However, in the ensuing years following these discoveries, there was relatively little progress in terms of the further refinement of the definition of Pi, with Western Europe largely entering the medieval period, which was met with comparatively little advancement. This continued until around the Renaissance and thereafter, during which time mathematicians made some significant advancements in the understanding of Pi.
The advancement of Pi that occurred during the period of around 1500-1800 was due to a number of factors, the largest of which was most likely the discovery and use of infinite series by mathematicians such as Isaac Newton and Gottfried Leibniz. The infinite series method allowed mathematicians to more quickly sum up the integers in a sequence of numbers, which allowed these mathematicians to make significant advancements in understanding the value of Pi, due to the fact that they could sum up the value of many diameter and circumference relationships. Isaac Newton, for example, was able to approximate a 15-digit value for Pi using an infinite sequence, which was considered a significant advancement at the time (Arndt & Haenel, 2001). It was also around this time that the symbol for Pi, or π, was first adopted. This was proposed by the mathematician William Jones, who wrote the book A New Introduction to the Mathematics in 1706, in which he mentions the idea of the pi symbol (Arndt & Haenel, 2001). The discoveries related to Pi during this time were certainly ground-breaking, and they led to a number of other mathematical advancements in fields such as calculus, which was discovered by Newton around this time. However, the real advancement in terms of our understanding of Pi has come in the age of computers, as we now have machines that are able to do millions of computations per second, and can much more easily approximate values for Pi than humans ever could.
