Renee Descartes, a French mathematician, scientist, and philosopher, had a revelation that the structure of the universe was mathematical and that nature obeyed mathematical rules. In 1637, he invented analytic [Cartesian] geometry, in which lines and geometric shapes can be described by algebraic equations and vice-versa. All conic sections: circles, ellipses, parabolas, and hyperbolas, could be represented by equations with two unknowns, or variables, on a coordinate system in which each point is represented by a pair of numbers representing distances from the two axis lines. An algebraic equation with two unknowns, could be represented as a shape thereon. An algebraic equation with one unknown represented a straight line thereon. The points of intersection geometrically were equivalent to the common solution of the associated algebraic equations. He started the convention of representing unknown quantities by x, y, and z and known quantities by a, b, and c. So, for instance, a circle with center at point 2,3 and a radius of 4 was represented by the equation: (x-2) squared + (y-3) squared = 4. He pioneered the standard exponential notation for cubes and higher powers of numbers. Analytic geometry aided in making good lenses for eyeglasses. The glass was first manufactured with attention to quality. Then, after it cooled and solidified, the clearest pieces were picked and their surfaces ground into the proper curvature.
Descartes formulated the law of refraction of light, which deduces the angle of refraction [deflection] of light through a medium from the lights' angle of incidence and the speed of light in each media in which the light passes. This explained why a rainbow is circular. In 1644, he described the universe in terms of matter and motion and suggested that there were universal laws and an evolutionary explanation for such. He opined that all effects in nature could be explained by spatial extension and motion laws that 1) each part of matter retains the shape, size, motion, or rest unless collision with another part occurs; 2) one part of matter can only gain as much motion through collision as is lost by the part colliding with it; and 3) motion tends to be in a straight line. Descartes feared persecution by the church because his ideas did not correlate with the Biblical notion of God's creation of the universe in the order of light, then sky and oceans and dry land, then plants, then seasons and the sun and moon and stars, then fish and birds, then all animals, and finally man. Descartes believed in a good and perfect God, and thought of the world as divided into matter and spirit. The human mind was spirit and could exist outside the human body. Without the mind, human body was a machine. The human mind had knowledge without sense experience, e.g. the truths of mathematics and physics. Ideas and imagination were innate. His observation that sensory appearances are often misleading, such as in dreams or hallucinations, led him to the conclusion that he could only conclude that: "I think, therefore I am." He rejected the doctrine that things had a proper behavior according to their natures, e.g. the nature of acorns is to develop into oak trees. As an example of erroneous forming of conceptions of substance with our senses alone, he pointed out that honeycomb has a certain taste, scent, and texture, but if exposed to fire, it loses all these forms and assumes others. He considered to be erroneous the belief that there are no bodies around us except those perceivable by our senses. He was a strong proponent of the deductive method of finding truths, e.g. arguing logically from a very few self- evident principles, known by intuition, to determine the nature of the universe.
Christian Huygens, a Dutch physicist, used the melting and the boiling point of water as fixed points in a scale of measurements, which first gave definiteness to thermometric tests.
There was much mining of coal, tin, copper, lead, and iron in the 1600s. Coal was transported from the coal pits down to the rivers to be loaded onto ships on coal wagons riding on wooden rails. The full coal cars could then be sent down by gravity and the empty wagons pulled up by horses. Sheet metal, e.g. lead, was used for roofing. Coal was much used for heating houses, and for laundry, cooking, and industrial use, such as extraction of salt, soap boilers, and manufacture of glass, bricks and tiles for buildings, anchors for ships, and tobacco pipes. It was used in the trades: bakers, confectioners, brewers, dyers, sugar refiners, coopers, starch makers, copper workers, alum makers, and iron workers.
In 1604 the Haberdashers, who sold imported felt for hats, got a charter of incorporation.
A tapestry factory was established in 1619.
Flax-working machines came into existence.
The royal postal system carried private as well as royal letters, to increase income to the Crown. Postmasters got regular pay for handling without charge the mail of letters that came from or went to the letter office in London. The postmaster kept horses which he let, with horn and guide, to persons riding "in post" at 3d. per mile. The post was to travel 7 mph in summer and 5 mph in winter and sound his horn four times in every mile or whenever he met travelers.
Wool and animals for butchering were sold in London with the sellers' agent in London taking the proceeds and paying out to their order, the origin of check writing.
Scriveners drew up legal documents, arranged mortgages, handled property transactions, and put borrowers in touch with lenders. They and the goldsmiths and merchants developed promissory notes, checks, and private paper money.