Poincaré was, with the possible exception of Hilbert, the deepest, most prolific, and most versatile mathematician of his time. His collected works fill eleven large volumes, and that does not include several volumes on mathematical physics and another several volumes of essays on science and philosophy for the educated reader. ...It is funny how these authors go out of their way to praise Einstein, even when the comments do not make any sense. Any discussion of special relativity always credits Einstein as the discoverer.
How did Poincaré find himself in non-Euclidean geometry? Bolyai and Lobachevskii developed nonEuclidean geometry in the 1820s, and Beltrami put it on a firm foundation (using Riemann’s differential geometry) in 1868. So non-Euclidean geometry was already old news, in some sense, when Poincaré began his research in the late 1870s. But in another sense it wasn’t. Non-Euclidean geometry was still a fringe topic in the 1870s, and Poincaré brought it into the mainstream by noticing that non-Euclidean geometry was already present in classical mathematics. ...
For most of his career, Poincaré was as much a physicist as a mathematician. He taught courses on mechanics, optics, electromagnetism, thermodynamics, and elasticity, and contributed to the early development of relativity and quantum theory. He was even nominated for the Nobel Prize in physics and garnered a respectable number of votes. ...
Another interesting thread that runs through the book is Poincaré’s interest in physics, particularly his near-discovery of special relativity. Gray shows how Poincaré took many of the right steps, starting from Maxwell’s equations and getting as far as introducing the Lorentz group. But he lacked Einstein’s physical insight, and the mathematical insight that could have made up for this, Minkowski’s space-time, was not yet available. As Gray memorably puts it (p. 378):For Poincaré ... to have grasped the full implications of special relativity he would have had to be not Einstein, but Minkowski.
While he says that Poincare lacked Eintein's insight, he also says that Poincare grasped it all except for Minkowski’s space-time.
Since Poincare had more of the theory than Einstein, the only way to credit Einstein is to claim that Poincare was deficient in some way. Sometimes the claim is that he did not understand what he wrote. In this review, the argument is that he did not explain what Minkowski wrote 3 years later, and which Einstein did not even understand until about 5 years later.
Poincare and Einstein both published their big special relativity papers in 1905, with Poincare announcing his results first. Minkowski published his famous papers in 1907 and 1908, based on Poincare, not Einstein. Minkowski's 1908 paper emphasized non-Euclidean geometry, and that was what caused relativity to catch on among physicists.
Poincare's 1905 paper had space and time united in a 4-dimensional spacetime, the Lorentz group and algebra as a 4D symmetry, 4-vectors, and the covariance of Maxwell's equations. In short, he presented relativity as a non-Euclidean geometry. Einstein had none of this and did not even mention it in a relativity survey paper he wrote a year later.
Minkowski extended Poincare's geometry with Minkowski diagrams and worldlines. Again, Einstein had none of this, and admitted that he did not understand it.
The book credits Poincare with seeing the gravitational implications in 1905, but suggests that he might have had an epistemological confusion by failing to distinguish between the length of a measuring rod, and what he measures the length of a measuring rod to be. That is, distinguishing the actual length from the apparent length. Overbye's Einstein book also mentioned this distinction.
But you can check the papers yourself for this distinction. Einstein does not make it in his famous 1905 paper. Poincare does in his long 1905 paper:
Or this part which would be, so to speak, common to all the physical phenomena, would be only apparent, something which would be due to our methods of measurement.Poincare explains how this new view is different from Lorentz's, but Einstein never made any such claim.
Poincare's special relativity was vastly superior to Einstein's work in every aspect. This is detailed in my book, How Einstein Ruined Physics. Einstein is only credited with nonsensical arguments.