Continuing to make false statements like all “Roman mathematical work doesn’t generalize anything” and bombarding this blog with vacuous time-wasting wordwalls that don’t present any true evidence for the thesis proposed signals the end of this conversation.

It’s wrong to dismiss Russo’s claims as “Bullshit he made up”. There is something ineffable and extremely important Russo has which nearly all historians lack, namely a deep education in mathematical science. This allows him to appreciate the network of ideas, and to judge the level of mathematical sophistication required to develop them. The historian’s tools are also important, and he is weaker in those, but it is important to appreciate the unique point of view he brings, being mathematically literate. There are important things that become obvious in this case which are not obvious otherwise, two of them being the “level” of Archimedes, Euclid, and Appolonius, and the other being the “obviousness” of Aristarchus’s model.

I don’t know about Hipparchus, but I am 100% certain about Archimedes’ cosmology, because I read everything of his that survives. Archimedes never talks about anything other than heliocentrism. The only reference to cosmology is in the sand reckoner and the heliocentrism isn’t incidental, it’s the foundation of the calculation.

The beginning of the sand reckoner, he says (paraphrase) “We are used to thinking of the universe as small, but recently, Aristarchus has put forward a theory in which the universe is much much larger, so that the size of the universe is basically infinite. In this model, the Earth goes around the sun, the sun is at the center, and the stars are enormously far away, like the ratio of a sphere to it’s center point. That ratio is infinite and meaningless, so we’ll take it to mean a ratio of the astronomical-unit to the radius of the Earth. etc, etc”. He then proceeds to fill Aristarchus’s universe with sand, and name a number larger than the number of grains. His goal is popularization, this is not a technical treatise, as is evident from the style. This is for the “lay public”, in this case some king or other, to introduce them to recent scientific ideas. He says so explicitly at the start, so this is not speculation.

The point is, Archimedes is introducing a new foreign and probably counterintuitive idea, heliocentrism, right at the beginning of a work for the lay-public, and then assumes it to be true without comment in the rest of the work, it’s at the very foundation of his calculation. His work is popularizing two things simultaneously: naming enormous numbers using exponential notation, and a heliocentric view of an enormous universe. It is also a kind of motivation for the “Archimedian Axiom” he introduces in technical work, that given any magnitude epsilon, no matter how tiny, there is an integer N such that epsilon times N is bigger than or equal to any given magnitude. In this case, epsilon is the sand, the magnitude is the universe, and he calculates an upper bound for N.

There’s no “We know this Aristarchus model is totally incorrect” (as you find in Aristotle when talking about atoms and void), there’s no “it’s totally absurd, but let’s assume…”, it’s just stated and assumed without question, there’s no disclaimer.

In Bayesian terms, if you look at popularizations which talk about a not-yet-universally-accepted scientific idea which the author believes, a fortiori, lets look at popularizations of ideas that turned out to be less than fruitful, for example, in Dancing Wu-Li Masters talking about the now defunct “Nuclear Democracy”, or Brian Green talking about “Large Extra Dimensions”, you will find a discussion that parallels Archimedes’ sand-reckoner. The author introduces the idea, says it is a hypothesis that is not yet universally accepted, and then runs with it. That’s the universal sign that the author is happy with the hypothesis. I don’t know of any case where an author introduces and runs with an idea without caveat or hedging when he doesn’t believe that idea.

On the other hand, when an author is talking about a new speculative idea which the author does not believe, for example, there are umpteen references to Everett’s “Many Worlds” in the popularizations of the 80s, they are all accompanied by some pushback, the author says “this idea is implausible, it does not work”, or something.

Given that this always happens, in Bayesian terms, my confidence that Archimedes believes Heliocentrism shoots to as close to 100% as possible. You could speculate that Archimedes was secretly Geocentrist, but this would be unsupported by any documents, and contradicted by the Sand Reckoner.

This conclusion is reinforced by knowing heliocentrism is “obviously” true (I mean obvious to any scientist, not to a lay person), and that Archimedes could evaluate science. The reason that the heliocentric model is to be preferred even if you are a complete relativist regarding motion I will explain later.

