the wiener memorandum on the mechanical solution of partial differential equations

Annals Hist Comput (1987) 9:183-197 0 American Federation of Information Processing Societies

The Wiener Memorandum on the Mechanical Solution of

artial Differential Equations P. MASANI B. RANDELL D. K. FERRY R. SAEKS

Categories and Subject Descriptors: K.2 [Computing Mileux]: History of Computing-hardware. G. 1.8 [Mathematics of Computing]: Numerical Analysis-partial differential equations; difference methods.

General Terms: Design, Theory Additional Key Words and Phrases: Norbert Wiener, Conceptual

Machines

Norbert Wiener and the Computer

P. Masani

In 1940, with war raging in Europe, Vannevar Bush sought from his colleagues at MIT research proposals that might contribute to national defense. The Wiener letter and memorandum that follow were written in response to Bush’s query.’ In this introduction we trace the intellectual an- tecedents behind these two rather remarkable documents.

In 1908, during his second year at Tuft’s Col- lege, Wiener took courses in philosophy and has his first fascinating glimpse of Leibniz, the great philosopher, mathematician, and computer builder. Two years later (at age 16) Wiener wrote a paper entitled “The Phylogenetic Development of the Brain,” a cellular organ, but computer none the less.’ Only in the late 193Os, however, long after

‘Though written in 1940, they were printed only in 1985 in the Wzener Collected Works, Vol. IV, and accordingly bear the numbers [85a], [85b] in the Wiener Bibliography con- tained therein. All references in square brackets are to papers in the Wiener Bibliography. All references in braces are to papers by other authors.

‘Unpublished manuscript. An earlier essay “The Theory of Ignorance,” written in 1906 (at age 111, concerned the chaotic aspects of reality, to the understanding of which Wiener went on to make profound contributions, but to which Leibniz seems to have been singularly blind.

completing his studies in mathematical logic and philosophy under Josiah Royce at Harvard (1911- 13) and Bertrand Russell at Cambridge (1913- 15), and after his own work in computing and engineering, was Wiener able to grasp the pro- fundity of Leibniz’s reflections on computers. He realized that

. . . the step from a system of deduction to a de- ductive machine is short. The calculus ratiocina- tor of Leibniz merely needs to have an engine put into it to become a mmhina ratiocinatrix. The first step in this direction is to proceed from the calculus to a system of ideal reasoning machines, and this was taken several years ago by Turing. [53h, p. 1931

To this insight derived from Leibniz, Wiener added one of his own. Whereas to the average philosopher the Russellian type hierarchy is a technicality, in Wiener’s philosophy such hierar- chies are fundamental to all forms of communication.3 Wiener assigned types 1, 2, 3, . . . to computers and learning machines, and felt that here, as in logic, trouble arose when type lines

‘And this included all social intercourse, peaceful or war- like. See [60d] for the introduction of types in the military field, and remarks on von Neumann games.

Annals of the History of Computing, Volume 9, Number 2, 1987 l 183

P. Masani and others l Wiener Memorandum on Mechanical Solution of PDEs

are uncritically transgressed (cf. e.g., 160bl). We must turn to the practical experiences that ac- companied such theoretical wisdom.

Ardently patriotic, Wiener, like H. W. Fowler (author of King’s English), made futile attempts to enlist in World War I, although he was pat- ently unsuited for military service. Failing one army exam after another, he did a short stint in 1916 as a technician with General Electric in Lynn, Massachusetts, and in 1917 became a staff writer for the Encyclopedia Americana in Albany, New York. In July 1918, he received an offer from Os- wald Veblen (then a major) to serve as Computer at the U.S. Army Proving Grounds in Aberdeen, Maryland. This was a godsend to Wiener, and he readily accepted. He joined in the construction of range tables for the new type of artillery and am- munition being designed. The demand for speed and accuracy necessitated the use of computers. Wiener’s year at Aberdeen was one of the hap- piest in his life. He has written:

* * . I am sure that this opportunity to live for a protracted period with mathematics and mathe- maticians greatly contributed to the devotion of all of us to our science. Curiously enough, it fur- nished a certain equivalent to that cloistered but enthusiastic intellectual life which I had previously experienced at the English Cambridge, but at no American university. [53h, pp. 258491

This early sustained exposure to practical computing was an important ingredient in Wiener’s intellectual evolution.

During the early 192Os, Wiener’s energies were devoted to postulate theory, the Brownian motion, and potential theory. But he was alive to the connection of his own ideas with those of his fa- vorite electrical engineer, Oliver Heaviside, and wrote the paper on the operational calculus 126~1. Wiener’s more tangible engineering predilections showed up, however, in his excitement over the differential analyzer and analog machines that Bush had set out to build.

Bush’s early continuous integraph, designed to trace the curve y = r f(t)g(t)dt, was unable to deal with the curve

In early 1926, while watching a show at the Cop- ley Theater in Boston, Wiener saw in a flash how this might be done: measure I(X) as the intensity

of light after it has passed through two apertures, A and B, cut out in the shapes of the curves y = f(x) and y = g(x) (interposed between the light- source and screen in planes parallel to the screen), with the B aperture rotated through 180” about the y-axis, and then displaced x units along the r-axis. Wiener, like Leibniz, liked to see his ideas worked out in-the-metal, but being myopic and manually clumsy, he had to rely on colleagues for this. Nearly a dozen papers and theses were written on his machine by Bush’s younger colleagues, starting with K. E. Gould (1929) and ending with Hazen and Brown (1940), by which time the apparatus was equipped with a moving film and called the cinema intergraph.

