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The short story is that based on Kepler's and Newton's physics, in 1846 Le Verrier mathematically predicted the existence and current location of Neptune within a single angular degree, and it was observed at first attempt. It not only demonstrated the power of science, it taught us something about heaven and the planets, both historically the domain of religion.

Was the successful prediction of Neptune generally perceived as such a triumph for science that it caused a paradigm shift as to the role of science in society? Was astronomy until then maybe widely viewed as a leftover from astrology, and since the 17th century reduced to only meaninglessly predicting the movements of the planets without it having any significance for our destiny.

Did it substantially help the science side against the Church during the debate about evolution that began 14 years later? Could Le Verrier's prediction even have stimulated the evolution debate to have come about?

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First, the prediction of Neptune was a big win for science in the eyes of the general public. It was not exactly spotless though, especially in the eyes of the scientists. Here is from Kelley's How was Neptune Discovered?:

"The world was excited by the find, for never before had mathematics predicted a natural object. This confidence in the results was lessened, however, when discrepancies were noticed in the calculated values and the actual ones (Lyttleton 227). For example, Adams calculated an orbital period of 227 years and Le Verrier found it to be 218 years using Kepler’s Third Law (Period squared is proportional to average distance cubed). The actual value of the orbit is 165 years. This discrepancy was not a result of using Kepler’s Third Law but because of using Bode’s Law for the average distance (229). The only actual value they were close to, if one looks at the table, is the location in the sky it would be found. It is possible that both men were simply lucky with this. We shall never truly know (233)."

Second, debate over evolution did not begin 14 years later, it was going on for a while. Lamarck proposed his version of evolution in 1801, and even the dominant in the early 19-th century Cuvier's catastrophism posited a kind of evolution, although in discrete leaps. Lyell's gradualistic geology defended in Principles of Geology (1830-33) that displaced it was far more influential in the acceptance of Darwin's ideas than astronomical events (and it was a direct inspiration for Darwin himself). In 1860 Leverrier himself moved on to a new prediction, Vulcan, blamed for the anomalous precession of the perihelion of Mercury. A curious report about the annual meeting of the British Association for the Advancement of Science in Manchester, 1861 (soon after the Oxford debate) indicates that Neptune was not much of a factor:

"In 1861, the city showcased both its wealth and brains as it hosted Britain’s largest celebration of knowledge, the annual meeting of the British Association for the Advancement of Science. Charles Darwin had published The Origin of Species less than two years before, and that explosion continued to reverberate through every gathering of the learned. At the Manchester meeting Darwin’s defenders prepared to battle religious doubters. One speaker, the “blind economist” Henry Fawcett, made the ultimate claim: Darwin was a true scientific hero, one who solved his problem by the same methods, the same approach to experiment, observation, and generalization that the great Isaac Newton himself had used in his physics. Much else was discussed, of course — advances in dredging engineering, a report on birds of New Zealand, news from the balloon committee. The astronomy section was relatively quiet..."

Finally, in contrast to 17th and 18th centuries the role of the church in scientific debates was much diminished, and its reaction to Darwin (and more specifically to human origins, not evolution in general), was ambivalent. The nature and significance of the Oxford evolution debate of 1860 might also have been glossed:

"Though the debate is frequently depicted as a clash between religion and science, the British Association at the time had a number of clergymen occupying high positions (including Presidents of two of its seven sections)... a case could be made for saying that for the many clerics in the audience, the underlying conflict was between traditional Anglicanism (Wilberforce) and liberal Anglicanism (Essays and Reviews)...

Many of the opponents of Darwin's theory were respected men of science: Owen was one of the most influential British biologists of his generation; Adam Sedgwick was a leading geologist; Wilberforce was a Fellow of the Royal Society (though at that time about half of the Fellows were well-placed amateurs)... The debate has been called "one of the great stories of the history of science" and it is often regarded as a key moment in the acceptance of evolution. However, at the time it received only a few passing references in newspapers, and Brooke argues that "the event almost completely disappeared from public awareness until it was resurrected in the 1890s as an appropriate tribute to a recently deceased hero of scientific education"".