There is no other statement about the motion of the Earth anywhere in the Archimedes works. The only other place which you could possibly by a long stretch misconstrue that is in “On Floating Bodies” where Archimedes talks of the shape of the oceans, and makes the (correct) claim that the oceans are spherical “In a situation where everything is at rest”. This is a locution we still use today when describing the equilibrium of fluids, even though none of us are Geocentrists, so it tells you nothing about his cosmology, even assuming that this work postdates him reading or talking to Aristarchus. This bit of evidence has zero Bayesian weight one way or another, as the sentence is equally unsurprising assuming Archimedes is a Heliocentrist or a Geocentrist.

That’s the end of statements in Archimedes. How could you possibly conclude he is geocentrist? He’s not. With as near to certainty as history allows.

Since Appolonius is in correspondence with Archimedes, since he is in the same school, I can make the reasonable assumption that he agrees with Archimedes about this. This is evidence of some kind, because in Bayesian terms, academics in correspondence tend to believe the same true ideas, they talk to one another, and share the stuff they learned. It’s the same reason we can be confident that Babylonian cosmology and Genesis cosmology are similar. Aristarchus worked on planetary orbits, and Hipparchus’s model is just a refinement of Appolonius. There is no evidence that Appolonius was Geocentrist, this is a speculation you are making. The only evidence we have, coming from Archimedes, suggest reasonably that he is the same as Archimedes, a heliocentrist.

The equant idea is due to Appolonius. The idea is important, approximating a motion as uniform in angle around a different point than the center. It is also easy to come up with in the heliocentric model, and extremely difficult to imagine in geocentric models, because you need equants for both the deferents and the epicycles, not just the simple orbits. That’s a lot of equants.

The equant model is what you would get if you take a Kepler orbit, and do a “leading order in eccentricity” approximation. This is possibly what Appollonius did, but not with any confidence. One thing one can hypothesize with some confidence, simply on the structure of the model itself, is that when Appolonius invented the equant, he stuck them into a sun-centered model, because it is dead easy to do this. You just need 5 circular orbits with 5 equants, and you get a great model, nearly as good as Kepler’s. Whether he considered this as an approximation to an ellipse, I have no idea. He could have, because that was his specialty. People working in astronomy who get interested in conics for no reason are suspicious.

The claim Russo makes for a centuries long gap in observations is in a plot in his book, where he shows you the date of the observations used by Ptolmey. There is a huge gap between 100BC for the latest Hellenistic era observations and 100 AD for the earliest Roman era observations. The graph is on p. 283.

The claim that the ancients thought all motion is relative is likely true, and it is discussed by Russo. It’s why they thought you could transfer the motion to the Earth without side effects.

But even then, if you ask them what is actually going on, assuming it was possible to discriminate in some way, the answer any half-way competent scientist would be forced to give is “Heliocentrism”, whatever they believed about “strong winds” or “we’d feel it” or whatever. The reason is that the parameters of the epicycles and deferents are derived from an Earth year, but not uniformly, in a crazy pattern.

The inner planet deferents are one year in orbit, so that they track the sun. That means they look like they are going around the sun to the naked eye. Their epicycles are their true orbital period, their deferents are equal to one Earth year. Remember that the educated folks in Copernicus’s time believed him regarding the inner planets.

The outer planet epicycles are also all derived from the Earth year, which is the period of the sun’s deferent! So the pattern is this: Sun, Mercury Venus deferents are a year, the Saturn Jupiter, Mars epicycles are “1 year”.

I put “1 year” in quotes, because they aren’t equal to 1 year in Ptolmey. The epicycle periods only asymptote to “1 year” as the planet gets farther and father away from the sun. In real life, there is a correction from the relative motion of the planet to the Earth, giving an epicycle period which is almost exactly 1 year in the case of Saturn, because it’s slow, still close to 1 year for Jupiter because it’s faster, and not at all 1 year for Mars, because Mars’s year is not that much longer than ours.