In the early 1930s Bush raised the question of the mechanical integration of partial differential equations. The numerical data to be computed are now spread over a planar or higher dimensional region. Wiener soon sensed the impracticality of carrying out such integration by analog devices. The idea of discretizing the data over a grid and then averaging came naturally to him, for he had done something similar in his 1923 work on Nets and the Dirichlet problem [23b]. Wiener surmised that the data could be essentially read off and av- eraged by a line-by-line scanning of the grid, as in television, and that the data had to be represented digitally rather than intensively (cf. [56g,

Pesi Rustom Masani is a University Professor in Mathematics at the University of Pittsburgh. He formerly held professorships in Bombay and at Indiana University in

Bloomington. He received his B.S. from the University of Bombay, and his A.M. and Ph. 0. from Harvard University in 1946. He was a member of the institute for Advanced Studies, Fullbright-Hays Senior Scholar at the University of Tbilisi, Alexander von Humboldt Senior Visiting Scientist to the GFR, and has held many other visiting positions. His main interest is in probabilistic functional analysis and cybernetics, specifically in prediction theory, vector measures and positive definiteness. He has collaborated with Norbert Wiener, edited the Wiener Collected Works in four volumes, and written a short biography of Wiener in the Encyclopedia of Computer Science and Technology.

184 l Annals of the History of Computing, Volume 9, Number 2, 1987

P. Masani and others

p. 138]).4 In 1935, working with Y. W. Lee at Tsing Hua University in China, Wiener became aware of the speed advantages of electrical relays over mechanical ones. Since the time required to scan an n-dimensional cube of edge-length I is the same as that required to scan a line-segment of length I”, Wiener saw the imperative necessity of high speed, and the importance of using valves and electronic gear to ensure this. He also realized the importance of using the binary scale after he had seen G. R. Stibitz’s machine exhibited at the sum- mer meeting of the American Mathematical So- ciety at Dartmouth. On the way home he told his colleague, N. Levinson, of his growing conviction that when equipped with proper scanning devices, “electronic binary machines would be pre- cisely the devices required for the high-speed computation needed in partial-differential-equation problems.” [56g, p. 2311

Thus, when Bush’s inquiry came in 1940, Wie- ner had merely to articulate in detail thoughts that had been incubating in his mind over a long period. The letter and memorandum were the results. What Wiener proposed was a high-speed electronic special-purpose computer, modern in all respects except for a stored program. Bush, however, sensing no quick defense payoff in the Wie- ner proposal, decided to set it aside for the du- ration of the war.5 Wiener then turned his intellectual energies to the antiaircraft fire-control problem.6

But Wiener’s interests in the computer did not dwindle. Under the influence of his neurophysio- logical friend, Arturo Rosenblueth, they merely shifted to the brain, the cellular computer, which he had written about at age 16, and continued to do so till his death in 1964.7 The circle around Wiener in the 1940s also included Walter Pitts, and later Warren McCulloch. It was the proxim- ity of the ideas expressed in their important paper on nervous nets (1943) with those in C. Shan-

41n the Wiener memorandum, reprinted below, this pro- cedure essentially amounted to the relaxation technique of Christopherson and Southwell, as B. Randell points out in his comment.

%iome of the possible reasons for this decision are alluded to in Randell’s comments in this issue.

‘For the enormous ramifications of this last work on time- series analysis, noise-filtration, teleological and learning machines, communication and automation theory, and sensory prosthesis, see Masani and Phillips (1985) and other articles in the Wiener Collected Works, Vols. IZI and IV, and Wiener’s paper 162bl.

7For this phase of Wiener’s activity, see his Cybernetics 161~1 and the papers and commentaries in the Wiener Collected Works, Vol. IV.

l Wiener Memorandum on Mechanical Solution of PDEs

non’s earlier paper on switching systems (1938) that convinced Wiener that the operating principles of the brain and the electronic computer are fundamentally the same.

To Wiener, the electronic computer was a pros- thetic aid to enable the human mind to transcend its limitations and better perform its duties. (To him, science and religion themselves were pros- thetic devices that enable the human race to live symbiotically and in rapport with nature.) In his 1953 paper on Gestalt [53d], for instance, Wiener suggested how the computer might help in the visualization of four-dimensional surfaces that appear in the study of several complex variables. This thought was definitively articulated by H. H. Goldstine and J. von Neumann in 1946 (in John von Neumann 1963). But, as the reader will see, a clear inkling of the same thought occurs on the last page of Wiener’s 1940 memorandum. Wiener attributed great significance to the computer, and saw in it an instrument that could do enormous good if properly used, and colossal harm if mis- used. In God and Golem, Inc. [64el he character- ized its misuse for the aggrandizement of human greed, murderousness, and vulgarity as simony; that is, the modern version of the practice of the Black Mass in the Middle Ages.

* * *

I first saw the Wiener letter and memorandum in the Wiener Archives at MIT in 1981. As editor of the Wiener Collected Works, I felt they were important and should be published because they were not mentioned in the standard literature, such as H. H. Goldstine (1972), and only a hand- ful of individuals were aware of them. Some ex- perts in the field also favored publication. This decided, the policy of the Collected Works-to supplement all Wiener papers with elucidating comments by one or more contemporary schol- ars-had to be met. Because numerical analysis, hardware, and software are all involved in the Wiener memorandum, I approached B. Randell, D. K. Ferry, and R. E. Saeks for their comments on its different aspects. I am grateful to them for their contributions, which follow the Wiener memorandum in this issue, as they do in Vol. IV of the Collected Works.

