By 1650 AD, the combination of a new understanding of Islamic algebra and trigonometry with the excitement of the invention of the telescope and microscope were leading to a lot of new scientific discoveries. The Wars of Religion also got a lot of people thinking about what they really believed and how you could know for sure. Most of these people lived in Europe, where a tremendous amount of wealth was flowing in from all over the world thanks to colonization. As Europe got richer and everywhere else got poorer, more people got good educations and could grow up to be scientists in Europe than in the Ottoman Empire, India, or China. For this reason, many important new discoveries happened in Western Europe during the Enlightenment period.
New awareness of algebra and trigonometry in Western Europe led to the invention of calculus about 1665, independently both in Germany, by Gottfried Leibniz, and in England, by Isaac Newton. Calculus is a mathematical method for calculating the area of more complicated shapes; it's very important for modern engineering. Leibniz also did a lot of work on binary number systems and mechanical calculators.
As an astronomer, naturally Newton was also very much interested in the new glass lenses, and he was also able to use these lenses to develop the science of optics. In 1675, Anton van Leeuwenhoek was the first to use the new microscope to see bacteria.
Newton also studied the work of the earlier European astronomers Copernicus, Galileo, and Kepler. Like earlier scientists, Newton saw mathematics mainly as a way to answer astronomical questions. Newton brought the new calculus together with optics and astronomy to work out the law of gravitation. In 1687, Newton also published the laws of motion.
About 1700, Bernardino Ramazzini figured out that some people's occupations made them sick: miners got cancer from working with coal, lead, arsenic, and iron, and people who worked with dust and fire, like bakers, blacksmiths, printers, and glass-workers, tended to get coughs and lung cancer. Ramazzini also saw that nuns didn't get cervical cancer, but they did get more breast cancer, so both kinds of cancer must be related to whether women had babies.
By 1745, scientists were beginning to do experiments with electricity: Ewald von Kleist invented the Leyden Jar, and European scientists did many experiments with lightning like those of Benjamin Franklin in North America.
Astronomical observations also kept pouring in as people built bigger and more powerful telescopes: astronomers first observed the atmosphere of Venus in 1761, the existence of galaxies and nebulae in 1771, and the planet Uranus in 1781.