You only figure out that all these epicycles are really “1 year” after you transition to heliocentrism, where the different crazy epicycle periods then are obvious from the true planetary period and the Earth’s period. You’ve just “predicted” the epicycle periods, you’ve reduced the indeterminacy in the model. That’s a sign you have the right model.

So you can’t say that the ancients believed that the epicycles are all a year by magic, because they aren’t equal to 1 year exactly, until you switch to Heliocentrism.

But it’s not only that. You also know the equants for all the outer planet epicycles from the Earth’s equant (it’s eccentricity). So you’ve traded in 6 outer planet equants for 3, you’ve predicted 3 equant parameters perfectly, and 3 epicycle periods. You’ve gotten all the equants and all the epicycles from 5 orbits, each of which have a single circle and equant.

Further, the epicycles regularly get smaller, so that Jupiter’s epicycle is smaller than Mars’s and Saturn’s is tinier still. On the other hand, venus’s epicycle is it’s orbit, and it’s larger than Mercury’s. When you transition to Heliocentrism, the size of the epicycle just tells you the distance of the planet in astronomical units to the sun, the smaller it is, the farther the planet, for outer planets, and for inner planets, the smaller it is, the closer the planet is to the sun.

The moment you “go heliocentric”, all the planets orbital periods are inversely proportional to the 3/2 power of their distance to the sun. That means the length of the year goes up smoothly, according to an obvious regular law, even if you don’t know that it’s 3/2 power. The Earth’s orbit falls right where you expect it to be on the interpolation of periods for Venus and Mars, the outer planets become no different from the inner planets, and the distances make complete sense. It’s really obviously right. There’s no justification for opposing it, other than the brain-damaged “why don’t we feel it”.

That’s a total no-brainer for a scientist, but notice how long a proper explanation took! Notice how it has nothing to do with any of the arguments you read. It’s all about the details of the stupid parameters, and it would be incomprehensible to a lay-person.

This is why every scientist worth their salt believed Copernicus immediately even when he had a crappy model with lots of epicycles and worse predictions. That’s why they were able to reject their idea that we would feel Earth rotation, or revolution. Because the heliocentric model is just extremely compelling just from the gross data, without regard to the details. Because the main economy is that the distances to the planets are predicted, and all the planet periods and equants become uniformly comprehensible.

Since I know Archimedes from his work, I know that he would understand all this in the blink of an eye, so it is inconceivable to me that he could be a Geocentrist even if he didn’t explicitly say he wasn’t. But one needs mathematical training to appreciate just how good he was at math, and just how obvious heliocentrism is, once you are exposed to the idea, even if you are a complete relativist regarding dynamical effects (which it is not clear they were— Russo mentions the ancient “Newton’s bucket”, turned upside-down and rotating about it’s axis)

Comparing “best to best” is indeed misleading, but in this case, this is not the point. The point is that the content of the treatises by Appolonius and Archimedes include ideas (coordinate systems, summing the squares of the integers from 1 to N in On the Sphere and Cylinder, infinitesimals in The Method, appreciation of the number of parameters in a quadratic form, knowledge of the defining Cartesian equation of the sphere and parabola, and how to algebraically shift the equation for the sphere) that simply don’t reappear until Descartes and Cavalieri.

The Europeans in fact did not understand the works until Cavalieri rediscovered infinitesimals, and Descartes figured out how to do coordinate geometry and define the equations of curves.

Newton’s work would be completely incomprehensible to ancient writers, despite the superficial impression you get from looking at geometric looking demonstrations, no matter how long they stared at it. It is actually mostly incomprehensible to modern readers, because he uses silly ancient methods, but modern readers just look at the statement of the theorems and reproduce the results by themselves using their own calculus education. An ancient reader wouldn’t be able to do that, so they would need to discover a whole bunch of stuff first. The reason is that Newton was computing with infinite polynomials (Taylor series), which were an innovation in medieval India and modern Europe, and Newton had a table of integrals, which he used in private, which we have, because his notebooks are published. His theorems about the Earth’s oblateness for example is evaluating an extremely difficult integral, by comparison, Archimedes could only integrate polynomials of low order, quadratics and cubics. Newton’s cycloid calculations would be too trigonometry heavy for ancients. There’s a reason why Newton’s Principia is the very last book written in the ancient geometric style. It’s really awful compared to modern algebraic notation that you find in any later physics work.