References

Goldstine, H. H. 1972. The Computer from Pascal to von Neumann. Princeton, N.J., Princeton Univ. Press.



Goldstine, H. H. and J. von Neumann. 1963. “On the Principles of Large-Scale Computing Machine, 1946” (unpublished). In John von Neumann, Collected Works, Vol. 5, New York, Pergamon Press, pp. l-32.

Gould, K. E. 1929. “A New Machine for Integrating a Functional Product.” J. Math. Physics 17 pp. 305- 316.

September 21, 1940

Hazen, H. L. and G. S. Brown, 1940. “The Cinema In- tegraph. A Machine for Integrating a Parametric Product Integral.” J. Franklin Institute 230 pp. 19- 44, 183-205.

Masani, P., and R. S. Phillips. 1985. Antiaircraft Fire- Control and the Emergence of Cybernetics, N. Wiener Coil. Works, Vol. ZV, Cambridge Mass., MIT Press, pp. 141-179.

McCulloch, W., and W. Pitts. 1943. “A Logical Calculus’ of the Ideas Immanent in Nervous Activity. Bull. Math. Biophys. 5, pp. 115-133.

Shannon, C. E. 1938. “A Symbolic Analysis of Relay and Switching Circuits. Trans. Am. Inst. Electr. Eng. 57, pp. 713-723.

Letter Covering the Memorandum on the Scope, etc., of a Suggested Computing Machine (September 21, 1940)

N. Wiener*

*Reprinted from pp. 122-124 of Norbert Wiener: Collected Works, Volume IV, edited by P. Masani, MIT copyright, 1985, with the kind permission of the publishers, MIT Press, Cam- bridge, MA 02142.

186 l Annals of the History of Computing, Volume 9, Number 2,

Dr. Vannevar Bush Carnegie Institution Washington, D .C.

Dear Bush:

In response to your suggestion of this morning, I am sending you a memorandum concerning my proposed computing machine for the solution of boundary value problems in partial differential equations.

This device solves a partial difference equation involving the time, and asymptotically equivalent to a partial differential equation involving the time, yielding for infinite time a purely space partial differential equation which may be of very different forms. This partial difference equation is solved by an apparatus which repeatedly scans a collection of data recorded on some very inex- pensive device and replaces these data by new data. This replacement is done by an apparatus which, as there is only one of it, may be reasonably elaborate without making the entire apparatus too costly.

I shall take up as a particular example a method for solving the boundary value problem for the Laplace equation

a2u a2u

iP+-=O* ay

We replace this by the difference equation

u(x + 1, y) + U(X - 1, yl + U(X, y + 1) + u(x, y - 1) = 0.

and obtain the solution of this difference equation as the limit for infinite values of time of the solution of the difference equation

;{u(x + 1, y; t) + U(3c - 1,y; t) + u(r,y + 1; t)

+ u(x,y - 1; t,} = z&x, y, t + 1).

It will be seen that in this last equation the step of moving forward one unit in time corresponds to the step of replacing numerical values at the meshes of a net by their average. In this replacement the boundary of the net is left untouched. We thus require an apparatus which will record a function at all the meshes of the net, combined with the scanning apparatus which will read the values at the four meshes surrounding a given

1987


point, average them, and replace the average value as the new value for this point. I propose that the values be recorded in the binary system on an endless steel tape in the usual magnetic manner. That is, this tape will contain ten parallel lines marked and scanned magnetically, and these lines will indicate a number between 0 and 1024, ac- cording to the lines which are printed or left blank at a particular cross-section of the tape. There are scanning apparatus placed slightly before and slightly after a given point as well as apparatus removed from this point by a fixed distance before and after it, to indicate points above it and below it when a pattern of equi-spaced lines on a two-dimensional figure are placed consecutively on the tape. These four values are read by an apparatus leading to a rapid electrical adding machine on the binary scale, and after discarding the last two digits the sum is printed magnetically on the binary system on the tape after a distance equivalent to the sum of the lengths of all the lines covering the figure. It will probably be ad- vantageous to slightly modify this arrangement by carrying printing and reading at the same level of the tape, printing on one side of a broad tape while reading on the other and vice-versa. The boundary values are to be carried on still a third width of tape and are to be switched in at the appropriate points by an appropriate signal, in place of the running values on the tape. The boundary values are never erased, but the rest of the tape is swept clean magnetically after it has been used.

The solution of this problem is

/ ^\ . TX . Try 5-P

s1n 100 sln 100 exp i ) 2( 100)Zt

so that the error is reduced by the factor e in a single scanning. If we are aiming at an accuracy of one part in 1,000, this means that 1.4 scannings are necessary and therefore that on each line 140,000,000 impulses must be recorded before the scanning is complete. Assuming a rate of scanning of 10,000 impulses per second, this yields a time not exceeding 3.89 hours for the final answer to be obtained.

If we can record 100 impulses per inch, the steel tape need not be over 100” long; furthermore, it must be wide enough to record, without confu- sion, thirty lines of impulses. Certainly a 6” tape will not be out of the question. The linear speed of the tape will be such that it makes a complete circuit in a second. When the scanning is complete, as the tape contains all the numerical values of the potential function for all the meshes inside the boundary, it will be perfectly practica- ble to devise a machine for printing the results in the form of a table.

