While diving deep into the proofs of the Fundamental Theorem of Calculus by Barrow, Leibniz, and Newton, I stumbled upon an intriguing topic – the historical development of calculus. I found myself wishing that my teachers had taken this historical approach, which brings added depth and interest to the cold precision of math. In this blog post, I’ll give you a taste of this journey as I delve into C.H. Edwards, Jr.’s book, ‘The Historical Development of the Calculus’.

It all began with the Greeks and their investigative forays into geometry. They calculated areas and volumes of geometric forms, such as rectangles, triangles, and cones, employing methods like dissection and exhaustion. The latter method, akin to our modern concept of taking a limit, sought to calculate an area by inscribing a sequence of polygons within a shape, the total areas of which would converge to the area of the containing shape. In addition, they proved numerous theorems relating to geometry.

The Greek method of exhaustion was evolved into a technique of compression by the renowned Archimedes. He applied this method to a range of forms – circles, parabolas, spheres, cylinders, spirals, conoids, and spheroids – to calculate their areas and volumes. To illustrate, Archimedes didn’t limit himself to just inscribed polygons; he considered both inscribed and circumscribed polygons. Noted how similar it is to take the limit.

The Measurement of a Circle

Despite their pioneering work, the Greeks faced limitations in their approach to mathematics. They insisted on absolute logical rigour, often sidestepping concepts that they couldn’t formulate precisely. As such, irrational numbers and infinity were absent from their mathematics. Moreover, they treated geometry and arithmetic as distinct disciplines, lacking an abstract language for performing general symbolic manipulations. Their approach leaned towards visual graphs and ratios to solve problems.

Fast forward to the Middle Ages, when scholars began studying motion. The Merton Rule of uniform acceleration ({\displaystyle s={\frac {1}{2}}(v_{0}+v_{\rm {f}})t}) – which you may remember from high school physics – was a key outcome of this era. A subsequent scholar, Nicole Oresme, expanded on this work by introducing graphical representations for varying intensities.

“Every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing,…Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally.”

You might recognize this representation – it bears a striking resemblance to integration. Oresme’s innovative ideas included:

  • Measuring diverse types of physical variables using line segments
  • Recognizing functional relationships between variables
  • Graphically representing these functional relationships, a precursor to modern coordinate systems
  • Conceptualizing a process akin to “integration” or continuous summation to calculate distance as the area under a velocity-time graph

If you’ve looked at Barrow’s graphical proof of the fundamental theorem of calculus, you may appreciate how he leveraged the mathematical tools and methods available in his time. He extensively utilized geometry, similar triangles, and ratios while avoiding the use of heterogeneous quantities in computing ratios. In my view, our journey through math history reveals a fascinating shift from static geometric properties to a dynamic understanding of quantities, requiring new tools to navigate. This shift illuminates the connection between dynamic and static views – for instance, the accumulation of velocity (dynamic) corresponding to the total distance travelled (static).

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