Direct observation of mathematical relationships almost never occur in nature; usually, experimental observation is necessary to reveal them; one exception is the simpler movements of the sun, moon and planets. Astronomy is the first exact science to make decisive progress.

At first, astronomy was used to time observance of religious rituals and to time agricultural seasons. Development of astronomy enabled navigation from point to point on the surface of the earth, a secular use of science that grew to ever greater importance. From simple observation of the fixed position of the pole-star, each successive observation that adds to the knowledge of the observer complicates science. As with any science, new scientists build on the work of their predecessors.

With most observations, the observer needs a place to stand. Early astronomers standing on the earth noted the movement of the heavenly bodies and assumed that the earth was in the centre of the universe. Observation indicated that movements of the stars and planets were constant, explained as fixed on the surface of a series of spheres with the earth at the centre — the circle being considered a perfect figure.

Data accumulated from observations began to disagree with this circular theory of perfection. In an effort to reconcile the difference, astonomers, and Hipparchus in particular, in the time around 130 BC, adjusted theory to fit observed fact by supposing ‘that the planets revolve in circles not actually about the earth but about points themselves revolving circularly about the earth;’ these epicycles Claudius Ptolemy recorded in his Almagest and ‘published a table of chords for astonomical calculation.’

Zenith of ancient astronomy was Hipparchus who drew together Babylonian observation and Greek geometry to discover the precession of the equinoxes, initiating trigonometry, creating and perfecting instrumentation, and started the practice of locating positions on the surface of the earth by coordinates of latitude and longitude. Heliocentric theory had been posited earlier than Hipparchus, as a result of exploration down the coast of Africa when the shape of the earth was tentatively suggested as round instead of flat. Aristarchus took this revolutionary information, combined it with lunar eclipses and proposed that the sun was vastly larger than the earth; in addition, he suggested that the stars were much farther away than was commonly thought and that their distance might account for their apparent immobility. Aristarchus didn’t develop his ideas enough to show that the epicycles were only the orbital motion of the earth transferred separately to each of the planets.

Much classical technique was lost during the Dark Ages, but Arabic and Jewish computing methods enabled construction of tables to assist navigation by the stars from 1000 to 1500 AD. Belief in a spherical earth led to dreams of an alternative route to India that would bypass Moslem power. Improvements in shipping since the 12th century along with laying and keeping a course and developments in the magnetic compass enabled exploration over ever greater distances; with this came improvements in instrumentation and chart making. Henry the Navigator, the Portugese prince, actively encouraged research into and development of navigation. Charts were a closely guarded secret. All navigation was hampered by an inability to determine longitude, whereas latitude could be determined with tables of solar declination or of pole-star altitude corrections. Accurate navigation required meticulous record keeping, observation, and dead reckoning.

The theory of a spherical earth made possible a merchant marine that produced funds for further scientific research. Notion of a sperical earth forced changes in mapping techniques and research into the best method of showing the surface of a sphere on flat sheets of parchment or paper. The work of Numez (1502-1578) was instrumental in pushing this forward; he invented a cumbersome precursor to the vernier that was eventually followed by Mercator's projection in 1568. Dutchman Regnier Gemma Frisius (1508-1555) suggested finding longitude by means of difference of times; this simplified the problem to construction of accurate clocks able to withstand the shipboard motion; a successful clock didn't appear until Harrison succeeded in making one two hundred years later.

Interest in astronomy was prevalent in the 15th century, especially as it applied to the Julian calender, which, since its inception fifteen hundered years ago, Easter had moved too far out of place. Schism and delay in the church prevented reform of the calender until 1752. Philosophers of this time, such as Nicholas of Cusa (1401-1464) and Leonardo da Vinci, concluded that the earth moved, that no two bodies are exactly alike, that exact knowledge is impossible, that planetary motion is not exactly circular, and that the earth is not the centre of the universe.

Cusa pointed out that continuity of motion can be envisaged yet cannot be shown in practice; in so doing he implied the possiblity of measurement, thereby implying discreteness and the replacement of absolute motion by a number, meaning that all motion can be reduced to a mathenatical equation that can be manipulated until the results match observed data. Philosophers had to be very careful to avoid falling foul of religious dogma and being declared heretic; they knew that dogma didn’t match their observations. Philosophers didn’t have clear dynamic physical concepts, no benefit of the telescope, nor great computational facility: they had discovered the empirical laws of planetary motion.