Fermat's Library | The Education of a Computer annotated/explained version.

Blaise Pascal, Gottfried Wilhelm Leibniz, and Charles Babbage all m...

Grace Hopper worked with Howard Aiken on the Harvard Mark I compute...

John Mauchly (1907-1980) was an American physicist who, along with ...

The UNIVAC (UNIVersal Automatic Computer) was a significant advance...

- **UNITYPER:** An input device for the UNIVAC I, it featured a mo...

Hopper is describing a scenario in which high-level programming lan...

THE EDUCATRON OF A COMPUTER

Grace Murray Hopper

Remington Rand Corp.

While the materialization is new, the idea

of mechanizing mathematical thinking is not new.

Its lineage starts with the abacus and descends

through Pascal,Leibnit% and Babbage. Fore

immediately, the ideas here presented originate

from Professor Howard K. Aiken of Harvard

University, Dr. John W. Nauehly of Eckert-

Hauchly and Dr. H. V. Wilkes of the University

of Cambridge. From Professor Aiken came, in

1946, the idea of a library of routines

described in the Hark I manual~ and the concepts

embodied in the Hark III coding machined from

Dr. Hauchly, the basic principles of the "short-

order code" and suggestions, criticisms, and

untiring patience in listening to these present

attempts; from Dr. Wilkes, the greatest help

of all, a book on the subject. For those

of their ideas which are included herein, I

most earnestly express my debt and my

appreciation.

Introd~Qtion

start at the beginning, Fig. I

represents the configuration of the elements

required by an operation: input to the

operations; controls~ even if they be only

start and stop; previously prepared tools

supplied to the operation I and output of

products, which may, in turn, become the

input of another operation. This is the basic

element of a production lines input of raw mate-

rial% controlled by human beings, possibly

through instruments~ supplied with machine

tools; the operation produces an automobile,

a rail~ or a san of tomatoes.

The armed services, government, and

INPUT

I CONTROL

~--'~OPERATION ~--~

industry are interested not only in creating

new operations to produce new results, but

also in increasing the efficiency of old

operations. A very old operation, Fig. 2?

is the solution of a mathematical problem.

It fits the operational configuration: input

of mathematical data~ control by the

mathematician~ supplied with memory, formulas,

tables, pencil? and papers the brain carries

on the arithmetic, and produces results.

It is the current aim to replace, as far

as possible, the human brain by an electronic

digital computer. That such computers

themselves fit this configuration may be seen

in Fig.

(With your permission, I shall use

UNIVAC * as synonymous with electronic digital

computer; primarily because I think that way?

but also because it is convenient.)

Adding together the configurations of the

human being and the electronic computer? Fig.

4 shows the solution of a problem in two

levels of operation. The arithmetical chore

has been removed from the mathematician, who

has become a programmer, and this duty

assigned to the UNIVAC. The programmer has

been supplied with a "code" into which he

translates his instructions to the computer.

The "standard knowledge" designed into the

UNIVAC by its engineers, consists of its

elementary arithmetic and logic.

This situation remains static until the

novelty of inventing programs wears off and

degenerates into the dull labor of writing and

checking programs. This duty now looms as an

imposition on the human brain. Also, with the

computer paid for, the cost of programming and

the time consumed, comes to the notice of vice-

presidents and project directors. Co,son sense

dictates the insertion of a third level of

operation, Fig. 5.

OUTPUT

* Registered trade mark.

TOOLS

Fig. I -

OPERATION

Fig, 2- SOLUTION OF

J PROBLEM H

PROBLEM

MATHEMATICIAN I

BRAIN

~-~

FORMULAS

TABLES

RESULTS

Page 243

~! ii~ ¸ i ¸

UNITYPER

PROGRAM

UNIPRINTER

Fig,

3- UNIVAC SYSTEM

The programmer may return to being a

mathematician. He is supplied with a catalogue

of subroutines. No longer does he need to have

available formulas or tables of elementary

functions. He does not even need to know the

particular instruction code used by the

computer. He needs only to be able to use the

catalogue to supply information to the computer

about his problem. The UNIVAC~ on the basis of

the information supplied by the mathematician,

under the control of a "compiling routine of

type A"? using subroutines and its own in-

struction code? produces a program. This

program~ in turn directs the UNIVAC through

the computation on the input data and the

desired results are produced. A major

reduction in time consumed and in sources of

error has been made. If the library is well-

stocked~ programming has been reduced to a

matter of hours? rather than weeks. The

program is no longer subject either to errors

of transcription or of untested routines.

