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 pioneered the scientific method of theory building by observation of phenomena instead of resort to sources such as Aristotle. 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 orbiting 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 God 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 glass 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 space above the water was a vacuum: an empty space. The notion of a vacuum, a space where there is nothing or void, was difficult for philosophers to accept. They believed that nature abhored 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 assert 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 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 than 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.