Reckoners, ASCC, page 0043

3
Bessie 
The Automatic Sequence
Controlled Calculator
.

... if it should ever turn out that the basic logics of a machine
designed for the numerical solution of differential equations coincide with the logics
of a machine intended to make bills for a department store, I would regard this
as the most amazing coincidence that I have ever encountered.
 Howard Aiken, 1956^{1}


In the summer of 1944, the American press reported the waging of the war:
grim news of battles in France, casualties, sacrifices and shortages at home. Among the
stories on August 7 was another report that probably did not arouse much attention. On that
day a large electromechanical computing machine was publicly unveiled at Harvard
University, in a ceremony attended by the presidents of both Harvard and the IBM
Corporation. The machine they dedicated that day was called the ASCC, short for Automatic
Sequence Controlled Calculator; later it would be known as the Harvard Mark I as
other similar devices appeared there. It was not really the first automatic computer.
Zuse's Z3 was already in use by then, and there were other topsecret projects in America
and Britain that also could have vied for that claim. But nonetheless its dedication is as
good a moment as any to mark the beginning of the computer age, for it was on that summer
day that the existence of the computer became public knowledge.^{2}
The Mark I was well publicized, and the details of its construction were
duly reported in the press. It was a generalpurpose calculator that could carry out a
sequence of operations specified by a program punched in a paper tape. It was not a
test machine like the Zuse computer but a fullscale device that was put to work as soon as
it was completed (well before its dedication that summer). The public dedication
brought news of computing machines to a world still at war, while it also marked the
entry of both the IBM Corporation and Harvard University into the business of building and
using large scale digital computers.
Reckoners, ASCC, page 0044

Like Konrad Zuse's relay computers, the Mark I was the product of the
vision and hard work of one man, although its enormous size and scale required the
cooperation of a large team of skilled collaborators. That man was Howard H.
Aiken, Commander in the U.S. Naval Reserve and Professor of Applied Mathematics at
Harvard.
Aiken grew up in Indianapolis, Indiana, where he attended the Arsenal
Technical High School. He went on to the University of Wisconsin, where he received a B.S.
in 1923, and then he worked as a power engineer for the next twelve years. But he
returned to school as a doctoral candidate in physics, first at the University of Chicago and
then at Harvard, where he submitted a thesis on "The Theory of Space Charge Conduction" in
1939.
He got interested in automatic calculating machinery while working on that
thesis. Like Zuse, he found himself spending hours computing sequences of sums and
products, aided only by a desk calculator and some scratch paper. What was worse, for many
of the problems he was trying to solve (involving the numerical solution of
differential equations) he found that the sheer number of individual calculations rendered any
solution impossible. He just could not do them all in a reasonable time without making errors.
It was not for any theoretical reason concerning the numerical methods themselves.
NUMERICAL METHODS OF SOLUTION
The problems that stumped him were nonlinear differential equations which
only numerical methods could solve. To solve those kinds of equations required
evaluating an integral: finding the area that would lie under the curve of an equation if
it were plotted on graph paper. For many equations one can find a corresponding formula that
gives the area directly. They are called analytical solutions; for example, the area under
the curve described by
y = sin(x), between two points, x1 and x2,
is given by the formula
A = cos(x2)  cos (x1)
But for many equations, especially those of the complexity that Aiken
faced, it is not possible to find such an equation. The area must be found by breaking it up
into small rectangular "slivers," whose height and width are known. (The person
selects the width in advance; the height of each sliver is given by plugging the appropriate
value into the original equation.)
So evaluating an integral by such numerical methods is straightforward; it
is the sum of the areas of each individual narrow sliver, whose areas are in turn found
by a simple multiplication of a constant width by a varying height.
Reckoners, ASCC, page 0045

The difficulty with numerical methods lies in the need to divide the area
under the curve into a large number of narrow slices to ensure enough accuracy in the
answer. Mathematicians had long ago established guidelines for just how fine this
mesh had to be for a given type of integral, but the logistics of carrying out
those elementary sums of products became a severe bottleneck as Aiken
worked on his thesis: not just the calculations themselves but the writing down,
labeling, and accurate recopying of intermediate answers mushroomed to
unmanageable proportions.^{4}
Aiken saw no reason why the whole process could not be carried out by
machinery: specifically by suitable combinations of commercial accounting and
tabulating equipment. Especially if intermediate results could be punched into cards rather than
written down, the errors of copying and reentering numbers would be reduced.
The first element of the system would be a card reader that would supply
the values of the function to be integrated. (Someone would have to punch the cards, but
by the 1930's many tables of mathematical functions were already being supplied as decks
of punched cards in addition to the more familiar form of books of printed tables.)
The values of the function would be taken from the card reader to a multiplying punch, where
the area of each sliver would be computed and punched onto another deck of cards. This
deck would be taken to an accumulating punch that would sum up the areas, and punch
the results onto cards. Finally those cards would be fed into a printer that would
print the results automatically.
In this method of computing, each part of the whole sequence of operations
is performed for all the values of the independent variable before the deck of
cards is transferred to the next station: first the values of f(x) are supplied,
then all the multiplications are done, then all the additions, and so on. Intermediate
results take the form of decks of punched cards that are physically carried (by hand) from
one station to another.
But Aiken felt that the intermediate steps of punching cards should also be
eliminated. Why not connect the various machines to one another by wires and not use
cards or printing except for the first and last steps? But to do that would require a change
in the way the computations were carried out. Instead of doing first all the
multiplications, then all the additions, and so on, the machines would have to do the
entire sequence of calculations for
each value of the independent variable: for each x, find f(x), then find
the area of the narrow segment under the curve at f(x), then add that value to the sum of the
areas obtained thus far. The machine would have to perform sequences of
elementary calculations, in contrast to the
parallel method suggested by the way that existing machinery operated.
Reckoners, ASCC, page 0046

(Consider evaluating f(x) = x^{3} + 3x^{2} + 4x + 4, for x = 1, 2, 3,
. . . ,10. Using punched card equipment, a multiplier would first be set up to produce the
cubes of the x's: 1, 8, 27,..., 1000, punched into a deck of cards. Another multiplier would be
set up to produce the next term: 3, 12, 27, . . . , 300, and so on. Finally the decks of
cards representing the values of those terms for all values of x would be fed into an accumulating
punch, which would sum their values to produce and punch the list of values of f(x) on a
deck of cards. That deck could be fed into a printer if desired. By contrast, a sequential
calculator would compute f(x) by performing the individual operations for each value of x,
one at a
time: f(1) = 12, f(2) = 32, . . . , f(10) = 1344. Hence the name "Automatic
Sequence Controlled Calculator" that he gave to his project.)
In 1937, while still a graduate student, he wrote a proposal for such a
machine.^{5} This proposal began by summarizing the reasons such a
machine with its novel way of computing sequentially was needed, and the
various attempts that had been made in the past to construct such a device. It
discussed in particular the unsuccessful attempt by Charles Babbage a century earlier to
build an Analytical Enginea machine Aiken recognized as a genuine ancestor to his
proposed sequence calculator.
Most of his proposal dealt with how the calculator differed from the
punched card machinery of the day. The major difference was of course that the devices
for addition, multiplication, reading cards, and punching or printing results would each
be connected to one another and would sequentially carry out a computation. (Aiken called
the method computing in "lines" instead of "columns," thinking of how a numerical
solution would be arranged on a page.)
Other features his machine would have were the ability to handle positive
and negative numbers equally well; an ability to supply, automatically, tables of
commonly used functions such as sines, cosines, logarithms, and probability functions;
and the ability to automatically increment the value of the independent variable each time a
sequence is carried out.
That last requirement is useful for numerical integration, where the width
of each rectangle is a small constant. There is no reason why the machine should
not supply those values automatically once it has been given a starting value and the size
of the constant increment. Implicit in that requirement is the need for the machine to stop
incrementing the xvalue when it reaches the endpoint of the integration.
The remainder of the proposal describes how a generalpurpose sequential
calculator could be built out of only a few basic components: out of machines that
could perform the four basic arithmetic operations, that could read selected values of
functions from punched cards or tape, and that could store and retrieve intermediate results on
mechanical registers.
The picture painted by Aiken's 1937 proposal is one of equipment racks on
which are mounted individual subassembliesthe innardsof standard business
equipment. Each of these assemblies would be connected to the others by a switchboard of
wires. Other equipment would supply constants such as pi or e, or specific constants
pertaining to a given computation. A standard printing device would print the results
directly, thus avoiding copying errors.
The sequence of operations would be controlled by punched cards or by a
long punched tape, as well as by physically wiring the individual units to one another
through a plugboard. That was the way that Babbage wanted to control the operation of
his Analytical Engineborrowing the idea from the silkweaving industry, where
by the 1830's punched cards were widely used to control the pattern woven into the cloth.
(The idea is usually attributed to J. M. Jacquard,
who successfully employed the method in his looms in Lyon in 1805; cruder
versions had been known in France since the 1720's.)
Reckoners, ASCC, page 0047

