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The discrepancy between the observed precession of the perihelion of Mercury and the value predicted by Newtonian theory was known in the 19th century to be approximately 43 arcseconds per century.

Maybe I totally misunderstand what this value means, but if it is what I think, then this value seems to be absolutely tiny, 43" approximately corresponds to the width of a needle held at an arm's length in front of you.

How can this value be obtained from data that cannot have had an extremely high precision and that cannot have spanned more than a couple of centuries of observations?

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    $\begingroup$ about half a kilometre in distance actually. I personally was pretty amazed after learning about the precision of measurements even back in Newton's day (see 3 and 4). maa.org/sites/default/files/pdf/cms_upload/Nguyen18979.pdf Turns out this was one of his favourite arguments to support the inverse square theorem $\endgroup$ – lucky-guess Apr 6 '17 at 22:48
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Some of the most accurate measurements of the solar system come from event timing. (Relatively) small changes in position can lead to significant differences in eclipse or occultation timing.

In fact the article on Tests of general relativity mentions that this was the method used to first notice the discrepancy.

This anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. His reanalysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton's theory

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    $\begingroup$ An additional link which mentions Le Verrier's work. $\endgroup$ – StephenG Apr 6 '17 at 22:35
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Transits of Mercury as observed from Earth occur in either November or May, with roughly 2/3 of them occurring in November. The November transits occur days before Mercury's perihelion passage while the May transits occur about a month after Mercury's aphelion passage. That tiny 43 arc seconds per century means that over the course of 150 years, there will be a four minute discrepancy in the timing of the November transits, and an even greater discrepancy in the timing of the May transits.

Clocks were accurate to within a few seconds per day by the end of the 17th century. That four minute cumulative discrepancy between the timing of the November 1697 transit and the November 1848 transit was well within the accuracy of late 17th century clocks to mid 19th century clocks.

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    $\begingroup$ With regard to the accuracy of clocks, it's worth noting that Harrison's H4 lost about 5 seconds in 81 days on its first trial in 1761-1762 (this is 5 seconds compared to its corrected rate). This was in a ship, and H4 was a watch: a good pendulum clock running in less horrible conditions could probably do quite a lot better even by then. $\endgroup$ – tfb Apr 7 '17 at 14:15

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