Physics 111	Prof. P. Berman
Evolution of Scientific Thought

Renaissance and 17^th Century Mathematics

There is a general discussion of Renaissance mathematics in the Boas book. Unlike the other sciences, mathematics did not require any technological advances for it development. We have already seen this in the level of mathematics achieved in ancient Greece. However, in some sense there were technical advances that were needed for mathematics to move to a new level. Thus, the ancient Greeks developed their system of geometric algebra to deal with irrational numbers. They were also hindered by the lack of a place value system, until they adopted the sexagesimal system of the Babylonians. The Babylonians were strong in algebra because of their number system. In the Renaissance and 17^th centuries, the foundations of modern mathematics were developed. To accomplish this, it was necessary to replace mechanical means of computation (abacus) with algorithms for carrying out the calculations, to systematize a place value method for writing numbers with an appropriate notation, and to develop an effective notation for expressing equations. All these tasks were accomplished between 1450 and 1700, allowing for the introduction of negative and imaginary numbers, for the development of calculus, and the first use of functions. These accomplishments form the foundation of our current theories of algebra, differential and integral calculus, functional analysis, and analytical geometry.

I am not going to give you a list of names and dates, but these can be found in many of the references on the reading list. The following is a list of some highlights:

· Simple mathematics needed in commerce. Simple books and summaries of Boethius, Pappus, and Nicomachus remain popular.

· With rise of humanism books of Euclid, Archimedes, Apollonius become available.

· Arabic numbers and methods for calculating arise in the West but are slow to catch on – abacus is still used until as late as 17^th century.

· Peurbach (1423-1461) writes new book on algorithm (that is, methods for multiplying and dividing using paper and pencil).

· Nicolas of Cusa (1401-1464) extends his ideas of continuum by stating that a circle is the limit of a polygon having an infinite number of sides.

· Nicolas Chuquet (end of 15^th century) writes book with new notation – negative numbers were introduced and used regularly, but book remains in manuscript form.

· Luca Pacioli (1445-1517) Summary of ancient and medieval writers – introduces and for plus and minus.

· Leonardo (1452-1514) finds center-of-gravity of pyramid. More practical than theoretical work.

· Around 1500, Italy focuses on consolidation of knowledge, while Germany on new notation.

· Euclid translated into most languages by 1540 – chairs of mathematics in universities from about 1500.

· Tartaglia, Cardan, Ferrari (all about 1530) work on cubic equations, issuing challenges to each other to solve new equations.

· Bombelli in 1572 uses signs for square and cube roots. Admits square roots of negative numbers!

· Still no symbols for unknown quantities, and negative numbers not treated routinely.

· Simon Stevin (1545-1620), not first, but uses decimal fractions, but not quite in modern notation. Before this time, sexagesimal fractions still used routinely in calculations.

· Maggi in 1592 uses modern notation.

· A decimal system of weights and measure was also introduced but in took about 2oo years to be fully implanted (and is still not used in some backward countries like the US).

· Stevin uses + and – signs with M for multiply and D for division. Considers all roots on an equal footing but does not allow for imaginary numbers.

· At this point equations such as were considered as different equations; No abstraction of algebraic operations; Rules and examples rather than formulas; This would change in late 16^th and 17^th century

· Vieta (1540-1603) can be considered one of the founders of modern mathematics. In 1579 his Canon Mathematicus has trig tables using decimal rather that sexagesimal fractions. Calculated p to 10 places and then showed that p could by expressed as an infinite product. After this p always obtained from product or series representation rather than inscribing or circumscribing polygons. Writes several tracts on solving equations using symbols for unknowns, transformations of equations, and solutions of equations using both geometric and algebraic methods – establishes equivalence of ancient Greek geometrical and new algebraic methods.

· Girard (1595-1632) accepts negative and imaginary roots, shows equation of nth order has n roots.

· Descartes in 1637 writes theory of equations similar to Girard with lower case numbers such as x,y,z for unknowns. Essentially notation used today with equal sign used. Mosly examples rather than proofs given. Also introduces algebraic equations for geometrical curves – Cartesian coordinates, beginning of analytical geometry. Influential.

· Fermat (1601-1665) also works on analytical geometry and theory of numbers. Theorems without proof such as has no integral solutions for a,b,c for n>2. This theorem was proved only about 5 years ago with a several hundred page proof using computer and analytical methods. Combinatorial analysis.

· Napier (1550-1617) realizes that logarithms combine arithmetical and geometrical progressions and could be used to multiply and divide large numbers using addition rather than multiplication of numbers. Finds first logarithms of sines and establishes a differential equation for logarithms – first differential equation. Briggs writes new tables. Extremely useful in calculations and basis of (now antique) slide rules.

· By 1630, Fermat had method for finding extrema of curves and tangents to curves – used geometrical means.

· Descartes had method for finding normals to curves.

· Cavalieri (1598-1647) “father of integral calculus”; 1629 filled curves with polygons – others had been using method of exhaustion, so this is not so new. Area equals sum of parallel lines of which it is made of. Restatement of method of exhaustion using indivisibles. Able to get integral of x^m using this method, for integral m. Fermat generalizes to arbitrary.

· Roberval proved theorems relating areas under different curves.

· Leibniz and Gregory (1638-1675) obtain series p/4=1-1/3+1/5- ……

· John Wallis (1616-1703) approximations based on infinite series; associates continued fractions with decimal fractions; convergence of series and expansions of functions.

· Newton by 1655 develops method of fluents and fluons related to integrals and derivatives. Thinks of space as continuous and views curves as points moving in space. Area under curve as limit of inscribed and circumscribed polygons. Obtained some algorithms for integrals, but returned to direct summation in many cases.

· Leibniz (1646-1716) worked with Huygens; Differential and integral calculus with notation that remains to today. Started with geometric curves, but this leads him to the concept of a function and enables him to obtain laws for differentiation and integation.