This stuff is not obvious from superficial look at the works, you need to put the book down and do it your own way. I think Russo is 100% justified in assuming even the best mathematicians of the Roman era, like Diophantus, would have an extremely difficult time with “On the Sphere and Cylinder”, because if they put the book down, they wouldn’t be able to do anything their own way.

The reason? We know what happens when you actually do understand and reproduce Archimedes! You generalize! The sum of the squares he did in “Sphere and Cylinder” generalizes to sums of powers, a generalization made in late medieval Europe. The integrals he does generalize to integrals of powers, also late medieval. The torque-balance and potential energy arguments generalize to dynamical ideas, like in Huygens, early modern. The hydrostatics gives impetus to study flow rates through pipes, and you can find a flow-rate law, and Pascal’s principle.The arguments he gives in “The Method” can be generalized (with difficulty) to integrals of polynomials, and the primitive algebra he does can be generalized into Cartesian stuff, if you invent a good algebraic notation. Europeans did all this stuff and more after Cavalieri and Kepler. But Cavalieri and Kepler are only as far along as Archimedes, which is consistent with Russo’s impression that Hellenistic science looks like late medieval work, sometimes early modern, while Roman science is early medieval. That’s centuries of backwards.

The Roman mathematical work doesn’t generalize anything. That means they didn’t get the hard math stuff. They generalized the medicine a little, they did something, but I can’t imagine any Romans summing a sequence of consecutive square numbers by a formula, let alone cubes.

I will buy and read your new book, and apologies for the long comments.

I reread Russo’s book to remember his arguments. Do you have the quote from Hipparchus that says he is geocentric? It would go a long way to establishing that heliocentrism wasn’t dominant in Hellenistic Alexandria, and it would refute Russo’s idea about Hipparchus.

That’s in my book The Scientist (pp. 74-75).

Your comment is assuming a notion of automatic progress in science, it is assuming that progress only requires preservation of literature, and not continuity of scientific culture.

The results are in the literature. Culture has nothing to do with this. One does not “forget” the discovery of Newtonian dynamics when one has all the texts that discoveries were written in. If that was discovered by Hipparchus or employed by him in any way whatever, everyone reading the works of Hipparchus two centuries later would know about it. Even if all scientists in between were abducted by aliens and no one became a scientist again for a hundred years.

Russo’s thesis is “regress”.

For which there is exactly zero evidence.

His claim is that to have scientific progress, people don’t just need access to the written work, they need a culture that allows them to be able to understand and expand upon it.

Yeah. Which happened in the Middle Ages. Not ancient Rome. Then, everyone had all the requisite skills to understand all the relevant mathematics and were actively building on it. This is very well attested. There is zero…literally zero…evidence that any relevant ability to understand the requisite mathematics was “lost” before the 4th century.

Even in the 1980s and 1990s, people in modern times forgot how to read some of the recent science of the 1960s.

Um. No. Not in any sense analogous to what we are talking about. Newtonian dynamics involves no mathematical understanding not abundantly present in Roman times and even employed by Ptolemy. So there is no possible way Ptolemy “wouldn’t understand” a treatise on the subject, much less any of the hundreds of other engineers, astronomers, and mathematicians of the day. Most specially as it was the particular custom of ancient scientific treatises to teach the reader the skills they needed to understand the treatise itself.

Russo’s thesis is that nobody in 100BC knew how to read Archimedes well, or Appolonius. They cited these works like you cite Newton.

And that’s bullshit he just made up. Based on no evidence whatever. Historians don’t accept things people just “make up” about the ancient world. That’s called pseudohistory.

…knowing the enormous gap between Hellenistic and Roman era mathematics by reading a sample

And who are you “comparing”? There are shit mathematicians in Hellenistic times, too. So are you comparing the best with the best? Because you might be starting to see where Russo is conning you with an invalid procedure.