Let it be noticed that the rapidity of the machine varies inversely as the fourth power of the number of meshes.

Now for an estimate of the speed of this apparatus. Taking a network of 100 by 100 points and replacing the difference equation of the apparatus by the closely related differential equation, the problem of determining the speed of the apparatus is equivalent to that of determining the rate at which a solution of the differential equa-

I have here assumed a magnetical printing and scanning. It is perfectly possible to replace this by a photoelectric scanning combined with a printing by some such device as an electric spark. The difficulty in this latter case is that erasure is im- possible and excessive quantities of paper tape must go to waste. However, you will see that the variations in method keeping the same idea are enormous.

To vary the setup for partial differential equations, whether linear or not, we have to modify the number of pickup points in accordance with the order of the equation, and the electrical apparatus between the scanning and the printing devices in accordance with the form and degree of the equation. Notice that no apparatus repeated more than a small number of times is changed by this. Thus the expense of the apparatus is only slightly increased. Of course, for more complicated boundary value questions a more complicated method for indicating and putting in the boundary values is needed, In any event this apparatus is primarily adapted for equations of the elliptic type.

tion

tends to 0 when the boundary values are 0 and the boundary is interior to the square

x= 0; x = 100; y = 0; y = 100.

We can estimate this by taking the extreme case of the complete square and initial values

TX . Try sin 100 sin 100 I repeat the main contribution which I have here



to make to the problem of the mechanical solution of partial differential equations is the con- sistent use of the idea of scanning; this is eco- nomically important as it allows expensive apparatus to be used repeatedly and efficiently.

I can think of many problems to which the method could be applied. For example, there are problems concerning the flow around airplane wings and problems concerning the field around condensers and there are hydrodynamical problems, problems from the theory of elasticity, problems concerning sound waves of finite amplitude, problems from interior ballistics, and so on. I can’t say from the outside just which, if any of these, are of more importance.

There is another quite different matter which I mentioned to you in private conversation and which I would like to repeat for purposes of record. It has to do with the idea of making an antiaircraft barrage by bursting in the air containers of liquified ethylene or propane or acetylene gases so that an appreciable region will be filled with an explosive mixture of these in the air. The idea is that in such a way a region of the air may be interdicted to enemy aircraft-a matter of tens of seconds or of minutes instead of the fraction of a second during which a high explosive burst is continuing, and that thereby the efficiency of a barrage from a small number of anti-aircraft guns may be appreciably increased.

I enjoyed very much the chance to talk with you on your visit here and hope you can find some corner of activity in which I may be of use during the emergency.

Very sincerely yours,

Norbert Wiener

Memorandum on the Mechanical Solution of Partial Differential Equations

N. Wiener*


This memorandum [85b] is printed from a copy of an il- legibly typed manuscript containing several errata as well. What follows is an edited version in which minor errors and omissions have been corrected, some ambiguous terminology altered, and a couple of cryptic sentences rewritten.

To understand the discussion following eq. (141, the reader might well refer back to the corresponding discussion in the middle of the preceding letter to Dr. Bush. -P. Masani

The projected machine will solve boundary value problems in the field of partial differential equations. In particular, it will determine the equipotential lines and lines of flow about an airfoil section given by determining about 200 points on its profile, to an accuracy of one part in a thousand, in from three to four hours. It will also solve three-dimensional potential problems, problems from the theory of elasticity, etc. It is not con- fined to linear problems, and may be used in direct attacks on hydrodynamics. It will also solve the problem of determining the natural modes of vibration, of a linear system.

1. Certain Typical Partial Differential Equation Problems

Partial differential equation problems of the second order belong to several different types, among which the following are particularly important:

A. Hyperbolic problems. Otherwise known as wave problem. Example:

2 2 2 i!-!+t+!Z;

ay at u and $ are given for t = 0.

If region studied is finite, some boundary condition such as u = 0 or au/an = 0 given along boundary.

B. Parabolic problem. Otherwise known as heat flow problem. Example:

a2u 2 + d” = d”*

ax2 ay2 at’ u alone is given for t = 0.

Boundary conditions as in A. C. Elliptic problem. Example:

2 2 g+q=o. ay Time does not enter into this problem. Boundary conditions as in A. Can be solved by letting t -+ 00 in B.

All of these, as well as many non-linear problems such as those of hydrodynamics, and many problems of higher order, such as those of elasticity, may be put in the following form:

a. We start with a time variable t, a number of space variables, x1, . .., x,, and a number of de- pendent variables, ui, . . ., u,.

b. We have a number of equations of the form



f$=f,(t,xl ,..., x.,,,,...,.,,~ )...) $2 ,... ). u(x + 1, y, t) - 2u(x, y, t) + u(x - 1, y, t)

1 1 + u(x, y + 1, t) - 2u(x, y, t) + u(x,y - 1, t) (4) (1)

= 4u(x, y, t + 1) - 4u(x, y, t),

c. We know the initial values of ul, . . ., u, as functions of x for t = 0.

d. We know certain auxiliary boundary conditions for the uJ in space.

