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We must first explain clearly the conception which is known to astronomers by the name of parallax; for it is by parallax that the distance of the sun, or, indeed, the distance of any other celestial body, must be determined. Let us take a simple illustration. Stand near a window whence you can look at buildings, or the trees, the clouds, or any distant objects. Place on the glass a thin strip of paper vertically in the middle of one of the panes. Close the right eye, and note with the left eye the position of the strip of paper relatively to the objects in the background. Then, while still remaining in the same position, close the left eye and again observe the position of the strip of paper with the right eye. You will find that the position of the paper on the background has changed. (181-82)

A recursive loop later in Lestrygonians indicates that Bloom may have grasped the idea here: different lines of sight will afford slightly different contextual views of an object. Standing in front of Yeates & Son, a shop that sold precision optics, he recalls that the timeball on the quayside corner of the Ballast Office was linked not to Dunsink but to the Greenwich Observatory southeast of London. Dunsink time was displayed on the clockface on the building's Westmoreland Street façade: "Now that I come to think of it, that ball falls at Greenwich time. It's the clock is worked by an electric wire from Dunsink. Must go out there some first Saturday of the month. If I could get an introduction to professor Joly or learn up something about his family. That would do to: man always feels complimented. Flattery where least expected. Nobleman proud to be descended from some king's mistress. His foremother. Lay it on with a trowel. Cap in hand goes through the land. Not go in and blurt out what you know you're not to: what's parallax? Show this gentleman the door. / Ah. / His hand fell again to his side. / Never know anything about it. Waste of time." More in a moment on Bloom's raised hand, which strongly suggests that he once tried Sir Robert's experiment.

In 1904 the Dunsink observatory was indeed open to the public on the first Saturday of every month, and another eminent Irish astronomer, Charles Joly, was the director. Bloom supposes that if he showed up for one of these open houses, flattery might induce Joly to disclose a mystery that learned scientists keep hidden from ordinary people. In addition to the divide between amateurs and professionals, social class plays a part in Bloom's feeling of being excluded from those in the know. In Circe he pretentiously inserts himself into the Anglo-Irish ruling elite: "I was just chatting this afternoon at the viceregal lodge to my old pals, sir Robert and lady Ball, astronomer royal at the levee. Sir Bob, I said..." Ball was married to a woman named Frances Elizabeth Steele.

But Bloom's scheming is comically, poignantly unnecessary. In a blog published by the Atlantic on 2 February 2012, "Joyce and the Internet: What Leopold Bloom Didn't Know," Alan Jacobs observes that specialized scientific knowledge was indeed much harder to come by in Bloom's time than it is now, when popular science writing abounds and everything under the sun can be found on the internet, but Bloom already has all he needs: a copy of Ball's book, and a prime example of parallax in the two discordant times.

Greenwich time was approximately 25 minutes ahead of Dunsink time. When the highly accurate timeball fell at 1 PM,  the highly accurate clock on the same building showed 12:35. Why? Because astronomers at the two observatories, sighting the sun at the same moment, saw it standing in two slightly different places in the sky. The novel's way of introducing these time differences suggests that Joyce did not include the disparity as a mere scientific curiosity, but rather to indicate that Bloom understands parallax better than he thinks he does. Why have parallax occur to him just after reflecting on the timeball's link to Dunsink, and then again just after remembering that it is linked to Greenwich, if not to imply some latent awareness of its relevance to the different times?

This inference from Joyce's textual juxtapositions is strengthened by the remarkable fact that Lestrygonians offers a second instance of Bloom almost glimpsing parallax, now in a more directly visual way. And here we return to the raised hand. In a personal communication,Vincent Van Wyk has suggested to me that the little experiment that Bloom performs while standing in front of Yeates & Son may be relevant to parallax: "He faced about and, standing between the awnings, held out his right hand at arm's length towards the sun. Wanted to try that often. Yes: completely. The tip of his little finger blotted out the sun's disk. Must be the focus where the rays cross. If I had black glasses." All of Bloom's thoughts about Professor Joly take place during this interval while he stands pointing a finger at the sun.

