Ok! So I know Newton and Leibniz invented Calculus independently and today a major part of maths requires calculus. So what mathematics was done BEFORE Newton?? (I mean the type of research.) Any history book suggestions would be priceless. Also there are methods of integration taught like integration by partial fraction and integration by parts etc. Later I found there was a topic in higher algebra by hall and knight about partial fractions. So that topic belongs to algebra first and later applied to evaluate integrals so I'm guessing whatever calculus we learn the basic algebraic and trigonometric (And geometric?) foundation was available to him so defining differentiation and integration was the most crucial part? Did he define integration as limit of sum?? More question on the same topic are welcome provided they don't drag the thread towards philosophy (moderators are there to take care of that ) . I will add more if If more questions pop in !!!
Much trigonometry (useful in both astronomy, mining (how to dig and orient shafts) and navigation). Also, a number of results that afterwards would be integrated into calculus. Various forms of equation solving were developed as well. The Wikipedia page on history of mathematics is quite good, with a number of cited scholarly works.
You should clarify your objective. Are you trying to understand what Newton knew? -or are you trying to understand what humanity knew (and perhaps forgot) before the time of Newton?
I typed a long reply and something bad happened to my phone. Sigh. I wanted to know what humanity knew! as you asked. Mathematicians din't knew any generalised methods to find area but did they know the importance of area as in area under the curve? For eg. today we know area under the uniform P-V curve will give us the work done. Lots of expressions of energies are found out by integration for eg. expression of kinetic energy is found by integrating momentum wrt velocity may it be m.v or relativistic momentum. Also my apologies for spelling and grammar blunders in the original post.
Morris Kline wrote one of the more detailed math history books that I've come across. It's three volumes and about 1200 pages, but he gets to Newton and Leibniz in the first volume. I've heard good things about Stillwell's book as well. They both have the desired trait of being mathematicians themselves.
They certainly knew the importance of area. Integration was around quite some time before differentiation, although integrals were hard to calculate. There are a few reasons why they would want to calculate integrals, one of these is the famous problem on the quadrature of the circle. Investigations in this direction required mathematicians to calculate areas of curves figures. Of course, the problem wasn't solved until the 19th century! Another reason was more practical. For example, fluids like wine and beer were stored in barrels. The traders liked to do the following: put a stick all the way to the bottom of the barrel, see where the stick is wet and where it is dry, from this informate calculate how much fluid was still in the barrel. However, due to the complicated form of the barrel (it wasn't a perfect cylinder!) this was a difficult problem at the time. I hate to be the one starting philosphy in this thread, but you should realize that the early mathematicians suffered from a lot of philosophical misconceptions which made development of the mathematics difficult. First, there was a strict division between algebra and geometry. This started the time of the Greeks. They thought the only numbers worth considering were natural numbers and propertions of natural numbers (= positive rational numbers). Nevertheless, irrational numbers show up very easily: indeed just take the diagonal of a square of unit length! This was a shock to the greeks and from then on they stopped trusting algebra. They certainly weren't ready to treat irrational numbers as numbers! It took quite some time for the notion of irrational numbers to become accepted. Also, there is this issue of notation. In the past, notation was simply horrible. For example, people in Europe used Roman numerals for a long time, even after Indian-Arabic numerals were introduced in Europe. Now, Roman numerals are horrible to calculate with. There are many other examples. I highly encourage you to investigate some notations that people used in the past. You'll see immediately how some bad notational habits really made it more difficult to do mathematics. What did the people knew before Newton? They were really big in geometry. The standard tomes were the Elements by Euclid and Conic Sections by Apollonius. Those heavily influenced mathematics. Note again that using algebra in geometry was a bit of a taboo, so parabolas (for example) had to be dealt with geometrically instead of our definition of ##y=x^2## which is truly easier and makes a lot of theorem trivial or simply computational! It was only when Descartes and Fermat appeared that the link between algebra and geometry was restored. Spherical Geometry was also big, especially due to the Arabs who wanted to find the direction to Mecca. Astronomy also provided a motivation to do geometry. Although they did not do integration, Archimedes was able to figure out the surface area of a sphere and special portions of the sphere. Other volumes and surface areas were known and were proven by the method of exhaustion (which looks very much like some kind of integration process!) As for algebra, the goal there was primarily solving equations. The Greeks considered Diophantine equations where you had to find natural numbers as solutions to equations. An example are Pythagorean triples. In the middle-ages, they managed to solve polynomials of third degree, but they had to make use of some weird number whose square is ##-1##. Those numbers were not understood or really accepted until Gauss in the 19th century. People before Newton knew various approximations to functions, like some special series. For example, the Taylor series decomposition for the sine function was already known by the Indians. These formulas were later rediscovered by the Europeans.
