Democritus Junior to the Reader, page 19

[p. 19]

Bilem sæpe, jocum verstri movere tumultus.

Shilleto: Horace, Epistles, Book 1, Ep. xix, line 20.

I like the Oxford World's Classics translation here, although less literal than Loeb's: 

[O, you imitators, servile herd,] how often have your antics stirred me to anger, how often to laughter!

Horace, Satire and Epistles, trans. John Davie, ed. Robert Cowan (Oxford University Press, 2011)

Although ostensibly, the thread here is laughter & tears (picking up again on the Heraclites/Democritus+Diogenes opposition), the context of Horace's Epistle is assertion of the poet's originality to refute accusations of plagiarism leveled against his Epodes and Odes. Horace also refuses to compromise and promote his poetry with social appearances and dinners.

Democritus Junior to the Reader, page 16

[p. 16]
My intent is no otherwise to use his name than Mercurius Gallobelgicus, Mercurius Britannicus, use the name of Mercury, Democritus Christians,* etc.;...

* R.B.: Auth. Pet. Besseo, edit. Coloniæ, 1616.

Mercurius was a brand of newspapers, or more properly, news books. The idea was copied after the Mercure française, launched in France in 1611. First to appear was the Mercurius  Britannicus, which in 1625 became the first English news periodical to carry that title.
(from Jonathon Green, The Vulgar Tongue: Green's History of Slang, Oxford University Press, 2011, p. 100)

The dates would seem to suggest that this name was thrown in in a later edition (Burton was a notorious re-writer).

Democritus Junior to the Reader, page 15

[p. 15]
[Edit. Feb. 29:]...that so insolently intrudes upon this common theatre to the world's view...

Edward Bensly notes that this comes from the dedication to a posthumous edition of Julius Caesar Scaliger's (1484–1558) Epistolae et orationes (Leyden, 1600), which read: "...aliqua scriptorum........quæ nondum communem theatri huius lucem aspexerant."

Jules César Scaliger was an interesting character: a page and protégé of Emperor Maximilian, he studied art under Albrecht Dürer. He fought with great valor in the battle of Ravenna in 1512, which claimed the lives of his brother and father. He then abandoned military life to pursue studies in Bologna. In 1525 he moved to Agen, as the physician to the noble Rovere family. He spent the rest of his life in that town, his reputation as a scholar and physician growing. In the 1520s, he was:

a majestic looking man of some forty years of age who was to become renowned as one of the greatest scholars of the Renaissance. ... [S]o great became his fame in all branches of learning that it was for long considered he was the greatest scholar who had ever dwelt in France.

Vernon Hall, Jr., Life of Julius Caesar Scaliger, in Transactions of the American Philosophical Society, new ser., vol. 40, part 2, 1950.

J.C. Scaliger, as well as his son, Joseph Juste, who managed to surpass his father in erudition, will be seen again and again in Anatomy...

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