Possible origin for the mad scientist.

The phrase and perhaps the concept, although images of stupid and errant scientists go back well into the 17th century.

Christopher Smart (1722-1771), English poet, wrote a lot of religious verse but was primarily known in his lifetime for his satirical work. In his "Temple of Dullness" (1745) he wrote the following (keep in mind that "Mathesis" is Greek for "science"):

Next to her, mad Mathesis; her feet all bare,
Ungirt, untrimm'd, with loose neglected hair;
No foreign object can her thoughts disjoint;
Reclin'd she sits, and ponders o'er a point
Before her, lo! inscrib'd upon the ground
Strange diagrams th'astonish'd sight confound,
Right lines and curves, with figures square and round.
With these the monster, arrogant and vain,
Boasts that she can all mysteries explain,
And treats the sacred sisters with disdain,
She, when great Newton sought his kindred skies,
Sprung high in air, and strove with him to rise
In vain--the mathematic mob restrains
Her flight, indignant, and on earth detains;
E'er since the captive wretch her brain employs
On trifling trinkets, and on gewgaw toys.

In this poem you've got: the phrase "mad Mathesis" (i.e., "mad science"); a scientist with "loose neglected hair" (and it's de rigueur for mad scientists to have unkempt, wild hair); a scientist "arrogant and vain" (although numerous previous poems had assigned anti-religious scientists these qualities) (Smart is attacking anti-religious scientists here, not scientists as a whole--note his invocation of "great Newton"); and a scientist creating "trifling trinkets" and "gewgaw toys" (not a lot of distance from a mad scientist creating those to creating a death ray).

Not that there was a rush of mad scientist figures in poetry or prose following "The Temple of Dullness," of course, and I'm now wondering if the many French and German novels of the time had something similar to "mad Mathesis." But the phrase "mad scientist" was in use in the 1890s, well before the classic pulp mad scientist, and Smart seems as likely a source for the image and phrase as any other.