Specifications for computer information?

a catalogue~ compiling routines? and

subroutines will be given after adding

another level to the block diagram. As Fig. 5

stands the mathematician must still perform

all mathematical operations? relegating to

the UNIVAC programming and computational

operations. However~ the computer information

delivered by the mathematician no longer deals

with numerical quantities as such. It treats

of variables and constants in symbolic form

together with operations upon them. The

insertion of a fourth level of operation is

now possible, Fig. 6. Suppose? for example~

the mathematician wishes to evaluate a function

and its first = derivatives. He sends the

information defining the function itself to

the UNIVAC. Under control of a "compiling

routine of type B"~ in this case a

differentiator~ using task routines? the

UNIVAC delivers the information necessary to

program the computation of the function and

its derivatives. From the formula for the

function? the UNIVAC derives the formulas of

the successive derivatives. This information

processed under a compiling routine of Type

A yields a program to direct the computation.

Expansion makes this procedure look?

and seem? long end complicated. It is not.

Peducing again to the two-component system?

the mathematician and the computer? Fig. 7

presents a more accurate picture of the

computing system.

Presuming that cod% program? input data~

and results are familiar terms~ it remains

to define and specify the forms of infor-

mation and routines acceptable to this system.

These include

catalogued

computer information?

subroutine?

compiling routines~ type A and B~

and task routines.

PROBLEM

PROGRAMMER

•

-~, BRAIN

FORMULAS

TABLES

CODE

DATA

OR I

PROGRAM

•

--~, UNIVAC

STANDARD

KNOWLEDGE ]

OR II

RESULTS

Fig. 4-SOLUTION OF A PROBLEM

Page ~I!I~

OR g

PROIBLEM

OR II

MATHEMATICIA~

•

BRAIN

CATALOGUE

COMPUTER ~__~

INFORMATION

COMPILING

ROUTINE

TYPE

UNIVAC

,,I

UBROUTINES J

I INPUT

I" DATA

PROGRAM

KNOWLEDGE

OP. III

Fig. 5-COMPILING ROUTINES AND SUBROUTINES

RESULTS

Catalogue and Com~ut~r Information

As soon as the purpose is stated to make

use of subroutines, two methods arise. In one,

the program refers to an intmediately available

subroutine, uses it, and continues computation.

For a limited number of subroutines, this

method is feasible and useful. Such a system

has been developed under the nick-name of

the "short-order code 'r by members of the staff

of the Computational Analysis Laboratory.

The second method not only looks up the

subroutine, but translates it, properly adjusted,

into a program. Thus, the completed program

may be run as a unit whenever desired, and

may itself be placed in the library as a more

advanced subroutine.

Each problem must be reduced to the level

of the available subroutines. Suppose a simple

problem, to compute

y - @ sin cx,

using elementary subrou%ines. Each step of the

formula falls into the operational pattern,

Fig. 8 ; that is~

U=e -u

v~ cx

V = s::L v

y=Ug.

ORI

[

PROBLEM

i,,,,.~ I I o~.~:'~'~ .....

LLI COMPUTER LLI

m .,,A.N ..,.,,OR,,ATO,. i. UN.VA° t

1 1

' OR III

I COMPILING

TYPE A , 1

:- !i Lo i , '

i I ~TAN'~D 1

KNOWLEDGE]

[

OR IV

RESULTS

Fig. 6.- COMPILING TYPE B AND TASK ROUTINES

Page 245

PROBLEM

I MATHEMATICIAN]

COMPUTER

INFORMATION

I INFORMATION

INPUT DATA

COMPILING

COMP~LING A

PROGRAM

UNIVAC

TASK ROUTINES

SUBROUTINES

STANDARD

KNOWLEDGE

PRO "%I

RESULTS __~

Fig. 7- COMPUTING SYSTEM

As presented in Fig. 9~ however~ this

information is not yet sufficiently standardized

to he acceptable to a compiling routine.

Several problems must be considered and

procedures defined.

The operations are numbered in normal

sequence and this number becomes part of the

computer information. Thus when it is desired

to change the normal sequence, the alternate

destination is readily identified. The

compiling routine translates these operation

numbers into instructions in the coded program.

