In this lesson, Auguste Comte delivers a comprehensive comparative analysis of the three major viewpoints underpinning transcendent analysisâor the calculus of indirect functionsâas established by Gottfried Wilhelm Leibniz, Sir Isaac Newton, and Joseph-Louis Lagrange.
Comte argues that while all three perspectives are mathematically equivalent and yield identical results, each possesses distinct advantages. Currently, mathematics remains in a provisional state because it has not yet unified these approaches into a single system, though Comte predicts that Lagrangeâs purely algebraic formulation will eventually achieve this integration. Historically, the root idea of this methodology dates back to the "method of exhaustion" used by ancient Greek geometers to understand curves via inscribed polygons. However, the ancients lacked abstract, systematic rules for determining limits. It was Pierre de Fermat who first sketched the direct formation of transcendent analysis through his method for tangents and maxima/minima, which introduced auxiliary increments that were later suppressed as zero. This approach was later refined into a distinct, systematic calculus with its own notation by Leibniz, while Newton simultaneously developed an equivalent but less adaptable framework.
Focusing extensively on Leibniz's infinitesimal method, Comte explains that it facilitates the creation of equations by decomposing variables into infinitely small elements, or differentials. The relations between these differentials are inherently simpler to discover than the equations linking the original finite quantities. Through specialized analytical processes, these auxiliary variables are ultimately eliminated to solve the problem. To simplify equations, the infinitesimal calculus allows mathematicians to constantly neglect smaller-order infinitesimals relative to larger ones. Comte illustrates this with several historical examples across multiple disciplines:
Geometry (Tangents): A tangent is treated as a secant connecting two infinitely close points. By representing the infinitely small differences in coordinates as dx and dy, mathematicians establish the trigonometric tangent equation t=
dx
dy
â
. The problem is then reduced to a standard analytical procedure to eliminate the differentials.
Geometry (Rectification and Quadrature): The arc length s of a curve is treated as a straight line, yielding the differential equation ds
2
=dx
2
+dy
2
. For squaring curvilinear areas (A), the curve's area is treated as a series of rectangles, leading to the simple differential formula dA=ydx.
Dynamics (Velocity): Varied movement is treated as uniform movement over an infinitely small element of time (t), establishing the differential relation de=vdt to deduce velocity (v) or space (e).
Thermology (Heat Distribution): Spearheaded by Joseph Fourier, the variable distribution of heat in a complex body is determined by treating the object as an assembly of infinitely small rectangular parallelepipeds where the heat flux is constant over an split second of time.
Comte praises Leibniz's framework for its extreme generality, as a single differential equation can represent entire classes of phenomena across diverse objects.
Addressing the logical foundations of Leibniz's method, Comte notes that early mathematicians like the Bernoulli brothers prioritized expanding its applications over proving its baseline logic. Leibniz's own explanationâthat infinitesimals are simply "incomparable" quantities that can be neglected like grains of sand in the oceanâwas profoundly flawed, as it reduced the calculus to a mere approximation. Later mathematicians like Leonhard Euler and Jean le Rond d'Alembert defended the method by showing its consistency with Newton's indisputable logic. Ultimately, Lazare Carnot provided the definitive logical justification for the infinitesimal method through his principle of the "necessary compensation of errors." Carnot demonstrated that while treating curves as straight lines introduces an infinitesimal error into the initial differential equations, the subsequent algebraic and analytical operations introduce an equal and opposite error. These errors systematically cancel each other out, ensuring that the final calculated results are rigorously exact.