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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... as he said,* Primum si noluero, non respondebo, qui coacturus est? I am a free man born, and may choose whether I will tell; who can compel me?

* R.B.: Seneca, in ludo in mortem Claudii Caesaris.

The quote is from Seneca the Younger's Apocolocyntosis (divi) Claudii, or the Gourdification [or Pumpkinification] of (the divine) Cladius.

Seneca had good reason to hone his quill on this deified subject: while alive Cladius had banished the Stoic philosopher (c.4 BCE – 65 CE) to Corsica. Not a bad place to spend one's exile, if you ask me. The Greeks knew the island under the name of Cirné, derived from the Greek word for Sirens. Although the Sirens are more commonly placed on the southern tip of the adjacent island of Sardinia, their song may well have been heard at one point or another on the Corsican shores. Although, to Seneca, Corsica may have just been a pile of granite which from its highest point he could take in at one glance. 

The speech read by Nero on the apotheosis of Claudius was, Tacitus reports, composed by the new Emperor's mentor: 

“A public funeral was to come first. On the day of the funeral the emperor pronounced his predecessor’s praises. While he recounted the consulships and Triumphs of the dead man’s ancestors, he and his audience were serious. References to Claudius’ literary accomplishments too, and to the absence of disasters in the field during his reign, were favourably received. But when Nero began to talk of his stepfather’s foresight and wisdom, nobody could help laughing.”

Tacitus, The Annals of Imperial Rome, trans.  Michael Grant (Penguin, 1956), ch. 13

Now lost, the speech may have served as a basis for Apocolocyntosis. This authorship must have been taken as common knowledge in Burton's time: it is curious that in the opening lines of his preface, Democritus Junior does not see it necessary to cite Seneca by name, using only a pronoun. The quoted line appears in the first paragraph of the satire:

“What happened in heaven on the thirteenth of October in the new year, the beginning of a most prosperous age, is my task to record. I yield nothing to offense or favor. This is the utter truth. If someone asks how I know these things, well, if I don’t want to, I needn’t respond. Who is going to make me? . . . I shall say whatever comes into my mouth. Whoever required sworn witnesses for a historian?”

Apocolocyntosis, cited  in Daniel M. Hooley, Roman Satire  (Blackwell, 2007), p. 145.

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... if I be urged, I will as readily reply that as that Egyptian in Plutarch,* when a curious fellow would needs know what he had in his basket, Quum vides velatam, quid inquiris in rem absconditam? It was therefore covered, because he should not know what was in it.

* R.B.: Lib. de Curiositate.

The quote is from Plutarch's Moralia (vol. VI in the Loeb edition), from an essay entitled De Curiositate, or, as Loeb translates it, On Being a Busybody. The anecdote about the Egyptian falls in the middle of discussion of the bad habit of meddling. Busybodies are like backyard chickens, Plutarch says, that will go off scratching in a remote corner of the yard rather than eat what has been laid in front of them. And so 

“...busybodies, passing over topics and narratives which are in plain view and matters concerning which no one prevents their inquiring or is vexed if inquiry is made, Epick out the hidden and obscure troubles of every household. And yet it was surely a clever answer that the Egyptian gave to the man who asked him what he was carrying wrapped up: "That's why it is wrapped up." And why, if you please, are you inquisitive about what is concealed? If it were not something bad, it would not be concealed.”

Plutarch, "On Being a Busybody," in Moralia, vol. VI, trans. W.C. Helmbold (Loeb Classical Library, 1962),  516d–e, p. 481.  

[p. 15]
Seek not after that which is hid; if the contents please thee, "and be for thy use, suppose the Man on the Moon, or whom thou wilt, to be the author";*

* R.B.: Wecker.

This quote is from Johann Jacob (aka Jean-Jacques) Wecker (1528–1586), a Swiss physician and chemist, author, among other things, of a Book of Secrets, a compendium of natural science. I first supposed the quote to come from this work, which would aptly reflect Democritus Junior's secret identity (a secret finally revealed -- it's hard not to sign one's work! -- in The Conclusion of the Author to the Reader which is omitted from the NYRB edition). Jean Starobinski, in his L'Encre de la Mélancolie (Seuil, 2012, e-book p. 163 n.25) traces the source of the Wecker quote to Medical Syntaxes, p. 11 (although the reference still remains mysterious as he never cites the exact edition. The book is better known under the title, A compendious chyrurgerie...).

[p. 15]
... it hath been always an ordinary custom, as Gellius observes,* "for later writers and impostors to broach many absurd and insolent fictions under the name of so noble a philosopher as Democritus, to get themselves credit, and by that means the more to be respected,"...

* R.B.: Lib. 10, cap. 12.

This quote comes from the said book and chapter of Aulus Gellius's Attic Nights (vol. II in the Loeb Classical Library edition, 1960, p. 245). This is a wonderful passage, worthy of Thomas Browne's Pseudodoxia Epidemica, in which Gellius enumerates fabulous phenomena cited by Pliny the Elder in his Natural History on the authority of Democritus (Book XXXVIII, ch. 29). 

One thing that strikes me about all these quotations thus far is that Burton chooses text fragments that, to a reader seeking the "substance" or the "gist" of the text, would be the least memorable. In other words, if you were reading Gellius as a school assignment, this is probably not the sentence you would highlight. And yet, once it is quoted, it shines like a hidden gem you feel you have missed. (As Walter Benjamin well knew: this is the power of citations!)

[p. 15]
... Novo qui marmori ascribunt Praxitelen suo [who sign the name of Praxiteles on a new statue of their own].

This quote, as we learn from the A.R. Shilleto's 3-volume edition of Anatomy of Melancholy, comes from Phaedrus (that is, Gaius Julius Phaedrus, known as the Fabulist, 1st c. CE), Book V, Prolog [lines 5–6]. A ninth-century manuscript of Phaedrus's fables was discovered in France only in the fifteenth century, and was published by Pierre Pithou [Petrus Pithœus] in 1596 under the title Fabularum Aesopiarum libri quinque. This relatively recent edition must be what Burton was referring to.