**Obsolete**

Definition:

1 No longer produced; out of date.

2 Biology Less developed (formly or related species);

rudimentary; vestigial.

•
I just bought a computer and now it is obsolete.

•
The dinosaurs disappeared from the historical record.

•
This antique is valuable because they stopped making

them.

• Vestigial organs like the appendix.

Steven

**Which is larger?**

Which of e^x or π^e is larger?

Found as the problem of math riddles book in the 1960’s

Perhaps a Math Puzzle book by Martin Gardner

Perhaps The Unexpected Hanging and Other Mathematical

Diversions 1969.

**Log-Log Duplex Deci-Trig
**

**Slide Rule**

•
invented by Oughtred in 1622 just a few years after

logarithms.

•
Mannheim (1851) Modern slide rule with A, B, C and D

scales.

•
Made the trip to the moon with Apollo.

• essentially disappeared in 1970’s when all the slide rule

manufactors silently quit, perhaps in response to the HP

35.

•
now a collectors item.

•
variations in size, circular, with microscope.

•
Over 40 million produced.

**Virtues of the Slide Rule**

• Scientific notation, estimation, no batteries required.

•
Develops analog reading abilities. (Gauges, dials)

•
Parallel computing.

• Wonderfull applicaton of Logs. Intermediate value

theorem.

•
Intermediate results were stored.

•
Stable design:

**Slide Rule Scales**

• Napier’s Bones 1617

•
Doubled the life of astronomers.

•
Great application of the laws of exponents.

•
Linear interpolation

•
Table lookup also used for trig, factors, integrals, Laplace

transforms . . .

•
Which is larger log log π + log e or log log e + log π ?

**Math Wars**

Math wars was roughly speaking a term used over the battle

over pencil and paper calculation vs calculators.

•
Which was the correct way to introduce elementary age

children to arithmetic. (US education 1990’s)

•
Which is the correct way to do arithmetic. (Algorists vs

Abacists, the 400 years war)

**400 Years War**

From roughly 1100 to 1500 there was a battle between:

Abacists Use Roman numerals along with the abacus.

Algorists Use Hindu-Arabic numerals along with the appropiate

algorithms for calculation.

Highlight: Florence 1299 forbid the use of the new numbers in

financial procedures.

**Roman Numerals**

• Letter stand for values: I for 1, V for 5, X for 10, L for
50, C

for 100, D for 500, M for 1000.

•
Arranged in decreasing size. (MCV and not CVM)

•
Letters are repeated for missing values: III for 3, XX for 20,

etc

•
Sometimes shorthands: IV for 4, CM for 900.

•
Other extensions.

**Virtues of Roman Numerals**

• Forced one to use an abacus (matched too)

•
Much harder to alter entries (try changing an id’s XVIII to

XXI)

•
numeral vs number

•
Look important? Use on clocks and movie copyright dates.

**Virtues of Abacus**

• Counting table ancestor: some lines in the sand and some

stones to count with. Indeed calculate comes from calculus

which is latin for stone. Abacus comes from the greek

abax, table covered with sand or fine dust

•
positional representation. (1950’s netherlands used them

in education)

•
paperless office (paper wasn’t cheap in dark ages)

•
add, subtract, multiply, divide

There is a on-line book on how to use the abacus

**Euler turns 300**

Euler didn’t have the technology to prove his analytical
results

with the same rigor required by today’s standards. Or he lived

before Cauchy invented
technology.

• The Basel problem
was first done

numerically.

•
Euler knew π= 3.141592653589793238 . . .

•
Euler computed S = 1.6449340668482264364 . . .

•
Euler came up with an agruement that S = π^2/6 and the

numbers above proved it. (It is better than the current

experiments of quantum mechanics)

•
Euler eventually answered all the weaknesses of his

agruement, — some many years later.

**Summing ζ(2)**

• Integral test Euler needed to

sum 10^{19} terms.

• The Euler-MacClaurin formula.

•
Accelerating series convergence, Euler transformation

x = y/(1 − y) maps y = 1/2 to x = 1.

•
Is this like turing completeness? Namely the fastest

computer can’t do anything a turing machine can do, nor

any faster mod polynomial time.

**Divergent Series**

Abel on divergent series (1828):

. . . Divergent series are in general the work of the devil and it is

shameful to base any demonstration whatever on them . . . For

the most part the results are valid, it is true, but it is a curious

thing. I am looking for a reason and it is a very interesting

problem.

Modern theory starts with Frobenius(1880).A good source is

Hardy’s book (1948) [Now printed by AMS (1992).]

**Human Calculators – Idiot Savant Dase**

A person who is mentally handicapped (or otherwise is

ordinary) but displays brilliance in one area, especially one

involving memory.

•
Computed π to 200 digits

•
Hired by Gauss (first person to buy computer time?) to

factor the intergers from X to Y.

**Human Calculators – WW2**

Calculator was a job title during World War II. Often
female, she

would do calculations for the military.

**Slide Rule in High School Education circa 1964**

• Taught in Chemistry class. Along with scientific notation,

the ideal gas laws and molarity calculations.

•
Getting the decimal point correct. (Estimation.)

•
Percentage error, significant digits.

•
Low cost items, roughly $3 ($15-$20 in current money)

•
No S or T scales, so the technology wasn’t used in

Trigonometry class. (Most Trig students had already taken

Chemistry.) Trigonometry classes used tables of trig and

log functions.

•
Drawing and reading graphs.

**Success of the Calculus Reform**

The rule of four:

•
Graphs to replace reading scales (gauges)

•
(Numerically) Tables to replace using log/trig tables

•
Verbally to replace what?

•
Formulas to replace what?

**Programing Assignment #7**

**Wag the dog**

• Graphics Processors double in speed every 12 months (vs

18 for CPUs).

•
But they are for games, hence only single percision.

•
Slide rules disappeared because they were a small and not

very profitable part of scientific instruments.

•
HP quit making graphing calculators briefly. The TI-89

hasn’t improved mathematically since 1999 and the price

as gone up.

**Victorian Internet**

**Larger – revisited**

Need to maximize which by calculus happens when

or ln x = 1 or x = e So a^e < e^a for all
a ≠ e

including π.

**e^π Wins**

• Does the history of obsolete technology give insight into

the rapid change of today’s techonology?

•
The analog world had virtues.