The hypothesis he is testing is fairly trivial. His finger is vastly less wide than the sun, but it is also vastly closer, so sun and finger appear to be the same size, and the one is "blotted out" by the other. This is plausible enough. The writer of a 2009 blog called "Bloom's little finger" on The peacocks' tail, a website devoted to mathematics and culture, points out that the moon can eclipse the sun because the distances involved are coincidentally perfect for making their diameters appear equal. In the same way, a soccer ball will cover the sun at a distance of 25 meters from the observer, and an orange will do the trick at a distance of 10 meters.

But two odd details in Bloom's experiment suggest a different scientific interest. One is the conceptual apparatus that he proposes. He thinks that his finger makes a "focus" (i.e., a focal point) where "rays cross," coming from far edges of the sun and slicing past his eyes on the opposite sides, as light rays focused by a lens will do. This may be true, but such a model is not needed to explain his finger's eclipse of the sun, and, as will be seen in a moment, it has considerable relevance to the phenomenon of parallax. A second problematic detail, which the writer of the blog fails to observe, is that a finger held at arm's length from the eyes will not eclipse the sun quite so perfectly as the moon does. The sun disappears if one eye is kept closed, but not if both are open. (Try this at home, carefully!) Bloom must be looking with one eye closed, then. His wish for "black glasses" indicates awareness of the pain he would suffer if he opened the other.

And this means that he is almost but not quite performing the experiment he read about in Sir Robert Ball's book. The book proposes fixing a thin strip of paper to a window pane, but raising a finger is just as effective. The illustration here shows how a raised thumb will appear to move across the background when one eye is closed and the other opened. The line of sight from the left eye to the finger affords one view of the background, while the line from the right eye gives a slightly different view. Bloom is perfectly positioned to confirm this finding (at the risk of burning his retina), but he does not perform the crucial step of switching eyes.

The idea of rays crossing at a focus also evokes parallax. Scientists had noted as early as the 16th century that nearby stars might be seen moving across a field of more distant stars, because every six months the earth's revolution around the sun takes it to a new vantage point similar to that of a second eye. Viewing a nearby star in January and again in July should make it appear to move across the background, because it forms a kind of focal point for rays coming from two different parts of the sky. Just as animal brains process the angle described by slightly different images to gauge distance, astronomers should be able to measure the angle at this focus and mathematically calculate the star's distance. The greater the distance to the star (d in the first astronomical sketch), the less it would "move" (as shown in the bottom figure of the second sketch), yielding a smaller measurable angle (p). With more movement and a larger angle (as in the top figure), the distance would be less. Since all stars but one are so very far away in comparison to the earth's solar orbit, these differences would be small, and measuring the angles would require good instruments. Robert Hooke made a heroic attempt in the 1670s, but not until 1838 did better instruments give scientists reliable ways of gauging the distances.

Parallax is a valuable tool because astronomical sightings are inherently limited. A celestial body's location can never be more than relative or apparent. The universe has no center, no boundary, no fixed reference points, no universal positioning grid. Planets, stars, and galaxies spin toward or away from one another in a vast cosmic flux that astronomers call "stellar drift," a term echoed in Ithaca: "the parallax or parallactic drift of socalled fixed stars, in reality evermoving wanderers from immeasurably remote eons to infinitely remote futures." We peek into this complexity like people viewing a huge whirling dance through a keyhole. Stars may have ceased to exist by the time their light reaches our eyes. Constellations obtain their shapes only from our spatial location, and in a few thousand years those shapes will be utterly transformed.

Such indeterminacy can set the mind reeling, as Bloom's does in Lestrygonians just after he decides that trying to understand parallax is a waste of time: "Gasballs spinning about, crossing each other, passing. Same old dingdong always. Gas: then solid: then world: then cold: then dead shell drifting around, frozen rock, like that pineapple rock." Even observations here on earth enjoy no real immunity from relativity. Until 19th century railroads created a need for temporal uniformity, Greenwich and Dunsink time differed not only from one another but from that of most other English and Irish localities. Spatial relations too are relative: the driver of a pickup truck sees a tree first on the right flank of a mountain, and then on its left flank, but the tree stands in no such relation to the hill. Its apparent location is determined by the location of the motorist.