The difference looks like what Russo describes it, as the difference between a science-aware Hellenistic society and a Roman society that is only aware of technology, and doesn’t understand the subtleties of scientific modeling.

No one who actually read Ptolemy’s Mathematical Syntaxis or the spherical trigonometry of Menelaus or the catoptics of Hero could make such a ridiculous claim.

The reification of visual-rays is a case-in-point for Russo, Euclid is clearly using these to model units of visual perception associated to light propagation, not in the absurd view that visual stuff is emitted by the eyes (although they might have thought light travels instantaneously).

Ptolemy actually discusses that very point, and why the debate continues in his day. Read his Optics. And there were still astronomers in his day who agreed with you on the fact. And they were well aware, for example, that Hipparchus advocated an atomic rather than a visual ray theory of light, and astronomers still debated his theory (see Plutarch’s On the Face in the Moon). Yet another example refuting Russo’s ridiculous claim that the theories of Hipparchus were forgotten. Again, zero evidence of any loss of knowledge here.

Similarly, the gap of centuries in astronomical precision measurement between Hipparchus and Ptolmey is evidence.

Zero evidence of that.

The evidence that Ptolmey is copying something else is that he says the epicycle-deferent-equant model is due to Appolonius

Nope. We have multiple attestation Ptolemy is using the model developed by Hipparchus. Hipparchus corrected the Appolonian model. And Ptolemy’s own model is partly his own innovation. He is very explicit about all of this. The discussion on Wikipedia is actually pretty good. And it’s based on peer reviewed literature.

Archimedes, who is definitely a heliocentrist (he popularized heliocentrism in “The Sand Reckoner”).

Nope. He merely mentions that Aristarchus posited heliocentrism. Archimedes himself did not say it was correct, but that even if it were, his own result regarding volume would hold. There is no evidence Archimedes ever adopted, endorsed, or argued for heliocentrism himself.

The only persons we can confirm by name as heliocentrists are Aristarchus and his pupil Seleucus. It was well known in Roman times that Seleucus actually presented an argument for heliocentrism that was considered in need of address (Plutarch cites the fact as being debated in his own day); and we can infer it was probably the same tidal theory Galileo fell in love with, only to eventually discover it was invalid (since Seleucus famously discovered lunisolar tide theory and posited universal gravitation to explain it, a fact attested and discussed, again, in Plutarch).

We don’t get to hear the names of the heliocentrists in the Roman era, but they were still around, as attested by Seneca, and Ptolemy himself, whose opening chapters of the Syntaxis take aim at two camps of competitors: heliocentrists, and dynamic geocentrists (for whom we have abundant independent attestation as well: proponents that the earth spins, but is still the center). Which is how we know no one invented Newtonian dynamics. Because that would have settled the argument for heliocentrism, and Ptolemy would be forced to address it in his first book of the Syntaxis.

It would be most natural for Appolonius to invent equant-deferent-epicycle models to mechanically describe true elliptical sun-centered orbits using Earth-centered clockwork gears.

That’s what Ptolemy is doing, too. He is literally describing the construction of a physical device. Which requires combined circular motions. In Planetary Hypotheses Ptolemy explicitly says that in the real world, you can do without the circular mechanisms and just propose the planets move in quasi-elliptical orbits, and he speculates on how we could find out which it was (which way nature actually produced the motions) and he settles for crystalline gearing only because he considered it parsimonious.

Appolonius…was a heliocentrist

Zero evidence of that.

…he for some reason knew everything there is to know about conics. He looks like an ancient Kepler, although perhaps he wasn’t as far along.

Nope. No evidence he ever linked his conics with planets at all, much less planetary laws of motion. We don’t have any evidence anyone ever did. Not Hipparchus. Not anyone. Russo is just making that up. Thereby confusing hypothesis with evidence. Which is why he fails so badly at this part of his work.