Two fundamental situations arise? the alternate

destination either precedes the operation

under consideration or follows it, by-passing

several intermediate operations. In both

cases? it is necessary only to have the

compiling routine remember where it has

placed each subroutine or that a transfer of

control to operation k has been indicated.

In any event the mathematician need only

state? "go to operation k", and the compiling

routine does the rest.

The symbols to be used for the arguments

and results~ as well as for the operations~

are of next concern. One mathematician

might write

I x Xl 6 q x 6

2 2~ x 2 7 v x 7

3 LX ~c 3 8 V x 8

4 u x 4 9 y x 9

5 U x 5 iO n xlO

As symbols for the operations and sub-

routines, a system of "call-numbers" is used.

These alphabetic characters represent the

class of subroutines. Following Dr. Wilkes~

example~ these symbols are partially

phonetic; that is~ a = arithmetic~ t =

trigonometric, and x = exponential~ amc =

arithmetic~ multiplication by a constan%

x-e = e-U,tsO = trigonometric~ sine. Placed

with the call-numbers~ n~ f~ or s~ indicates

normal? floating~ or stated (flxed) decimal

point. Other letters and digits indicate

radians or degrees for angles~ complex

numbers~ etc. These call-numbers are listed

in the catalogue together with the order in

which arguments~ controls? and results are

to be stated. The general rules for the

description of an operation are:

.x 2

y = e sin cx

and another

_v 2

u=e sin gv.

The obvious solution proves best. Nake a list

of arguments and results and number them.

(This amounts to writing all constants and

variables as xi.) The order is i~naterial?

so that forgotten quantities can be added at

the end.

i. call-numbers~

2. number of operation~

3. arguments in order of appearance

in formula~ variables preceding

constants~

4. controls~ normal exit if altered?

followed by alternate exits in

order of appearance in subroutine~

5. results~ in order of appearance.

All exceptions to the general rules are listed

in the catalogue.

The problem has been reduced to computer

information. The exact positions of characters

Page 24.6

in words as submitted to the UNIVAC has been

omitted since it hardly seems of general

interest. The preparation of information might

be called creating a "multiple-address code",

by which any number of arguments may enter an

operation~ to produce any number of results~

and to proceed directly to the next operation

unless routed to any one of several other

operations.

Each subroutine in the library is expressed

in coding relative to its entrance line

considered as OO1. They are~ in general~

programmed and coded for maximum accurary and

minimum computing time. They may store within

themselves constants peculiar to themselves.

They may also make use of certain "permanent

constants" read in with every program. These

permanent constants occupy a reserved section

of the memory and are called for by alphabetic

memory locations~ a trick~ at present peculiar

to UNIVAC. Thus, these addresses are not

modified in the course of positioning the

subroutine in a program. They include such

quantities as 1/22~ ~/4, logloe~ ~0, .2, .5,

and the like.

Each subroutine is preceded by certain

information, matching and supplementing that

supplied by the mathematician:

I. call-number~

2. arguments, the destination of the

arguments within the subroutine,

expressed in the relative coding

of the subroutines

3. non-modification indicators

locating constants embedded in the

subroutine which are not to be

altered~

4. results~ the positions of the

results within the subroutine,

expressed in relative coding.

Each subroutine is arranged in a standard

pattern.

~l_a~f~_l~_~ - The first line of a subroutine

is its entrance line~ thus in relative coding

it is number one. It is the first line of the

subroutine transferred to a program~ and it

contains an instruction transferring control to

the first action line.

E~it ~ines - The second line of a subroutine is

its normal exit line. This contains an

instruction transferring control to the line

following the last line of the subroutine.

Unless an alternate transfer of control is

desired, all e~its from the subroutine are

referred to the normal exit line. Alternate

exit lines~ involving transfers of control

from the usual sequence? follow the normal exit

llne in a predetermined order as listed in the

catslogue.

Ar~umen%s - The exit lines are followed by

spaces reserved for the arguments arranged

in predetermined order.

Results - The results~ also in specified

order, follow the arguments.

Constants - The results are followed, when

possible~ by any arbitrary constants peculiar

to the subroutine. When the subroutine has

been compounded from other subroutines~

groups of constants may also appear

embedded in the subroutine. These are cared

for by the non-modification information.

The first action line appears next in the

subroutine. Its position in the relative

coding is defined by the entrance line. No

instruction line may precede this line.

The sequence assigned to the entrance

and exit lines, arguments, results, and

constants is arbitrary. It is convenient.

