Not quite; astronomers knew it, and a few people Hermann of Reichenau dared to suggest that the Church dates were wrong even the 11th century. That is, until Roger Bacon, who did not only suggest, but made quite a loud ruckus about in the mid-13th century. Bacon wanted calendar reform, which of course didn't happen until the Gregorian reforms of the mid-16th century. However, he ultimately ignored by the papacy.In any case, they established the fact that a year is not equal to 365 1/4 days was not practical accepted until the introduction of the Gregorian calendar.
This is issue took quite a while to sort out because European scholars had trouble with complex fractions. However, the estimates Bacon relied on put the tropical year at 365+63/260 days, which is actually better than the Gregorian calendar's 365+97/400 days, although an even better value would have been 365+31/128, which would also have much more convenient as an extension of the Julian calendar: leap years would those divisible by 4 but not 128.
No, that's not what's going on physically. If you quotient out the reduced mass, the rate of swept area is one-half the specific angular momentum, which is an independent parameter. So it's not correct to say that the reduced mass determines the orbit. There are actually six degrees of freedom: the orientation of the orbital plane (2), the rate of swept area (1), and three more are equivalent to the conservation of the interfocal vector (3-1) and eccentricity (1), which after the preceding parameters are fixed is equivalent to specific energy.I gather that Kepler himself was able to make observations precise enough to worry him, in that it turned out that different planets appeared to be following slightly different laws--that is, the numerical constant associated with the equal-area rule was different for each. This is because the actual motion of a pair of objects relative to each other is determined by the reduced mass of the pair, not the individual masses of each (reduced mass does of course result from combining each so there is a relationship, just not a linear one).
It's almost certainly impossible without significant advances in mathematics. Other than just very good observations he had available, the two main that made Kepler's discoveries possible were the mathematical understanding of how to calculate the quadrature of an ellipse, and the invention of logarithms. The former for Kepler's second law, and the latter for the third.What one needs, to get a heliocentric view adopted, is to have someone (either Hipparchus himself, or someone else) reach the rather obvious conclusion that if Earth must be “off-centre” for his model to work, the Aristotelian notion of perfect circles and relative positions is disturbed anyway! To be precise, you’d need someone to take the premises of Aristarchus en Seleucus, the knowledge of Hipparchus, and use this to work out (an equivalent of) Hipparchus’ unfinished heliocentric model ...
It is possible that someone of Archimedian calibre could solve the elliptical quadrature problem in the ancient world, but logarithms take considerably more conceptual leaps.
For a Kepler orbit, the velocity or momentum vector moves along a circle. The Greek obsession with perfect symmetries wasn't wrong per se; it was just completely misdirected. The Kepler problem actually has the symmetry of the 3-sphere, described by the group SO(4), and rotations in four dimensions have six degrees of freedom. Cutely, this fact was also used for the first correct derivation of the energy levels of the hydrogen atom (by Pauli, 1926), and explain why their degeneracy is so large.(which would do away with the whole “perfect circle” nonsense and accept that orbits may be elliptical).
Physically, the Kepler orbits are actually way more ‘perfect’ than the Greeks could imagine. A circle, after all, is just a 1-sphere, whereas the Kepler problem has a full 3-sphere symmetry.
Completely agree. Galileo's conceptual analysis of constant acceleration and Newton's concept of momentum were both found in the scholastic commentaries to Aristotelian physics. Any scholar reading them at the time would immediately recognise Galileo's reliance on the conceptual innovations of the Merton rule of the Oxford calculators, his law of motion to Oresme in the 14th century, and the concept of inertia and momentum as refinements of Buridan and older Arabic scholars that themselves were commentaries on Aristotle.The "dogmatic clinging to Aristotle" was mostly a rhetorical invention rather than an actual fact. Even during the heyday of Aristotelianism in the thirteenth and fourteenth centuries his theories weren't just uncritically accepted, and philosophers and theologians could and did argue over whether he was correct, or what the implications of his theories were.
Eh, this is indeed an ideological supposition; Newtonian mechanics does not actually rule out vitalistic conceptions of the world and similar things. You have some system of differential equation to solve, but it does not automatically follow that they need to have a unique solution.The real ideological challenge of the Newtonian system comes in when someone like Laplace comes along and asserts that everything can in principle be worked out from the first motions of matter; once set in motion a certain way gravitational systems, and by ideological supposition thus all physical systems which must have similar types of laws, will be determined to play out a certain way and no other, as one cause causes a definite set of effects in lockstep. This might be reconcilable with some Christian worldview of OTL, say Calvinist predestination, but it tends to give anyone who thinks about a nasty chill and might seem like license for total nihilism to some.
For example, suppose a point-particle with exactly specified initial position and initial velocity is subject to some exactly known force. If that force is locally Lipschitz-continuous, the Newtonian equations of motions have a unique solution locally. If it's not, though, it is possible (but not necessary) for them to have have multiple, and thus non-unique, solutions that diverge from one another. Which path does the particle pick then? It's not determined by Newton's laws.
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