I know Ptolmey isn’t doing things right, because he isn’t a heliocentrist, and even the most modest exposure to the data will turn any honest scientist into one

False. Almost all experts in antiquity were geocentrists, including Hipparchus. It’s actually not so obvious that heliocentrism is true. It requires a lot of research to establish numerous counter-intuitive premises to get there. This is why everyone rejected Copernicus. He had no evidence. And his system actually under-performed against Ptolemy’s. Ptolemy’s system made more accurate predictions; because Copernicus obsessively rejected noncircular orbits and inconstant velocities, two features abundantly employed in Ptolemy’s model (and as it happens necessary to any correct model). Until Kepler advanced on Ptolemy by updating Ptolemy’s planetary motion law and linking it to conics. And even then most experts still rejected the idea; it didn’t actually win consensus approval until Newton. Read Kühn on this point.

(the three outer planet epicycles all have a period derived from the Earth year, a ridiculous coincidence — why should Saturn retrograde once a year? Jupiter?).

It’s not ridiculous when the earth is the center of the system and doesn’t spin. The year then only exists in that system precisely because of linked planetary annual motions.

Remember classical relativity: phenomenologically, geocentrism and heliocentrism are actually mutually identical. Because motion is relative. Whether the earth or the universe is spinning, is a question of not-directly-observable physics that can’t be determined by merely observing the relative motion between the two. You need some underlying physical reasoning; hence Newtonian dynamics, which finally showed that you can explain two observations (rate of fall of objects on earth; and the motion of the earth relative to the planets and sun) with a single theory. Which coincidence is indeed too improbable to posit; but to get to that point, was enormously difficult. Even Kepler and Galileo didn’t see it. Kepler couldn’t even imagine it. Galileo at least thought it might have to do. But he never produced a demonstration. That didn’t happen until Newton uncovered the mathematical proof. Galileo never discovered anything about this that wasn’t already discussed in Roman times, e.g. Plutarch attests his friend and astronomer Menelaus was still debating whether universal gravitation could explain, for example, the lunar orbit, and hence all other orbits (with their motion pushing them out, and gravity pulling them in, producing a combined orbital motion). To get to the additional step that the sun’s gravity is so much larger as to be a better explanation of planets orbiting it, rather than they and it orbiting the earth, would be a long time in coming. Because it’s not obvious at all.

I reread Russo’s book to remember his arguments. Do you have the quote from Hipparchus that says he is geocentric? It would go a long way to establishing that heliocentrism wasn’t dominant in Hellenistic Alexandria, and it would refute Russo’s idea about Hipparchus.

Your comment is assuming a notion of automatic progress in science, it is assuming that progress only requires preservation of literature, and not continuity of scientific culture. This is an unjustified assumption. Russo’s thesis is “regress”. His claim is that to have scientific progress, people don’t just need access to the written work, they need a culture that allows them to be able to understand and expand upon it.

Even in the 1980s and 1990s, people in modern times forgot how to read some of the recent science of the 1960s. Things like “Forcing” and “S-matrix theory” were still cited and occasionally used in a ritualized form, forcing was used in Solovay’s ritualized poset form as a standard method, but without reference to Cohen’s philosophy, which only reappeared in print a year or so ago, right after Cohen’s death, in the book “Forcing for Mathematicians”. To see what the problem was, you can read about the notion of an “open exposition problem” in the article “Forcing for dummies” of more recent years. S-matrix theory was used in string theory, but as a ritualized ill-understood “Regge limit” part of a precise model, so, for example, Mandelstam’s double-dispersion relations are a major cited thing in the 60s, and disappear from the literature entirely from 1980 until 2015.

The main philosophical ideas motivating both, which were clearly evident in the literature of the 1960s, completely left the mainstream of what scientists were trained in or understood, just because of the way scientific philosophy regressed (slightly). “The Pomeron” (something everyone in physics understood in 1960 which dominated high energy physics until quarks took over) disappears from the literature entirely around 1980, only to reappear in the late 2000s. In that time, which was my time, nobody talked about Pomerons, nobody even remembered what a Pomeron was, aside from moderately bad analogies to closed strings in string theory, and we not only had the original literature, some of the people who discovered it were still alive! They also stopped talking about it, mostly, except for a brief period when p-p and p-pbar total scattering were shown to be equal at high energies in the mid 90s, the central prediction of Pomeron theory. It’s really weird how forgetting happens when a dominant social order suppresses another one, in this case, anti-positivist mystical philosophies replacing logical positivism.