Joyce had no use for mathematical computation of distances, but he found almost endless literary analogues for parallax itself—the different views of an object obtained from different positions. Ulysses extends the idea from optics into the mental realm by a kind of implied analogy: just as eyes take in the shifting spatial relationships of the universe from different locations, yielding different views, objects of thought take on different appearances according to the angle from which they are viewed. One's subject-position, which varies from individual to individual and even from moment to moment in a single consciousness, affects what one sees.

In Lestrygonians food looks different to Bloom depending on whether he is in Burton's slovenly restaurant or Davy Byrne's pristine pub. Lemon sole seems elegant in a fancy hotel, and "Still it's the same fish perhaps old Micky Hanlon of Moore street ripped the guts out of making money hand over fist finger in fishes' gills can't write his name on a cheque." Bloom characteristically flops back and forth in this way, seeing things first from one vantage point and then from another. It marks him as a practitioner of parallax. In Hades he listens to Simon Dedalus fulminating about Buck Mulligan and thinks, "Noisy selfwilled man. Full of his son. He is right. Something to hand on." Between the second sentence and the third, Bloom shifts from the perspective of an outsider who is barely tolerated by men like Simon to the empathic view of a father who has lost a son. From this angle, Simon's angry pride appears very different.

In Cyclops, the Homeric parallel turns the denizens of Barney Kiernan's pub into one-eyed troglodytes who scorn binocular vision: Alf Landon says of "Poor old sir Frederick" Falconer that "you can cod him up to the two eyes." (According to the OED, "up to the eyes" means "to the limit." Alf adds the number.) In this monocular environment Bloom's tendency to see things from multiple angles invites ridicule. A discussion of reviving manly native Irish sports prompts him to observe that strenuous exercise can be bad for someone with a bad heart, which excites the narrator's contempt: "I declare to my antimacassar if you took up a straw from the bloody floor and if you said to Bloom: Look at, Bloom. Do you see that straw? That's a straw. Declare to my aunt he'd talk about it for an hour so he would and talk steady." But a straw is not just a straw. To a famished horse it is one thing, to a phlegmy drinker another, to the man who sweeps the floor still another. Bloom's inclination to look at things from different points of view identifies him as a complex life form, one with stereoscopic depth of understanding.

Molly performs the same perpetual adjustments, regularly using "still" and "but" locutions like those of her husband. At the beginning of her chapter she criticizes Bloom for pandering to old Mrs. Riordan in hopes of getting an inheritance from her, but then thinks, "still I like that in him polite to old women like that and waiters and beggars too." Such flip-flops occur throughout Penelope, especially in Molly's attitudes toward her husband. They are contained in her endless repetitions of the word Yes, which open the chapter in a spirit of suspicious skepticism and close it with joyful affirmation. Stephen too practices parallax: his insistence that Haines be evicted from the tower dissipates when Mulligan proposes giving him a violent "ragging," and his fantasy of blowing an insolent doorman to bits with a shotgun flips over into a recuperative fantasy of reconstituting the bloody bits into a person. As he thinks later in Proteus, "Ah, see now! Falls back suddenly, frozen in stereoscope. Click does the trick."

