Gifted to Beethoven by Maelzel, as sourced from the following.

Antique-Metronomes.com, "The Original Beethoven Metronome c1815"

We are pleased to have in our possession a totally original ( including the original upper & lower pendulum weight's ) and fully working example of a 12 inch French manufactured metronome. identical to the alleged metronome gifted to Beethoven in 1815 by Johann Maelzel, the inventor of the mechanical metronome. We also have a slightly later 9 inch version of the 1818 London manufactured metronome he received after his initial metronome had 'broken' or was dropped.

Smithsonian Magazine SmartNews - "Was Beethoven’s Metronome Wrong?"

When great musicians play the classics, they often like to recreate the exact feel of a piece of music. But when playing Beethoven, many musicians completely disregard the tempo markings on his original sheet music. Sixty-six out of 135 of them have been regarded as “absurdly fast and thus possibly wrong,” writes Sture Forsen in a new paper published in the American Mathematical Society. Now, mathematic and musical detectives have discovered that perhaps Beethoven’s tempo was so strange for a simple reason—his metronome was broken.

It’s worth checking out the entire paper, but the premise of their work is to figure out the “possible mathematical explanations for the “curious” tempo markings.” Here’s the story they tell.

Beethoven got his metronome from a man named Johann Nepomuk Mälzel, who was something of a mechanical wizard. He made little musical automatons, tiny robots that could play music that the public very much enjoyed. Beethoven and Mälzel connected when Beethoven was looking for help in dealing with his hearing loss, and Mälzel made him several ear trumpets. The two most likely also discussed the issue of timekeeping, as Mälzel had been working on metronomes.

Mälzel went on to invent more automatons, like the famous Mechanical Turk who played chess, but he continued his work on metronomes, as well. In 1812 he heard about an invention by Dietrich Winkel, who had created a double pendulum device. Mälzel hurried to Amsterdam to meet Winkel and realized that his rival had a metronome far superior to his own. He tried to buy the invention, but Winkel refused. So Mälzel simply made a copy and patented it in London, Paris and Vienna.

Around the same time, Mälzel was trying to swindle Beethoven. There was a later debate between them over who owned the rights to a piece of music Mälzel suggested and Beethoven composed. They went to court over it. Historians think that, around 1815, Mälzel might have sent Beethoven a metronome as a sign of forgiveness and peace, and by 1817 Beethoven certainly had one of Mälzel’s devices—the one he used to write all the crazily timed pieces.

Fast forward to today, and music historian Peter Stadlen has actually located Beethoven’s metronome. But the heavy weight was gone so he couldn’t test its operation. Which brings us to the mathematicians on this paper. They looked at the mechanical properties of the double pendulum metronome, to figure out which parts alter the device’s performance the most.

What they found, in looking into the history of metronomes, the mathematics of their behavior and the music of Beethoven is that the master’s metronome was probably not working so well:

How could Beethoven not note the occasional odd behavior of his metronome? A thorough account by Peter Stadlen gives the impression that the master was not entirely comfortable with the new device, most especially in the process of converting from beat frequencies to actual tempi markings for half-notes, quarter-notes, etc. Obviously, it would be very helpful if we knew more about the actual design of his metronome(s). We suggest that one or more of the devices could have been damaged, perhaps accidentally during one of his well-known violent temper tantrums. Whatever the case, our mathematical analysis shows that a damaged double pendulum metronome could indeed yield tempi consistent with Beethoven’s markings.