The Discovery of Uranus
 Prior to the late 1700's, no planet had been "discovered". The nature of the planets and the fact that the Earth was one of them wasn't fully realized until the Copernican revolution, but the fact that they existed, whatever their nature, was known since antiquity.
 Uranus was discovered on March 13, 1781 by a music teacher and composer, Frederick William Herschel, who had become an avid amateur astronomer and telescope maker. He at first thought it a comet, naming it Georgium Sidus (George's Star) in honor of George III, but its motion was unlike that of a comet, and the astronomer royal, Nevil Maskelyne, suggested that it might be a planet. Mathematical confirmation of this proposal created a sensation, making Herschel the most celebrated astronomer of his day (and a professional to boot, as he was appointed King's astronomer at what was then a handsome rate of pay and was thereafter able to devote all his time to astronomy).
 Other than in Britain, where the new planet was referred to as the Georgian Planet for nearly sixty years, there was great resistance to Herschel's suggested name for the planet. France and the newly created United States, at that time at war with Britain, particularly found the name anathema, and it was quickly christened Herschel. Eventually, a multitude of suggestions based on mythological figures was resolved in favor of Ouranos (Ούρανός), the Greek god of the heavens, transliterated to the present Uranus.

For Heaven's Sake, Say It Right  Due to the tittering of people with infantile notions of humor, many English-speaking astronomers have been reluctant in the age of visual media to pronounce Uranus the way it should be pronounced, and most English verbal references to the planet use incorrect pronunciation. However, aside from the way the God of the Heavens' name (Ούρανός) is pronounced ("OOR-ahn-nohss") in (the more or less original) Greek, the correct way to say the seventh planet's name is how you pronounce "uranium", which is named after Uranus (just as neptunium is named after Neptune, and plutonium after Pluto). So in Germany, where uranium is called Uran and pronounced "OOHR-ahn", the planet is called Uranus, pronounced "OOHR-ahn-uss", while in the the United States, where uranium is called "yoo-RAIN-ee-umm", Uranus should be pronounced "Yoo-RAIN-uss", not the bastardized "YOOR-uhn-uss" all too often used. And if you can't say that and ignore the titters that may result, then you shouldn't say anything at all.

Earlier Observations of Uranus
 As it happens, Uranus was actually observed on many occasions prior to its discovery as a planet, as it is just visible to the unaided eye under dark-sky conditions. The earliest known observation, by John Flamsteed, was in 1690, and Flamsteed made at least half a dozen other observations of the planet, on each occasion mistakenly recording it as a faint star (e.g., in the first instance as 34 Tauri). Many other observers recorded the planet's position, unfortunately usually when it was near a stationary point and hardly moving relative to the stars, so that it appeared merely a star itself. Although these observations failed to reveal Uranus' planetary status, they were immensely helpful in early calculations of its orbital motion, and crucial in discovering Neptune.

