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History Forum / General / What If / July 2009Archimedes invents calculus.
Is this even remotely feasible considering the state of analytical geometry in the classical world? If so, what form might an Archimedean calculus take? What changes would result to the later Roman period and the middle ages as a result? > Is this even remotely feasible considering the > state of analytical geometry in the classical [quoted text clipped - 3 lines] > take? What changes would result to the later > Roman period and the middle ages as a result? My favorite source on Hellenistic sciences says"Archimese does not employ limits only in the sense that he fails to use a ***word*** that matches ours exactly; a proof in modern analysis needs only to have ***limit*** replaced by a definition to become equivalent to his in every way." Russo The Forgotten Revolution >> Is this even remotely feasible considering the >> state of analytical geometry in the classical [quoted text clipped - 9 lines] > ***limit*** replaced by a definition to become equivalent to his in > every way." Russo The Forgotten Revolution The work is still in progress on the Archimedes Palimpsest, but increasingly it looks like he *may* have invented calculus. So I suppose it's a matter of his having developed it further. More to the point, I don't think algebra was developed very far in the Roman period. Maybe the answer is that Archimedes didn't provide calculus in a form that was very useful in practice. Dennis > >> Is this even remotely feasible considering the > >> state of analytical geometry in the classical [quoted text clipped - 20 lines] > > Dennis Russo mentions Heron and Diophantus (c 200 to 298) as using "algebraic methods" in Alexandria. Enough that Diophantus is called the "Father of Algebra", a title that he probably should share with Persian mathematician al-Khwārizmī, born some 500 years after Diophantus) > The work is still in progress on the Archimedes Palimpsest, but > increasingly it looks like he *may* have invented calculus. > > So I suppose it's a matter of his having developed it further. And spread it around more, so that it survives. David  Signature_______________________________________________________________________ David Johnson home.earthlink.net/~trolleyfan "So many of you come time and time again to watch this final end of everything which I think is really wonderful and then to return home to your own eras and raise families and strive for new and better societies and fight terrible wars for what you know is right, it gives one real hope for the whole future of lifekind... ...Except of course we know it hasn't got one." On Jul 4, 2:01 pm, David Johnson <trolleyfan_spamf...@earthlink.net> wrote: > > The work is still in progress on the Archimedes Palimpsest, but > > increasingly it looks like he *may* have invented calculus. [quoted text clipped - 16 lines] > > ...Except of course we know it hasn't got one." At least gotten it accepted by a few "writing" mathematicians. Hero is one I would hit. In article <50bb4562-abed-479d-bd5f-ec2b7fb6ff00@a37g2000prf.googlegroups.com> , > Is this even remotely feasible considering the > state of analytical geometry in the classical > world? This is right on the boundary of secret history and alternate history, see (< ttp://www.sciencenews.org/view/generic/id/8974/title/Math_Trek__A_Pr ayer_for_Archimedes>) which discusses a previously unknown Archimedes work that flirts with integral calculus. > If so, what form might an Archimedean calculus > take? What changes would result to the later > Roman period and the middle ages as a result? The problem is that Archimedes could have developed Calculus to the extent done by Newton and Leibniz, but it would have been completely forgotten if the few copies of this work had disappeared (a lot of his work did disappear). SignatureRobert Woodward <robertaw@drizzle.com> <http://www.drizzle.com/~robertaw> On Jun 27, 12:29 pm, "Robert A. Woodward" <rober...@drizzle.com> wrote: > In article > <50bb4562-abed-479d-bd5f-ec2b7fb6f...@a37g2000prf.googlegroups.com> [quoted text clipped - 22 lines] > Robert Woodward <rober...@drizzle.com> > <http://www.drizzle.com/~robertaw> Example: Quadrature of the parabola. A single copy, made in Byzantium in the 9th century, thence to Sicily in the 12th Century,to Frederich II HRE in 1266 to the Vatican Library. Copies were made in France and Italy in the 15th century and then the manuscript disappears. Some brigand or noble used it to light a fire or wipe his a.s and bye bye Archimedes. > Is this even remotely feasible considering the > state of analytical geometry in the classical > world? Yes it is very much so. In fact the recently found and published Archimedes Codex References: http://en.wikipedia.org/wiki/Archimedes_Palimpsest http://www.amazon.com/Archimedes-Codex-Revealing-Antiquitys-Scientist/dp/030681580X shows clearly that he had already been using the foundations of the Calculus in that he was using infinities in proofs as in the limit as x -> is y sort of proofs which is the mathematical step that lead straight to the calculus for both Newton and Leibnitz. That he was murdered not too long after writing this indicates that it is quite possible we could have had calculus being discovered and taught by Archimedes in the ancient world. > If so, what form might an Archimedean calculus > take? Different nomenclature for sure, the Greeks used a form of Geometric notation that is no longer