So Aiken proposed to use perforated media for both numerical storage (as
business machines used cards), and for control (as Jacquard looms used
them), in a single machine. The idea was a novel one (actually it originated with
Babbage, but it was not picked up again until Aiken's proposal). The business
machines industry had developed the use of punched cards from the work of Herman
Hollerith, who first applied them for the U.S. Census of 1890. But
Hollerith was probably not influenced by either Jacquard or Babbage, and the industry that
grew up around his invention never used the cards for anything but data storage.
Whatever control those machines had was done by plugboards. It was left to
Howard Aiken to see what had been clear to Babbage but strangely had been
neglected since: that punched card equipment could not only calculate: it
could also direct the sequence of those calculations automatically, using
perforated cards or tape for each.
To ensure enough accuracy of the results of any calculations done on such a
calculator, Aiken proposed that the machine be able to handle numbers of up
to twentythree decimal places in its individual registers. That was over
twice what commercial business equipment handled, but Aiken recognized a fundamental
difference in the kinds of problems his machine would be solving. Business
calculations are carried out to dollars and cents: two and only two digits
to the right of the decimal point. Few businesses dealt with amounts of money in
the range of millions of dollars, so seven or eight decimal places to the left
of the decimal point were almost always more than adequate. Business equipment and
desk calculators therefore had a capacity of eight to ten decimal digits,
with the decimal point fixed two places from the right of the register.
But scientific calculations could range throughout the number fields,
positive and negative, with very small quantities just as likely to occur as very
large ones. Zuse had met that problem by adopting floatingpoint arithmetic, with part
of each number set aside to store the position of the radix point. But that
complicated the construction of his machines; it would have been intolerably
complex had he not taken advantage of the simplicity of the binary scale of
enumeration.
Reckoners, ASCC, page 0048

But Aiken was committed to modifying commercial business equipment, which
operated in the decimal system and had a fixed decimal point. So he
proposed a register that could hold twentythree decimal digits, with the decimal
point fixed at the middle, giving a decimal range of at least 10^{±}.
The breakdown of the digits was to be:
0 000 000 000 000 . 000 000 000 00
^ ^ ^
  
sign decimal point two "guard digits" that would be rounded
0 = + , 9 =  off in the final answer
So Aiken retained the simplicity of registers with a fixed decimal point
while giving them a much longer "word length" to allow his machine to handle quantities
more typical of scientific calculations.
If Aiken drew the inspiration for his proposed automatic machine from
Charles Babbage's Analytical Engine a hundred years earlier, he also knew that
Babbage never completed more than fragments of that Engine.^{6}
Babbage was known to some
in the 1930's not as a man of great vision but rather as a failure: an impossible,
irascible dreamer who could not anchor his dreams in the reality of iron, brass, and steam. Aiken
felt that Babbage's "principles were theoretically sound," but he felt that the
problem lay in the Englishman's having to rely on a machinetool industry which was not up to
the demands of the project. The Analytical Engine was to have been powered by a steam
engine. Today that seems faintly ludicrous; if he had been successful, would we see
today's computer designer doing his business with an oilcan and an adjustable wrench instead
of with a voltmeter and an oscilloscope? But Babbage had no alternative to steam: in
1830 electricity was still a mysterious force, and practical motors that harnessed it were
still half a century away. Aiken of course could draw on the experience of forty years of
business machines and commercial calculators (both industries were established after 1890),
but he still had to face skepticism from his peers. His proposed machine was much more
complicated than the business machines he was to build it out of. There certainly was the
risk that he would fail, just as Babbage had failed, in getting all those individual pieces to
fit and work together as a system.
But Aiken succeeded. The Mark I was completed only a few years after he
proposed building it in 1937. Whatever else may be said about its design, it worked.
It succeeded where Babbage had failed, and its dedication in 1944 served as an existence
theorem for the largescale digital computer. (By 1944 there was at least one other
largescale computing machine that had been built that was the Differential Analyzer which was
built under the direction of Vannevar Bush at MIT. But it was an analog, not a digital
computer, and so its existence did not resolve the question of whether or not Babbage's
centuryold dream of a generalpurpose largescale digital computer would ever be more than just
thata dream).
Aiken first sought assistance from the Monroe Calculating Machine Company
to build the computer. But they turned him down. The Harvard astronomer Harlow
Shapley suggested to Aiken that he go to the IBM Corporation. Shapley knew that
Professor T. H. Brown of the Harvard Business School was on good terms with President
Thomas Watson of IBM. Through Brown, Aiken was introduced to Watson. So began not only
Harvard's but also IBM's involvement in computers.^{7}
Reckoners, ASCC, page 0049

Watson had already begun exploring the use of his company's computers for
scientific applications, with the establishment of a Computing Bureau at Columbia
University (under Wallace Eckert) in 1934. He did not believe that scientific
work would ever displace the more mundane business data processing as IBM's
bread and butter. But he was sufficiently impressed by Aiken's proposal to commit his
company's resources to the building of the calculator, although most of the money
used for its construction came from the U.S. Navy.^{8} What IBM contributed most
was the experience and skill of a veteran team of engineers from its Endicott,
New York, plant, where its punched card equipment was designed and built. The senior
engineer put in charge of the project was James W. Bryce, who had been with the company
over thirty years; during that time he amassed a long string of patents covering many
aspects of the company's line of punched card equipment. Bryce in turn assigned Clair D.
Lake, Frank E. Hamilton, and Benjamin M. Durfee to the project at the Endicott plant.
Aiken's own role was not as an engineer, even though he had experience in engineering. It
was rather as a consultant to IBM's own engineers: Aiken would sketch out what he wanted
the machine to do; Hamilton, Lake, and Durfee would match those needs with what they
knew existing IBM machinery could do. Aiken spent the summers of 1941 and 1942 in
Endicott, but in the early years of the war he also entered active duty as a commander in
the Navy, stationed at the Navy School of Mines in Virginia. At least some of that
time was spent performing duties entirely unrelated to computers. Nevertheless, Aiken
seems to have fitted himself well into that role; he was a selfdisciplined man who wore
his Navy uniform well. And though he had very specific ideas about how his computer
would function, he was glad to be able to use the resources of IBM and that
company's engineers to realize those objectives. Unlike Konrad Zuse, Aiken did not immerse
himself in the details of building the components of the machine.^{9}
The Automatic Sequence Controlled Calculator was completed by January,
1943, in Endicott, where it secretly demonstrated its abilities. It was then
disassembled and moved to Cambridge, to a room specially prepared for it in the basement of
Harvard's Cruft Laboratory (a few years later the Harvard Computation Laboratoryand the
Mark Igot their own building). It began doing serious work for the Navy's Bureau of
Ships in May, 1944, three months before its public dedication.
Before that unveiling, the ASCC was covered with a polished stainless steel
and glass enclosure: not an easy matter given the wartime shortages of those
materials. Watson insisted on it, although Aiken preferred leaving the case off to make the
computer easier to maintain and modify. Watson's decision reflected his attitude toward the
project and the IBM Corporations's general approach to computing machinery.^{10} He
did not see much future in making scientific computers, but that is not to say
that in the 1940's IBM was content to provide machines that only did simple tabulating
and counting. In fact, IBM had already developed machines that had quite a capability for
advanced numerical processing by 1944.
Reckoners, ASCC, page 0050