For example, B is already in this form, and A becomes

Ol-

u(x + 1, y, t) 9 U(X - 1, y, t) + u(x, y + 1, t) + u(x, y - 1, t) = 4u(x, y, t + 1). (5)

The initial conditions are of the form

au au a% a2u -==;-= 2+2. at \ at ax ay

24x, y, 0) = UC% Y), (6)

C is merely the limit of B as t + ~0, and may SO be solved. We ask for ul, . . ., ZL, as functions of x1, . . ., x, and t.

and the boundary conditions may be u = 0 at a certain specified set of points (p, u) of the net.

If we consider the solution of this problem for large values of t, then irrespective of the function assumed for u(x, y), if we put

2. The Approximation to Differential Equations by Difference Equations dx, y) = lim u(x, y, t), (7)

t--Y= A system of equations very close to (11, and yielding (2) as a limiting case, is we shall have

Atuj = fi(t,xl, . ., xn, UI, . . .) urn, (2) u(x + 1, y) + u(x - 1, y) + u(x, y + 1)

hlU1, . . ., AWL, A2xlul, . . .I, + dx, y - 1) = 4u(x, y), (8)

where the quantities uj are determined only for values of t, X1, . .., X, lying in arithmetical pro- gressions, which we may restrict without loss of generality to the arithmetical progression of the integers. This system of equations is equivalent to a system

Uj(Xl, . . .a Xn, t + 1) = $j(t, Xl, *. *) Xn,

Ul(X1, *. 9, x,, t), . . ., (3) U,(Xl, .., x,, t), Ul(Xl + 1, x2, . . ., x,, t1, *. .I.

corresponding to the differential equation &/&r2 + a2u/ay2 = 0.

Again, let us consider the equation

(-$+$)($+$) = -205

The corresponding difference equation will be

u(x+ 2, y, t) + 4x - 2, y, t) + u(x, y + 2, t) + z&(x, y - 2, t) + 2{u(x + 1, y + 1, t) + u(x + 1, y - 1, t) + u(x - 1, y + 1, t) + 24x - 1, y - 1, t)} - 8{u(x + 1, y, t) + u(x - 1, y, t) + 24x, y + 1, t) + ucx, y - 1, t,> + 2Ou(x, y, t + 1) = 0.

Conditions c are not much altered by the transi- tion from differential to difference equations, and conditions d become conditions holding on the dis- Crete points forming the boundary of a region of n-dimensional meshwork.

We propose to obtain an approximate solution to system of equations (1,2) by approximating to them by a system (3) of a finite number of si- multaneous equations, which we shall solve by an appropriate mechanism, taking full account of boundary and initial conditions.

An example is the equation a”u/ax” + a2u/ay2

As t -+ co, this will give the equilibrium equation of the elastic plate

($+-$($.+$) =o.

= 4&/d& which corresponds to the difference Appropriate boundary conditions will fix the val- equation ues of u around two consecutive lines of boundary



nodes. This will correspond to fixing u and its normal derivative on the boundary in the differential equation, and to a fixed edge in the sense of elasticity theory. As an example, in the following network, u is given at the points marked by circles and is to be determined at the points marked by crosses:

averaging by which we obtain UP,, t + 1) from u(P,, t) will be represented as follows:

I. If P,, is an interior point of the mesh region over which we wish to determine the potential function,

up,, t + 1) = i {20,-I, t) + UP,-,, a (11) 000

00000 oooxxxoo

ooxxxxoo 2.

ooxxoo

+ uPn+1, t1 + on+,, a.

If P, is a boundary point,

00x00

00

0 3.

up,, t + 1) = up,, t). (12)

If P is neither an internal point nor a boundary point, how we determine u(P,, t + 1) is in- different.

Thus if we have a record of the values of u(P,, t) on a linear tape, if we can scan these values so as to obtain u(P,,~, t), u(P+,, t), if we have a rapid mechanism for adding these values together and dividing by four, and if we can imprint this on the tape for the next run, leaving all boundary values unaltered, say by some switch-off mechanism, we can solve the system of equations (5) mechanically.

3. The Component Elements Required

A. A quick mechanism for imprinting numerical values on a running tape. I suggest that these values be carried in the binary scale, as a number of lines on which a signal may be turned on or off, to represent a digit 1 or 0 in the corresponding place in the binary scale. The signal might be magnetic-either DC mark or an AC hum; mechanical-a puncture in paper made by a spark; phosphorescent-stimulated by light, cathode rays, or X-rays; a state of ionization in the tape-stimulated by cathode rays, light, or an electrostatic field; or it might be none of these.

B. A mechanism for reading four such im- printed values simultaneously at fixed stations on the tape as it moves by. Again, the reading may be magnetic, photoelectric, dielectric, or something still different.

C. A rapid adding mechanism for adding the values given in B to a single number for reim- printing on the tape as in A. I am told that vac- uum tube-capacitance mechanisms of this type exist with an overall speed of l/50,000 sec. per operation.

