Science had been hampered by the confusion raised by the discussions relating to forms. Aristotle realized that proper progress in logic, physics, and ethics, the leading sciences of his time, could not be made unless the essential nature of science were kept in view. He saw that knowledge of the forms of correct thinking can be understood only by keeping in view the object of thought and this requires definite ideas of the general relations of knowledge and its objects. The study of general relationships led to the study of particular or special relations. The connection of general with particular ideas was unfolded, and Aristotle saw that conceiving, understanding, and proving result from the deduction of particular from universal, or general, ideas. Therefore science consists in deriving or deducing facts acquired through perception from their general grounds or phenomena. The logical form of the syllogism naturally suggested itself to Aristotle when engaged with these thoughts and the invention of the syllogism was one of the most brilliant contributions to knowledge made by the Greeks. The logical results of the invention of syllogistic forms suggested a solution of the problem of true reality which Aristotle showed was the essence that unfolds in phenomena themselves. This led to fruitful scientific results. Plato and his contemporaries unified mathematics, formulated the definitions logically, and demonstrated correct methods of criticism and proof. A point was shown to be the boundary of a line; while a line is the boundary of a surface, and a surface the boundary of a solid. This concrete definition of scientific elements progressed through the use of analytic methods, by proceeding from the known to the unknown, and led to the discovery of tests for scientific assumptions and of synthetic proof. None of the earlier philosophers possessed anything like the progressive tools Aristotle placed in the hands of scientists. Their use quickly led to a general review of knowledge and a great increase in the number of sciences. 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.