“Two hundred years ago there were two distinct and separate streams of human life running through this country. They stood at opposite extremes of ethnological classification: all black on the one side, all white on the other. Now, between these two extremes, an intermediate race has arisen, which is neither white nor black, neither Caucasian nor Ethiopian, and this intermediate race is constantly increasing. I know it is said that marital alliance between these races is unnatural, abhorrent and impossible; but exclamations of this kind only shake the air. They prove nothing against a stubborn fact like that which confronts us daily and which is open to the observation of all. If this blending of the two races were impossible we should not have at least one-fourth of our colored population composed of persons of mixed blood, ranging all the way from a dark-brown color to the point where there is no visible admixture. Besides, it is obvious to common sense that there is no need of the passage of laws, or the adoption of other devices, to prevent what is in itself impossible.”

Fourth Lincoln-Douglas Debate (Charleston, 18 September 1858)
1850s, Lincoln–Douglas debates (1858)
Context: While I was at the hotel to-day, an elderly gentleman called upon me to know whether I was really in favor of producing perfect equality between the negroes and white people. While I had not proposed to myself on this occasion to say much on that subject, yet as the question was asked me, I thought I would occupy perhaps five minutes in saying something in regard to it. I will say then that I am not, nor ever have been, in favor of bringing about in any way the social and political equality of the white and black races, that I am not, nor ever have been, in favor of making voters or jurors of negroes, nor of qualifying them to hold office, nor to intermarry with white people; and I will say in addition to this that there is a physical difference between the white and black races which I believe will forever forbid the two races living together on terms of social and political equality. And inasmuch as they cannot so live, while they do remain together there must be the position of superior and inferior, and I as much as any other man am in favor of having the superior position assigned to the white race. I say upon this occasion I do not perceive that because the white man is to have the superior position the negro should be denied every thing. I do not understand that because I do not want a negro woman for a slave I must necessarily want her for a wife. My understanding is that I can just let her alone. I am now in my fiftieth year, and I certainly never had a black woman for either a slave or a wife. So it seems to me quite possible for us to get along without making either slaves or wives of negroes. I will add to this that I have never seen, to my knowledge, a man, woman, or child who was in favor of producing a perfect equality, social and political, between negroes and white men... I have never had the least apprehension that I or my friends would marry negroes if there was no law to keep them from it, but as Judge Douglas and his friends seem to be in great apprehension that they might, if there were no law to keep them from it, I give him the most solemn pledge that I will to the very last stand by the law of this State, which forbids the marrying of white people with negroes.

1860s, Our Composite Nationality (1869)

Source: 1930s, On my Painting (1938), pp. 13-14

This is from a fictional speech by Lincoln which occurs in The Clansman : An Historical Romance of the Ku Klux Klan (1905) by Thomas Dixon, Jr.. On some sites this has been declared to be something Lincoln said "soon after signing" the Emancipation Proclamation, but without any date or other indications of to whom it was stated, and there are no actual historical records of Lincoln ever saying this.
Misattributed

The Rhythm of Time
Context: It is found in every light of hope,
It knows no bounds nor space
It has risen in red and black and white,
It is there in every race. It lies in the hearts of heroes dead,
It screams in tyrants’ eyes,
It has reached the peak of mountains high,
It comes searing ‘cross the skies. It lights the dark of this prison cell,
It thunders forth its might,
It is "the undauntable thought", my friend,
That thought that says "I'm right!"

1860s, Our Composite Nationality (1869)

Source: Black Theology and Black Power (1969), p. 14-16

Interview http://www.malcolm-x.org/docs/int_pbert.htm for the Pierre Berton Show. Toronto, Ontario, (19 January 1965)
Attributed
Context: I believe in recognizing every human being as a human being, neither white, black, brown nor red. When you are dealing with humanity as one family, there's no question of integration or intermarriage. It's just one human being marrying another human being, or one human being living around and with another human being.

1880s, The Future of the Colored Race (1886)