By this time, what was known about mathematics included fractional exponents, trigonometry in terms of arcs of angles, long division, square root symbol, decimal fractions, methods for solving cubic equations, trigonometry in terms of ratios of sides of a right triangle, equal sign, plus and minus signs, and a consistent theory of imaginary numbers.
John Napier, a large Calvinist landholder in Scotland who had built his own castle, did mathematics in his older years. He explored imaginary numbers, which involve 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 with a common base could be done by 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. In 1622, Willliam Oughtred invented the slide rule for calculations.
Galileo Galilei was a professor of mathematics at the University of Padua in Italy and was later a protege of the powerful Medici family. He conducted experiments, e.g. throwing objects off the tower of Pisa in 1590 to show that all, whether light or heavy, fall at the same rate.
This disproved the widely held belief that heavier objects fall faster than light objects. He reasoned by induction from experiments that the force of gravity has the same effect on all objects regardless of their size or weight. His law stated that the speed of their descent increases uniformly with the time of the fall, i.e. speed [velocity] = gravity"s acceleration multiplied by time. This was a pioneering mathematization of a physical phenomenon.
From his observation that an object sliding along a plane travels increasingly farther and slows down at a decreasing rate as the surfaces become smoother and more lubricated, he opined that the natural state of a body in motion is to stay in motion, and that it is slowed down by a resistant force, which he called ?friction?. He conceived of the air giving a frictional force to an object moving through the air. From his experiments showing that a rolling ball rolls up a plane farther the lesser the slope of the plane, he intuited that if the plane were horizontal, the ball would never stop rolling except for friction. He opined that bodies that are at rest stay at rest and bodies that are in motion stay in uniform motion (?inertia?), unless and until acted upon by some force. This was a radical departure from Aristotle"s theory that any horizontal motion requires a prime mover. Galileo drew a graph of distance versus time for the rolling ball, which indicated that the distance traveled was proportional to the square of the time elapsed.
He put his ideas of vertical and horizontal motion together to explain the movement of projectiles, which travel horizontally, but also fall downward vertically. He realized that the movement of a projectile involved a horizontal impetus of projection and a vertical force of gravity, each being independent of the other, but acting simultaneously, instead of sequentially. He demonstrated that a projectile follows the path of a parabola, instead of a straight line, and that it descends a vertical distance which is proportional to the square of the time taken to fall. That is, a thrown object will strike the ground in the same amount of time as an object simply dropped from the same height. This suggested that gravity was a constant force.
Galilieo described mathematically the motion of a lever such as a seesaw in which the weight on one side multiplied by its distance from the fulcrum is equal to the weight on the other side multiplied by its distance from the fulcrum.
Galileo determined that a pendulum, such as a hanging lamp, swings back and forth in equal intervals of time. He measured this time with water running through a tube; the weight of the water was proportional to the time elapsed. Also, pendulums with equal cord length swing at the same rate, regardless of the substance, weight, or shape of the material at the end. So a pendulum could be a mechanical clock. Galileo knew that ice floated on water because ice is less dense and therefore lighter than water. It had formerly been thought that ice was heavier than water, but floated on water because of its shape, especially broad, flat-bottomed pieces of ice.
The telescope was invented in 1608. The next year, Galileo built a greatly improved telescope to observe bodies in the skies. He observed that the spots on the moon had shifting illumination and that the moon"s perimeter had a jagged outline. From this he deduced that the surface of the moon had mountains, valleys, and craters much like the earth, and was illuminated by reflected light. He noticed that the planet Jupiter had moons...o...b..ting it in a manner similar to the orbit of the Earth"s moon. He observed that when the planet Venus was very small it had a round shape and when it was very large, and therefore nearer the earth, it had a crescent shape. Also, Venus progressed through periodic phases of increasingly wide crescent shapes in a manner similar to the phases of crescent shapes of the Earth"s moon. He realized that these features of Venus could be explained only if Venus revolved around the sun, rather than around the earth. This finding added credence to the Copernican theory that the earth and all planets revolve around the sun.
But church doctrine that the sun revolved around the earth was supported by the Biblical story of G.o.d making the sun stand still to give additional sunlight on a certain day so a certain task could be completed that day. Galileo argued against a literal interpretation of the Bible, so he was denounced by the church. His finding of sunspots on the sun conflicted with church doctrine that the celestial bodies such as the sun were perfect and unblemished. His observation that certain sun spots were on certain locations of the sun, but changed location over time, suggested that the sun might be rotating. He observed that when air was withdrawn by a suction pump from the top of a long gla.s.s tube whose lower open end was submerged in a pan of water, the water rose to a height of 34 feet and no higher. This result indicated that the evacuated s.p.a.ce above the water was a vacuum: an empty s.p.a.ce. The notion of a vacuum, a s.p.a.ce where there is nothing or void, was difficult for philosophers to accept. They believed that nature abh.o.r.ed a vacuum and would prevent it. About 1600, Galileo invented the first thermometer by heating air at the top of a tube whose open end was in a bowl of water; as the top end cooled, the air contracted and water rose part way up the tube; the column of water rose or fell with every change of temperature. Galileo invented the compound refracting microscope, which used more than one lens, about 1612.
Galileo"s book on the arguments for and against the Copernican theory was unexpectedly popular when published in 1632. The general public was so persuaded by the arguments that the earth revolved around the sun that Papal authority felt threatened. So Galileo was tried and convicted of heresy and sentenced to house arrest as an example to others who might question church doctrine, even though the seventy year old Galileo recanted and some of the inquisition judges who convicted him believed the Copernican theory and their decision did not a.s.sert the contrary.
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...o...b..t around the earth. Thirdly, in 1619, he found that the square of the time for each planet"s...o...b..t about the sun is proportional to the cube of that planet"s mean distance from the sun, so that the farther planets...o...b..t 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 than the moon. This contradicted the church"s explanation that what lies within the moon"s...o...b..t pertains to the earth and is essentially transitory and evil, while what lies beyond belongs to the heavens and is permanent and pure.
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 a.n.a.lytic [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 a.s.sociated algebraic equations. He started the convention of representing unknown quant.i.ties by x, y, and z and known quant.i.ties 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. a.n.a.lytic geometry aided in making good lenses for eyegla.s.ses. The gla.s.s 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 pa.s.ses. 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 G.o.d"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 G.o.d, 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 a.s.sumes 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 gla.s.s, 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 doc.u.ments, 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.
The influx of silver from the New World was a major factor in the second great inflation in England and in the devaluation of money to about one third of what it had been. Also contributing to the inflation was an outracing of demand over supply, and a debas.e.m.e.nt of the coinage.
This inflation benefited tenants to the detriment of their lords because their rents could not be adjusted upward.
There was an increase in bankruptcies. Houses of Correction were built.
As Attorney General, Edward c.o.ke was impa.s.sioned 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. c.o.ke was subsequently elevated to the position of Chief of Common Pleas and then to Chief of the King"s Bench. But there c.o.ke propounded a doctrine of the supremacy of the law over the king as well as over Parliament. For instance, c.o.ke 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 c.o.ke 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.