collaborated with other savants, intellectually speaking, their best work was carried out alone.

Archytas' Proof of Doubling the Cube

Archytas' proof as found in Eutocius' commentary.

Just over a decade ago, Luc Brisson (2013) raised some serious doubts about Archytas' supposed solution to the Delian problem: How to find the side of one cube that has twice the volume of another. This was one of the three celebrated problems of ancient Greek mathematics, though whether these problems were highly regarded from the beginning or only became so in Late Antiquity is an open question.

Brisson's qualms came up as part of his criticism levied at Carl Huffman's Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician King (2005), a 600+ page volume devoted to Archytas, the first—and so far only—of its kind. Brisson calls out Huffman's for not discussing "the context of transmission of the fragments and the testimonies he uses," a charge he believes other are also guilty of. The piece in question is a passage from the commentary on Archimedes' On the Sphere and the Cylinder by sixth century AD author Eutocius, who, in the words of Brisson, "lived a millennium later than Archytas (a contemporary of Plato) and who, I will try to prove, was dependent on compilations whose sources it is impossible to determine, insofar as we can now refer only fragmentarily to works that were copied more or less carefully" (2013).

The reader is invited to inspect Brisson's judicious article for details on the varied testimonies on Archytas and what these tell us about the man and his work. This reply will focus instead on addressing the questions Brisson raises regarding the veracity of Archytas' proof as found in Eutocius' commentary.

Tracing the proof trail

Brisson first order of business is to sketch the history of how the proof now found in Eutocius came to be. After a short summary of the earliest sources for Archytas (i.e., Plato, Aristotle, Aristoxenus) and whether the label "Pythagorean" can even be ascribed to him, Brisson turns to the tradition regarding Archytas' proof. Eutocius tells us the proof comes from Eudemus' History of Geometry, no longer extant. It is here where Brisson asks:

As a first hand informant Eudemus of Rhodes would be very reliable; but what if Eutocius read only compilations giving second hand information? The question is sound, because, as we will see, many problems, internal as well as external, arise in the case of Archytas’ solution. Archytas’ solution in Eutocius seems peculiar, while Eutocius and Plutarch give opposing accounts of the solution. What can this mean?

Brisson is right to point out that Archytas' solution is odd and that conflicting reports exist regarding its character. He then proposes a historical account on how Archytas' proof found its way to Eutocius—outlined here it broad strokes (details can be found in Brisson's article):

• Archytas and his alleged student Eudoxus were interested in the problem of doubling the cube and tried, with little success, to find a concrete method of constructing the cube in question. Let's call this Archytas' Original Proof (OP).
• Eudemus reports on the work of Archytas in his History of Geometry, possibly making mention of Archytas' OP.
• Many years and much mathematics later, Eratosthenes read about Archytas' proof somewhere but, rather than presenting Archytas' OP, embellishes the account as one where Archytas "construct the two proportional means through semi-cylinders" only to chastise him (contra Archytas' OP) for being so abstract!
• Later, someone (Menelaus?) decides to devise a proof that will fit Eratosthenes' earlier description, using a complicated construction with semi-cylinders. Let's call this Archytas' Fake Proof (FP).
• At some point, Eudemus' History of Geometry gets compiled and Archytas' OP is replaced by Archytas' FP. In this form it reaches late commentators like Proclus, although Plutarch seems to be aware of Archytas' OP somehow.
• Finally, Eutocius—being none the wiser—reports Archytas' FP as the original, betraying no awareness of the convoluted history.

This reconstruction reads at best as a comedy of errors, and at worst as a conspiracy spanning centuries, but given the paucity of sources regarding Archytas, it is at least plausible. However, that still does not make it likely, so Brisson needs to show that Archytas' FP as found in Eutocius could not be from Archytas—for, as Brisson says, if Archytas had already found such a solution, "why would anyone look for another solution, less ingenious and less sophisticated?"

Brisson furnishes two pieces of evidence that Archytas' FP is not by Archytas:

• Plato, a contemporary of Archytas, was highly interested in the development of solid geometry and yet says nothing about anyone solving the duplication of the cube.
• Archytas' FP relies on conic sections, which were not developed until Menaechmus, a generation after Archytas.

I will show that neither argument holds much water below.