Mr Casby lived in a street in the Gray’s Inn Road, which had set off from that thoroughfare with the intention of running at one heat down into the valley.
Little Dorrit, chapter 13
What does the term "one heat down" mean?
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Per World Wide Words, heat once had a meaning of
a single burst of intense physical activity of any sort, often in the phrase at a heat, at one go, in one continuous operation
(World Wide Words cites the Oxford English Dictionary for this, but I do not have an unabridged OED handy.)
Note that the particular phrase Dickens uses—at a heat—is specifically called out here. And World Wide Words goes on to note that this meaning can be found “as late as 1855,” citing one John Lothrop Motley. Little Dorritt was published serially from 1855-57, so at the same time.
So this is the sense the word has in the quoted sentence: the point of the street was to be able to run down from the thoroughfare and into the valley in one burst of effort, rather than a prolonged journey that might possibly involve stops along the way. In other words, a direct route, where the route otherwise would be meandering, with larger detours.
Put another way, the quoted sentence means
Mr Casby lived in a street […] which [led directly down into the valley from Gray’s Inn Road].
Note that this meaning of heat is no longer used, hence “as late as 1855.” However, modern English does bear one vestige of that meaning: the modern use of heat for a single race within a tournament (where the earlier heats might be used as qualifiers for the finals, or competitors’ final standing is determined by some kind of averaging across their various heats, or what have you). According to World Wide Words, this sense came about by combining the “one continuous operation” sense of heat with another usage of heat, specifically in horse racing, to mean what we would call a warm-up today.