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

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

3. 3 §9-1 The Seventeenth Century Student Discussion.

4. 4 §9-2 John Napier Student Discussion.

5. 5 §9-3 Logarithms Student Discussion.

6. 6 §9-3 Logarithms More from me later

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

8. 8 §9-5 Harriot and Oughtred Student Discussion.

9. 9 §9-6 Galileo Galilei Student Discussion.

10. 10 §9-7 Johann Kepler Student Discussion.

11. 11 §9–8 Gérard Desargues Student Discussion.

12. 12 §9–9 Blaise Pascal Student Discussion.

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

14. 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 ) = 0.1227845 · 0.9925462 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. 15 Logarithms 2 0.9925462 0.1227845 · 0.9925462 This becomes our multiplication problem. 0.1227845 49627310 397018480 7940369600 69478234000 198509240000 1985092400000 9925462000000 0.12186928889390 Let’s multiply!

16. 16 Logarithms 3 2cosAcosB = cos (A – B) + cos (A + B) sin 7º = 0.1227845 · 0.9925462 Let 2cos A = 0.1227845, then A = 86.4802681 cos B = 0.9925462, then B = 7.0000000 A – B = 79.4802681 and A + B = 93.4802681 cos (A – B) = 0.1825741 + cos (A + B) = - 0.0607048 0.1218693 Which is the sin 7º

17. 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. 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 1660. as an infinite series. Mercator 1670.

19. 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. 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 42 83 164 325 646 1287 2568 5129 102410...

21. 21 Logarithms Artificial Numbers 3 Logarithms were developed for plane and spherical trig calculations in astronomy. Napier chose as his base 0.9999999. To avoid decimals he multiplied by 10 7. 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. 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 90100000000 809848078153089 6086602541438410 4570710683465736 3050000006931472 10 7 1736482 1218693 17507240 885264 234899533552829 117452440482777

23. 23 Logarithms Artificial Numbers 5 From the previous problem by Prostaphaeresis sin 7º = tan 7º · cos 7º tan 7º · 10 7 = 1227846 and the Log = 20973240 The sum = 21048058 cos 7º · 10 7 = 9925462 and the Log = 74818 And 10 7 (1 – 1/10 7 ) 21048058 = 0.1218693 = sin 7º

24. 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 1624. These tables were used until 1920 when they were replaced by a set of 20 place tables.

25. 25 Logarithms Artificial Numbers 7 Modern method - sin 7º = tan 7º · cos 7º Log tan 7º = - 0.9108562 And inverse log sin ( – 0.9141055) = 7º. The sum = - 0.9141055 Log cos 7º = - 0.0032493

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

27. 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. 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. 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. 30 Assignment Papers presented from Chapters 5 and 6.