John Napier, a large Calvinist landholder in Scotland who had built his own castle, did mathematics in his older years. He explored imaginary numbers such as the square roots of negative numbers. By 1614, he had started and developed the theory of logarithms: the relationships among positive and negative exponents of numbers. This simplified calculations because the multiplication and division of numbers would be equivalent to addition and subtraction of their exponents. His table of logarithms, which took him twenty years to compile, was used in trigonometry, navigation, and astronomy. It reduced the enormous labor involved in trigonometric calculations.
Johannes Kepler was a mathematician from Germany who made his living as an astrologer. He was in contact with Galileo by letter, as most scientists of Europe were with each other. Kepler was fascinated with perfect geometric shapes, which he tried to relate to celestial phenomenon. He discerned that the orbit of Mars was not perfectly circular. He knew that the apparent path of the sun with respect to the constellation of fixed stars differed in speed at different times of the year. He opined that this showed that the speed of the earth revolving around the sun varied according to the time of year. Then he measured the angles between the earth and the sun and the earth and Mars as they changed through the Martian year. He noted when the earth, Mars, and the Sun were on the same straight line. Then he deduced the earth's true orbit, and from this the true orbits of the other planets. Then by trial and error, he attempted to match this empirical data with regular mathematically defined shapes, until he discovered in 1609 that these paths were elliptical. Also, the planets each move faster when they are nearer the sun and more slowly when they are farther from the sun so that in equal time intervals, a line from the planet to the sun will sweep out equal areas. This observation led him to opine that there is a force between the sun and each planet, and that this force is the same as that which keeps the moon in its orbit around the earth. Thirdly, in 1619, he found that the square of the time for each planet's orbit about the sun is proportional to the cube of that planet's mean distance from the sun, so that the farther planets orbit at a slower speed. He connected the earth's tides with the gravitational pull of the moon. Kepler also confirmed that the paths of comets were governed by a law and were farther from the earth from the moon. This contradicted the church's explanation that what lies within the moon's orbit pertains to the earth and is essentially transitory and evil, while what lies beyond belongs to the heavens and is permanent and pure.
In 1637, Renee Descartes, a philosopher, mathematician, and scientist from France, invented analytic [Cartesian] geometry, in which lines and geometric shapes can be described by algebraic equations and vice-versa. An algebraic equation with two unknowns, or variables, could be represented as a shape on a coordinate system in which each point is represented by a pair of numbers representing distances from the two axis lines. 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. All conic sections, e.g. ellipses, parabolas, and hyperbolas, could be represented by equations. 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 pioneered the standard exponential notation for cubes and higher powers of numbers. He formulated the sine-law of refraction, which determines in general the way a light ray is deflected, according to the density of the media through which it 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 rectilinear. These ideas did not correlate with the Biblical notion of the creation of the universe by God in seven days, so Descartes feared persecution by the church. 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. 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 expressed that it was error to believe 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.
In 1600 William Gilbert, son of a gentleman, and physician to Queen Elizabeth, wrote a book on terrestrial magnetism which founded the science of electricity. He cultivated the method of experiment and of inductive reasoning from observation and insisted on the need for a search for knowledge not in books but in things themselves. He showed that the earth was a great magnet with a north pole and a south pole, by comparing it to loadstones made into spheres in which a north and south pole could be found by intersecting lines of magnetism indicated by a needle on the stone. The vertical dip of the needle was explained by the magnetic attraction of the north pole. He showed how a loadstone's declination could be used to determine latitude at sea. He showed how the charge of a body could be retained some time by covering the body with some non-conducting substance, such as silk. He distinguished magnetism from electricity, giving the latter its name. He discovered that atmospheric conditions affected the production of electricity, dryness decreasing it, and moisture increasing it. He expounded the idea of Copernicus that the earth revolves around the sun in a solar system. However, the prevailing belief was still that the earth was at the center of the universe.
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.
As Attorney General, Edward Coke was impassioned and melodramatic. He once described the parts of the penalty of treason as follows: being drawn to the place of execution reflected the person's not being worthy any more to tread upon the face of the earth; being drawn backward at a horse tail was due to his retrograde nature; being drawn head downward on the ground indicated that he was unfit to breathe the common air; being hanged by the neck between heaven and earth indicated that he was unworthy of either; being cut down alive and his privy parts cut off and burnt before his face indicated he was unworthily begotten and unfit to leave any generation after him; having his bowels and inners taken out and burnt indicated he had inwardly conceived and harbored such horrible treason; his head cut off, which had imagined the treason, and his body to be quartered and the quarters set up to the view and detestation of men a prey for the fowls of the air. Coke was subsequently elevated to the position of Chief of Common Pleas and then to Chief of the King's Bench. But there Coke propounded a doctrine of the supremacy of the law over the king as well as over Parliament. For instance, Coke would not agree to stay any case in which the king had a concern in power or profit, to consult with him. But the other eleven justices did agree. Since James I believed in the divine right of kings, he therefore dismissed Coke from his position as Chief Justice of the King's Bench. James even believed that he could suspend any law for reasons known only to him and issue proclamations that were not limited to the reinforcement of old laws, but made new offenses with punishment of fine and/or imprisonment.