E. Cobham Brewer 1810–1897. Dictionary of Phrase and Fable. 1898.
Hitch.
There is some hitch. Some impediment. A horse is said to have a hitch in his gait when he is lame. (Welsh, hecian, to halt or limp.)
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To hitch. To get on smoothly; to fit in consistently: as, “You and I hitch on well together;” “These two accounts do not hitch in with each other.” A lame horse goes about jumping, and to jump together is to be in accord. So the two meanings apparently contradictory hitch together. Compare prevent, meaning to aid and to resist.
