Answer :

Final answer:

Euclid's 'rigorous' study of geometry refers to the systematic and precise derivation of geometric principles from a set of basic axioms, demonstrating the science of 'exact thought or rigorous thinking' central to mathematics.

Explanation:

To say that Euclid's textbook was a 'rigorous' study of geometry means that it provided a meticulous and precise approach to the subject.

Euclid's work was grounded on a set of axioms or postulates, and from these, he logically derived further geometric propositions and theorems.

This rigorous approach forms the basis of deductive reasoning in mathematics and ensures that geometry is not just a collection of random facts, but a coherent and logically structured discipline.

The term 'rigorous' implies exactness and adherence to strong logical methods. It demonstrates precision, sharpness and completeness of definitions.

By introducing theorems in a systematic way, starting from basic principles and building upon them, Euclid established a model not just for geometry but for rigorous thinking in all of mathematics.

The notion of rigor in Euclid's work also emphasizes that despite geometry's theoretical nature, it has practical applications, such as in engineering.

Hence, one appreciates the reliability and depth of Euclid's geometry when it is understood as a system that can be applied to solve real-world problems.

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