In the equation J2u/&c2 + #u/dy’ = 0, a com- mon form of boundary condition is

au au au au+b--CC, orau+bcosa-+bsinu--CC.

ax ax ay

The analogue in the difference equation case is a condition of the form

au(x, y) + b cos a{u(x + 1, y) - U(X, y)} + b sin a{u(x, y + 1) - u(x, y)} = C (9)

or

Au(x, y) + Bu(x + 1, y) + Cu(x, y + 1) = D,

for a boundary point u(x, y) such that u(x + 1, y) and u(x, y + 1) are interior or boundary points of the net. Similar conditions may be found when u(x - 1, y) and u(x, y + 1) are interior or boundary points, etc. Every effective boundary point has either u(3c - 1, y) or u(x + 1, y) and either u(x, y - 1) or u(x, y + 1) on the net. Thus in scanning a boundary point of the type for which (9) is ap- plicable, we put

u(x, y, t + 1) = - 4 u(x + 1, y, t)

-~u(x,y+l,t)+~. (10)

Let us now consider a net of u2 points, some of which are boundary points. Let us develop this set, row after row, into a single sequence. Calling the nth point in this sequence P,, the process of



D. The number given by C should be reim- printed as in A, with the last two digits dropped, to signify division by four.

E. A special switching apparatus should be provided which will exempt the values at the boundary points from any alteration under the operations of the mechanism.

of the complete square. The solution of the differential equation problem for the complete square is

m7Fx CC A,, sin m n

u sin ~ ~-[(~2+~2)/41(~2t/~2) (14)

F. It is highly desirable in the interest of con- serving material that after the data on the tape have been read, the tape can be cleanly erased to receive new numerical data. Otherwise the amount even of cheap paper tape consumed may be excessive.

Almost the complete set of data for u(P,, t> should be retained while u(P,, t + 1) is printing. Perhaps a desirable tape should be in three widths, one for boundary values, and the other two alter- nating between reading on one width and erasing and printing on the other.

With magnetic scanning and printing, it does not seem too much to hope that an entire set of reading, adding, and printing operations may be completed in 10m4 sec. To avoid blurring, the sep- arate channels in the tape had better be wires of magnetic material, held together in a non-magnetic matrix. Under these circumstances, it does not seem hopeless to get a sharp signal in from one hundredth to one tenth of an inch of length of the tape. Thus with ten thousand points to be scanned, the length of tape would be from eight to eighty feet. If readings are required to a tenth of a percent, each of the three sections of the tape should contain ten channels, or thirty in all-perhaps three to six inches of tape in width. For an accuracy of a hundredth of a percent, the tape need only be a third wider.

where the initial values are given by

m7Nc CC Tsiny. A,, sin m n

(15)

The number of complete scannings needed is easy to compute roughly. Subtracting the function which fits the boundary values and goes ini- tially on the tape, which may be arbitrary, from the final value, we obtain a function vanishing on the boundary, and serving as the initial value for the difference equation system

w(P,, t + 1) = ; {w(P,-“, t) (13)

+ w(P,+1, t) + w(P,+,, t,>

with zero boundary values. The number of scannings needed is the value of t for which the solution of this latter system becomes less than some agreed standard of accuracy. The most unfavor- able case is that in which the boundary is that

The slowest term in (14) to vanish [as t * ~1 is the term m = n = 1, and we are justified in taking een212 as the factor by which the error is reduced in u2 time-steps. Thus in (1.4)~’ time-steps the original error is divided by a factor of more than 1000. In other words, this will take place after 1.4~~ scannings of individual points, or on our time assumption,t in .00014u4 sec. If u = 100, this will be about 3.9 hr.

Mutatis mutandis, the argument given here applies to the three dimensional potential problem. For a cube of u3 points, the number of points to be scanned to reduce the original error by a factor of 1000 is about 2u5.

4. The Use of the Machine to Determine Flow about an Airplane Wing

The flow of air about an airplane wing has lines of flow which are equipotential lines of the following function u:

2 2

1. 2 + $ = 0 outside the wing.

2. u = 0 on the wing. 3. u is asymptotically y + const. at infinity.

Now let us put

p-z.-. Y x2 + y2’ -iI=-.---- x2 + y2’

u(x, Y) = UC<, 4;

then

5 rl x=m; Y = 52 + 92’

+In which a single set of reading, adding, and printing takes 10e4 seconds.-Ed.



The contour of the wing maps into a certain curve in (5, n) space, and the exterior of the contour to the interior of this curve. The point at infinity goes into the origin, and it is well known that within the boundary curve except at the origin, u([, +q) satisfies the differential equation

2 2

d”+!L()* at2 aq2

Thus if w(.& n) is the solution of this equation free from singularities in the interior and satis- fying

5. General Considerations Concerning Mechanical Solution of Partial Differential Equations

The fundamental difficulty in the solution of partial differential equations by mechanical methods lies in the fact that they presuppose a method of representing functions of two or more variables. Here television technique has shown the proper way: scanning, or the approximate mapping of such functions as functions of a single variable, the time. This technique depends on very rapid methods of recording, operating on, and reading quantities or numbers.

Y All computing machines are composed of parts w=pqi to perform certain specified operations. These parts

may be used in conjunction in such a way as to

on the curve, then carry on a process of successive approximation, in which number of parts is proportional to accu-

u([, 7)) = i!l-- - w. t2 + r12

racy, measured on a logarithmic scale, or to solve more problems of a given sort. The speeding up of the pace of a machine leads to a directly pro-

In other words, we may replace the external potential problem of flow about a wing by an internal problem much more suitable for setting up on the machine, in which the number of boundary points for a net of 100 X 100 points will easily exceed 200.

The operation of inversion and the determi- nation of the boundary in the inverted problem will be greatly facilitated by the use of paper on which the families of curves

X -zz 5, z=, x2 + y2 x2 + y2

portional increase in the number of operations which can be performed in a given time, and is thus indirectly equivalent to an improvement in accuracy.