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    $\begingroup$ Great! It should be avoided according the rules here, but: +1! You have opened several doors of depth to the history which I widely speculated about. (Debating Leonard Kelly's objections are OT here, but I've heard about them through those who in turn object against those objections). Already Isaac Newton was highly celebrated, with a great funeral. The science debate was maybe settled already then. $\endgroup$ – LocalFluff Mar 21 '17 at 22:35
  • $\begingroup$ @LocalFluff Thank you. Actually, I'd appreciate if you could give me names of/references to Kelley's critics. I was trying to find out more about the physics behind the Leverrier-Adams prediction in the light of Kelley's discussion, and even asked a question on Physics SE, see Was Leverrier-Adams prediction of Neptune a lucky coincidence? But unfortunately no one could answer. $\endgroup$ – Conifold Mar 22 '17 at 1:41
  • $\begingroup$ I'm afraid I might've read the very answer you got in the question you link to! Or some similar speculation. I wonder what's documented of Le Verrier's calculations. Since it got such a huge attention, it should be more than "This there now because I says so". Isn't science all about getting lucky and then changing the laws of nature so that it doesn't look lucky, but predestined and unavoidable? $\endgroup$ – LocalFluff Mar 22 '17 at 8:25
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Discovery of Neptune was indeed a significant step which increased the prestige of science but it was only one of a long chain of such events starting in 17th century. Let me give a short sample: discovery of Jupiter satellites with the newly discovered telescope (unlike the discovery of Neptune, this had a great practical application!), Newton's gravitation theory and celestial mechanics, it predicted the correct shape of the Earth, verified by the French expeditions, it gave the first working theory of tides, and it permitted Halley to predict the return if the Halley comet. Mathematical theory of Moon's motion solved the centuries old practical problem of finding longitude at sea.

The discovery of spectral lines made it possible to determine chemical composition of stars and the Sun.

This short list only mentions some discoveries in physics and astronomy. But remarkable discoveries were made in other areas as well, let me only mention smallpox vaccination which saved millions of lives.

So by 1846, science had already sufficiently high prestige among educated public. All discoveries that I listed had more influence on people's lives than the discovery of Neptune. By the way, discovery of new planets (large asteroids, Ceres, Pallas, Juno and Vesta) before Neptune, also had enormous public resonance, now forgotten.

EDIT. To address the comments, let me explain more about Jupiter satellites for example. The main thing about this discovery was not that they just exist, but that they obey Kepler laws. Kepler laws had to be discovered in advance to determine this. Now the fact that they obey Kepler's laws permitted to predict their motion. And this gave the first practical method of determining longitude, one of the most pressing scientific problems of 17th century. Jupiter satellites made possible determination of longitude only on land, because you need a telescope to see them, and a telescope cannot be installed on a ship. Solution of longitude problem at sea had to wait for another century, and was obtained in two ways: a) mathematicians managed to predict Moon's motion, using Newton's laws and advances in mathematics, and b) chronometer was invented. (A lot of mathematics was used in both cases, especially for the Moon motion).

Now about asteroids. You look at the sky through a telescope, and you see a new star. But you do not know what is this (a Nova, a comet, etc.) When you look again and again, you notice that it moves. So a Nova is excluded. But then you have to compute its orbit, to be sure that when you look in a year from now you see the same object. If the orbit turns to be elliptic with small excentricity, this is a new planet. In the case of Ceres, computation was performed by Gauss, the greatest mathematician of that time. He had to develop new methods which enriched mathematics. This discovery has an enormous public resonance.

So in no way these milestones in the development of science can be called "stamp collecting".

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    $\begingroup$ Yes, but that is all only what theoretical astronomers today call "stamp collecting". One observes something one doesn't understand and writes down its position and time in a catalog. It is something completely different to totally logically and impersonally make calculations that PREDICT the existence and position of a new unknown planet. And so perfectly precisely and directly. That's being an oracle, without the riddles. $\endgroup$ – LocalFluff Mar 21 '17 at 21:03
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    $\begingroup$ @LocalFluff: You have not read my answer attentively: Halley PREDICTED the return of the comet, Newton's theory PREDICTED the shape of the Earth, and permitted to PREDICT tides. And so on. What "stamp collecting" are you talking about? $\endgroup$ – Alexandre Eremenko Mar 22 '17 at 4:21
  • $\begingroup$ Of course astronomy is about predictions. Until Le Verrier it was about predicting the location of known objects based on direct observations of them. But anyone can predict that the Sun will rise tomorrow. I do agree that Halley's prediction is at least en par with Le Verrier's, because it wasn't known that comets are recurring individual objects. Anyway, this is about how it was publicly perceived when it happened, not how I rank discoveries. $\endgroup$ – LocalFluff Mar 22 '17 at 8:01
  • $\begingroup$ I said that I do NOT rank discoveries. You do! It has always been obvious to every traveler that Earth is round. What has that got to do with the science versus religion debate in the mid 19th century? $\endgroup$ – LocalFluff Mar 22 '17 at 13:25
  • $\begingroup$ @LocalFluff: Of curse it was known since the time immemorial that Earth is round. Newton predicted that it is flattened at the poles (approximately an ellipsoid). And this was dramatically confirmed by French expeditions. $\endgroup$ – Alexandre Eremenko Mar 22 '17 at 13:40

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