The Mathers, Cotton and Increase, are particularly vexed by something called "spectral evidence." What in the world is that?
It's when someone sees the "spectre," or apparition, of another person, doing some evil act. The actual person can be miles away at the time, but witches were believed to lend the devil their bodily form, and further, people believed that the devil could not assume their appearances unless they gave permission.
This is how people could be convicted of acts of witchcraft despite being able to prove that they were nowhere near the scene of the acts. Their "spectres" were there, and that was enough. Sometimes these spectres showed up in other people's dreams, and this was also admissable as evidence. A huge number of the accused in Salem were imprisoned on the basis of spectral evidence, against which there was no defense. If someone says they saw your apparition doing something evil, you would not be able to prove that it wasn't.
Increase Mather was so deeply troubled by the abuses inherent in this, that he wrote a letter urging judges to admit spectral evidence only as support for stronger, empirical evidence. By itself, Mather decreed, the evidence was worthless. The devil could certainly assume the form of an innocent person as well as a guilty one, and therefore innocent Christians would suffer.
The end of spectral evidence put an end to the witch trials. It seems that actual, physical proof wasn't thick on the ground, and in the years following, judges like Samuel Sewell would repent of their use of it.