There are thus three equivalent ways of im- proving computing apparatus: improvement in accuracy, number of parts, and speed. Of these, at our present stage of progress, improvement in speed is incomparably the cheapest. By giving up our present dependence on mechanical parts of high inertia and friction, and resorting to electrical devices of low impedance, it is easy to perform arithmetical operations at several thousand times the present speed, with but a slight increase over the present cost. Where operations are

are printed. so multiplied in number as is the case with par- In this formulation of the problem, we have tial differential equations as distinguished from

availed ourselves of the duality between the lines ordinary differential equations, the economic ad- of flow and the equipotential lines of a two-di- vantage of high-speed electronic arithmetical ma- mensional potential problem. The direct attack on chines, combined with scanning processes, over the wing problem would be to solve the boundary the multiplication of mechanical parts, becomes value problem for zero potential gradient normal so great that it is imperative. to the wing and asymptotic potential of x at in- Every such apparatus must combine with the finity. Like the other problem, this may be re- high speed arithmetical machines adequate high- duced to an internal problem by an inversion. speed devices for the storage and reproductions of Unlike the method of the other problem, this data. Steel tape and other similar devices for this method may be extended directly to three-dimen- purpose are almost independent of the particular sional problems, such as that of the flow of air nature of the problem to be solved. The data taken about an airplane model. In this case, the bound- off these recording devices must be recombined ary value conditions, though linear, are like those by fast arithmetical machines adapted to the par- in (101, and need more elaborate methods to take titular equation or set of equations in question. them off the tape. If machines of this sort can be devised, they



will be of particular use in many domains in which the present theory is computationally so complex as to be nearly useless. This is true of all but a few of the simplest problems in hydrodynamics. Turbulence theory, the study of waves of shock, the theory of explosions, internal ballistics, the study of the motion of a projectile above the speed of sound, etc., suffer greatly for the lack of computational tools. There are many cases where our computational control is so incomplete that we have no way of telling whether our theory agrees with our practice.

However, even where existing machines function well, there is scope for machines such as are described in this memorandum. Ordinary differential equations may be solved by scanning as well as partial differential equations, and for boundary value or characteristic function problems, these methods are preferable to the methods of search which are necessary on the differential analyser until conditions at the two ends of an interval can be pieced together.

For a machine of general applicability, we must have:

equations in two dimensions, though somewhat sketchy, are quite novel. The actual “television scanning” technique Wiener proposes for the it- erative solution of Laplace’s equation is essen- tiaily the method of systematic relaxation that had been published two years earlier by Christopher- son and Southwell {2}, though there is no evi- dence that Wiener knew of this paper. The suggestion that the technique be mechanised-and in particular by means of a binary electronic digital computer-is believed to be original, though in fact it was not to be special-purpose machines but program-controlled general-purpose computers that were to be used for the purpose a de- cade later. Similarly it was to be a number of years before Wiener’s suggestions concerning the use of magnetic tape data storage were to be realized in practice.

Wiener’s long-standing interest in computers and the circumstances that led him to prepare this memorandum and submit it to Dr. Bush in Sep- tember 1940 for possible use in the war are de-

1. a tape apparatus like that of the potential machine, only perhaps with more wires for more internal functions and boundary data;

2. scanning and recording apparatus for scanning and recording in more places; and

3. electronic machines capable of performing rapid sequences of operations such as addition and multiplication on the data read off, before printing the result on the binary scale.

Brian Randell graduated in 1957 with a degree in ma thema tics from Imperial College. He joined the English Electric Company, where

\ he led a team which implemented a number of compilers, including

The machines in (3), while perhaps expensive, are not often repeated, and their total cost need not be excessive.

Comments on Wiener’s Memorandum

B. Randell*

This memorandum is of interest both for its tech- nical content and for the circumstances in which it was written and sent to Vannevar Bush. The proposals it contains for a special-purpose electronic digital device to solve partial differential

the Whetstone KDf9 Algol compiler. From 1964-1969 he held a position with IBM at the IBM Research Center,

working on operating systems, the design of ultra- high-speed computers and system design methodology. In 1969 he became Professor of Computing Science at the University of Newcastle upon Tyne, where in 1971 he initiated a programme which now encompasses several major research projects sponsored by the Science and Engineering Research Council and the Ministry of Defense. His interest in the history of computing, a subject on which he has published one book and a number of papers, has been

*Reprinted from pp. 135-136 of Norber Works, Volume IV, edited by P. Masani, M. with the kind permission of the publishers bridge, MA 02142.

t Wiener: Collected IT copyright, 1985, :, MIT Press, Cam-

concentrated on the period from Babbage’s work on calculating engines up to the invention of the earliest electronic computers.

Annals of the ‘History of Computing, Volume 9, Number 2, 1987 l 193


scribed in his article [58f, pp. 4-71,” his book Cy- bernetics [48f], [61c, pp. 3-41, and more fully in his autobiography [56g, pp. 231-238.1

It appears that Dr. Bush felt that immediate work on such a project would not be worthwhile because it would involve a drain on manpower needed for war work of higher priority. The memorandum was filed away for possible postwar use, and Wiener’s attention turned to antiaircraft fire control. As a result the memorandum has only recently come to light, and did not influence the course of development of the electronic computer. (For Wiener’s own views on this project via-ci-uis those which began later at Harvard and else- where, and on the subsequent history of computers, the reader may see his books [61c, pp. 4, 41-511, [56g, pp. 238-2391, [58f, pp. 6-71.

