Have you ever wondered about the origins of Calculus? About the role of Gottfried Wilhelm Leibniz? Do you want to learn about more than just the facts on Leibniz’s life, but also think about his results the way he did? Then The Tangled Origins of Leibnizian Calculus might be the book for you. This book contains the historical details on Leibniz’s life and many of his mathematical results. What sets it apart from many other texts on the subject is the inclusion of proofs, presented using the mathematical notation that was common for the period. The author argues that by presenting material in this manner he is giving the reader better insight into how Leibniz and his contemporaries thought about Calculus and practiced mathematics in his time.
The author argues that using modern notation, as many texts on this subject do, is a distortion of the facts. He cites examples where the use of modern notation makes it seem like individuals had discovered mathematics well before its time. As an example, the author shows that while one of Leibniz’s results appears to be the formula for integration by parts, Leibniz’s reasoning had nothing to do with integrals. I saw some benefit in including the original notation in this book; it did help me have a better grasp of the way mathematicians of this time thought about Calculus. That being said, it was also quite burdensome at times, since I was not familiar with this notation and often had to look back several times to be sure I correctly understood an argument. This is fine for someone who is comfortable with the results of Calculus but might not be the best approach for a student still struggling with the concepts of Calculus that we teach today.
I think Tangled Origins could be useful for educators who teach Calculus on a regular rotation and would like to enrich their lectures by being able to provide their students with insights into why we study Calculus the way we do today. For many of my students, the algebraic computations of derivative and integrals are completely detached from the geometric concepts of tangent lines and areas under a curve. They often forget that the derivative is associated with the slope of the tangent line, because for them this is just another memorized fact. In fact, the author argues, one of Leibniz’s own contributions to Calculus is what has enabled us to teach a first year college student to obtain, with an algorithmic computation, results that were difficult for Leibniz and his contemporaries. This has helped me emphasize to my students the importance of geometry as it relates to the historical development and original motivation for the essential concepts of Calculus.
The author states that he has five goals: to give a reasonably accurate account of Leibniz’s creation and development of Calculus, to present results the way Leibniz thought about them himself, to illustrate how Leibniz motivated a paradigm shift in mathematics, to show that the technical characteristics of Leibniz’s Calculus were more profound than his actual results, and to present another view on the priority dispute between Leibniz and Newton on the discovery of Calculus. This makes the book different from many other historical studies of this topic. It does not focus solely on mathematics but delves into Leibniz’s contributions to logic and philosophy. These endeavors significantly shaped Leibniz’s approach and contributed to his thoughts on Calculus.
For comparison, I looked at Leibniz in Paris 1672–1676, by Joseph E Hofmann, a classic study of Leibniz’s development. Hofmann’s approach is significantly different; he focuses primarily on contributions to Calculus. I felt that Hofmann’s text seemed a bit more focused. He presents Leibniz’s mathematical contributions to Calculus by studying Leibniz’s communications with other mathematicians at the time.
By comparison, one shortcoming of Brown’s text was that the narrative seems to jump around a bit, going from historical development to the invention of Calculus and then on to long discussions on logic and the culture of Leibniz’s time. Many times the author would begin an argument only to defer the conclusion to a later chapter. It was not until I had finished the book that I was able to reflect and see how the various topics fitted into Brown’s central thesis.
Overall, Brown does a good job of achieving his five goals. His presentation of material is effective at helping the reader understand (as much as one could) how mathematicians in Leibniz’s time thought about Calculus. It is strange to think about being constrained in the ways that mathematicians in Leibniz’s time were constrained. All of their calculations were related to a geometric object. For example, it did not make sense to multiply two quantities that did not have the same units. It was Leibniz’s developments that led to mathematicians study how to manipulate algebraic objects in a logically sound way, without the necessity of a connection to a geometric object. In fact the author argues that perhaps Leibniz’s contributions to symbolism and logic were more profound than his contributions to Calculus.
It was interesting to learn that many of Leibniz’s results contained errors. His errors may have stemmed from the fact that, unlike many of his partners in dialog, Leibniz was not originally trained as a mathematician. The author states that Leibniz’s Calculus is not distinguished by its mathematical ingenuity, but rather by its ability to simplify and generalize previous results.
Finally, the author presents an interesting argument on the priority dispute between Leibniz and Newton on who was the founder of Calculus. The author claims that although Leibniz had invented the differential and integral symbols before he went to London, where he learned of Newton’s work and had access to his manuscripts, it was not until after these trips that the use of these symbols was perfected and the Calculus we use today was invented. This argument is different from the general consensus that Newton and Leibniz developed Calculus independently.
Overall, I think this text is beneficial to anyone who is interested in learning a historically accurate description of the development of Calculus. I think it is beneficial especially to anyone who teaches this material to see how Calculus was motivated and studied in Leibniz’s time as a purely geometric endeavor. By learning about the past we can help to motivate why the geometric interpretations of the algebraic manipulations we teach in Calculus are the foundation of this field.
Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory. More recently she has become interested in ways to introduce undergraduate students to research in all levels of their education, which includes studying the history of the development of calculus.