It seems to be a slight exaggeration to say that Newton and Leibniz invented calculus. They developed it into a more coherent theory, but they didn't start from scratch. Fermat basically already had some stuff about derivatives in some form and solving for the max or min by finding a horizontal tangent line. Some versions of integration had already been done, too. Apparently, Newton's teacher, Barrow, already knew about the fundamental theorem of calculus.
The development of calculus like you see in modern textbooks didn't come about until Cauchy wrote a calculus text in the early 19th century. The concept of limits was developed rather haphazardly until then, and calculus had a very uncertain logical basis as a result. It wasn't until the work of Weierstrass that notions like limit and continuity were put into a rigorous basis, and calculus was re-cast in these terms, discarding or de-emphasizing the previous approach of using 'infinitesimals'. In fact, Newton's "Principia" was based mostly on geometrical reasoning, and it wasn't until later that Newton wrote other papers which laid out his approach to "fluxions" (differential calculus) and integration. A lot of the theoretical basis for these methods had apparently been done by Newton as a young undergrad at Cambridge, but he had kept his work to himself and didn't publish anything, which is one reason why the great controversy between Newton and Leibniz broke out over priority in the discovery of calculus.
Thank you. I found them on the net. That explains a lot thank you. Can you please recommend further reading on this topic? How do you know so much? He sure deserves more credit than this. hehe. Thank you He had conflict with someone I think. I have incomplete knowledge. Many people after Newton perfected his work but that was after they knew in what direction to think. But after reading the replies I'm starting to think even Newton knew in which direction to think. But that was most difficult job at that time! This was helpful so I should read more history after the Newton's age too!!
What is impressive about Newton's early work is that he did it on what essentially was a long break due to the closure of Cambridge U after an outbreak of the plague. At this time, Newton had gained his B.A. and had already discovered the generalized binomial theorem. During his long break, Newton developed his ideas about the calculus, gravitation, and optics. At this time, Cambridge was no research university like modern institutions, it was primarily run to turn out vicars for the Anglican Church, and graduate students were required to take holy orders upon graduation. Newton was also secretly interested in various aspects of heretical theology and alchemy, and if these interests had become known to the masters at Cambridge, Newton would have faced expulsion, or worse. However, Newton was granted an exception by Charles II and was able to forgo ordination. Given that Newton spent a great deal of his time on theological studies and various alchemical experiments, it is even more remarkable that the quality of his mathematical and physical work was at the exceedingly high level that it was.
Calculus was "in the air," as they say. The first proof of the Fundamental Theorem of Calculus was provided before Newton, by Newton's teacher Isaac Barrow. Wikipedia has a good writeup. http://en.wikipedia.org/wiki/Isaac_Barrow
Not for calculus. The key elements of calculus were already there. One of his big contributions to calculus was power series. Also, his binomial theorem. But calculus isn't the only thing Newton did. To my mind, his most important work was in physics.
Just about everything was "in the air" before someone finally manages to nail it down. When we give someone credit "for" a scientific discovery, we're nearly always giving them credit for expressing it clearly for the first time, so that other scientists can use it as a springboard for further investigations: "Hey - look at this! This may be the way forward!". This is one reason why Nobel prizes are usually only awarded long after the fact... It takes time to see whether the insight really has legs.
William Durham's book Journey through Genius illustrates historical mathematical development before (and after) Newton. My favorite chapter before Newton outlines the development of algebra, specifically solving the general cubic and quartic equation by Italian mathematicians, culminating in Ars Magna in the sixteenth century. I could not do the topic justice relating these stories. Please read chapter 5 in Journey through Genius. It is amusing today, although the contentiousness of the Italian mathematicians would prevent them in seeing the humor. Also I think early arguments in probability theory and the sharing of winnings in interrupted contests may have preceded Newton. Names like Fermat, Chevelier De Mer, Pascal, come to mind.