James Burgh, like Joseph Priestley and Richard Price was a British (Burgh was of Scottish origin) dissenting divine, a Whig, and apparently a unitarian. And like Revs. Priestley and Price, Burgh tremendously influenced the American Founding.
Priestley, Price and Burgh were with Ben Franklin members of the Club of Honest Whigs. When writing Ezra Stiles about "Jesus of Nazareth," Franklin said, "I have, with most of the present dissenters in England some doubts as to his divinity." No doubt Franklin had his friends Priestley, Price and Burgh in mind as those "dissenters in England." They presented their dissent as "rational Christianity" or "unitarian Christianity."
John Adams sought to make Burgh's writings "more known and attended to in several parts of America," and stated Burgh's writings were "held in as high estimation by all my friends as they are by me."
Google books has uploaded originals of Burgh's 1766-67 "Crito," excerpts of which we will see below. Crito may also be where Jefferson lifted "wall of separation between church and state" from. (Yes Roger Williams said it first; but Jefferson likely learned the phrase from Burgh, not Williams.)
With that, let's look at Burgh discussing the dynamic of unitarians worshipping in Trinitarian churches in late 18th Century England. Note, this was published in 1767 and it was a crime to publicly deny the Trinity until 1813. That explains why the Unitarian Burgh didn't come out and deny the Trinity, but rather writes as though he didn't believe in the doctrine. His advice seems to be for unitarians to break away and start their own churches.
Beginning on page 240, Burgh writes:
We see some few among us do still make a point of attending solemnly a place of public worship - once in seven days. If there be any meaning in this practice (which they best know, who observe it) one would imagine it should be of some consequence, that people worship what they, at least, believe has a being.
It is notorious that many who statedly attend Athanasian worship do hold the Athanasian doctrine in abhorrence. (Many whole parishes constantly sit down whenever that celebrated creed is read.) And that those, who do not believe it, do constantly give this reason for their disbelief of it. That it appears to them flatly self-contradictory.
I am not here setting myself to enter into the question, whether the Athanasian doctrine be true or false. I am only observing, that many among us, who (with Newton, Clarke, Locke, Whitby, Emlyn, &c.) are satisfied, that it neither is, nor can be true, do constantly pay solemn worship to H--y, bl----d and gl-----s Tr---ty.
Quaeritur, therefore, the rationale of worshipping, or seeming to worship, what we are persuaded, has no existence? The papists have thought proper to put the Virgin Mary into the Tr---ty, and call her the complement, or completing of it. That is, the F----r, the S-n, the H--y Gh--t, and the Virgin Mary, the undivided mystical four, or three, which is the same (for in a mystery, three is the same as four, and four the same as one; finite the same as infinite; human the same as divine) the mystical four, I say, are the tr---ty, or rather quaternity, that is, four different beings, some infinite, some finite, some mortal, some immortal, are only three beings, and these three-four beings, are the One, indivisible, simple, unoriginated Spirit, the first cause and fountain of being.
No Protestant holds the Virgin Mary, who has these many ages been dead and rotten, to be any part of the immortal God. This is out of the question. But I would imagine, that to a person who denies the Athanasian doctrine, it should not appear a whit more absurd to put the Virgin Mary into the Tr---ty, or Godhead, than any other being whatever. All beings are equally different from and inferior to the Supreme; the S-n as much as the virgin; the virgin as much as a worm. For all beings, but the One Supreme only, are finite; and there must ever be an infinite distance between finite and infinite. The question, therefore, is, how any rational and pious person satisfies himself that it is lawful for him to attend constantly a species of worship, which he himself holds to be absurd; and this, while he has it in his power to withdraw himself from such worship, and give support and countenance to what is, according to his own notions, rational as to the Object worshipped.
Will it be said,We freely declare our sentiment. We do not dissemble. We publicly discountenance the Athanasian creed, by refusing to join in the reading of it. Whenever ecclesiastical authority insists on our joining in the recital of that famous creed, we will immediately turn our backs upon those places of worship, which support absurdity by power. Till then, we see no impropriety in attending on a species of worship not modified to our perfect approbation; as, perhaps none can be found altogether irreprehensible.
If this apology should be offered, let it be considered, how, on such principles, religious truth would ever have prevailed over error; and how a Protestant's constant and exclusive attendance, in a Protestant country, on popish worship, could be proved culpable; which yet would meet with the universal disapprobation of all conscientious persons. I will urge this no farther; though much more might be said. Only, I beg leave to add, that to those, who disbelieve the Athanasian doctrine, it should, in my opinion, be a much weightier cause of dissenting, that a certain establishment is formed upon what they look upon as absurd, and idolatrous, than upon usurped human power. And that, therefore, to the opposers of the tr--------n opinion, it ought to be very desirable to see religious societies formed professedly on unitarian principles, and denominated accordingly, rather than, by the general appellation of dissenters, which leaves the grand point, viz. What object of worship they hold, undetermined; as it is known, that some among them are tr--------n, some Unitarian, in principle, and in worship, and most too inexplicit in declaring themselves.
[Editorial edition -- Jonathan Rowe's blogging: I worked from the original edition and turned what looked like "f's" to "s's." In addition, I added some paragraph breaks and made some punctuation changes that made it look more modern. I also failed to reproduce all but one of Burgh's italics. I left the overwhelming majority of it alone, however. Click on the link and read pages 240-43 to see the originals from which I worked.]
You can smell the anti-Roman Catholic bigotry in Burgh's words (bigotry that prevailed in Protestant American as well). I don't think Burgh's sentiments accurately represented Catholic doctrine on Mary. However, it does explain John Adams' quotation:
“The Trinity was carried in a general council by one vote against a quaternity; the Virgin Mary lost an equality with the Father, Son, and Spirit only by a single suffrage.”
-- John Adams to Benjamin Rush, June 12, 1812.
The unitarian dissenters believed that doctrines such as original sin, trinity, incarnation, atonement, eternal damnation were corrupted inventions of ecclesiastical authorities, mainly the Roman Church. Therefore, truly reformed "Protestant Christianity" (with their preferred adjectives "liberal," "rational," and "unitarian") rejected all of those orthodox doctrines as fraudulent "Popish" inventions.