The above is an example of an “alphabetical whim”, a sort of word game. This particular sentence succeeds in using every letter of the alphabet, using only 48 letters in total. There are other kinds of alphabetical whim, like a lipogram, in which the goal is to omit a certain letter throughout the entire text.

“Alphabetical whims” is one of the first sections in “Gleanings from the harvest-fields of literature, science and art — A Melange of Excerpta, curious, humorous and instructive”, collated by C. C. Bombaugh, A.M, M.D. (1860). What a mouthful!

From the same chapter of the book:

The stanza subjoined is a specimen of both lipogrammatic and pangrammatic ingenuity, containing every letter of the alphabet except e. Those who remember that e is the most indispensable letter, being much more frequently used than any other,* will perceive the difficulty of such composition.

A jovial swain may rack his brain,
And tax his fancy’s might,
To quiz in rain, for ’tis most plain,
That what I say is right .

A generous preview of the book is on google books. Thanks very much to Alice for the heads up — great find!