Russo’s thesis is that nobody in 100BC knew how to read Archimedes well, or Appolonius. They cited these works like you cite Newton.

Having witnessed an analogous event of (very minor) regress (which was fixed by the internet), and knowing the enormous gap between Hellenistic and Roman era mathematics by reading a sample, it is much less persuasive that the difference is small or that there is any continuity. The difference looks like what Russo describes it, as the difference between a science-aware Hellenistic society and a Roman society that is only aware of technology, and doesn’t understand the subtleties of scientific modeling. The reification of visual-rays is a case-in-point for Russo, Euclid is clearly using these to model units of visual perception associated to light propagation, not in the absurd view that visual stuff is emitted by the eyes (although they might have thought light travels instantaneously). Similarly, the gap of centuries in astronomical precision measurement between Hipparchus and Ptolmey is evidence.

The evidence that Ptolmey is copying something else is that he says the epicycle-deferent-equant model is due to Appolonius, and Appolonius is in correspondence with Archimedes, who is definitely a heliocentrist (he popularized heliocentrism in “The Sand Reckoner”). It would be most natural for Appolonius to invent equant-deferent-epicycle models to mechanically describe true elliptical sun-centered orbits using Earth-centered clockwork gears. It’s a good approximation. Then Ptolmey would just be reifying the clockwork Earth centered model, while jettisoning the challenging heliocentric philosophy. This is a normal process when you don’t understand, it’s like the Pomeron or Forcing. It is very difficult to analyze this in a Bayesian way, because it involved the internal logic of the models, which demands certain features, but also properties of the people involved, who are not the perfect logicians of mathematical puzzles, they make logical mistakes.

I don’t know if Appolonius knew elliptical orbits, he might have had the orbits around the sun be equant-deferent circles (that’s good enough for government work). But he studied astronomy, he was a heliocentrist, and he for some reason knew everything there is to know about conics. He looks like an ancient Kepler, although perhaps he wasn’t as far along.

I know Ptolmey isn’t doing things right, because he isn’t a heliocentrist, and even the most modest exposure to the data will turn any honest scientist into one (the three outer planet epicycles all have a period derived from the Earth year, a ridiculous coincidence — why should Saturn retrograde once a year? Jupiter?).

The difference between “ellipse” and “approximate ellipse” is that the ellipse has a focus, and that’s where the body causing the motion is. The ellipse associates motion with a gravitational cause. The “approximate ellipse” is just a squooshed circle, and it’s not really close philosophically (although it’s better than a bad fit to data). A circle on a deferent with an equant is an approximate ellipse too.

I only reject Russo’s claim that Newtonian dynamics was invented by Hipparchus and forgotten within a hundred years. There is no evidence of any loss of texts from the Hellenistic to the Roman periods, and many people were clearly reading them, quoting them, and well aware of what was in them. So Russo’s theory is implausible. In Bayesian terms, his thesis entails it is improbable that we should observe so widespread a familiarity with Hellenistic treatises (such as by Hipparchus) in the Roman period. It thus has a small likelihood ratio. This, in addition to the fact that there actually is no evidence Hipparchus had developed Newtonian dynamics. He wasn’t even a heliocentrist (and we have a direct quote from him on that point).

Note that we have Roman authors discussing heliocentrism and universal gravitation. And Ptolemy himself was constructing elliptical orbits in his model, e.g. for the moon. Russo seems to overlook the fact that Ptolemy is literally describing the construction of physical machines, and giving instructions on how to build them, such that they’d produce complex non-circular motions, as for the moon nearly indeed elliptical. And he introduces planetary motion laws (equal angles in equal times). There is no evidence he was downgrading from some prior model (like equal areas in equal times). Nor is there any evidence this was invented by someone and “forgotten.”

Yes, they had all the tools in place, and someone responding to Ptolemy (had science been allowed to continue) in defense of the heliocentrists (who still existed in Ptolemy’s day; we just don’t get to read them) may indeed have made those conceptual leaps. But we have no evidence anyone ever did that. And the evidence there is, is against anyone having done that.

