# The tangled origins of the Leibnizian Calculus

During the first decades of the seventeenth century mathematics experienced a period of intense activity. There were two problems that attracted the attention of the mathematicians of that time: the determination of tangents and the calculation of areas. Descartes and Fermat contributed to the development of these issues, initiating the Analytical Geometry in the 1630s. It was Descartes who gave a method for determining a normal line to a curve, a technique that was later improved with the algebraic methods of Huddle in 1659. Fermat also dealt with these issues, determining tangents and quadratures as well as the determination of maxima and minima. Previously, Cavalieri had computed some areas using the method of indivisibles, perhaps inspired by the work of Oresme in the fourteenth century. The culmination of all these works took place when Isaac Barrow managed to prove that these two problems (tangents and areas) were reciprocal: the Fundamental Theorem of Calculus. All this prompted the study of infinitely small magnitudes, that is, what we know today as Differential Calculus. Isaac Newton was the first main builder of the theory with the introduction of the concept of fluxion. A few years later, Gottfried Wilhelm Leibniz followed the works of Newton by introducing the concept of differential, based on what he called the differential triangle. This idea was possibly inspired by some medieval geometric works, particularly on discussions about contingency angle, or the angle between a curve and its tangent at a point, which happen to be always lower than any rectilinear angle at that point. This serves to identify any continuous curve with its tangent at a certain neighbourhood of each point. Although at the time a fierce controversy arose about the inventor (or discoverer) of the differential calculus, it is now universally acknowledge that its paternity is due to Newton. However, Leibniz’ notation and techniques are the ones who are really used today. In any case, Leibniz himself acknowledged the priority of Newton from the first moment.

The work of Leibniz and his research, carried out between 1676 and 1680, along with its historical background, is the object of study of this book. With special emphasis on the many vicissitudes that took place around Leibnizian calculus. The book is organized into 11 chapters, where both the historical context that led to the birth of the Calculus and its revolutionary consequence are studied. Chapter 8 deserves special attention. In this chapter, entitled The Universal Characteristic, the author shows that the spirit and motivation of the new ideas of Leibniz follow the guidelines of the Majorcan philosopher of the thirteenth century, Ramón Llull, who designed a "universal" system of symbols, created to interpret different human situations. This fact presents Leibniz as a precursor of logic symbolic.

The book concludes with six appendices, analyzing some concrete aspects of the mathematical work of Leibniz. Among them, the fourth (Appendix D) deals with the controversy between Leibniz and Newton on the invention of calculus, a controversy raised by Newton, as Leibniz himself admitted the authorship of the former. In Appendix E some applications of Leibniz’ techniques are applied to some geometric problems of the ancient Greeks: conic sections and mechanical curves, as well as problems associated with them (squaring the circle, doubling the cube, among others). The reading of the book requires some concentration, but the fruits obtained are worthwhile. It is a deep and thorough treatise on the origins of calculus of Leibniz as well as its consequences (some of them revolutionary, using the words of the author) in the mathematics of his time. The issue is very carefully edited, what makes the reading even more enjoyable.

**Submitted by Juan Tarres Freixenet |

**20 / Jun / 2016