A decade earlier the company had designed and marketed an electromechanical
multiplier, the Type 601, that accepted two eightdigit decimal numbers
Reckoners, ASCC, page 0051

punched into a card and computed and punched their product in the remaining
columns of the card. (What was interesting about the 601 was that it computed products
not by repeated addition, as most mechanical calculators did, but rather by
looking up an internally stored multiplication table for the 10 x 10 decimal digits.) By
the late 1940's they were working on replacing the electromechanical parts of the 601 with
vacuumtube circuits that could operate much faster. In 1942, under the direction of R.
L. Palmer, they had developed a prototype "crossfooting" punch that multiplied
electronically.^{11} Work on electronic computing circuits continued at a separate
lab in New York City;
after 1945, research and development of electronic devices was centered at a laboratory
in Poughkeepsie, New York. (Endicott remained a center for electromechanical
research and development.) When IBM unveiled their Type 603 Multiplier in 1946, it was
one of the first commercial devices anywhere that calculated with electronic circuits.
(The 603 could perform up to sixty multiplications per second, which was about a hundred
times faster than the 601. )
Now a suitable combination of IBM's commercial equipment, including the 601
multiplier, coupled with tabulating, card reading, and printing machines,
could certainly do some scientific calculations just as easily as Aiken's Sequence Controlled
Calculator. Indeed, Wallace Eckert had been using those machines at Columbia for
astronomical calculations with great success (where, among other things, he calculated
the orbit of the moon).^{12} These standard punched card machines that IBM produced
could be
programmed to do short sequences of arithmetic, using plugboards to determine which
operations would be performed on selected fields of the punched cards. What they could not
do was execute an arbitrarily long sequence of operations, the kind Aiken wanted his
proposed machine to do. But of course that was precisely the pointthe Mark I's completely
flexible programming made it a generalpurpose computernot just a fancier
calculator, but something qualitatively different. The engineers and executives at IBM, and
elsewhere too for that matter, did not see that distinction as clearly in 1944 as it is
seen today.
So while IBM engineers took on building the ASCC, they did so because of
its status as a military project, not as a potential prototype for a future line of
commercial products. The U.S. Navy supplied most of the estimated $300,000 construction cost of
the machine, with IBM providing additional money for maintenance and support. The total
cost was probably over a million dollars; hardly in the same range as the rental
costs of IBM's existing lines of business equipment.^{13}
The gulf between Aiken's and Watson's perception of the future of computing
was hardly evident at the dedication ceremony, but it was deep. In accepting
the machine from Watson, Aiken and President Conant of Harvard graciously acknowledged their
debt to Watson and IBM, but although Aiken would go on to build other large scale
computers at Harvard, he did so without IBM's help. And for the company that was to
dominate the computer industry after 1955, the Mark I was really a reluctant step into a
new world.
Reckoners, ASCC, page 0052

DESCRIPTION OF THE MARK I
The public dedication of the Mark I in 1944 brought a wide response from
other mathematicians and engineers, who saw in it a way to solve problems they
too were facing. So in contrast to other wartime projects that were kept strictly
classified, the Harvard Computation Laboratory felt the need to provide a detailed
description of the Mark I and how it was programmed. Only the specific military problems for
which it was first used remained secret. Of course any publicity about the machine would
also reflect upon the expertise of the IBM Corporation, even if they were not offering
that kind of product to their customers.
The staff of the Computation Laboratory prepared a detailed Manual of
Operation for the Automatic Sequence Controlled Calculator in 1946, listing Aiken and
Lieutenant Grace Murray Hopper as principal authors. It is from that manual that we can get
a detailed picture of how the machine worked and what it could do.^{14} The Manual
describes the machine as it existed on September 1, 1945; after the war it was fitted
with a socalled Subsidiary Sequence Mechanism, which extended its powers, but by that time
it was already fast becoming obsolete.
The fundamental calculating element of the Mark I was a wheel that could
assume one of ten positions, for the ten decimal digits, just like a desk calculator.
(Some wheels had up to twenty positions: ten decimal digits, and others for control, carry, and
transmission of numbers from one wheel to another.) A mechanical desk calculator indicates
numbers by digits printed on the wheels themselves, but that was not the case with the
Mark I. It stored and counted numbers mechanically, but it transmitted and read those
numbers electrically. A wiper attached to each wheel made contact with one of ten
wires, one for each decimal digit, and that indicated the wheel's position. And the wheels
were turned not by a mechanical linkage but by an electric motor that advanced the wheel to
a position that corresponded to the number it was supposed to represent.^{15}
Hence the
description of the Mark I as an electromechanical computer.
In a desk calculator, number wheels are engaged by their contact with the
keys that the operator presses. If the operator keys the number 6 in a certain column,
the wheel corresponding to that column will move six and only six additional places
from its initial position (it may also initiate a carry if the wheel passes through zero).
In a mechanical device first the keys are pressed, then the number is "loaded" onto the
wheels by pulling a lever that moves each wheel the appropriate number of places.
In the Mark I that transfer was accomplished electrically, by a clutch
attached to each wheel. A 5hp electric motor drove a long shaft that ran the full
fiftyfoot length of the machine. When, say, the number 6 was to be loaded onto a wheel, the clutch
would couple that wheel to the shaft for a precise interval of time, until the wheel had
moved the six places. Then the clutch would disconnect the wheel from the shaft (see
Figure 3.1).
Reckoners, ASCC, page 0053

The timing pulses for the clutch relays were supplied by a series of cams
connected to the same rotating shaft. The cams could have up to twenty lobes
depending on their function. To add the number 6 to a register, an
electrical signal would be routed through a sixlobed cam, which would pick up the
clutch on the wheel for six units of time before dropping it off, while also
keeping track of any carry digit produced.
Should a number be subtracted from an accumulator (as the cells were
called), an electrical signal was sent to another cam having the "complement" of the
number's lobes (the difference between the number and nine: for example, the
complement of six is three). The shaft and the wheels always turned in the
same direction, but by adding the complements of numbers, with an appropriate
carry adjustment, the machine could subtract any number from an accumulator. (An
explanation of subtraction by complements is given in Glossary B.)
Note that when a number is added to a wheel, that wheel passes through all
the digits in between its original position and the sum. If the wheel was at the
position for number 3, and the number 6 was added to it, it would be
connected to the motor for six units of time, during which the wheel would pass
through the positions 3, 4, 5, 6, 7, 8, and 9, where it would come to rest. That is the
way all mechanical calculators work (at least all that use decimal numbers). When
human beings perform addition they usually proceed directly to the sum of
two digits, previously having memorized the addition table. Occasionally a
person forgets that table and has to add the numbers unitbyunit on his fingers.
(From the Latin word for finger comes the word digital itself. A digital machine,
whether it be a computer or anything else, operates with discrete units.)
Reckoners, ASCC, page 0054

For that reason the storage cells of the Mark I were called accumulators,
since they accumulated numbers in them one unit at a time. An accumulator further
more does not distinguish between storing a number and adding a number;
when a number was transmitted to a cell of the Mark I, it was automatically added to
whatever quantity was already in that cell from a previous operation. That property follows from
the way electrical impulses advanced its counting wheels. (Also, any carry digits produced
must be transmitted from one column to another; the cells of the Mark I, built from pieces of
standard punched card equipment, had that ability as well.) Simply to store a number in an
accumulator required that it be set to zero beforehand; that was done by subtracting
whatever number was in it from itself just before instructing the accumulator to accept a
new number for storage.
Modern terminology is often not as strict about preserving this distinction
between an accumulator and a register, which is the name given to a memory cell that
simply writes over any number in that cell when a new number is sent to it. Computers today
use binary arithmetic; their design favors doing all arithmetic in a central processor
(as the Zuse machines did). With few exceptions, modern memory units can store but not
accumulate sums; hence the term register to describe them. The main difference between
those registers and the "accumulators" also found in the processors of modern computers is
that the former do not contain carry circuits. But nowadays that distinction is not
strictly adhered to. Aiken often referred to the accumulators of the Mark I as registers,
when they were being used only
for storage of a number. (Both terms should be further distinguished from a
counter, which can also accumulate sums, but which can only be fed numbers from its
rightmost column.
An automobile odometer is a counter; miles are added to it only a tenth of
a mile at a time. An accumulator can add quantities to any column. These terms are further
explained in Glossary B.)
The fact that addition (and subtraction) was done in the accumulators had
an important bearing on the overall architecture of the machine. The Mark I had no real
"central processor" like the Zuse computers or like most modern computers. Any
memory cell could be used for addition and subtractionand there were seventytwo such cells.
Only multiplication and division were done in a separate processor unit. That
design reflects the Mark I's legacy: punched card accounting equipment that was mainly used to
tabulate and accumulate totals.
The Mark I's seventytwo accumulators could perform addition and
subtraction, as well as ordinary storage, by first clearing their contents to zero. Accumulator
number 71 (they were numbered from 1 to 72) was constructed so that it could be split in half to
handle two twelvedigit numbers. Transferring the contents of the other accumulators
through it allowed the machine to store twice as many numbers, each having half the
precision. Aiken noted that tradeoff was useful for statistical problems "where the
quantities dealt with are large in number and of low accuracy."^{}16
That such a feature
was built into the Mark I
suggests that the twentythree digit word length was probably too long for
many applications, meaning that the full capacity of the accumulators often was
wasted. But at the same time, the Mark I also had the ability to link accumulators 68 and 69
to each other, giving the machine a double precision of fortysix digits plus sign.
The need for that kind of numerical precision arose in one type of work
that the Mark I was especially designed to doto compute and print tables of
mathematical functions to many decimal places of accuracy.
Reckoners, ASCC, page 0055

