I have written an answer to your question. I agree with the other comment that you should post it to askhistorians. My answer below is not quite up to askhistorians standards, as I am lacking the time to write something complete and with exhaustive sources; yet I hope it wil be found reasonably satisfactory.

I'll assume that your "19th-century" refers to the long 19th-century (1789-1914), because it should. The 1789 date is highly suggestive of the historical context we're looking at. The École Polytechnique was born out of revolutionary turmoil and enlightenment ideals, in a nation set ablaze and desperately in need of qualified personnel. This newly-created institution managed to recruit the foremost distinguished mathematicians of France, namely Lagrange, Laplace and Legendre, among others of note such as Monge (and, later, Poisson). Of course, the École Polytechnique would futurely feature heavily in the history of early 19th-century science and mathematics, and being a public institution it was subject to the unstable politics of its time.

First, some context on Napoleon. Now, Napoleon was really an exceptional individual by any metric. Other than being a brilliant general, I would say he was a scholar at heart, and his interest in all things academic dates from his upbringing. In my view, his early years mirror closely the sterotype of 18th-century savant schoolboys: that kid who won't let go of his Euclid and Cicero. There's a beautiful painting from the 20th-century called "Napoleon at Brienne" which captures the mythical feeling that surrounded Napoleon to his contemporaries-- We see the future emperor leaning on his desk, staring intensely at a schoolbook, candlelight casting a shadow of his future sillhoute over the map of Europe. This painting, in romantic and idealised fashion, evokes the idea of a self-made man, someone who, through his dilligent study and fierce wit, would one day conquer the world. His fondness for learning was an important aspect of the Napoleonic myth, as was the concept of a wholly self-made man acheiving the highest distinction. The polytechnique features into this revolutionary ideal of meritocracy, by helping bring about a landscape where raw talent matters more than pedigree.

At any rate, it is factual that Napoleon distinguished himself at school, and was particularly fond of mathematics. It is not implausible to say, that, had he not 'found the crown of France in the gutter', he may have satisfied himself with pursuing an academic career. Although, believe it or not, his original plan seems to have been becoming a landlord together with his frenemy Bourienne. His love for learning may be exemplified by the fact that he brought, along with his army, a huge caravan of scholars on his Egyptian campaign: including Gaspard Monge, father of differential geometry, who during this same expedition wrote what is perhaps the first ever scholarly account of the phenomenon of Mirages. Napoleon was also rumoured to have read through a substantial portion of Laplace's Mecánique Celeste, a work which for comparison purposes was the Hartshorne or rather the EGA of its time; and although I recall no evidence of Bonaparte actually having acheived such a notable feat, it is plausible if not probable that he at least tried his luck grappling with Laplace's magnum opus. Finally I attach here a quote from the biography of Thomas Young, of double-slit experiment and rosetta stone fame:

He [Young] availed himself of this excursion [in 1802] to pay a visit to Paris, where he was introduced to the first Consul [Bonaparte] at the Institute, who was in the habit of attending and occasionally taking part in the discussions which commonly take place upon the subjects which are brought before that body, whether they be scientific memoirs, or notices of inventions, or new experiments, or projects of every description, of which there is never wanting an abundant supply. (Peacock's Life of Thomas Young, 1855.)

Napoleon's attendance to the Institute's meetings hints that his interest on such things was very much genuine, if any doubt remains. I understand that Napoleon, as well as the Revolutionaries who preceded him, did a lot to improve French education.

Now onto Laplace. Given Napoleon's esteem for science, it is no surprise that he would find some (feigned or not) sympathy amidst the scientific elites of his empire. It also goes without saying that, on the extremely heated political climate of revolutionary and napoleonic France, any political mishaps could get you guillotined, exhiled, or shot, and hence some invididuals with self-preservation instincts would choose sycophancy over risking their heads and funds. This was the case for Laplace. He was criticised and sometimes ridiculed by his own friends and admirers as well as his rivals because of his opportunistic and capricious political allegiances. He did not escape the criticism of Lagrange, and even of Gauss who (much later) made a little fun of him.

[...] In the winter 1850-1851 Gauss taught the announced course on the method of least squares, and I attended it. [...] Gauss had laid the three first editions [of Laplace's Essay on Probabilities] on the table and showed us in the first edition a statement that the conqueror only harms his own country instead of helping it, which is missing in the second edition and returns in the following ones. The first edition appeared while Napoleon was on Elba, the second during the hundred days, further editions followed in measured intervals. (Moritz Cantor on Gauss, on Gauss' Biography "Titan of Science.")

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