All that is required is that a sequence be

established and that the computer recognize

this sequence.

For convenience in manipulation, a

certain number of elementary subroutines

have been combined to form a sub-library.

These include

a = arithmetic

b = transfer of data

c ~ counters

h = hyperbolic functions

i = input routines

1 = logarithmic functions

o = output routines

polynomials

r = roots and fractional exponents

t = trigonometric functions

u ~ control transfers

w =~ storage routines

x = exponential functions

y = editing routines

As subroutines are added to extend the

library~ it becomes more useful and

programming time is further reduced.

ARGUMENTS

Xjn

I OONT OL

•__.[SU

BROUTINE~L__J

u.x,' I-I

i ND'RDi

OONSTANTSJ

RESULTSu ']

Fig. 8 - OPERATION

Page 247

y- ~ "x= sin cx

OPERATION

NUMBER

OPERATION

TRANSFER

bOi

apn

X-6

¢®

omc

sin v

t$O

omO

ARGUMENTS

Q Dr,99, 2, ,5

X (1, Z, 3, 4, 5)

x, 2

1,t0

6, '1

U, V

5, B

EDIT x, y

6 yrs I, 9

~)---'L x,&x,L R

7 @eL t, 2, 3

STOP

8 ust

RESULTS

X, ~X, Lx, n,

1, 2, 3,10, 6

Lj=X 2

u=e ~

U=CX.

V " ~irl v

y- UV

CONTROL

l,y

0 (t, 2)

x • Ax--~x x tLx-~ xtL~,'8

1 8, t

Fig g - EXAMPLE

Indeed~ the day may come when the elementary

subroutines are rarely used and the conputer

information will contain but seven or eight

items calling into play powerful subroutines.

Construction of Subroutines

It is not neeessary~ nor is ~t advisable?

that the inexperienced programmer tamper with

the coding within a subroutine. It is usually

minimum latency coding using every trick and

device knov to the experienced programmer.

It has been tested by operation on the

computer. However, in order to speed the

original construction~ on paper? of the

elementary routines~ kernel routines and

threading routines have been devised.

A kernel routine computes a mathematical

function or carries out an elementary

process for a limited range of the variable

concerned; for exampl% sin x~ for 0 < x <

~/% and i0 -x for 0 < x < I. A kernel routine

is always entered and left by way of a

threading routine.

Threading routines, incomplete without

kernels~ remove from the arguments and stor%

such quantities as algebraic signs~ integral

parts~ and exponents~ deliver the reduced

arguments to the kernel routin% receive

results from the kernel, and adjust algebraic

signs and exponents. For exampl% the

threading routines for sin y remove the

algebraic sign of y, reduce y by multiples

of 22~, reduce the remainder to a quantity

x less than ~/%~ store the information and

select the sin x or cos x kernel routine.

Page 248

The kernel routine returns sin x or cos x.

The threading routine adjusts the sign?

exponent? and decimal point completing the

computation.

Threading routines recognize and give

special treatment to such values as zero

and infinity, and provide signals and

printed information when the capacity of

the computer is exceeded.

An elementary subroutine consists of

a threading routine accompanied by one or

more kernel routines. Hence, the threading

routines are similar to the subroutines in

form having at the beginning an entrance line?

exit lines~ (usually undetermined until the

kernel routine is supplied)~ arguments~

results~ and constants. At the end of a

threading routine are certain lines prepared

to "overlap" the first section of the kernel

routine. This overlap contains

I. the entrance line of the kernel

routine;

2. the exit line of the kernel routine

set-up by threading routines

3. arguments~ and

4. results.

Compiling Poutines Type A are designed to select

and arrange Subroutines according to information

supplied by the mathematician or by the

computer. Basically~ there is but one Type A

routine. However~ since the UNIVAC code

contains instructions transferring two

neighboring quantities simultaneously, a

second compiling routine has been designed to

care for floating decimal? complex number? and

double precision programs. For each operation

listed by the mathematician~ a type A routine

will perform the following services:

I. locate the subreutine indicated

by the call-number~

2. from the computer and subroutine

information combined with its record of the

program~ fabricate and enter in the program

the instructions transferring the arguments

from working storage to the subroutines

3. adjust the entrance and normal exit

lines to the position of the subroutine in the

program and enter them in the program~

~. according to the control infor-

mation supplied by the programmer~ adjust

alternate exit lines and enter them in the

program (this process involves reference to

the record);

5. according to the control infor-

mation supplied with previous operations adjust

auxiliary entrance lines and enter them in

the program~

6. modify all addresses in the

subroutine instructions and enter these

instructions in the programs

7. according to information supplied

by the subroutine, leave unaltered all constants

embedded in the subroutine and transfer them to

the program~

8. from the computer and the subroutine

information fabricate and enter in the program

the instructions transferring the results to

9. maintain and produce a record of

the program including the call-number of each

subroutine and the position of its entrance

line in the program.

