Chapter 5: Eternal Triangles: Trigonometry and logarithms
Trigonometry (measuring triangles).
Hipparchus: first trigonometric tables, around 150 BC.
Menelaus: Spherica, AD100. Astronomical calculations.
Ptolemy: Mathematical Syntaxis, AD150. Trigonometric tables in terms of cords, methods to calculate them, and a catalogue of star positions. System of epicycles.
Early Trigonometry (Hindu mathematicians and astronomers): Varahamihira, 500. Brahmagupta, 628. Bhaskaracharya, 1150.
15th century: German Hanseatic League, Johannes Müller, George Joachim Rhaeticus, Vieta.
17th to 20th century: creation and constant use of logarithms. John Napier (Napier’s rods, Napierian logarithms), Henry Briggs (Base ten logarithms), John Speidell, Jobst Bürgi.
“Their efforts paved the way to a quantitative scientific understanding of the natural world, and enabled worldwide travel and commerce by improving navigation and map-making… Logarithms made it possible for scientists to do multiplication quickly and accurately… Science could never have advanced without some such method.”
Hipparchus: first trigonometric tables, around 150 BC.
Menelaus: Spherica, AD100. Astronomical calculations.
Ptolemy: Mathematical Syntaxis, AD150. Trigonometric tables in terms of cords, methods to calculate them, and a catalogue of star positions. System of epicycles.
Early Trigonometry (Hindu mathematicians and astronomers): Varahamihira, 500. Brahmagupta, 628. Bhaskaracharya, 1150.
15th century: German Hanseatic League, Johannes Müller, George Joachim Rhaeticus, Vieta.
17th to 20th century: creation and constant use of logarithms. John Napier (Napier’s rods, Napierian logarithms), Henry Briggs (Base ten logarithms), John Speidell, Jobst Bürgi.
“Their efforts paved the way to a quantitative scientific understanding of the natural world, and enabled worldwide travel and commerce by improving navigation and map-making… Logarithms made it possible for scientists to do multiplication quickly and accurately… Science could never have advanced without some such method.”
Chapter 6: Curves and Coordinates: Geometry is algebra is geometry
“In mathematics, there are no hard and fast boundaries between apparently distinct areas, and problems that seem to belong to one area may be solved using methods from another. In fact, the greatest breakthroughs often hinge upon making some unexpected connection between previously distinct topics.”
Pierre de Fermat: first person to describe coordinates. 1620. Introduction to Plane and Solid Loci.
He discovered a general principle: if the conditions imposed on the point can be expressed as a single equation involving two unknowns, the corresponding locus is a curve – or a straight line.
Descartes: Discours de le Méthode. Geometry of the plane can be reinterpreted in algebraic terms. Cartesian planes and coordinates.
“A function is not a number, but a recipe that starts from some number and calculates an associated number.”
“Mathematics is the ultimate in technology transfer. And it is those cross-connections, revealed to us over the past 4000 years, that make mathematics a single, unified subject.” (page 116)
Pierre de Fermat: first person to describe coordinates. 1620. Introduction to Plane and Solid Loci.
He discovered a general principle: if the conditions imposed on the point can be expressed as a single equation involving two unknowns, the corresponding locus is a curve – or a straight line.
Descartes: Discours de le Méthode. Geometry of the plane can be reinterpreted in algebraic terms. Cartesian planes and coordinates.
“A function is not a number, but a recipe that starts from some number and calculates an associated number.”
“Mathematics is the ultimate in technology transfer. And it is those cross-connections, revealed to us over the past 4000 years, that make mathematics a single, unified subject.” (page 116)