Accumulator 72 was fitted with another special circuit that gave it
interesting properties. I have already mentioned that the Mark I performed subtraction
by adding complementsby adding the difference between the subtrahend and
999999. . . , then adding the digit 1 (the socalled endaround carry) to
the unit's place, viz.:
45793 45793
15342 +84657 the 9's complement of 15342
 
30451 130450
+ 1 the endaround carry; the 1 at
 the leftmost column drops
30451 away
But if the result of a subtraction is a negative number, there is no
endaround carry produced at the leftmost column. What happens instead is that the
digit 9 (which stood for  in the Mark 1) would appear in the signfield, followed
by the complement of the true answer:
5793 5793
15342 +84657 complement of 15342
 
9549 90450 last four digits are the complement
of the difference = 9549
sign digit = 9 no endaround carry
Accumulator 72 could detect when no endaround carry took place, and then it
would stop the machine. In that way, a function could be evaluated by
repeated iterations; after each cycle the value would be tested against a tolerance
value stored in cell 72. When the numerical accuracy of the Mark I exceeded that
tolerance, the machine would stop, "knowing" that further evaluation of the
function would not increase the accuracy of the result.^{17}
Thus the Mark I could automatically direct the course of its sequence of
calculations, at least in this limited way. The question of whether or not
the early computers could "choose" their paths of computation is a controversial one,
central to the distinction between what a computer is and what it is not,
even today. The Mark I had that capability, if only in a limited sense: it could
sense the results of its previous calculations and act on that knowledge (that
is, stop if programmed to do so by its operator). What it could not do was go to a
different sequence of operations depending on the number in cell 72. That capability,
which is really what is meant by the term conditional branching as it is applied
to computers, was only added to the Mark I after 1945 (as it was added to
Zuse's Z4 computer when it was installed in Zurich after 1949).
Reckoners, ASCC, page 0056

Aiken does not seem to have recognized the importance of that kind of
logical decision making beyond his use of cell 72. In another description of the Mark I by
Richard Bloch, the "register" is said to "permit the choice of one function if the value
of the independent variable is less than a given quantity."^{18}
But it is not clear if Bloch
meant that the computer could turn to a completely different sequence of operations
automatically depending on the value in register 72.
Given the way that the sequence of operations was fed into the machineby a
long strip of punched paper tapeit is clear that full conditional branching
capability could not have been built into it without some difficulty. There was only one paper tape
reader on the Mark I to read operations (three others read tapes containing data). The
readers ran the tapes in only one direction. Branching to a different sequence would have
required the tape to skip over instructions without executing them. It probably would
have also required that the tape be able to back up and repeat a group of
instructions in a loop. There would have to be a way for the tape to sense both where it was in a
sequence and where the next part of the sequence was locatedtwo pieces of information
not part of the Mark I's repertoire. In short, coding instructions by a single continuous
strip of tape does not lend itself well to conditional branching. When the Mark I was
fitted with that capability after the war, it was done by adding a completely separate tape
reader, on which were mounted coding tapes that were repeatedly executed.
Aiken wanted a machine that could evaluate long sequences of operations in
which only the argument is changed each time by a small increment. For that work there
is less need for flexible branching. Even today many computer programs do little more
choosing of their operations besides stopping the machine when the sequence is finished.
At least one of Babbage's writings on his proposed Analytical Engine
mentions conditional branching capability. The Analytical Engine was to be capable
of sensing when a computed value exceeded a certain tolerance, after which the machine
would advance or back up to another part of the instruction sequence (delivered to the
Engine by special "operation cards," not by a tape).^{19}
In 1841 Ada Augusta, Countess of Lovelace, a daughter of Lord Byron and a
gifted mathematician, learned of Babbage's computer and quickly mastered its
operating principles. Two years later she translated into English a description of
the Engine written by an Italian named L. F. Menabrea. To that translation she added a long
set of notes, longer than the original description itself, in which she discussed the
importance of the machine's ability to choose its own course of computations depending on
what it had just computed.^{20} In particular, she stressed the ability of the Analytical
Engine to reverse its operation cards and repeat a loop of operations again and again, until a
certain limit was metprecisely the ability added to the Mark I and to the Z4 after the
Second World War. Countess Lovelace's notes on Menabrea's description of the Analytical
Engine stand as one of the first thorough studies of the nature and power of digital
computers, written a hundred years before any working computer existed.
Reckoners, ASCC, page 0057

Aiken was familiar with Babbage's writings, although he may not have been
aware of Ada Augusta's commentary on the Analytical Engine. But in any case
Aiken did not feel that a full branching capability was a necessary part of his Sequence
Controlled Calculator.
The Mark I was built with the technology of the 1930's business machines
industry, but that industry was by no means a static one. Research was constantly going
on at the Endicott plant, and no doubt the Mark I benefited from that. The accumulators were
based on the accumulators used in standard IBM equipment. Other components of the Mark I
were taken directly from IBM's product line, modified only to the extent that they had
to be in order to be coupled to the rest of the machine. Card readers, card punches, and
electric typewriters could be plugged into the machine at various places. The multiplier unit of
the machine computed the partial products directly, one digit at a time, and was based
on the ingenious design of the IBM Type 601 multiplier first marketed in 1933.
Other parts of the computer were especially designed for it. The tape
readers, for example, supplied numbers or instructions by sensing the presence of holes
punched into twentyfour columns of stiff paper tape. If the tape was to supply a
number, each digit was coded into a group of four holes: four consecutive lines of tape for each
digit, by twentyfour columns for the full twentythreedigit number plus sign. The four
lines were punched as follows for the ten decimal digits:^{2l}
Decimal DigitsUp to 24 Columns
0 1 2 3 4 5 6 7 8 9
Line # 
4 0 0 0 0 1 0 0 1 0 1 v direction of tape movement [down]
3 0 0 0 1 0 0 1 0 1 0
2 0 0 1 0 0 1 0 0 1 1 1 = hole in tape; 0 = no hole
1 0 1 0 0 0 1 1 1 0 0
The machine read the bottom line first, twentyfour columns across.^{22}
If the tape was to supply an instruction, a different coding scheme was
used; it will be described later. The readers themselves were physically identical to one
another despite their being used for different purposes. It is conceivable that the Mark I's
designers could have recognized that instructions and data in a computer are fundamentally
equivalent to one another. That is of course the essence of the stored program principle; it
will be discussed at greater length in Chapter 6. But there is no mention of that concept in any
contemporary descriptions of the Mark I. In later years, as the stored program principle
became more widespread, Aiken was reluctant to acknowledge its advantages for computer
design.^{23}
Reckoners, ASCC, page 0058

In addition to the instruction and the value tape readers, the Mark I also
had a tape reader that read in values of functions for selected arguments. Those values could
be connected to a device called an Interpolator, which could calculate the value of the
function for any arguments in between the values punched into the tape, using familiar
methods of interpolation.^{24} In this way the Mark I could
supplyor generatetables of the most common logarithmic and trigonometric
functions, and their inverses, as needed in more advanced calculations.
The physical layout of the computer was determined by the common driveshaft
that synchronized the accumulators. Like a 19th century New England textile
mill, all the components of the Mark I were aligned on a common axis to be driven by the
rotating shaft. That layout ensured a precise synchronization in a straightforward
way. So the machine was long and narrow: fiftyone feet long, eight feet tall, and only
two feet deep. Photographs of the Mark I always show it polished and neat, occupying a
back wall of an equally wellpolished room at Harvard.
Finally it must be said that the Mark I had a distinctive sound unlike that
of any modern computer. It did its computing by electromechanical relays, not by
invisible and silent electrons in a microscopic silicon chip. There was no mistaking when
the Mark I was running, although it did not make a deafening clatter. It sounded
rather more, as one observer put it, "like a roomful of ladies knitting.^{25}
Table 3.1 summarizes the specifications of the Mark I in 1945.
Table 3.1 THE IBM ASCC (HARVARD MARK I)
Inventors:  Howard Aiken, C. D. Lake, F. E. Hamilton, B. M. Durfee.
 Place:  Endicott, New York; moved to Cruft Laboratory, Harvard University;
then to the Harvard Computation Laboratory.
 Dates:  Proposed 1937; construction begun 1939; completed January, 1943;
placed in operation spring, 1944; dedicated August 7, 1944. In use until 1959.
 Cost:  $300,000$500,000, exclusive of R & D.
 Technology:  Electromechanical, 3,000 decimal storage wheels. 1,400 rotary
dial switches, 500 miles of wire.
 Size:  51' x 8' x 2'; 19 panels; weight 5 tons.
 Number
Registers:  Decimal system, 23 digits plus sign (0 = + , 9 =  );
provision for half or doubleprecision arithmetic; subtraction by 9's
complements; parallel transfer of all numbers.
 Memory:  72 accumulators; 60 constant registers (set by 10position
switches); 3 valuetape readers.
 Processor:  Multiplier and divider, multiplication by table lookup; all
addition and subtraction done in accumulators; special circuits for log and
antilog (to base 10), sine, and interpolation of functions.
 Programming:  1 sequence tape reader, using twoaddress code: outfield,
infield, miscellaneous, other sequencing by plugboard; 1 "choice register"
and 1 "check register," allowing internal logical control.
 Speed:  Cycle time: 300msec. (one rotation of shaft); multiplication time
approx. 6
sec.
 Input:  3 valuetape feeds, 2 card feeds, plugboard, switches.
 Output:  1 card punch, 1 tape punch, 2 automatic typewriters.