The compiling routines also contain certain

instr~ctions concerning input taoes? tape

library? and program tapes~ peculiar to the

UNIVAC. All counting operations such allo-

cation of temporary storage and program space?

"and control of input and output are carried

on steadily by the compiling routine. Stated

bluntly~ the compiling routine is the

programmer and performs all those services

necessary to the production of a finished

program.

Compiling Poutines of Type B? will for each

operatlon~ by means of "task routines"? replace

orsupplement the given computer information

with new information. Thus? compiling routine

B-1 will? for each operation~ copy the infor-

mation concerning that operation and call in

the corresponding task routine. The task

routine will generate the formula~ and derive

the information? necessary to compute the

derivative of the operation. Compiling

routine B-I then records this information in

a form suitable for submission to a T~fpe A

routine.

Since information may be re-submltted to

a type B routine? it is obvious that in order

to obtain a program to compute f(x) and its

first n derivatives? only the information

defining f(x) and the value of n need be

given. The formulas for the derivatives of

f(x) will be derived by repeated applications

of B-1 and programmed by a type A routine.

It is here that the question can best

be answered concerning a liking for or an

aversion to subroutines. Since the use of

subroutines in this fashion increases the

abilities of the computer? the question

becomes meaningless and transforms into

a question of how to produce better

subroutines faster. However~ balancing

the advantages and disadvantages of using

subroutines? among the advantages are

Page 2~9

i. relegation of mechanical jobs

such as memory allocation~ address modifi-

cation? and transcription to the UNIVAC~

2. removal of error sources such

as programming errors and transcription

errors~

3. i conservation of programming time~

~. ability to operate on operations;

5. duplication of effort is avoided?

since each program in turn may become a

subroutine.

Only two disadvantages are immediately

evident. Because of standardization~ a small

amount of time is lost in performing duplicate

data transfers which could be eliminated in a

tailor-made routine. In base load prob!ems~

this could become serious. Even in this case~

however~ it is worthwhile to have UNIVAC produce

the original program and then eliminate such

duplication before rerunning the problem. The

second disadvantage should not long remain

serious. It is the fact that? if a desired

subroutine does not exist? it must be

programmed and added to the library. This will

be most likely to occur in the case of input

and output editing routines until a large

variety is accumulated. This situation also

emphasizes the need for the greatest gener-

ality in the construction of subroutines.

Several directions of future developments

in this field can be pointed out. It is to

be hoped that reports will be presented on

some of them next September.

More type A compiling routines will be

devised! those handling commercial rather than

mathematical programs; some special purpose

compiling routines such as a routine which

will compute approximate magnitudes as it

proceeds and select sub-routines accordingly.

Compiling routines must be informed of the

average time required for each sub-routlne

so that they can supply estimates of running

time with each program. Compiling routines

can be devised which will correct the

computational procedure submitted to produce

the most efficient program. For example~ if

both sin @ and cos ~ are called for in a

routine~ they will be computed more rapidly

simultaneously. This will involve sweeping

the computer information once to examine its

structure.

Type B routines at present include

linear operators. More type B routines

must be designed. It can scarcely be

denied that type C and D routines will

be found to exist adding higher levels of

operation. Work is already in progress to

produce the formulas developed by type B

routines in algebraic form in addition to

producing their computational programs.

Thus by considering the professional

programmer (not the mathematician); as an

integral part of the computer~ it is evident

that the memory of the programmer and all

information and data to which he can refer is

available to the computer subject only to

translation into suitable language. And it is

further evident that the computer is fully

capable of remembering and acting upon any

instructions once presented to it by the

progrmmner.

With some specialized knowledge of more

advanced topics~ UNIrqAC at present has a well

grounded mathematical education fully

equivalent to that of a college sophomore? and

it does not forget and does not make mistakes.

It is hoped that its undergraduate course will

be completed shortly and it will be accepted

as a candidate for a graduate degree.