used in mathematics that is in some ways superior to the modern in that it is easier to follow. >What changes would result to the later > Roman period and the middle ages as a result? It would make the science of mechanics easier to understand, possible industrial revolution much earlier. > > Is this even remotely feasible considering the > > state of analytical geometry in the classical [quoted text clipped - 30 lines] > It would make the science of mechanics easier to understand, possible > industrial revolution much earlier. I would argue that the idea of calculus would be interesting but not applied in any practical way. The Romans had slaves and little interest in mechanics beyond big projects like aquaducts. However much the Greeks and others in the Hellenistic age liked the toys that Hero and others made they did not look to use those toys in meaningful industrial way. > > > Is this even remotely feasible considering the > > > state of analytical geometry in the classical [quoted text clipped - 37 lines] > and others made they did not look to use those toys in meaningful > industrial way. On the contrary! The Romans were very very much interested in effective military hardware, especially artillery to which this sort of mathematics is directly applicable. In fact Archimedes main help for his home city of Syracuse was to devise means to shoot further and more accuratly with their artillery, which anyone who understands the math can tell you that thrown objects follow parabolas in free trajectory 100% if you can ignore air drag, which for more massive objects (~ 20 -30 kg stones for example) at smaller velocities (< 150 m/s) you can. Such engines are very, very useful to an empire of that technological level. That would be the first and principle use for a significant amount of time, but the use would spread. > > > > Is this even remotely feasible considering the > > > > state of analytical geometry in the classical [quoted text clipped - 54 lines] > That would be the first and principle use for a significant amount of > time, but the use would spread. Never did. These are Romans, theory is for Greeks, performance for Romans Euclid's Elements wasn't translated into Latin until the 6th Century, Varros' opinion of Theophrastus was that he wasn't suitable for people who wanted cultivate land , despite the many examples of improving viticulture. Varros is anti-science, later writers like Pliny and Seneca are interested in science but can't understand it enough to write coherently. There's more but you should get the picture. > > > > > Is this even remotely feasible considering the > > > > > state of analytical geometry in the classical [quoted text clipped - 56 lines] > > Never did. Never did what? The Romans were bloody damn well interested in Artillery and it's effective use and wanted to capture Archimedes alive for that very reason. http://en.wikipedia.org/wiki/Archimedes FYI his age at the time of his death is strongly disputed by the authors of the book about the Archimedes codex, they hold that he was a much younger man at that time. > These are Romans, theory is for Greeks, performance for > Romans Euclid's Elements wasn't translated into Latin until the 6th > Century, Varros' If it works, they copy it. They had little use for theory that they could not see as useful, but adopted the hell out of ideas that were. Like his various other inventions like the Archimedes screw. > opinion of Theophrastus was that he wasn't suitable for people who > wanted cultivate land , despite the many examples of improving > viticulture. Varros is anti-science, later writers like Pliny and > Seneca are interested in science but can't understand it enough to > write coherently. There's more but you should get the picture. > > > > > > Is this even remotely feasible considering the > > > > > > state of analytical geometry in the classical [quoted text clipped - 83 lines] > > Seneca are interested in science but can't understand it enough to > > write coherently. There's more but you should get the picture. Okay, try this What IF: Archimedes is not 75 but some younger age, you choose. He is taken prisoner by the Romans and is forced to do "contract research" on military subjects. What does he come up with and what effect does it have on ensuing Roman history? Archimedes dedicated works to Eratosthenes and Conon, both with similar birth dates recorded. Four years slop as the "year" is usually an Olympiad. > > > > > > > Is this even remotely feasible considering the > > > > > > > state of analytical geometry in the classical [quoted text clipped - 92 lines] > similar birth dates recorded. Four years slop as the "year" is usually > an Olympiad. I do not like that time line. I like better the time line where the Romans lose (quite possible, they only took Syracuse by treachery) and Syracuse and so Carthage wins, and for the sake of argument Rome is sown with salt and the Carthaginian civilization rises. Archimedes finishes his work on the calculus and learned men in both the Greek and Carthaginian/Phoenician world adopt his mathematics and apply it to nature and the science of mechanics and hydraulics (physics) advance at pretty much the same dizzy pace they did after Newton and Leibniz invent and publish the calculus in our time line. Some one, probably after telescopes are invented, and the fact of elliptical shaped orbits with the area swept out being constant twigs to the nature of gravity. |
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