Reckoners, ASCC, page 0059

THE MARK I IN OPERATION
How the Mark I actually solved problems may be considered separately from
what it was physically made of; the latter is the machine's "anatomy,"
while the former is its "physiology." Although the Mark I's components were similar to
those of commercial business equipment, it obviously worked in a different
way. In fact, its physiology was remarkably similar to other contemporary
computers, even if they had completely different physical anatomies.
A paper tape, called the sequence tape, supplied the instructions that
commanded the computer. This tape had twentyfour channels across its widththe
same as one line of a number tape. But unlike the number tape, a complete
program step could be coded into one line of tape (recall that a number tape
needed four lines to code a full twentythreedigit number). The twentyfour
channels were divided into three groups of eight channels each. The first
field specified from which part of the machine a number was to be taken
(the "outfield"), the second where a number would go (the "infield"), and the third
initiated an operation. Thus the Mark I was a twoaddress machine, but not
in the same sense as the term is used today. The operation specified by the code
in the last eight channels did not always specify an arithmetic operation to be
performed on the numbers given in the two preceding fields. Most of the outand
incodes referred to numbers in the accumulators; since those devices
automatically added quantities introduced into them, there was no need to give an
"add" command in the operation field to perform that most basic of the computer's
operations. The operation field would have to be punched with the command
for the machine to advance to the next line of coding, however; this was done by
punching a hole in the seventh place (from the right) of the opfield.
So the operation, "Take the contents of Unit 5, add it to the contents of
Unit 6, and go on to the next instruction," would be coded as follows:
outfield infield misc. or opfield
_ _ _ _ _ 0 _ 0 _ _ _ _ _ 0 0 _ _0_ _ _ _ _ _ _
8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1
Holes punched in columns three and one indicate the fifth accumulator; holes
punched in columns three and two indicate the sixth.^{}26 Note how this coding
is different from the coding of the number tapes. The eight columns for the in
and outfields were more than enough to address the seventytwo accumulators,
sixty constant registers, and other devices.
Reckoners, ASCC, page 0060

For multiplication, the third field functioned as a true operationcode
field, although multiplication was not done in a single step. First two steps
loaded the numbers to be multiplied from their locations (given in the outfield of two
successive lines of coding) into the multiplier unit of the machine (its
code given in the infield of the two lines). After the numbers were loaded, a command
was
given in the opfield to multiply, with the in and outfields left blank.
The next line of coding gave the address of a special register designed to handle up to the
fortysix digits of the product.
All in all, then, it required at least four lines of coding to perform a
multiplication. Depending on the length of the numbers themselves, it took from ten to
twenty machine cycles to perform the operation. Thus the maximum multiplication time was
20 x 300msec., or about 6 seconds.^{27}
If the sequence did not immediately require that product, the programmer
could specify other operations on the lines of coding between the command to multiply and
the command to store the product. That was, in fact, often done. So although
Aiken always emphasized the sequential nature of computations in his machine, the Mark I
was capable of a lot of parallel computation as wella consequence of the dual storage
and addition capabilities of the seventytwo accumulators.
Interpolation, consulting function tables, and other related functions were
handled in the same way; the function tables and constant switches were also given
addresses, although of course those addresses made sense only if punched into the outfield of
the tape. In the same way the output facilities such as the typewriters and card punches
were given addresses that could be punched into the infield.^{28}
The functioning of the Mark I is best illustrated by an example. The
following problem is probably too simple to justify using the computer for its solution, but
it is representative of the types of problems the Mark I handled. The example is
taken from the Manual of Operation, p. 292:
Evaluate the polynomial
F(x) = x^{4} + 3x^{3}  3x^{2}/4  22x + 3
by successive multiplication, in the interval 5 <= x <= 10, with
[Delta]x = 0.01 (that is, for x = 5.00, 5.01, 5.02,...10.00).
The first step is done mentally by the programmer. That is to rewrite the
above expression in such a way as to minimize the number of multiplications
required:
F(x) = (((x + 3)x  3/4)x  22)x + 3
This socalled nested parenthetical form reduces the number of
multiplications from ten to only four, including one division. Inasmuch as multiplication is a
timeconsuming operation, it is imperative that this form be used on any problem of
appreciable size.
The next step is to initialize the computer by storing the coefficients 3,
3/4, 22, and 3 in the switch (constant) registers. Other registers that are to be used in
the course of the computation have to be cleared to zeroa simple matter of coding the same
address in both the in and the outfield, plus the operation to
continue (the machine automatically interpreted this command to subtract
the number in the accumulator from itself, leaving zero).
Reckoners, ASCC, page 0061

Finally the starting value of the independent variable, 5.00, the ending
value 10.00, and the increment 0.01 are also stored.
After coding the tape to initialize the machine, codes for the actual
computation can begin. That consists of a sequence of elementary operations to evaluate the
polynomial for each value of x.^{29} Once calculated, the result is sent to an
output device, the value of x is incremented by 0.01, and the sequence is run through again.
When all values of x have been run, the operator would stop the machine manually by
pressing a button. Appendix section (2) lists the actual 36line coding for this problem on
the Mark I, followed by that same program rewritten for a pocket calculator, using the
same flow of data and instructions as the original.
ERROR CHECKING AND ACCURACY
Assuming the computer has successfully run through the sequence of
calculations and has printed out the successive values of x and f(x), the programmer now
faces the question of whether or not those results are accurate. The Mark I was an enormously
complicated device, containing threequarters of a million individual parts. Even a
simple program like the one just listed would exercise many of those parts, and they all had to
function correctly if the answers were to be correct. Actually there are several
different ways that errors can occur, and for each a separate treatment may be required. In his
analysis of the reliability and accuracy of the Mark I, Aiken recognized four different
types of errors:
 Errors inherent in the mathematical formulas themselves; the equations
yield only an approximation of the phenomena they describe.
 Roundoff errors caused by a repetition of the four basic operations of
arithmetic to only a finite decimal precision (23 digits on the Mark 1).
 Mistakes made by human operators: pressing the wrong buttons, setting
the constant registers improperly, etc.
 Errors resulting from a mechanical or electrical failure within the
calculator itself.^{30}
The first type of error is part of all mathematics, whether or not one is
using a computer. It was a problem that computer users faced, but then so so did
every other mathematician. The other sources of error are more directly consequences of
using a computer, although to a limited extent they appear even when simpler
mathematical instruments such as slide rules, planimeters, or desk calculators are used.
In 1944 there was no general agreement on what would be the best way to detect and minimize
these errors, and so the pioneers in computing each tried a number of different
approaches.
Reckoners, ASCC, page 0062

Errors of the second kind, roundoff errors, are inherent in any physical
process that must necessarily handle numbers of a finite length. Aiken sought to
minimize these errors by providing the Mark I with an extraordinarily long
word length of twentythree digits; yet there were times when he felt that even that was
not enough, and he resorted to a doubling up on accumulators to get fortysix digits. Later
generations of computer designers would find ways of minimizing roundoff errors while
carrying fewer digits in the machine's registersa careful analysis of the problem
beforehand often would reveal just how many digits were needed to give the desired accuracy, and
in many cases that proved to be far fewer than twentythree decimal digits.
Human errors of programming, setting switches, feeding stacks of cards, and
the like are much more difficult to detect the person who makes the error
often repeats it when he goes over his work again. By careful attention to
details, doublechecking by another person, and mechanical safeguards built into the
machine they may be kept to a minimum. One obvious test is to run the
program as set up with data that were known to give a certain answer beforehand; if
the machine does in fact deliver that answer, then it is safe to say that the
operation sequence was coded onto the tape correctly (this test does not always work,
of course, but it is good enough when combined with other checks). In all of
those approaches, Aiken and his staff were exploring unknown territory in what
would become an important branch of modern computer science.
The last type of error, due to machine malfunctions, was also difficult to
detect and prevent. A machine as complex as the Mark I had thousands of places where
something could go wrong at every step of a computation. Aiken stressed one approach
to that probleman approach which curiously reveals his knowledge of the history of
computing and his sense of his own place in that history.
The approach was to repeat the computation, using a subset of the input
data, but not using the same formulas as before. If the machine yielded the same results
as before while using different registers, accumulators, switches, and tape units, then it
was safe to say that the original computations were correct for all their input values. If
they did not agree, then further calculations could localize and evenutally isolate the
defective components. (A good computer programmer is like Sherlock Holmeswhen a program does not
work the programmer has to be able to piece together bits of strange and seemingly
incomprehensible evidence to find the culprit.)
The alternate method Aiken chose to evaluate polynomials was that of finite
differences the same method that started Charles Babbage on his long, frustrating road
to computers in 1822. So Aiken not only acknowledged Babbage as the inspiration for his
plans to build a sequence controlled calculator, but he also incorporated some of Babbage's
very methods into his machine as a way to check its reliability.^{31}
The link between Babbage and the modern computer age is a complex one. Other
computer pioneers from the 19351945 era knew of Babbage: Zuse in Germany,
Vannevar Bush at MIT, perhaps a few others. But Aiken, and probably only he,
consciously saw his own role as one of picking up where Babbage had left off a century
before.^{32} In his published descriptions of the Mark 1, Aiken always gave
generous credit to Babbage as the true pioneer in computing.
Reckoners, ASCC, page 0063