Discussion

Grace Hopper worked with Howard Aiken on the Harvard Mark I computer. The Harvard Mark I was an electromechanical computer designed and built by IBM under the direction of Howard Aiken, a Harvard University mathematician and physicist. The Mark I relied on a combination of over 3,000 electromechanical relays and approximately 765,000 mechanical components. This massive machine was 51 feet long and weighed around 5 tons. It had limited memory capacity, storing only 72 numbers of 23 decimal digits each. The Mark I was programmed using punched paper tape, which contained a sequence of instructions for the machine to execute. Programming the machine was a tedious process that required a deep understanding of its architecture. Initially used for military purposes during World War II, the Mark I was employed primarily for calculating ballistics tables and other complex mathematical computations. Later, it was used for various scientific and engineering applications, including atomic energy research and aerodynamics. You can find the Harvard Mark I manual [here](https://chsi.harvard.edu/harvard-ibm-mark-1-manual). ![](https://i.imgur.com/EPaR6U5.jpg) *Harvard Mark I* - **UNITYPER:** An input device for the UNIVAC I, it featured a modified Remington typewriter keyboard for user input. This device accepted data entered through the keyboard and then recorded it onto a metal magnetic tape using its built-in tape drive. - **UNISERVO:** A magnetic tape drive used with the UNIVAC I for data storage and retrieval. - **UNIPRINTER:** An output device for the UNIVAC I, the Uniprinter was a printer that produced printed output from the computer's processed data. John Mauchly (1907-1980) was an American physicist who, along with J. Presper Eckert, made significant contributions to the development of early electronic computers. The duo designed and built the ENIAC (Electronic Numerical Integrator and Computer), the first general-purpose electronic computer, completed in 1945. They later founded the Eckert-Mauchly Computer Corporation and developed the UNIVAC I (UNIVersal Automatic Computer I), the first commercially available computer in the United States. Grace Hopper worked at the Eckert-Mauchly Computer Corporation as a Senior Mathematician. ![](https://i.imgur.com/JeVgrWs.jpg) *Mauchly and Eckert* Hopper is describing a scenario in which high-level programming languages and compilers make programming more accessible and abstract. Instead of working directly with machine code or assembly language, programmers can use a more intuitive language to express their algorithms and solutions. By using a catalog of subroutines, they can build upon existing code, making it easier to develop complex programs without needing to know every detail of the underlying computer architecture. The UNIVAC (UNIVersal Automatic Computer) was a significant advancement in the field of computing, as it was the first commercially available computer in the United States. Developed by Mauchly and Eckert at the Eckert-Mauchly Computer Corporation, the UNIVAC was designed to handle both scientific and business applications, making it a versatile and influential piece of early computing history. The UNIVAC used vacuum tubes for its logic circuits, magnetic tape for data storage, and mercury delay lines for memory. It could perform around 1,000 calculations per second, which was a significant improvement over its predecessor, the ENIAC. The machine's input and output system utilized punched cards, and it was capable of executing both arithmetic and logical operations. ![](https://i.imgur.com/F5QDHGZ.jpg) *UNIVAC* Blaise Pascal, Gottfried Wilhelm Leibniz, and Charles Babbage all made significant contributions to the endeavor of mechanizing mathematical logic: - **Blaise Pascal (1623-1662):** Pascal was a French mathematician, physicist, and philosopher. His major contribution to the field of computing was the development of the Pascaline (also known as the Arithmetic Machine) in 1642. The Pascaline was an early mechanical calculator that could perform addition and subtraction. It used a system of gears and dials to represent and manipulate numbers, making it easier for people to perform basic arithmetic operations. - **Gottfried Wilhelm Leibniz (1646-1716):** Leibniz was a German mathematician, philosopher, and polymath who made numerous contributions to various fields, including mathematics, logic, and computing. In the late 17th century, he developed the Stepped Reckoner, a mechanical calculator that was capable of performing addition, subtraction, multiplication, and division. Relevant quote by Leibniz: “It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used.” - **Charles Babbage (1791-1871):** Babbage was an English mathematician and inventor, often regarded as the "father of the computer" due to his work on the design of early mechanical computers. He conceived two groundbreaking machines: the Difference Engine and the Analytical Engine. The Difference Engine was designed to compute and print mathematical tables, while the more advanced Analytical Engine was intended to be a general-purpose computing machine that could be programmed using punch cards. Babbage never completed the construction of these machines.

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