The Discovery of Neptune
 The first accurate predictions of Uranus' motion were published in 1792. Within a few years it was obvious that there was something wrong with the motion of the planet, as it did not follow the predictions. Alexis Bouvard, the director of the Paris Observatory, attempted to calculate improved tables using the latest mathematical techniques, but was unable to fit all the observations to a single orbit, and finally decided to rely only on the most modern observations, while suggesting, in his 1821 publication of his results, that perhaps there was some unknown factor that prevented better agreement with the older observations. A commonly accepted suspicion during the next few years was that Newton's Law of Gravity, although accurate out to the orbit of Saturn, might not work in the same way at greater distances; but by the late 1830's it seemed at least equally likely that there was an unknown body lying beyond the orbit of Uranus, which it most likely passed in the early 1800's, so that it was accelerated prior to its passage of the outer planet, and decelerated afterwards.
 The story of the calculations and searches carried out by various individuals, and the fame and infamy attached to each of them at the time and by recent historians has been the subject of whole books, and will therefore not be covered here. Suffice it to say that as a result of those calculations, Neptune was discovered on September 23, 1846 by the astronomer Johann Galle and his student Heinrich d'Arrest, after only thirty minutes of searching the sky, within a degree of the position predicted by Urbain Le Verrier. This was the high-water mark of Newtonian physics -- to be able, given the laws of physics and the peculiar motion of one object, to reach out into the depths of space and uncover a previously hidden object -- and caused an even greater sensation than the discovery of Uranus.
 Rather ironically, as in the case of Uranus, there were a number of observations of Neptune prior to its discovery as a planet, but once again, mostly at times when it was near a stationary point, and hence nearly motionless relative to the stellar background. In fact the first observations were made by Galileo in December of 1612 and January of 1613, when he was following the motion of Jupiter and its newly discovered moons (now called the Galilean satellites in honor of their discoverer). He even noticed that the object which we now know as Neptune seemed to have moved relative to the nearest star, but failed to follow up his observations, and hence lost the opportunity to discover Neptune 168 years before Herschel discovered Uranus.
 Thanks to the almost immediate discovery of Triton, the largest moon of Neptune, it was possible to calculate its mass and gravitational effect on Uranus, and find that it had indeed been responsible for the errors in the motion of the previously discovered planet, thus ending more than half a century of confusion about Uranus' orbit. Another century of confusion followed, however, due to the understandable desire of others to share in the glory of discovering a new planet (as discussed below).

The Search Goes On
 A new planet, Neptune, had been found merely by using errors in the calculated positions of another planet. Was it possible that other planets might be found in this way? And if so, where?
 Both ends of the Solar System were studied in the subsequent search for a Planet X, whose existence might or might not be proven by careful investigation of the motions of other bodies. In the inner Solar System, there was a minor oddity in the motion of Mercury that held out the promise of an infra-Mercurian object, and in the outer Solar System there were still some minor errors in the observations of Uranus that could be used to look for planets beyond Neptune.
 In the inner Solar System,efforts centered on the precession of the line of apsides of Mercury. The line of apsides is the imaginary line that runs from the perihelion position of the planet's orbit, through the Sun, to the aphelion position of the planet; in other words, the major axis of the orbit. If only the Sun and the planet existed, Newtonian physics requires that the planet's orbit, once determined by initial conditions (its position and motion relative to the Sun), should remain essentially the same for all eternity; but in the presence of other planets (primarily Venus and the Earth) the repeated passage of one planet by another, as each goes around the Sun at differing rates, causes perturbations (small changes) in the orbit. Each time round the Sun the effects of the perturbations are small, and over time they tend to cancel each other out, but they can cause "wobbles" of one sort or another in the orbit, and one of these, in the case of Mercury, is a slow change in the direction of the line of apsides of about 2/3 of a degree per century.
 Even in the 1840's it was obvious that the calculated rate of precession for the line of apsides (based on the effects of Venus and the Earth on Mercury's orbit) was too small, by about 1/100 of a degree per century (more accurately, 44 seconds of arc per century). Given the discovery of Neptune and its effects on Uranus' orbit, it seemed reasonable to suppose that there might be an infra-Mercurian planet (one inside the orbit of Mercury) which was responsible for the excess in its apsidal rotation.
 As it happens, that planet, christened Vulcan (whence the name for the hot planet that Spock supposedly came from in the Star Trek series) does not exist, though it was "observed" on at least two occasions. (Presumably, a small asteroid or comet or some such object happened by coincidence to be located in the direction expected for the planet.) We know that Vulcan does not exist, not because we never found it, but because in the early 1900's Albert Einstein proposed new theories of motion and gravity, the special and general theories of relativity, which correctly predicted the excess motion of Mercury's line of apsides without resorting to an infra-Mercurian planet.
 (Note to self: should put discussion or link here, for Einstein's solution?)
 In retrospect it is ironic, given the early supposition that Uranus' orbital errors were due to a flaw in Newtonian physics, and the subsequent discovery that they were due to an unknown planet (Neptune), that in the case of Mercury the situation was exactly opposite: initial speculations centering on an unknown planet, and the solution involving errors in the laws of physics...