There is an intellectual thread that runs through all of these advances: measurement and calculation. Geometric calculations led to breakthroughs in painting, astronomy, cartography, surveying, and physics. The introduction of mathematics in human affairs led to advancements in accounting, finance, fiscal affairs, demography, and economics – a kind of social mathematics. All reflect an underlying ‘calculating paradigm’ – the idea that measurement, calculation, and mathematics can be successfully applied to virtually every domain. This paradigm spread across Europe through education, which we can observe by the proliferation of mathematics textbooks and schools. It was this paradigm, more than science itself, that drove progress. It was this mathematical revolution that created modernity.

**The geometric innovations**

Advances in geometry began with the rediscovery of Euclid. The earliest known Medieval Latin translation of Euclid’s *Elements* was completed in manuscript by Adelard of Bath around 1120 using an Arabic source from Muslim Spain. A Latin printed version was published in 1482. After the mathematician Tartaglia translated Euclid’s work into Italian in 1543, translations into other vernacular languages quickly followed: German in 1558, French in 1564, English in 1570, Spanish in 1576, and Dutch in 1606.

Beyond Euclid, the German mathematician Regiomontanus penned the first European trigonometry textbook, *De Triangulis Omnimodis* (*On Triangles of All Kinds*), in 1464. In the sixteenth century, François Viètehelped replace the verbal method of doing algebra with the modern symbolism in which unknown variables are denoted by symbols like x, y, and z. René Descartes and Pierre de Fermat built on Viète’s innovations to develop analytic geometry, where curves and surfaces are described by algebraic equations. In the late seventeenth century, Isaac Newton and Gottfried Leibniz extended the methods of analytic geometry to the study of motion and change through the development of calculus.