“I am enclosing two tickets to the first night of my new play; bring a friend… if you have one.""Cannot possibly attend first night; will attend second, if there is one."”

Version given in Irrepressible Churchill: A Treasury of Winston Churchill’s Wit by Kay Halle, 1966
Apocryphal, originally featured Noël Coward and Randolph Churchill (Winston’s son); attested 1946 (columnist Walter Winchell, attributed to anonymous United Press journalist in London). Originally only featured first half about lack of friend; second half (retort about lack of second performance) attested 1948, as was replacement of personages by George Bernard Shaw and Winston Churchill. Specific plays added in later variants, ranging from Man and Superman (1903) to Saint Joan (1923), and appeared in biographies and quote collections from the 1960s.
The quote is presumably apocryphal due to earliest attestations being too different, less famous personages (easily replaced by more famous ones), the quotation becoming more elaborate in later versions, the 20+ year gap between putative utterance and first attestation, and the approximately 50 year gap between putative utterance and appearance in reference works, all as undocumented hearsay.
Detailed discussion at “ Here are Two Tickets for the Opening of My Play. Bring a Friend—If You Have One http://quoteinvestigator.com/2012/03/25/two-tickets-shaw/”, Garson O’Toole, Quote Investigator http://quoteinvestigator.com/, March 25, 2012.
Misattributed

Version given in Irrepressible Churchill: A Treasury of Winston Churchill’s Wit by Kay Halle, 1966
Apocryphal, from 1946. See discussion at Winston Churchill#Misattributed, and detailed discussion at “ Here are Two Tickets for the Opening of My Play. Bring a Friend—If You Have One http://quoteinvestigator.com/2012/03/25/two-tickets-shaw/”, Garson O’Toole, Quote Investigator http://quoteinvestigator.com/, (March 25, 2012)
Misattributed

§ 5
From Lives and Opinions of the Eminent Philosophers by Diogenes Laërtius

Source: 1930s, On my Painting (1938), p. 19

New Scientist interview (2004)
Context: There is nothing more to this than a simple iterative formula. It is so simple that most children can program their home computers to produce the Mandelbrot set. … Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it. It was as if somehow I had seen it before. Of course I hadn't. No one had seen it. No one had described it. The fact that a certain aspect of its mathematical nature remains mysterious, despite hundreds of brilliant people working on it, is the icing on the cake to me.

The Confession (c. 452?)

Source: The Prince (1513), Ch. 18
Variant translations of portions of this passage:
Every one admits how praiseworthy it is in a prince to keep faith, and to live with integrity and not with craft. Nevertheless our experience has been that those princes who have done great things have held good faith of little account, and have known how to circumvent the intellect of men by craft, and in the end have overcome those who have relied on their word.
Ch. 18. Concerning the Way in which Princes should keep Faith (as translated by W. K. Marriott)
A prince being thus obliged to know well how to act as a beast must imitate the fox and the lion, for the lion cannot protect himself from traps, and the fox cannot defend himself from wolves. One must therefore be a fox to recognize traps, and a lion to frighten wolves.
You must know there are two ways of contesting, the one by the law, the other by force; the first method is proper to men, the second to beasts; but because the first is frequently not sufficient, it is necessary to have recourse to the second.
Context: How laudable it is for a prince to keep good faith and live with integrity, and not with astuteness, every one knows. Still the experience of our times shows those princes to have done great things who have had little regard for good faith, and have been able by astuteness to confuse men's brains, and who have ultimately overcome those who have made loyalty their foundation. You must know, then, that there are two methods of fighting, the one by law, the other by force: the first method is that of men, the second of beasts; but as the first method is often insufficient, one must have recourse to the second. It is therefore necessary to know well how to use both the beast and the man. This was covertly taught to princes by ancient writers, who relate how Achilles and many others of those princes were given to Chiron the centaur to be brought up, who kept them under his discipline; this system of having for teacher one who was half beast and half man is meant to indicate that a prince must know how to use both natures, and that the one without the other is not durable. A prince being thus obliged to know well how to act as a beast must imitate the fox and the lion, for the lion cannot protect himself from snares, and the fox cannot defend himself from wolves. One must therefore be a fox to recognise snares, and a lion to frighten wolves. Those that wish to be only lions do not understand this. Therefore, a prudent ruler ought not to keep faith when by so doing it would be against his interest, and when the reasons which made him bind himself no longer exist. If men were all good, this precept would not be a good one; but as they are bad, and would not observe their faith with you, so you are not bound to keep faith with them.... those that have been best able to imitate the fox have succeeded best. But it is necessary to be able to disguise this character well, and to be a great feigner and dissembler.