How did the mountain ranges in the northern and western regions india remain independent for thousands of years

The study of mathematics as a "demonstrative discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".[4] Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts

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the crusades are one of the most significant events in the history of europe and the middle east.   they were a series of religious wars carried out by christian crusaders from europe during the timeframe of the middle ages.   beginning in 1095 ce, the crusades saw european knights and noblemen travel to the middle east in an attempt to capture the holy land away from muslim people that had controlled the region for the previous centuries.   the term crusade means ‘cross’.   therefore, the europeans that became crusaders viewed themselves as ‘taking up the cross’.   in fact, many of the crusaders wore crosses on their clothing and armor as they made their pilgrimage to the holy land.

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