First, the “still ray”. The theory that sight works by rays that are emitted from the eye was held by some ancient philosophers, for example:
For [the gods] caused the pure fire within us, which is akin to that of day, to flow through the eyes in a smooth and dense stream; and they compressed the whole substance, and especially the center, of the eyes, so that they occluded all other fire that was coarser and allowed only this pure kind of fire to filter through. So whenever the stream of vision is surrounded by midday light, it flows out like unto like, and coalescing therewith it forms one kindred substance along the path of the eyes’ vision, wheresoever the fire which streams from within collides with an obstructing object without […] and brings about that sensation which we now term “seeing”.
Plato (c. 360 BCE). Timaeus 45. Translated by W. R. M. Lamb (1925).
Second, “pure reason” is a reference to the philosophy of Immanuel Kant, in whose Critique of Pure Reason (1781) the phrase is used in this sense:
reason, n. 5.c. Philosophy. A faculty transcending understanding, by which first principles are grasped a priori.
Oxford English Dictionary
So these lines from Aurora Leigh give us a metaphor in which Kant’s “pure reason” (glossed as “noble trust” or “instinct”) strikes out from you towards the truth, as fast and straight as Plato’s “stream of vision”, while “bare inference” (that is, practical reasoning and deduction) lags behind, so that you cannot explain logically how you came to your conclusion (“how, you cannot tell”).
Third, “elimination” and “analysis” are two kinds of operation in logical reasoning. “Analysis” is the philosophical technique of breaking a concept down into its components to reveal its logical structure and assumptions, and “elimination” is a principle of inductive logic corresponding roughly to the modern idea of rejection of a hypothesis.
Phenomena may be invariably concomitant and therefore be known to have a fixed connection, as antecedent and consequent, but the order of the sequence may not at once appear. Now, inasmuch as the causal influence acts through the antecedent to the production of the consequent, it follows that a consequent can be made to disappear, or be modified only by the elimination or modification of the antecedent.
Henry P. Tappan (1844). Elements of Logic: Together with an Introductory View of Philosophy in General and a Preliminary View of the Reason, p. 287. New York: Putnam.
So the parenthetical “did you eliminate, / That such as you, indeed, should analyse?” is a guess as to how you were unable to reach the truth through “bare inference”: that is, you made a mistake in your reasoning by wrongly ruling out a hypothesis.
How could Browning have come by this sense of the word “eliminate”? (It seems doubtful that she, or anyone, read Tappan’s turgid prose.) Perhaps she got it from John Stuart Mill:
The two methods which we have now stated have many features of resemblance, but there are also many distinctions between them. Both are methods of elimination. This term […] is well suited to express the operation […] which has been understood since the time of Bacon to be the foundation of experimental inquiry: namely, the successive exclusion of the various circumstances which are found to accompany a phenomenon in a given instance, in order to ascertain what are those among them which can be absent consistently with the existence of the phenomenon.
John Stuart Mill (1843). A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation, volume I, p. 456. London: John W. Parker.
We know that Browning read at least some works of Mill’s because she mentions him in a letter:
Robert and I shall like much to see anything of John Mill’s on the subject of Socialism or any other.
Elizabeth Barrett Browning (c. 1850). Letter to Isabella Blagden. In Frederic G. Kenyon, ed. (1898). The Letters of Elizabeth Barrett Browning, volume I, p. 467. London: Macmillan.