Most accounts of the history of the computer mention Babbage's Analytical
Engine as the machine that began the computer age. That is because in it one finds
all the functions that a true computing system requires: program control, memory, arithmetic
unit, and input/output facilities. The Analytical Engine was furthermore consciously
designed to be a generalpurpose computer, and that, too, is a central feature of today's
computers. But Babbage came to the idea of building the Analytical Engine after first
building a test model, then planning a fullscale model, of a more modest Difference Engine. (The
test model of the Difference Engine was the only machine that he ever finished.)
The Difference Engine was to have computed the values of polynomials by
summing up finite differences: by mechanizing only the operation of addition
(obviously much easier than mechanizing all the arithmetic operations). The method of finite
differences could nonetheless evaluate precisely the same kinds of functions that the Mark I
was planned to evaluate. Thus it was a good way to check the Mark I's circuits, as the
additions could be programmed in any of the seventytwo accumulators, checking the answers
yielded by the multiplier, divider, and other more complex parts of the machine during a
regular run of a program. Aiken conceived of the Mark I as the direct successor of the
Analytical Engine, and the Mark I was a generalpurpose computer. But the spirit of the
Difference Engine lived in it as well.
THE METHOD OF FINITE DIFFERENCES
Values of higherorder polynomials like the one given above may be
evaluated using only simple additions, once certain initial values have been calculated.
That was the secret of the Difference Engine: it could be mechanically simple, yet
mathematically sophisticated, by an ingenious linking of its wheels to one another.
The following table lists the values of the function given above:
F(x) = x4 + 3x3  3x2/4  22x + 3
for the integer values of x from 5 to 10:
x
 F(x)
 1st diff.
 2nd diff.
 3rd diff.
 4th diff.
 5th diff.
 5  874.25
 .  .  913.75
 6  .1788.00  .  540.5
 .  .  1454.25  .  174
 7  3242.25  .  714.5  .  24
 .  .  2168.75  .  198  .  0
 8  5411.00  .  . 912.5  .  24
 .  .  3081.25  .  222
 9  8492.25  .  . 1134.5
 .  .  4215.75
 10  12708.00

Reckoners, ASCC, page 0064

The numbers in the columns to the right of F(x) are obtained by subtracting
the two adjacent values in the column immediately to the left (for example, the
first entry in the column labeled "1 st diff.," 913.75, is the difference between F(6) and
F(5), the two numbers immediately to its left). The first entry in the column labeled
"2nd diff." is the difference of the first two entries in the firstdifference column, and so
on.
Note that the fifth difference is zero, and that the fourth difference is a
constant value (24). For any polynomial of the nth degree (the degree is the highest
exponent that appears in the expression), the nth difference will be constant. (It is in fact
n!([delta]x)^{n}a_{n}
Where n! is the product n(n1) (n2) (n3). . .(1), Ax is the increment
value of x, and a_{n} is
the coefficient of the highest exponent of x. In the above example the
increment of x is 1, the coefficient is 1, and the degree is 4, so the fourth difference is 4!,
or 24.)
So by calculating the constant nth difference, plus a few initial values of
the intermediate differences (that would have to be done by hand), one can
construct a table like the one above for more complex equations, using only the operation of
addition. Furthermore, those polynomials could approximate, as accurately as one
wished, any equations that one encountered in descriptions of realworld
phenomenaincluding the far more difficult and refractory transcendental functions such as
exponentials, logarithms, trigonometric functions, as well as the more exotic Bessel and elliptic
functions that would be the grist for the Mark I's mill throughout its lifetime.
Hence Babbage's Difference Engine would have been a powerful mathematical
tool had it been completed. Even though it was limited to ordinary polynomials, given
enough levels of differences it could have approximated just about any function a
mathematician would have requiredaccurately, tirelessly, mechanically. In Aiken's own words:
"Thus the calculus of finite differences has become the bridge between mathematical
analysis and numerical computation."^{33}
In 1864, long after Babbage had given up hope of ever completing either of
his computing engines, he wrote:
Half a century may probably elapse before any one without those aids which
I leave behind me, will attempt so unpromising a task. If, unwarned by my example,
any man shall undertake and shall succeed in really constructing an engine
embodying in itself the
whole of the executive department of mathematical analysis upon different
principles or by simpler mechanical means, I have no fear of leaving my reputation in his
charge, for he alone will be able to appreciate the nature of my efforts and the value of
their results.^{34}
Howard Aiken chose that quotation to introduce the Manual of Operation of
the Mark I.
Reckoners, ASCC, page 0065

The Mark I evaluated functions directly. The method of finite differences
was
used only as a check since it used different parts of the machine and was
coded in a different way without any multiplications. Results of the two methods were
compared in register 72, using its comparison feature: the machine stopped if the two
answers did not agree.^{35} Computations were checked frequently: as often as every twenty
minutes to ensure that the machine was not spewing out volumes of nonsense. Besides
using that check, problems were often recorded and run again using the same direct
evaluation method, but using different parts of the machine. In all, the checking of
the machine for errors was an important, timeconsuming part of the work of the Harvard
Computation Laboratory.
At its public unveiling, the Mark I was hailed as "Babbage's dream come
true." Aiken succeeded in building the Analytical Engine that Babbage failed to build.
But there were important differences between the two. Their anatomies were different: the
Analytical Engine was to have been powered by steam, the Mark I was driven by
electricity. Numbers in the Mark I were transmitted by wire, not by mechanical linkages.
But there were also more fundamental differences. Their physiologies were different
as well; had Babbage completed his machine it would have been programmed differently
(more like modern computers than like the Mark I, actually). That is the result of the
Mark I's arithmetic having been spread out through its memory units.^{36}
Howard Aiken probably did not perceive those differences, or if he did, he
did not think they were important. It is not clear exactly how much Aiken really drew
from his knowledge of Babbage besides the general idea. His 1937 proposal said that
the Analytical Engine was to have evaluated algebraic formulas "by the method of finite
differences," which was not true. Others have pointed out how, if Aiken had been familiar
with Babbage's writings, and especially with Ada Augusta's comments on his work,
he would certainly have recognized the importance of fitting his machine out with
full conditional branching capabilities  something he did not do until after the Second World
War when he learned of other computers that did have them.^{37}
But the spirit of Babbage lived in the clicking relays of the Mark 1. By
1959, when the machine was dismantled, those relays seemed as quaint and oldfashioned as
Babbage's "store," "mill," and Jacquard control. Babbage once remarked that he would
gladly give up the rest of his life for the opportunity to live a few days in the distant
future.^{38} Had he been granted his wish and come back to earth in the 1980's,
when executives
are stuffing computers into their attache cases while their children are stuffing
quarters into computer games at the local mall, he might be bewildered by it all. But if he could
have returned in 1944 to the Harvard Computation Laboratory, where he could have seen the
Automatic Sequence Controlled Calculator gobble up paper tape and whack out long
tables of Bessel functions on a typewriter, he would have felt right at home.
Reckoners, ASCC, page 0066

