Steakanator: The first two games in this mini match showed that neither player is very good at defending. Game 1 showed that Löwenthal's defenses often created weaknesses elsewhere(13. Bf4 defending h3 but losing d4, 26. Qe7 overloading his own Queen), and game 2 showed that Williams's defenses were weaknesses in themselves(17. f5, 23. g4, both making the pawns quite vulnerable). So it seems like, going into game 3, whichever player could first apply pressure would come out on top.
Below I've included the notes from Staunton in the TB(in quotes, square brackets for algebraic notation substituting descriptive), as well as some personal observations.
5... Bxc3+ is an idea Williams does often: trade the Bishop pair for an early doubled pawn. See game 1 of his mini match vs Staunton.
6... c5 "This can be safely played, since, if White take it, Black moves his [Qa5], and wins a Pawn in return, after having broken up the central phalanx of his adversary." - It's generally a bad idea to triple your pawns.
7... Qe7 is a move that doesn't mean much at the moment, but makes Löwenthal's life easier in a little bit.
9. a4 "Threatening [dxc5] and then play [Ba3]."
Suppose 9... Nc6
10. dxc5 Qxc5 11. Ba3 would win Löwenthal the exchange. If some other 9th move from Williams, then c6 and Ba3 would still be an option for Löwenthal. Strike one for Qe7.
16. d5 "This is all natural enough, and apparently the best play. If [16. Ng5] then [16... e5 17. f4 h6] and White does not seem to gain anything." - The idea of Ng5 will come up a couple times this game, and almost every time it seems to be a mediocre move. More on that later.
16... Bd7 - Note that the Queen on e7 has caused the e6 pawn to be pinned, and taken on the next move. Strike two for Qe7, though sadly it never gets strike three.
17. dxe6 - You can call this strike three if you really want.
18. Ng5 "If this Kt. could afterwards be played advantageously to [f7], then the present move is defensible; if, as Mr. Löwenthal appears to have thought, by retreating him, he could not, then was the move most blameable, for it involved no less, perhaps, than the loss of the match!" - To summarize, since Staunton's English can be a pain to read: Ng5 was only reasonable if Nf7 followed, otherwise this simply costs too much time.
19. Nh3 "Having once played this Kt. forward, I think White would have done better to have gone with him to [f7]. That might have cost the advanced Pawn, but retreating the Kt. gave an advantage, in time, of more consequence than a Pawn." - Yup.
19... Ng4 is the start of a long journey to h4, via Ne5, Ng6, Nh4. After Williams brings his Queen to the g file, Löwenthal will hopefully have to play g3, allowing Williams's Bishop to become extremely annoying.
20. Qe2 "I should have preferred playing [f4]." - Agreed.
22. g3 "To prevent the threatened move of [Ng4], but it was injurious to White, on account of the range it gave the adverse Bishop."
23. Qh5 "Why this move? Why give up so all-important a Pawn? Surely it was to be defended without much difficulty." - A lot of gambit theory back in the 1800's involved giving back the pawn as soon as you felt like the pressure was starting to build. Perhaps Löwenthal tried to apply that logic here? I'm not entirely sure.
25... Re8 "Actually threatening mate in two moves."
26... Qe3 "An excellent move. If White had taken the Kt., next move the Black Q. would have been played to [f3], etc." - Suppose 27. Qxg5 Qf3 - You need to block or displace the Bishop in order to prevent mate. One idea is Qxe8+, but for those of us who don't like sacrificing our Queen:
28. Ne4 Rxe4 29. Rf1 Qxc3 and the Bishop has nowhere to go. If 30. Bc1 then 30... Re2 and mate follows in at most 4 moves.
28. Rf1 - White gets mated if the Rook ever leaves the first rank, due to that lovely Bishop on c6. Unfortunate.
29. Kf2 - White can't win back the piece with 29. Qxg5 for similar reasons as previously stated. After 29... Qf3 30. Ne4 Qxe4 31. Qxe4 Rxe4. Although there are no immediate mating threats, a4 and c4 are hanging, and Williams will no doubt waste little time snatching them up.
37... Nxc4 - I never considered Williams to be an exciting player, but this is as flashy an ending as he could possibly come up with.
An unfortunate result in Löwenthal's otherwise excellent career, but a very solid performance by Williams. A shame that he had to catch cholera before international chess began to truly pick up - I would've liked to see more matches against people other than Harrwitz(not to discredit the German master, but variety is the spice of life, of course).