My uncle Toby would never offer to answer this by any other kind of argument than that of whistling half a dozen bars of Lillibullero.—You must know it was the usual channel through which his passions got vent, when anything shocked or surprised him;—but especially when anything which he deemed very absurd was offered.[Laurence Sterne, The Life and Opinions of Tristram Shandy, i. xxi.]
As not one of our logical writers, nor any of the commentators upon them, that I remember, have thought proper to give a name to this particular species of argument, I here take the liberty to do it myself, for two reasons: first, That, in order to prevent all confusion in disputes, it may stand as much distinguished for ever, from every other species of argument—as the Argumentum ad Vericundiam, ex Absurdo, ex Fortiori, or any other argument whatsoever:—and, secondly, that it may be said, by my children's children, when my head is laid to rest,— that their learned grandfather's head had been busied to as much purpose once as other people's;—that he had invented a name,—and generously thrown it into the TREASURY of the Ars Logica, for one of the most unanswerable arguments in the whole science. And, if the end of disputation is more to silence than convince,—they may add, if they please, one of the best arguments too.
I do, therefore, by these presents, strictly order and command, That it be known and distinguished by the name and title of the Argumentum Fistulatorium, and no other;—and that it rank hereafter with the Argumentum Baculinum and the Argumentum ad Crumenam, and for ever hereafter be treated of in the same chapter.
As for the Argumentum Tripodium, which is never used but by the woman against the man;—and the Argumentum ad Rem, which contrariwise, is made use of by the man only against the woman,—as these two are enough in conscience for one lecture —and, moreover, as the one is the best answer to the other—let them likewise be kept apart, and be treated of in a place by themselves.
Argumentum fistulatorium means, roughly, 'argument by piping'.
Ad verecundiam, ex absurdo, and ex Fortiori are, respectively, argument from authority, from the absurdity of the position, and from the truth of a stronger conclusion. They are legitimate, and would have been fairly standard argument classifications.
All the rest are jokes, but argumentum baculinum was already a very old joke by Sterne's time; it occurs when you resolve an argument by beating your opponent with a club or a stick. Sterne might have Henry Fielding's joke essay on argumentum baculinum in mind here, and another possible background source is Spectator 239 -- although, again, it is an old joke. Remarkably, later lists of fallacies (which are descendants of the argument-classifications Sterne is mocking) by humorless authors will often seriously list argumentum ad baculum as an actual fallacy, namely, one in which you try to end a conversation by threatening someone.*
Argumentum ad Crumenam would be appeal to the purse; it is also a joke argument-form that was later taken to be a serious one, and is later toaken to involve claiming that your view is better because you are richer. [In fact, Sterne's use of the phrase is to mean arguing by betting that something is true.] Fielding's essay makes a similar joke, but calls it argumentum pecuniarum; the Spectator does as well, without labeling it. I don't know of any source earlier than Sterne who uses this phrase, so it's possible that Sterne invented it on the spot.
Argumentum Tripodium and Argumentum ad Rem are bawdy jokes, involving references to genitalia. The second one is rather clever; literally it means something like, 'argument to the point or purpose', i.e., a relevant argument, and so he manages simultaneously to be bawdy and make the age-old and otherwise tired joke about how only men stick to the point.
* Sterne, however, is not listing fallacies; it was Whately in the nineteenth century who adapted earlier lists of argument-forms to discussion of fallacies. In Sterne's day ad verecundiam et al. are simply treated as kinds of arguments, with no implication that they are bad arguments. (Whately recognized this; later textbook authors following him did not.) If we look at the Logic of Isaac Watts, which is probably the best 18th-century logic textbook written in English, he takes argumentum ad verecundiam to be a logical common-place or topos -- i.e., one of the ways you would come up with the middle terms to build an argument is by drawing them from the sentiments of "some wise, good, or great man". The whole point of Sterne's joke is that there is no suggestion of fallacy at all.