The textbook on geometry compiled by Euclid, still used in many schools, gives us a good picture of the state of scientific methods in his time. Euclid, like Aristotle, Plato, Socrates and others, was a great systematizer. He collected the geometrical proofs of his mathematical predecessors, selected those which were logically correct and simple, and raised on a few axioms, or first principles, a great geometrical system.
Archimedes published textbooks on spherical and cylindrical geometry. He proved that the surface of a sphere is equal to four times a great circle. He showed the properties of spherical segments and methods for calculating surface areas and other parts of spherical forms.
This great scientist also developed mechanics and physics. He investigated the lever and demonstrated the principle upon which its power is based. He then studied hydrostatics and hydraulics, and discovered the theory of specific gravity and invented methods for determining it.
Apollonius began publishing scientific textbooks about forty years after Archimedes. His masterpiece was his textbook on conic sections.
The work done by Archimedes on the quadrature of curvilinear figures resulted, centuries later, in the discovery of the infinitesimal calculus, while the theory of conic sections published by Apollonius led to theories for the solution of problems relating to geometrical curves of all degrees. They placed the geometry of measurements and the geometry of forms and positions on strictly scientific bases.
Hipparchus applied the new mathematical and geometrical discoveries to astronomy. He found a method for representing the observed motions of the sun, moon, and planets by assumed uniform circular motions. His theory of the sun's motion assumed that the earth was not the center of the sun's orbit. He drew a line through the earth and the real center of the orbit and found where the sun's distance is least and where greatest. He then compiled a large set of solar tables giving the position of the sun among the stars at any time. He next turned his attention to the movements of the moon and prepared tables for determining eclipses.
Then the various planets were studied and their mean motions were calculated and recorded. The stars were mapped and catalogued. He described the apparent movements of 1,080 stars and comparing his observations and calculations with those of Aristyllus and Timocharis, made 150 years previously. He also discovered the precession of the equinoxes.
The astronomical calculations of Hipparchus led to a great improvement in trigonometrical methods. By using chords, as we use sines, and assuming the heavens to be a plane surface, he fixed the positions of stars (and similarly geographical points) by the intersections of lines of latitude and longitude.
A planosphere, an instrument for representing the mechanism of the heavens, was among the many scientific inventions of Hipparchus.