I was looking for some history and authenticity to this myself. In that search I have found your question. I had found this version quoted by Heber J. Grant, a Mormon Apostle in 1919 from where he got it except for a possible author at the bottom, I could find nothing much more about either. Most of this may be familiar, but interestingly different. I hope this helps your search.

Let each man learn to know himself:
To gain that knowledge, let him labor,
Improve those failings in himself,
Which he condemns so in his neighbor.
How lenient our own faults we view,
And conscience' voice adeptly smother;
But oh! how harshly we review
The self-same errors in another!

And if you meet an erring one
Whose deeds are blamable or thoughtless,
Consider, ere you cast the stone,
If you yourself be pure and faultless.
Oh! list to that small voice within,
Whose whisperings oft make men confounded,
And trumpet not another's sin,
You'd blush deep if your own were sounded.

And in self-judgment, if you find
Your deeds to others are superior;
To you has Providence been kind,
As you should be to those inferior;
Example sheds a genial ray
Of light, which men are apt to borrow;
So first, improve yourself today,
And then improve your friends tomorrow.

Philip De La Mere.