HOW THE MARK I WAS USED
The various magazine and newspaper articles which announced the dedication
of the Automatic Sequence Controlled Calculator in 1944 accurately described
its size, weight, storage capacity, and other physical specifications. But
they were necessarily vague as to the intended use for the computer. Most said only
that it would be used for classified problems by the Navy. Later accounts said that the Navy
used it "for ballistics and ship design. . . also... some problems in lens
design."^{39} No
doubt the ballistics work included the preparation of firing tables. The wartime need
for those tables was a major force behind the development of computers in the United States.
It was the sole purpose for which ENIAC was built. One other problem for which the
Mark I was used was "a highly secret, mathematical simulation of the first atomic
bomb." That calculation is still classified, but it might have been a calculation of
the bomb's blast effects.^{40}
But the main use for the computer was for the preparation and printing of
tables of mathematical functionsthe same use Babbage intended for his Difference
Engine. Time magazine devoted a cover story in 1950 to the new computers; in that story
the Mark I had acquired an affectionate nickname from its programmers"Bessie"because
she computed Bessel functions.^{41} By 1950 the urgent wartime problems were
gone; other military work was better done on the newer and faster computers at Harvard
and elsewhere. "Bessie" settled into a daily routine of computing and printing
tables of the various Bessel functions, and that would remain her work until she was
dismantled in 1959.
BESSEL FUNCTIONS
It was the desire to solve differential equations that started Howard Aiken
on his path to the design of an automatic computer. Such equations describe physical
phenomena in terms of a rate of change of a quantity that varies with time: hence the
name "differential." One particular differential equation arises frequently in the study of
vibration and periodic motion, and it involves not only rates of change but also second
derivatives, that is, rates of rates of change. This equation takes the form:
x^{2}(d^{2}yldx) + x(dyldx) + y(x^{2}  n^{2}) = 0
There are two families of solutions to this equation (since it contains a
second order derivative), and these are usually called solutions of the first and second
"kinds." For each kind there are many different solutions depending on the value of the
parameter n, which is usually a small positive integer but in fact may take on any positive real
value.^{42}
Besides the two basic families of solutions, there are also functions that
resemble a combination of the solutions of the first and second kinds. These are
sometimes called Hankel Functions of Order N, and I mention them because the first tables
that the Mark I computed were of those functions.^{43} There are also classes of
solutions defined for complex values of the independent variable, and finally,
integrals of all the various Bessel functions of the various kinds.
Reckoners, ASCC, page 0067

It was the job of the numerical analyst or engineer to determine which of
the above functions (if any) was applicable to a particular problem at hand;
appropriate values of n and x also had to be selected. That was often the most difficult part of
the analysis of a physical problem. But there still remained the problem of evaluating the
Bessel function itself. The Bessel functions resemble the trigonometric functions in that
they are periodic and often regularly pass through the zero value (the socalled zeros of the
Bessel functions). But there is no easy way to evaluate them. The analyst had to
select a polynomial approximation that fit the Bessel function closely enough for
the interval he or she was interested inusing techniques already well known in Babbage's
time. Obviously it would be advantageous if the engineer or mathematician could consult a
table of those values rather than compute them every time they were needed. It was the
task of the Mark I to compute (using appropriate polynomial approximations) and print tables
of the various kinds of Bessel functions for a wide range of arguments. The many
different kinds of functions and the range over which their values were tabulated ensured
that the Mark I would be busy for many yearseven after other more advanced computers had
long made its design and technology obsolete. The computations themselves were long
and involved, but they suited the abilities of the Mark I well, and they were not
hindered by the machine's limitations.
Bessel functions are used throughout the sciences and engineering. A
typical application might be the vibration of a plate that is fixed at its perimeter (like a
drumhead). The vibration at any point is a function of the elasticity of the material, its
tension, and the distance of that point from the fixed perimeter. Another common application
is the analysis of radio transmissions, especially FM signals, in which the
regular vibration of the carrier wave is modulated by a signal that carries the voice or other
information. Even the structure of the DNA molecule, consisting of a periodic repetition of a
basic chemical structure, was unraveled with the help of Bessel functions.^{44}
That was how the Mark I was used. In the years following the war, Harvard
University Press brought out a series of those tables, and they are available in
libraries throughout the world as the Annals of the Harvard Computation Laboratory, volumes 2
through 36. The first of them, the tables of Hankel Functions of Order OneThird, was coded
on the machine in August, 1944, and was run successfully that fall and winter. The
entire set of tables contained over 70,000 eighteendigit quantities and took the
equivalent of fortyfive days to be computed. Because that work had to be sandwiched in between the
other warrelated computations, the Harvard staff could not compute the tables in a
long continuous run, and so they had to retype the numbers by hand before they could be
published.
Reckoners, ASCC, page 0068

Subsequent tables were printed directly from the output of the Mark I's
output devices, which greatly reduced errors in copying and typing long strings of digits.
(Such errors were frequent in any tables set by handBabbage saw the elimination of that step
as a decisive advantage of his Difference Engine).^{45} The tables themselves were
computed by evaluating suitable polynomial
approximations for given intervals, and then checked by alternate methods
of direct evaluation (not, however, by the method of finite differences).
In his review of the speed and power of the machine, L. J. Comrie observed
that though the tables were useful and their evaluation took a lot of effort, the work
could nevertheless have been done by teams of human beings equipped with desk calculators. He
asked, "Does the calculator open up new fields in mathematical and numerical
analysis?"^{46} That is hard to say: it did do more than compute tables. But for the most part it
performed yeoman service for mathematicians and engineers. It never did anything
glamorous like crack codes or design secret weapons, but the Mark I represented a
breakthrough in computing nonetheless. The tables themselves were wellused, although by
1959 other tables of the same functions were also available. Sales of the volumes
brought in money for the Computation Lab, money that financed the construction of other
computers at Harvard.
Whatever else it did, it spelled the end of the era of computing tables by
hand. Even today, when computing power is cheap and widespread, there is still a need
for tables of the more complex functions. After the war the National Bureau of Standards
undertook a compilation of the most frequently used tables (using the Mark I's work for
the Bessel Functions). The NBS's Handbook of Mathematical Functions became a staple of
many a working mathematician's library. A numerical analyst today can compute the
desired values f one of the simpler Bessel functions at his desk, using a pocket
calculator or personal computer, but there are many times when that procedure is neither
practical nor accurate
enough. The computer itself has brought the field of numerical analysis to
a new level, where even more tables, more accurate and more elaborate, are needed. All
of that activity may be said to be part of the legacy of Howard Aiken and the Automatic
Sequence Controlled Calculator.^{47}
POSTWAR COMPUTING AT HARVARD
The rift between Aiken and IBM that surfaced before the dedication ceremony
grew after the war. It was not just the dispute over whether the ASCC was to be
covered or not. Thomas Watson felt that Aiken was unwilling to share any credit for the
success of the project. In 1944 IBM was not the industrial giant it is today. Aiken felt
that the computer was his idea; whoever happened to build it was of little import. The
Harvard Lab continued building computers under Aiken's directionbut not with IBM's
help. (The Harvard machines took on the names "Mark I," "Mark Il," and so on.)^{48}
Aiken was never convinced that vacuum tubes would make good computer
components, and so his machines continued to use relays long after much faster
electronic computers were completed elsewhere. He did use tubes in the Mark IV, but only for a
few units where speed was absolutely needed; everywhere else he used relays.^{49}
But by
1950 other pioneers were arguing that computers should use only vacuum tubes, handle
numbers in binary not decimal form, and store
their programs internally. On all those features Aiken swam against the
stream, and he did not prevail.
Reckoners, ASCC, page 0069

Meanwhile, Watson and IBM had a bitter first taste of largescale
computing, but they were smart enough to remain involved in the new technology. They of course
continued to develop and market faster punched card machines, like the 603 electronic
multiplier already mentioned. One of their customers, the Northrop Aircraft Company, hooked up
an improved type 604 multiplier to an IBM 402 accounting machine, and with
that they had a reliable automatic calculator of modest power. IBM recognized Northrop's
ingenuity and followed in 1949 by
marketing their own Card Programmed Calculator based on that
lashup.^{50}
But the CPC was not capable of fully general program control like the Mark
I, although its reliability and low cost made it a very successful product. The company
also went on to build a largescale computer along the same lines as the Mark I, which
they called the Selective Sequence Electronic Calculator, built in Endicott and dedicated
in 1948. As the name suggests, the SSEC could branch to different sequences in its
instruction set, but despite its name, it was more a relay calculator like the Mark I than a
true electronic computer. It used vacuum tubes only for a central highspeed processing
unit; electromechanical parts were used elsewhere. It handled numbers in the decimal system,
and despite the fact that it could store (and even modify) some of the instructions in its
highspeed memory, its architecture owed more to the punched card, plugboard technology
than to the new ideas about storedprogram computers that were current by 1948.
(Nonetheless, it must be given credit as the first machine to function with a stored program
capability.) The SSEC was dismantled in 1952, but it maintained IBM's
foothold in largescale computing at a critical time, and that foothold would be vital to the
company's entry into the mainframe computer business a few years later. (The company
marketed its first true storedprogram electronic computer, the 701, in 1952.)^{51}
The SSEC was installed in a prominent place at IBM's New York offices on
57th Street, in full view of passersby. (When the machine needed maintenance, curtains
had to be drawn to prevent the public from seeing the machine with its panels down.)
Like the Mark I it was well publicized, and the SSEC gave ordinary people their first
glimpse of what a real computer looked like. One littleknown fact is that it alone among the
early computers was available for private use, at a charge of $300 an hour. All the other
machines were used primarily by the military, with only "extra" time, if there was any,
allocated to civilians.^{52} IBM deserves credit for that innovation as well.
In conclusion, Aiken's work at Harvard was an impressive first step into
the computer age. But it did not contribute directly to the mainstream after 1950.
The fact that his laboratory was publicized meant that he became a clearinghouse for
correspondence and information about computers for those who were not privy
to other sources of knowledge. As a result he was always at the center of later
computer activity, even though his aversion to electronics and to the stored program principle
meant that his ideas had little influence.
Reckoners, ASCC, page 0070