Ulysses offers countless instances of intersubjective parallax—different individuals' dissimilar views of the same object. Molly and Bloom are always correcting each other in this way. So too, from a greater distance, are Bloom and Stephen. Joyce makes one such discrepancy particularly suggestive of parallax by having the two men view the same celestial object from slightly different angles. In Telemachus Stephen sees a cloud begin to cover the sun from Sandycove, at nearly the same time (but not quite) that Bloom sees it from north Dublin. The swallowing of the sun provokes dark thoughts of divine judgment and personal despair in both men, but in slightly different ways: Stephen thinks of his mother's ghoulish visitation while Bloom thinks about Sodom and Gomorrha, the Dead Sea, and sterility. In Ithaca, Stephen attributes the crisis he experienced in the brothel, precipitated by his mother's hallucinated reappearance, to the influence of this "matutinal cloud," while in Oxen of the Sun Bloom's thoughts of having no son give way to a nightmarish vision of Palestine as a wasteland: "And on the highway of the clouds they come, muttering thunder of rebellion, the ghosts of beasts. Huuh! Hark! Huuh! Parallax stalks behind and goads them."

Parallax also describes the bewildering variety of perspectives on the book's actions taken by its many modes of narrative presentation. From the first page, traditional third-person objective narration veers into the subjective orbits of certain characters. At times, notably in the third and last chapters, interior monologue swamps exterior narration, leaving the reader at the mercy of a character's shifting thoughts, uncertain of what exactly may be happening in the world at large. But objective narration too proves shifty. In an unending series of experimental presentations of the action—newspaper headlines, stage directions, parodic styles, catechistic questions and answers—the novel makes its readers jump from one viewpoint to another. It is filled with instants when new perspectives pop into view: the moment when free indirect style evocative of Gerty's thoughts suddenly gives way to interior monologue conveying Bloom's, the moment when a first-person narrator suddenly declares his existence in a third-person book, the moment when people in the library suddenly become dramatic characters in a script. In Wandering Rocks parallax returns to its unmetaphorical, purely visual roots, with sudden filmlike cuts from one scene to another.

In The Book As World (1976), Marilyn French observed that because of such constantly shifting views Parallax "could have served as a title for this novel" (106). She did not think, however, that the polymorphous subjectivity ruled out the possibility of objective knowledge: "The fact that various characters walk the same streets, see the same shops or the same 'matutinal cloud,' or meet the same people confers on these streets, shops, cloud, and people a solidity, a reality. One way to test the reality of an object is to compare one's perceptions with those of others . . . Joyce builds into his book the solid substance of things. This is all the more necessary since the modes of perceiving them are so spaced out, so private and eccentric" (27).

In his critical study Ulysses (1987, first published 1980), Hugh Kenner too observed that the novel's pointillist perspectivism has the effect of substantiating objective reality: "Parallax makes possible stereoscopic vision: 'In order to see that basket,' Stephen instructs Lynch in the Portrait, 'your mind first of all separates the basket from the rest of the visible universe which is not the basket' (212), something the mind can do more easily since two eyes have presented it with separate versions of the basket's location. Two different versions at least, that is Joyce's normal way; and the uncanny sense of reality that grows in readers of Ulysses page after page is fostered by the neatness with which versions of the same event, versions different in wording and often in constituent facts—separated, moreover, by tens or hundreds of pages—reliably render one another substantial" (75).

Kenner selects more or less randomly the example of Bloom seeing George Russell walking his bike on a route leading to Kildare Street, and Russell then appearing in the National Library in the next chapter: "This does much to assure us that Bloom really did see Russell, a substantial Russell in motion through Dublin's Newtonian space" (75). Bloom's recollection of Robert Ball's book in Lestrygonians, supplemented by its appearance on his bookshelf in Ithaca, prompts Kenner to observe that "each speck in this book has somewhere its complementary speck, in a cosmos we can trust" (76). The narrative presentation of Dlugacz in Calypso, followed by Bloom's memory of slightly different details about the butcher later in the chapter, illustrates his claim that "virtually every scene in Ulysses is narrated at least twice, and by varying what he tells and emphasises Joyce ensures that repetition shall not dilute but intensify" (76).

French's and Kenner's sensible critical responses to Joyce's fiction cohere with the scientific concept, which does not imply that celestial objects cannot be known, or that varied observations will involve irreconcilable discrepancies. Parallax implies only that stars will take on slightly different appearances against different backgrounds. The objects of human knowledge represented in Ulysses become more fully known when seen from new perspectives—the more the better.