History of asteroids discoveries
Discovery of each asteroid was a different story. Here you can read selected discovery stories of first 100 minor planets as they were written in book of J.J. von Litrow: Die Wunders des Himmels, published in 1878, Berlin.
Prelude to asteroid search
There were only 6 known planet in times of Copernicus, Galileo and Kepler. Uranus and Neptun as 7th and 8th planet of Solar system were discovered in subsequent 2 centuries. In those times when astronomers could compute orbits of planets it showed that between Mars and Jupiter is a gap. The idea that there should be another planet arose also from Titius - Bode law. According to this, there should be a planet orbiting between Mars and Jupiter. So astronomers started to search for missing planet.
In the end the missing planet was discovered by Italian astronomer G.Piazzi, who not belonged to "Celestial Police". He spotted hidden "planet" between Mars and Jupiter on 1. January 1801, later named as Ceres. In one year time was found Pallas, second "planet" in this area. After discoveries of another 2 objects, it looks like to many objects for a such space. When astronomers learned that those objects are much more smaller than other known planets, William Herschel proposed new name for this type of "planet" - asteroid or minor planet.
(1) Ceres
(2) Pallas
Second minor planet Pallas,
was discovered on 28. March 1802 by coincidence. German physicians and astronomer, Heinrich Wilhelm Matthias Olbers, noticed during his observation of Ceres nearby an unknown star of 7th magnitude. He was sure that this object was not at this position earlier as he used to observe this part of the sky. He wrote down its position and followed it for 2 hours. After that time it was clear that new object moved between the stars. Olbers was sure that he found a second "missing" planet between orbits of Mars and Jupiter.
Heinrich W. Olbers Although suggested name for second minor planet was Olbersiana, finally it was named after Greek Goddess Pallas Athena. Slovak astronomer F. X. Zach suggested also sign for the new asteroid as it was usual in that time. The preliminary orbit of newly discovered asteroid was computed by Friedrich Gauss. Orbital elements of Pallas proved that it moves like Ceres between orbits of Mars and Jupiter.
After the discovery of second planet in the gap between planets Mars nad Jupiter, Olbers suggested a hypothesis that both new object were remnant of bigger and somehow destroyed planet. So that was the main reason for astronomy community continuing their hunt.
(3) Juno
Karl L. Harding German astronomer Karl Ludwig Harding worked since 1796 at Johann Hieronymus Schröter at Lilienthal, the other member of the Celestial Police, as a teacher of his son. So he can use Schröter´s Observatory for another possible asteroid search. Success came on Sept. 1, 1804, when Harding found 8th magnitude object in Pisces. There was not any star in stellar atlas on that position and after three days he could confirmed its movement. His discovery was later assigned as asteroid (3)
(4) Vesta
Calculated paths of three new "planets" showed that they intersect in Virgo constellation. So W. Olbers considered, that new objects are remnants after collapse of their parent body. Hence other debris should pass through this space too. Therefore he decided to search this part of sky in constellation Virgo and opposite in Cetus. Olbers regularly observed both constellations. On March 29, 1807, 5 years after his discovery of Pallas, he found the fourth "planet" near the site, where he spotted Pallas first. This new object was called as
Members of the Celestial Police continued their search for other possible objects, but as another discovery did not come, in 1815 they stopped their effort.