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Newton suffered a mental breakdown in 1675 and was still recovering through 1679. In response to a letter from Hooke, he suggested that a particle, if released, would spiral in to the center of the Earth. Hooke wrote back, claiming that the path would not be a spiral, but an ellipse. Newton, who hated being bested, then proceeded to work out the mathematics of orbits. Again, he did not publish his calculations. Newton then began devoting his efforts to theological speculation and put the calculations on elliptical motion aside, telling Halley he had lost them (Westfall 1993, p. 403). Halley, who had become interested in orbits, finally convinced Newton to expand and publish his calculations. Newton devoted the period from August 1684 to spring 1686 to this task, and the result became one of the most important and influential works on physics of all times, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) (1687), often shortened to Principia Mathematica or simply "the Principia."

In Book I of Principia, Newton opened with definitions and the three laws of motion now known as Newton's laws (laws of inertia, action and reaction, and acceleration proportional to force). Book II presented Newton's new scientific philosophy which came to replace Cartesianism. Finally, Book III consisted of applications of his dynamics, including an explanation for tides and a theory of lunar motion. To test his hypothesis of universal gravitation, Newton wrote Flamsteed to ask if Saturn had been observed to slow down upon passing Jupiter. The surprised Flamsteed replied that an effect had indeed been observed, and it was closely predicted by the calculations Newton had provided. Newton's equations were further confirmed by observing the shape of the Earth to be oblate spheroidal, as Newton claimed it should be, rather than prolate spheroidal, as claimed by the Cartesians. Newton's equations also described the motion of Moon by successive approximations, and correctly predicted the return of Halley's Comet. Newton also correctly formulated and solved the first ever problem in the calculus of variations which involved finding the surface of revolution which would give minimum resistance to flow (assuming a specific drag law).

Newton enter Cambridge University study in, after graduation from Cambridge University 23 years old at the age of 18, because the plague spread at that time, hid from and went home the township. These two years in the hometown ( 23- 25 years old) He all one's life creativity period of summit most this, establish three great discovery - gravitation, optics, important foundation of calculus in the all one's life such as he too.
After the plague prevails, Newton returns to Cambridge University. One year, teacher bar of him is it retire to work, Newton take over mathematics lecture Professor offer as a gift scarce, become the professor of Cambridge University formally.
Newton, with his good friend Halley in 1684 (have tail bear Halley, calculate Halley's Comet cycle later, name after he scientist Halley of comet, such as right picture) Explanation, how to calculate out planetary sports and oval track by way of mathematics. Halley thinks Newton's research is a very great break-through, the principle announcement common people advising him to study at once, published and became a book later, this book is " mathematics principle of natural philosophy ", do unprecedentedly great in the history of science, usually abbreviate as " the mathematics principle ".

牛頓18歲時進入劍橋大學就讀，23歲從劍橋大學畢業後，當時由於黑死病蔓延，便躲回家鄉。在家鄉的這兩年（23～25歲）是他這一生創造力最顛峰的時期，也奠定了他一生中三大發現－萬有引力、光學及微積分的重要基礎。
黑死病流行過後，牛頓重回劍橋大學。第二年，他的老師巴勞退休，牛頓接任數學講座教授的遺缺，正式成為劍橋大學的教授。
牛頓在1684年跟他的好朋友哈雷（不是無尾熊哈雷，是後來算出哈雷彗星週期，並以他的名字命名彗星的科學家—哈雷、如右圖）解釋，如何用數學的方法，計算出行星的運動和橢圓形軌道。哈雷覺得牛頓的研究是很大的突破，馬上勸他將研究的原理公告世人，後來出版成為一本書，這本書就是《自然哲學的數學原理》，是科學史上空前偉大的著作，通常簡稱為《數學原理》。


Entered the Cambridge university to attend when Newton was 18 years old, 23 years old since the Cambridge university graduated, at that time because the Black Death spread, hiding the country of going home then.Are his this whole life creativities at this two years(23-25 years old) of home town the period of the top also established his whole life most in three big detection-the universal gravitation, optics and the important foundation of the calculus.
The Black Death spreads later on, Newton is heavy to return to Cambridge university.The second year, his teacher Ba's Lao retires, Newton takes office mathematics academic chair professor of the Yi lack, becoming the professor of the Cambridge university formally.
Newton explains in 1684 and his good friend's Halley(is not the koala Halley, calculate the Halley comet period afterwards, and assign name to the scientist-Halley of the comet by his name, as right diagram), how use mathematics of method, calculate planet of the exercise and the oval orbit.The Halley feels the Newtonian research is a very big breakthrough, advising the principle that he will study to announce people of this world right away, publishing to become a book afterwards, this book is 《the mathematics principle of the natural philosophy 》, before being science history sky great work, usually call for 《mathematics principle 》.

When Newton 18 years old enters Cambridge University study, 23 years old after Cambridge University graduation, at that time as a result of black death spread, then hid the hometown.In hometown these two years (23~25 years old) are his this life creativity most summit times, also has established in a his living three big discovery - gravitations, optics and the calculus important foundation. Black death popular from now on, Newton re-entry Cambridge University.The second year, his teacher the Pakistan fatigue retirement, Newton takes over mathematics course professor's vacancy, becomes Cambridge University officially professor. Newton in 1684 with his good friend Halley (was not koala Halley, was afterwards figures out Halley's Comet cycle, and by his name naming comet scientist - Halley, like figure to the right) explained, how used mathematics the method, calculated the planet the movement and the elliptic orbit.Halley thought Newton's research will be the very big breakthrough, immediately urges the principle announcement common people who he will study, afterwards published into a book, this book was "Natural philosophy Mathematics Principle", was in the science history the unprecedented great work, usually Jian Chengwei "Mathematics Principle".

When Newton 18 years old enters Cambridge University study, 23 years old after Cambridge University graduation, at that time as a result of black death spread, then hid the hometown.In hometown these two years (23~25 years old) are his this life creativity most summit times, also has established in a his living three big discovery - gravitations, optics and the calculus important foundation.Black death popular from now on, Newton re-entry Cambridge University.The second year, his teacher the Pakistan fatigue retirement, Newton takes over mathematics course professor's vacancy, becomes Cambridge University officially professor.Newton in 1684 with his good friend Halley (was not koala Halley, was afterwards figures out Halley's Comet cycle, and by his name naming comet scientist - Halley, like figure to the right) explained, how used mathematics the method, calculated the planet the movement and the elliptic orbit.Halley thought Newton's research will be the very big breakthrough, immediately urges the principle announcement common people who he will study, afterwards published into a book, this book was "Natural philosophy Mathematics Principle", was in the science history the unprecedented great work, usually Jian Chengwei "Mathematics Principle".