from the abacus to the iphone - cmc\^3-s · from the abacus to the iphone cmc south – spring...
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From the AbacusTo the iPhone
CMC South – Spring Conference 2017 John Martin
Santa Rosa Junior College
3
Abacus ENIAC
From the AbacusTo the iPhone
HP-35iPhone 1
Mechanical Calculators of the 17th CenturyThe Main Characters
Gottfried Wilhelm Leibniz
Blaise Pascal
Wilhelm Schickard
Wilhelm Schickard
Mechanical Calculators of the 17th Century
1592 – 1635
Herrenburg
The Calculating Clock
Mechanical Calculators of the 17th Century
Wilhelm Schickard
“What you have done in a logistical way, I have just tried to do by way of mechanics. I have constructed a machine consisting of eleven complete and six incomplete sprocket wheels, which can calculate. You would burst out laughing if you were present to see how it carries by itself from one column of tens to the next or borrows from them during subtraction.”
Wilhelm Schickard
Johannes Kepler
The Calculating Clock
Mechanical Calculators of the 17th CenturyThe Calculating Clock
Schickard’s drawings, 1624
Mechanical Calculators of the 17th CenturyThe Calculating Clock
Mountain View, CaliforniaComputer History Museum
Mechanical Calculators of the 17th CenturyThe Calculating Clock
Multiplying Unit
Storage for Intermediate
Results
Six digit, decimal adding machine
Mechanical Calculators of the 17th CenturyNapier’s Rods
Mechanical Calculators of the 17th CenturyNapier’s Rods
BOARD
SET OF RODS
Example: 423 X 6
6 24 2
18
1+
= 5 = 8
+
= 3= 2
423 X 6 = 2538 BOARD
Mechanical Calculators of the 17th CenturyThe Calculating Clock
Back View Sheet of Paper With 10 Rods
Mechanical Calculators of the 17th CenturyThe Calculating Clock
Dials and WheelsGears
Input Dials
Mechanical Calculators of the 17th Century
Blaise Pascal1623 – 1662
Les Machine Arithmétiques de Blaise Pascal
Blaise Pascal
“I submit to the public a small machine by my invention, by means of which you alone may, without any effort, perform all the operations of arithmetic, and may be relieved of the work which has often times fatigued your spirit.”
Blaise Pascal
Mechanical Calculators of the 17th Century
“Pascaline”
Clermont-Ferrand
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
Clermont-Ferrand
Musée Henri-LecoqClermont-Ferrand, France
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
The Marguerite Périer
Mechanical Calculators of the 17th Century
The Queen of Sweden
The PascalineMechanical Calculators of the 17th Century
The Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th CenturyThe Pascaline
Mechanical Calculators of the 17th Century
Output Drum
The Pascaline
Input Dial
Mechanical Calculators of the 17th CenturyThe Nines Complement Method
Definition: The nines complement of any one digit decimal number d is 9 — d.
Examples: The nines complement of 3 is 6. The nines complement of 9 is 0.
Notation: Let the nines complement of A be denoted by ncp (A).
—1 — 538
Mechanical Calculators of the 17th Century
Definition: The nines complement of any one digit decimal number d is 9 — d.
= 999 — 538= 461
The Nines Complement Method
So that: ncp (538) = 103
For any n digit decimal number A, we have:ncp (A) = 10 —1 — An
ncp (ncp (A) + B)
ncp (A) = 10 —1 — An
+ B
Mechanical Calculators of the 17th Century
= 10 —1 — (A — B)n
= 10 —1 — A + Bn
= ncp (A)
Thus:
The Nines Complement Method
Using:
Note: ncp (ncp (#)) = #
We have: ncp (A — B)
A — B =
ncp (ncp (A) + B)A — B =Thus:
Mechanical Calculators of the 17th Century
Standard Method 9’s Complement Method
Example: 538 — 64
461
525474
538— 64 + 64
ncp
ncp
The Nines Complement Method
Mechanical Calculators of the 17th CenturyThe Pascaline
The Scientific Machines
all other dials fourth dial third dial second dial first dial
DeniersSolsLivres
Mechanical Calculators of the 17th CenturyThe Pascaline
The Chancelier Séguier
The Accounting Machines
base 20 base 12base 10TensHundreds …
base 10base 10
all other dials fourth dial third dial second dial first dial
Mechanical Calculators of the 17th CenturyThe Pascaline
The Surveying Machines
Toises Piedsbase 6
Lignesbase 12
Tens …base 10base 10
The Léon Parcé Collection
Poucebase 12
Mechanical Calculators of the 17th CenturyThe Pascaline
Musée Henri-Lecoq Clermont-Ferrand
Musée Roger Quillot Clermont-Ferrand
Musée des Arts et Métiers
Paris
Mechanical Calculators of the 17th Century
Gottfried Wilhelm Leibniz
dxdy
∫ f (x) dx
The Derivative:
The Integral:
Mechanical Calculators of the 17th Century
“Several years ago I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian. It occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only counting but also addition and subtraction, multiplication and division could be accomplished by a suitably arranged machine easily, promptly, and with sure results.”
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz
Mechanical Calculators of the 17th Century
Gottfried Wilhelm Leibniz Library Hanover, Germany
Mechanical Calculators of the 17th CenturyHanover
Hanover
Mechanical Calculators of the 17th Century
Mechanical Calculators of the 17th Century
The Stepped ReckonerMechanical Calculators of the 17th Century
The Stepped ReckonerMechanical Calculators of the 17th Century
Staffelwalze — Stepped drum
The Stepped ReckonerMechanical Calculators of the 17th Century
Stepped drum
Input dial
Output wheel
Rotating rod
The Stepped Reckoner
RodStepped Drum
Drive Crank Input Dial
Mechanical Calculators of the 17th Century
Mechanical Calculators of the 17th Century
1690 — 1716: Under construction
1716 — 1764: Stored in Hanover
1764 — 1879: Stored in Göttingen
1879 — 1894: Returned to Hanover
1894 — 1896: Restored
1896 — present: Gottfried Wilhelm Leibniz Library
The Odyssey of the “Younger Machine”
Mechanical Calculatorsof the seventeenth century