The Computation Laboratory was perhaps his most lasting achievement. It provided a
place where a new generation of students could learn about computers, and many of his
students went on to become pioneers in their own rightto be sure, in ways that Aiken
might not have understood. (Ken Iverson, the designer of the language APL, and Fred
Brooks, who worked on the IBM 360, were two of Aiken's students.) He remained on the
staff until his retirement in July, 1961, after which he did some private consulting. He
died in 1973; by then his reputation as a pioneer was quite secure.
NOTES
 Howard Aiken, "The Future of Automatic Computing Machinery," Elektronische
Rechenanlage and Informationsverarbeitung (Darmstadt, 1956), p. 33.
 William Rodgers, Think: A Biography of the Watsons and IBM (New York: Stein and
Day, 1969), p. 172; "Giant New Calculator," Science News Letter, August 12,
1944, cover; August 19, 1944, p. 111; Volta Torrey, "Robot Mathematician Knows All the
Answers," Popular Science Monthly, 145 (October, 1944), 8689, 222, 226, 230.
 Henry Tropp, "The Effervescent Years: A Retrospective," IEEE Spectrum, 11 (February,
1964), 7081.
 Anthony Oettinger, "Howard Aiken," Communications ACM, 5 (June, 1962), 29.
 Howard Aiken, "Proposed Automatic Calculating Machine," IEEE Spectrum, 1 (August,
1964), 6269.
 Howard Aiken, "Historical Introduction," A Manual of Operation for the Automatic
Sequence Controlled Calculator, Annals of the Harvard Computation Laboratory, vol. 1
(Cambridge, Mass.: Harvard, 1946), p. 8; Tropp, "Effervescent Years," pp. 7273.
 Brian Randell, The Origins of Digital Computers: Selected Papers, 2nd ed. (New York:
Springer, 1973), p. 127; Rodgers, Think, pp. 166169; Charles and Ray
Eames, A Computer Perspective (Cambridge, Mass.: Harvard, 1973), pp. 114, 173.
 Rodgers, Think, pp. 169171.
 Manual of Operation, preface, forward; Tropp, "Effervescent Years," p. 73; Rodgers,
Think, p. 169.
 Rodgers, Think, pp. 169171.
 Byron E. Phelps, "The Beginnings of Electronic Computation," Report TR 00.2259, IBM
Corporation, Systems Development Division (Poughkeepsie, N.Y.: IBM, Dec. 9, 1971).
 Wallace J. Eckert, Punched Card Methods in Scientific Computation (New York:
Thomas J. Watson Astronomical Computing Bureau, Columbia University, 1940).
 Manual of Operation, preface; Richard M. Bloch, "Mark I Calculator," Annals of the
Harvard University Computation Laboratory, 16 (1947), 2330.
 "Giant New Calculator," p. 111; Thomas G. Belden and Marva R. Belden, The Lengthening
Shadow: The Life of Thomas J. Watson (Boston: Little, Brown, 1962) pp. 258261; Rodgers,
Think, pp. 169171.
 Edmund C. Berkely, Giant Brains, or Machines that Think (New York: Wiley, 1949), pp.
8993; J. H. Curtiss, "A Review of Government Requirements and Activities
in the Field of Automatic Digital Computing Machinery," Theory and Techniques for Design of
Electronic Digital Computers, Lectures Given at the Moore School, July
August 1946 (Philadelphia, 1947), lecture 29; Manual of Operation, p. 22;
Margaret Harmon, Stretching Man's Mind (New York: Mason/Charter, 1975), p. 118.
Reckoners, ASCC, page 0071

 Howard Aiken and Grace Hopper, "The Automatic Sequence Controlled Calculator I,"
Electrical Engineering 65 (1946), 384391; Bloch, "Mark I Calculator," pp. 2330.
 Aiken and Hopper, p. 205; Bloch, pp. 2425.
 Bloch, pp. 2425.
 Randell, Origins, pp. 11, 187188; Grace Hopper, "Computer Software," Computers and
Their Future (Lladudno: Richard Williams, 1970), pp. 7/37/26; Berkeley,
Giant Brains, pp. 90, 112.
 Philip and Emily Morrison, eds., Charles Babbage and His Calculating Engines:
Selected Writings by Charles Babbage and Others (New York: Dover, 1961), pp. 245295.
 Berkeley, Giant Brains, p. 96.
 Randell, Origins, p. 216.
 Manual of Operation, p. 45.
 Howard Aiken and Grace Hopper, "The Automatic Sequence Controlled CalculatorII,"
Electrical Engineering, 65 (1946), 449454.
 Jeremy Bernstein, The Analytical Engine (New York: Random House, 1964), p. 54.
 Berkeley, Giant Brains, p. 98.
 Manual of Operation, pp. 32, 111119, 296; "Mark I," Encyclopedia of Computer
Science" (New York: Petrocelli/Charter, 1976), pp. 852853.
 Howard Aiken and Grace Hopper, "The Automatic Sequence Controlled Calculator III,"
Electrical Engineering, 65 (1946), 522528.
 Manual of Operation, p. 16.
 Aiken and Hopper, "Automatic Sequence. . .III."
 Joseph 0. Harrison, Jr., "The Preparation of Programs for the Mark I Calculator,"
Proceedings of a Symposium on LargeScale Calculating Machinery, Annals of
the Harvard Computation Laboratory, vol. 16 (Cambridge, Mass.: Harvard, 1948), pp.
208210.
 N. Metropolis and J. Worlton, "A Trilogy on Errors in the History of Computing,"
Proceedings First USAJapan Computer Conference, Tokyo, 1972, pp. 684685.
 Manual of Operation, pp. 10, 296304.
 Morrison and Morrison, Charles Babbage, p. 142.
 Manual of Operation, pp. 10, 296304.
 L. J. Comrie, "Babbage's Dream Come True," Nature, 158 (October 26, 1946), 567568;
Tropp, "Effervescent Years," p. 73.
 Randell, Origins, pp. 187, 192; Metropolis and Worlton, "Trilogy," pp. 684685.
 Morrison and Morrison, Charles Babbage, p. xxxi.
 Science News Letter, August 12, 1944, p. 11; Time, August 14, 1944, p. 72; Eames,
Computer Perspective, p. 122.
 "Mark I," Encyclopedia, p. 853.
 Time, January 23, 1950, pp. 5460.
 M. Abramovitz and 1. Stegun, Handbook of Mathematical Functions (Washington, D.C.:
National Bureau of Standards, 1964), pp. 358494; Frank Bowman,
Introduction to Bessel Functions (New York: Dover, 1958).
 Published as the Annals of the Harvard Computation Laboratory, vol. 2, 1945.
 Bowman, Bessel Functions, pp. 122134; Francis Crick, "The Double Helix: A Personal
View," Nature, 248 (April 26, 1974), p. 766.
Reckoners, ASCC, page 0072

 The tables were reviewed in Mathematical Tables and Other Aids to Computation, 2
(1946), 176177.
 Annals of the Harvard Computation Laboratory, vol. 2, introduction; Comrie,
"Babbage's Dream," p. 568; Morrison and Morrison, Charles Babbage, p. 314.
 Herman H. Goldstine, The Computer from Pascal to von Newmann (Princeton:
Princeton University Press, 1972), p. 17.
 Rodgers, Think, p. 178; Randell, Origins, p. 188.
 Oettinger, "Howard Aiken," p. 298.
 John W. Sheldon and Liston Tatum, "The IBM CardProgrammed Calculator," in Randell,
Origins, pp. 229235.
 Cuthbert Hurd, "Early IBM Computers: Edited Testimony," Annals of the History of
Computing, 3 (1981), 168; Belden, Lengthening Shadow, pp. 258261.
 Time, January 23, 1950, p. 59.
