1

Cultural ConnectionPuritans 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 )

= 0.1227845 · 0.9925462This 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 2

0.9925462

0.1227845 · 0.9925462This becomes our multiplication problem.

0.1227845

49627310 397018480

7940369600 69478234000

198509240000 1985092400000 9925462000000

0.12186928889390

Let’s multiply!

16

Logarithms 32cosAcosB = cos (A – B) + cos (A + B)sin 7º = 0.1227845 · 0.9925462

Let 2cos A = 0.1227845, then A = 86.4802681cos 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

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Logarithms 4Napier 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.

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Logarithms 5Please 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.

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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.

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Logarithms Artificial Numbers 2The 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

2 1

4 2

8 3

16 4

32 5

64 6

128 7

256 8

512 9

1024 10

. . . . . .

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Logarithms Artificial Numbers 3Logarithms 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.

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Logarithms Artificial Numbers 4Hence 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

90 10000000 0

80 9848078 153089

60 8660254 1438410

45 7071068 3465736

30 5000000 6931472

10

7

1736482

1218693

17507240

885264

2 348995 33552829

1 174524 40482777

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Logarithms Artificial Numbers 5From 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

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Logarithms Artificial Numbers 6Henry 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.

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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

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Logarithms as Areas.

Log development as an area measure under a hyperbola (1660). Natural logs.

Consider y = 1/x

alndxx

1then

a

1

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Logarithms as a Series.

Log development as a series a la Mercator (1670). Natural logs.

Converges for - 1 < x 1.

...4

x

3

x

2

xx)x1(ln

432

Show convergence on a graphing calculator.

Let x = 1 and calculate the ln 2.

Show a slide rule calculation.

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§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?

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§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 1/8 1/16

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Assignment

Papers presented from Chapters 5 and 6.