1 Cultural Connection Puritans and Seadogs Student led discussion. The Expanse of Europe – 1492 –1700.

2 9 – The Dawn of Modern Mathematics The student will learn about Some European mathematical giants.

3 §9-1 The Seventeenth Century Student Discussion.

4 §9-2 John Napier Student Discussion.

5 §9-3 Logarithms Student Discussion.

6 §9-3 Logarithms More from me later

7 §9-4 The Savilian and Lucasian Professorships Student Discussion.

8 §9-5 Harriot and Oughtred Student Discussion.

9 §9-6 Galileo Galilei Student Discussion.

10 §9-7 Johann Kepler Student Discussion.

11 §9–8 Gérard Desargues Student Discussion.

12 §9–9 Blaise Pascal Student Discussion.

13 §9–9 Blaise Pascal “Problem of the Points” at end if time.

14 Logarithms Forerunner to logs was Prosthaphaeresis. Werner used it in astronomy for calculations. Find sin 7º = tan 7º cos 7º (1/2 chord of 14º ) = · This becomes our multiplication problem. 2cosAcosB = cos (A – B) + cos (A + B) 2sinAsinB = cos (A – B) - cos (A + B) 2sinAcosB = sin (A – B) + sin (A + B) 2cosAsinB = sin (A – B) - sin (A + B) Example from astronomy using 2cosAcosB = cos (A – B) + cos (A + B)

15 Logarithms · This becomes our multiplication problem Let’s multiply!

16 Logarithms 3 2cosAcosB = cos (A – B) + cos (A + B) sin 7º = · Let 2cos A = , then A = cos B = , then B = A – B = and A + B = cos (A – B) = cos (A + B) = Which is the sin 7º

17 Logarithms 4 Napier used “Geometry of Motion” to try to understand logarithms. -4a -3a -2a -a 0 a 2a 3a 4a b –4 b –3 b –2 b –1 1 b 1 b 2 b 3 b 4 P moves with constant velocity so it covers any unit interval in the same time. Q moves so that it covers each interval in the same time period. P Q Q moves so that it covers each interval in the same time period. What is implied about the velocity? Q moves so that it covers each interval in the same time period. What is implied about the velocity? The velocity is proportional to the distance from Q.

18 Logarithms 5 Please take out a sheet of paper and fold it in half ten times. Logarithms were developed in three ways - as an artificial number. Napier/Briggs 1614 as an area measure of a hyperbola. Newton as an infinite series. Mercator 1670.

19 Logarithms Artificial Numbers The previous discussion on on Prosthaphaeresis and the geometry of motion were contributing factors. Historically the comparison of Arithmetic and Geometric progressions like the one used by Napier was a contributing factor. Indeed there is evidence on Babylonian tablets of this comparison.

20 Logarithms Artificial Numbers 2 The Babylonians were close to developing logarithms. They had developed the following table! Note the first column is a geometric progression and the second is arithmetic. To multiply 16 times 64 from the left column they would add 4 and 6 from the right column to get 10 and look up the corresponding number 1024 on the left

21 Logarithms Artificial Numbers 3 Logarithms were developed for plane and spherical trig calculations in astronomy. Napier chose as his base To avoid decimals he multiplied by Hence N = 10 7 (1 – 1/ 10 7 ) L where N is a number and L is its Napier Log. They were used in developing a table of log sines using a circle of radius 10,000,000 = 10 7 since the best trig tables had seven digit accuracy. There was also a need to keep the base of the log system small to help in interpolation.

22 Logarithms Artificial Numbers 4 Hence N = 10 7 (1 – 1/ 10 7 ) L where N is a number and L is its Napier Log and a table follows:  sin  · 10 7 Napier’s Log

23 Logarithms Artificial Numbers 5 From the previous problem by Prostaphaeresis sin 7º = tan 7º · cos 7º tan 7º · 10 7 = and the Log = The sum = cos 7º · 10 7 = and the Log = And 10 7 (1 – 1/10 7 ) = = sin 7º

24 Logarithms Artificial Numbers 6 Henry Briggs traveled to Scotland to pay his respects to Napier. They became friends and Briggs convinced Napier that if log 1 = 0 and log 10 = 1 life would be better. Hence Briggsian or common logs were born. The first tables to 14 places were published in These tables were used until 1920 when they were replaced by a set of 20 place tables.

25 Logarithms Artificial Numbers 7 Modern method - sin 7º = tan 7º · cos 7º Log tan 7º = And inverse log sin ( – ) = 7º. The sum = Log cos 7º =

26 Logarithms as Areas. Log development as an area measure under a hyperbola (1660). Natural logs. Consider y = 1/x

27 Logarithms as a Series. Log development as a series a la Mercator (1670). Natural logs. Converges for - 1 < x  1. Show convergence on a graphing calculator. Let x = 1 and calculate the ln 2. Show a slide rule calculation.

28 §9–9 Problem of Points “Problem of the Points” at end, if time. Helen and Tom are playing a game with stakes of a “Grotto’s” pizza. They flip a fair coin and every time it comes up heads Helen wins a point and every time it comes up tails, Tom wins a point. The first one to get five points wins the pizza. Helen is ahead 3 to 2 when the bell rings for Math History class. How do they divide the pizza fairly?

29 §9–9 Problem of Points 2 H T H = ¼ + ¼ + 3/16 = 11/16 T = 1/8 + 3/16 = 5/16 H T H T H T H T T H T H T H H T H T 1/4 1/81/16

30 Assignment Papers presented from Chapters 5 and 6.