Mathematics as a science commenced when first someone, probably a Gree — Ancient Greek mathematics

"That the discovery of incommensurability of lines made a strong impression on Greek thought is indicated by the story of Hippasus... It is demonstrated more reliably by the prominence given to the theory of irrationals by Plato and his school [e.g., Eudoxus of Cnidus]. ...the Greeks were led by Zeno and Hippasus to abandon the pursuit of a full arithmetization of geometry... there was no such thing as algebraic analysis. Geometry was the domain of continuous magnitude, arithmetic was concerned with the discrete set of integers; and the two fields were irreconcilable."

"Logistic is the theory which deals with numerable objects and not with numbers; it does not, indeed, consider number in the proper sense of the term, but assumes 1 to be unity, and anything which can be numbered to be number (thus in place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic. ... It treats, then, on the one hand, that which Archimedes called The Cattle Problem, and on the other hand melite and phialite numbers, the one discussing vials (measures, containters) and the other flocks; and when dealing with other kinds of problems it has regard to the number of sensible bodies and makes its pronouncements as though it were for absolute objects. ... It has for material all numerable objects, and as subdivisions the so-called Greek and Egyptian methods for multiplication and division, as well as the summation and decomposition of fractions, whereby it investigates the secrets lurking in the subject-matter of the problems by means of the procedure that employs triangles and polygons. ... It has for its aim that which is useful in the relations of life in business, although it seems to pronounce upon sensible objects as if they were absolute."

"Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"The inspiration of Fermats discussion of the conic sections, and that is practically the whole of his analytic geometry, comes direct from Apollonius. The same had been true of Pappus, fourteen centuries before. His point of departure is the famous four-line problem... This question seems to have stumped both Euclid and Aristaeus, and to have been first solved by Apollonius. In Apolloniuss own work we find what is rather the converse of this problem. Almost the first piece of geometrical writing which Fermat did was to prove the three-line case."

"It is important to remember that the ancient Greeks did not have an abstract system of number symbols, and used the letters of the alphabet as number symbols. They also commonly manipulated pebbles to learn arithmetic and used small stones on calculating boards. In this case, number patterns were their common experience of arithmetic. From this use of pebbles, we have inherited the word calculation, from the Latin calculus, which means pebble."

"These three problems—the , the duplication of the cube, and the trisection of the angle—have since been known as the "three famous (or classical) problems" of antiquity. More than 2200 years later it was proved that all three... were unsolvable by means of straightedge and compass alone. ...the better part of Greek mathematics, and much of later mathematical thought, was suggested by efforts to achieve the impossible—or failing this, to modify the rules."