As to funding, that only affects the pace of progress, not progress itself. And it has no relation to Russo’s thesis, which requires widespread destruction of information, not the mere slowing of advancements upon it. On the funding of science under the Romans, my books cover that (Science Education Ch. 8, and The Scientist, Ch. 4.7).

There is no evidence anything about the Roman class system had any plausible effect on scientific research or the production of scientists. I thoroughly document the complete absence of such effect in Chs. 3 and 4 of Scientist.

Russo is an outlier because he understands the development of Hellenistic heliocentrism and some sort of Hellenisitc gravitational theory. He plausibly claims that this is evidenced by the remaining mangled citations to them in non-scientific works.

I know you read Russo, as you introduced me to his work (I suggested that Appolonius knew elliptical orbits, you told me “Read Russo’s book”). I don’t think you ever explained any non-ideological reason for rejecting Russo’s ideas, which bear a revolutionary character similar to your own ideas in Jesus studies, and, while I am not familiar enough with the background for Bayesian estimation, they seem naively plausible just from reading Archimedes and Appolonius.

It is extremely plausible that elliptical Keplerian planetary orbits and some sort of centripetal force were understood by Appolonius (and accepted by others), that ballistics and primitive dynamical ideas were understood (as you explain), and rudimentary infinitesimal calculations were understood by Archimedes. Russo argues for solar gravitation with inverse square law, but perhaps it is not likely. I do not have enough information to make a Bayesian estimate, but you certainly do. Do you rule out the Russo hypothesis because it is disfavored, and if so, with what confidence, and using what evidence? Perhaps you will say “read the book”.

The thesis of Roman stagnation can be simply explained by the introduction of a rigid heirarchical society, and the loss of state funding for science. The examples of Ptolmey and Hero are not very convincing, as Ptolmey was writing in Alexandria, with a state sponsored library, and probably city/state funding. Hero also comes from a Hellenistic region, where state funding might not have dried up. Is there evidence of consistent state funding for pure science in Roman times? It is dead certain in Hellenistic times.

Scientists were not class conscious, as you explain yourself in public recorded lectures. The rigid Roman hierarchy, much like modern American money hierarchy, is anaethema to scientific cullture. Scientific culture is lowbrow, but rigorous. It is not difficult to imagine how Roman culture could lead to a destruction of science, and it seems in evidence simply by comparing the quality of the mathematics between Appolonius’s time and Galen’s. For a similar period of (much mroe modest) decline, the 1970s, 80s, 90s, show a mini-dark-age compared to the 1950s, 60s and 2000-present, in that certain ideas like logical Forcing, modern Scheme-theoretic Algebraic Geometry, or S-matrix string theory are not as prominent as in the 1960s or today. The motive factor in that era is the lack of internet, coupled with the enormous explosion of literature after 1957 and the expansion of state funding.

Looking foward to reading the new book, but perhaps you can say why Russo is ridiculous. I apologize for not being a patron, I support your work by purchasing the books I can afford.

I’m not sure what you mean. Do you mean, stagnate during the Roman Empire, or after? I’m assuming you mean during.

Yeah, that’s what I meant. Thanks for clarification.

I’m not sure what you mean. Do you mean, stagnate during the Roman Empire, or after? I’m assuming you mean during.

Of those writing after 1990, and who have stated any opinion on the matter: All of them except Russo agree with me. Hence I extensively cite and discuss them. In fact a major component of ancient science history literature since 1990, is the universal rejection and refutation of the early 20th century stagnation thesis. Russo is a weird outlier. And I discuss that fact in the book.

(it’s heavily footnoted; the bibliography alone extends over sixty pages)

How many of those people on those sixty pages agree with your thesis that science didn’t stagnate?

That’s a broad subject. Scientific values are the focus of this book (so that’s part of what you are asking for). Political values (origin of rights, the founding principles of the US constitution, etc.), I treat in Christianity Is Not Great, and I link to that and expand on it here: https://www.richardcarrier.info/archives/3314.