Desargues:

(1591-1661) French mathematician & engineer, one of the founders of projective geometry, born in Lyon, his family were devoted servants to the crown; worked as an architect, and as a tutor, engineer & technical consultant for Richelieu.  He planned & built several buildings in Paris & Lyon; engineered a system for raising water in Paris.  His work in perspective & geometrical projections was a culmination of the works of many earlier mathematicians & provided a new synthesis incorporating Renaissance perspective theory & practice.

 

illimitable:

incapable of being limited; limitless; boundless

 

function: *  see Endnote 49

a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.  

 

logarithm:

the inverse operation to exponentiation, just as division is the inverse of multiplication and vice versa.  That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In the simplest case the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 10^3, the "logarithm to base 10" of 1000 is 3.

 

imaginary numbers:

a complex number that can be written as a real number multiplied by the imaginary unit i, ( i defined by its property i^2 = −1; square of an imaginary number bi = −b^2. (eg 5i is an imaginary number, its square is −25).  Zero is considered to be both real and imaginary.  Term coined in 17th century as derogatory, regarded as fictitious; the concept gained wide acceptance following the work of Euler and Gauss.

 

complex numbers: * see Endnote 50

number that can be expressed in the form a + bi, where a and b are real numbers, but i is a solution of the equation x^2 = −1;  called imaginary number because no real number will satisfy this equation.  For the complex number a + bi, a is called the real part, and b is called the imaginary part.  

 

Cardanus :

(1501-1576) Italian polymath, mathematician & gambler, physician, scientist, philosopher, writer; influential mathematician of the Renaissance, key figure in foundation of probability, produced earliest  binomial coefficients & binomial theorem in the west.  Invented/described numerous mechanical devices (including combination lock, gyroscope gimbals, universal joint).  In mathematics (algebra) he is first European to systematically use negative numbers and he acknowledged the existence of imaginary numbers.

 

Infinite series & Newton's binomial: * see Endnote 51

the sum of a series in which an operation (addition, subtraction, multiplication, division)  is conducted an infinite number of times, following a specific rule; the sum may either converge towards a limit or diverge (get larger & larger).  Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent which he discovered in 1665; it was valid for any arbitrary rational values of n.  With this formula he was able to find infinite series for many algebraic functions (functions y of x that satisfy a polynomial equation p(x, y) = 0). This led Newton to infinite series for integrals of algebraic functions.

 

differential geometry:

a mathematical discipline using techniques of differential & integral calculus, linear & and multilinear algebra, to study problems in geometry; the theories of plane & space curves and surfaces in 3D Euclidean space formed the basis for development of differential geometry during the 18th & 19th century.  It has grown into a field concerned with the geometric structures on differentiable manifolds, closely related to differential topology.

 

definite integral: * see Endnote 52

the evaluation of the indefinite integral between 2 limits, representing the area between the given function and the x- axis, between these two values of x; an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.  Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other.

 

aggregate (new number): * see Endnote 53

(aka set theory) branch of mathematical logic that studies collections of objects (or sets), its terms can be used to define nearly all mathematical objects; modern study of set theory initiated by G. Cantor & R. Dedekind in the 1870s.  Later developments include the discovery of Russell’s paradox, and the axioms to avoid such (best-known Zermelo–Fraenkel axioms).

 

general integral:

principles of integration formulated by Newton & Leibniz late 17th century, who visualised the integral as an infinite sum of rectangles of infinitesimal width; Riemann gave a rigorous mathematical definition (based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs); beginning in the 19th century, sophisticated notions of integrals appeared, where the type of the function & its domain were generalised.

 

functions into infinite series:
a series, where the summands are not just real or complex numbers but functions.  Examples of function series include the Laurent series (doubly infinite power series) which is a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.