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The purpose of this paper is to use historical textbooks to explore the number line, a mathematical object common to our time, contextualizing its use during its initial development period in the seventeenth century until its current use as in the nineteenth century. Relevant analytic geometry textbooks are explored, mainly from France and from Germany of that time, to understand when and how this notion was used. Additionally, pertinent entries from encyclopedias of that time are used to contrast with what was not sufficiently explained in the aforementioned textbooks, until, finally, we can identify this object as it is currently characterized. I analyze how the negative numbers were used to constitute the number line and survey several number line diagrams, which were sometimes similar, but most often quite different from each other. The intention is to understand the historical contexts that configure the emergence and development of the notion of the number line. [ABSTRACT FROM AUTHOR]

A biography of mathematician Gaston Darboux is presented. He was born on August 14, 1842 in France, began his studies in October 1861, completed his doctoral studies in 1872, and taught in a famous French secondary school. His unusual intellectual ability and his enthusiasm for mathematics made him outstanding in his field. Darboux was a prolific writer who wrote on various subjects which include differential geometry, analytical geometry, and mathematical physics.

The article discusses the familiar geometrical demonstrations of French and Italian engineers in the 16th century. It notes that military engineers did not have the societal standing of mathematicians though they need solid theoretical knowledge and practical experience. It suggests that engineers can lift their status and increase their power over the common soldier by preventing the understanding of Euclid on mathematics. It points out that the 16th century societal pressure to abstract significant mathematics contrasts the modern-day mathematicians' translation of math into everyday language.