Vincenzo Riccati (1724): De usu motus tractorii in constructione polygonorum ex dato peripheriae arcubus
In Vincenzo Riccati's seminal work "De usu motus tractorii in constructione polygonorum ex dato peripheriae arcubus" (On the Use of the Tractional Motion in the Construction of Polygons from Given Arcs of the Circumference), he introduced several novel mathematical calculations and concepts related to the hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions. Here are the key novel mathematical calculations and concepts presented in this work:
Infinite series definitions of hyperbolic functions
Riccati defined the hyperbolic sine function as the infinite series: sinh(x) = x + (x^3/3!) + (x^5/5!) + (x^7/7!) + ...
He defined the hyperbolic cosine function as the infinite series: cosh(x) = 1 + (x^2/2!) + (x^4/4!) + (x^6/6!) + ...
These infinite series representations were novel at the time and introduced a new class of transcendental functions.
Properties of hyperbolic functions
Riccati explored some basic properties of the hyperbolic sine and hyperbolic cosine functions, such as: sinh(-x) = -sinh(x) (odd function) cosh(-x) = cosh(x) (even function)
These properties were novel and demonstrated the unique characteristics of these functions.
Geometric interpretation and applications
Riccati introduced the concept of "tractional curves," which were curves defined by the hyperbolic functions.
He demonstrated how these curves could be used to construct polygons from given arcs of a circle, applying the hyperbolic functions to geometric problems.
This geometric interpretation and application of hyperbolic functions was a novel concept at the time.
Relationship between hyperbolic and circular functions
Riccati explored the relationship between the newly introduced hyperbolic functions and the well-known circular functions (sine and cosine).
He showed how the hyperbolic functions could be viewed as an extension of the circular functions beyond the unit circle.
This connection between hyperbolic and circular functions was a novel concept and laid the foundation for further study and exploration.
Expansion of the trigonometric realm
By introducing the hyperbolic functions, Riccati expanded the realm of trigonometric functions beyond the traditional circular domain.
This opened up new avenues for mathematical exploration and applications, particularly in areas such as geometry and trigonometry.
The expansion of the trigonometric realm was a novel and significant contribution.
While Riccati's work did not provide a comprehensive theoretical framework or explore the full extent of hyperbolic functions, the introduction of these novel mathematical calculations and concepts laid the groundwork for their further development and study by subsequent mathematicians.
In the context of hyperbolic geometry, as the number of sides of a regular polygon increases towards infinity, the polygon approaches the shape of an ideal hyperbolic circle or a horocycle, which can be considered an "infinite-sided polygon" or an "infinigon" in a theoretical sense.
Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate of Euclidean geometry is replaced by the hyperbolic parallel postulate, which states that through a point not on a given line, there are infinitely many lines parallel to the given line.
In hyperbolic geometry, the concept of a circle is different from the Euclidean notion of a circle. A hyperbolic circle is defined as the set of points equidistant from a given point (the center) and a given line (the polar line or absolute). As the radius of a hyperbolic circle tends to infinity, it approaches a horocycle, which is a curve that is everywhere equidistant from a given line (the absolute) but has no center.
Now, consider a sequence of regular polygons inscribed in a hyperbolic circle or a horocycle, where the number of sides increases towards infinity. As the number of sides increases, the polygon approaches the shape of the hyperbolic circle or the horocycle more and more closely. In the limit, as the number of sides approaches infinity, the polygon essentially becomes indistinguishable from the hyperbolic circle or the horocycle.
This limiting case of an "infinite-sided polygon" is sometimes referred to as an "infinigon" or an "apeirogon" in the context of hyperbolic geometry. It represents a theoretical construct that helps understand the properties and behavior of hyperbolic circles and horocycles, which are fundamental concepts in hyperbolic geometry.
However, it is important to note that while this "infinite-sided polygon" or "infinigon" is a useful theoretical construct in hyperbolic geometry, it does not represent a true polygon in the strict Euclidean sense, as it violates the definition of a polygon having a finite number of sides and vertices.
Hyperbolic geometry is indeed used to understand and conceptualize an "infinite circle" or an "infinigon," which can be thought of as a theoretical construct or a limiting case of a regular polygon with an infinite number of sides.
In hyperbolic geometry, the concept of a circle is different from the Euclidean notion of a circle. A hyperbolic circle is defined as the set of points equidistant from a given point (the center) and a given line (the polar line or absolute). As the radius of a hyperbolic circle tends to infinity, it approaches a horocycle, which is a curve that is everywhere equidistant from a given line (the absolute) but has no center.
By considering a sequence of regular polygons inscribed in a hyperbolic circle or a horocycle, where the number of sides increases towards infinity, mathematicians can study the properties and behavior of these "infinite circles" or "infinigons" in the context of hyperbolic geometry.
As the number of sides of the polygon increases, it becomes an increasingly better approximation of the hyperbolic circle or horocycle. In the limit, as the number of sides approaches infinity, the polygon essentially becomes indistinguishable from the hyperbolic circle or horocycle, allowing mathematicians to understand and work with these "infinite circles" as theoretical constructs.
This concept of an "infinite circle" or "infinigon" in hyperbolic geometry is not just an abstract idea but has practical applications in various fields, including:
Relativity theory and cosmology, where hyperbolic geometry is used to model the geometry of space-time.
Computer graphics and computer-aided design (CAD), where hyperbolic circles and their approximations are used for modeling and rendering curved surfaces.
Geometry processing and mesh generation, where hyperbolic circles and their approximations are used for creating high-quality meshes and representations of complex shapes.
So, in essence, hyperbolic geometry provides a mathematical framework for understanding and working with the concept of an "infinite circle" or "infinigon," which is a useful theoretical construct with practical applications in various domains.
