Giant said:
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.
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.
) . I will add more if If more questions pop in !