Dr. Goldstine (3, p. 91} has written about MIT’s possible overcommitment to the analog point of view as opposed to the digital. It would, however, be misleading to attribute Dr. Bush’s decision to defer consideration of the Wiener memorandum to this factor. For it now turns out that Bush had been involved in a program of research into electronic digital devices since 1936, namely the Rapid Arithmetical Machine Project. The earliest mem- oranda from this little-known project cannot be traced, but one by Bush himself, dated March 1940, has recently been found and published (1). This shows that the project was by this time quite well advanced, and that Bush was already considering techniques of program control (for evaluating simple arithmetic expressions), although apparently planning to use just decimal, and not binary, forms of number representation.

The Wiener memorandum is nevertheless a fascinating document that raises a number of in- triguing questions. There is no question that it is of considerable historical interest and well de- serves being rescued from obscurity and being published now alongside reprints of his many better-known papers.

References

1. V. Bush, Arithmetical Machine (memorandum dated 2 March 1940), pp. 337-343 in The Origins of Dig- ital Computers: Selected Papers, B. Randell, ed., Springer-Verlag, Berlin, 1982.

2. D. G. Christopherson and R. V. Southwell, Reloxa- tion methods applied to engineering problems: III,

*The article [58f] is reprinted in Norbert Wiener: Collected Works, Volume IV.

problems involving two independent variables, Proc. Roy. Sot. London A 168 (19381, pp. 317-350.

3. H. H. Goldstine, The Computer from Pascal to von Neumann, Princeton University Press, Princeton, N.J., 1972.

Comments on Wiener’s Memorandum

D. K. Ferry and R. E. Sacks”

This September 1940 letter to Vannevar Bush and attached memorandum represents a tour de force in the computing art of the day. Indeed, Wiener synthesizes the best ideas of the time, together with some new innovations of his own, into a proposal to build a digital partial difference analyzer that exhibits every characteristic of the modern digital computer except for the stored program. The proposed machine employs:

A discrete quantized numerical algorithm for solution of the PDE. A classical Turing machine architecture. Binary arithmetic and data storage. An electronic arithmetic logic unit. A multitrack magnetic tape.

1. Numerical Considerations

Wiener was most likely familiar with the con- cepts of discretization in finite mathematics from his time spent on ballistic calculations during the first world war at the U.S. Army Proving Grounds in Aberdeen, Maryland. Indeed, the approach proposed here is basically a Gauss iteration similar to that which Wiener had employed in his 1923 paper with Phillips on the Dirichlet problem 123b1, whose stability was examined a little later by Courant, Friedrichs, and Lewry in 1928, {2}. Ele- ments of this approach can also be found in the work of Lagrange.

Given the sequential nature of the computational process to implement a numerical algorithm, one must impose some type of linear ordering on the discrete (space-time) grid over which the problem is defined. Indeed, such an ordering would be implicit in the programming process even if it were not explicitly imposed in the algorithm. As such, Wiener’s proposal differs significantly




from the literature cited in the preceding para- estimated that each computational step in his al- graph in that he uses a “television scanning” pro- gorithm could be done in 100 microseconds (i.e. in cess to define a (lexigraphical) ordering on the two- 100 x 10e6 seconds) if the electronics could per- dimensional array, z&r, y), associated with the form each of the required arithmetic operations PDE. Although this “television scanning” ap- in 20 microseconds. In fact a modern (&bit) mi- preach has to a great extent been replaced in nu- croprocessor can perform a complete step of the merical analysis by finite element methods, cf algorithm in about 20 microseconds, and Wiener’s Ciarlet {l}, it dominated the field for a quarter century and is still widely used in signal processing.

David K. Ferry is the director of the Center for Solid State Electronics Research (CSSER) and

2. Architecture professor of Electrical and Computer Engineering Architecturally Wiener’s proposed partial differ- at Arizona State University. He is the author or ence analyzer is a textbook example of a Turing co-author of some 200 scientific works. Dr. Ferry machine. Wiener has left no record of any prewar received his B.S. and M.S. degrees from Texas meeting with Turing, though he certainly must Tech University and a Ph.D. from the University have known of Turing’s classic 1936 paper (6). In- of Texas, Austin. His career includes fellowships deed, he may have met Turing in April 1937, when at the University of Vienna and the Boltzmann his visit to Princeton overlapped with Turing’s 14, Institute of Solid-State Physics in Vienna, and p. 1761 and/or became aware of the work through positions with Texas Tech University and the his 1940 discussions with Turing’s Kings College Office of Naval Research. Until recently, he was colleague A. E. Ingham in New Hampshire, cf. professor and head of Electrical Engineering at [56g, p. 2281. Colorado State University. Dr. Ferry was involved

Whether or not discussions on computation had with the NAS/NRC study on Thin Film been held during this period will never be known. Microstructure Science and Technology and two We are inclined to think not, since Wiener still NATO Advanced Study Institutes. Since 1982, he talks about external apparatus to carry out the has been a member of DARPAs Materials actual numerics. In this he does not make the log- Research Council. He is a Fellow of the ical conclusion of a stored program machine that American Physical Society, and a Fellow of the can be drawn from Turing’s work. Von Neumann IEEE.

of course later made a significant advance in this regard.*

Significantly Wiener immediately recognized Richard Saeks was born two major factors after hearing Stibitz’s talk at in Chicago in 194 1. He the Dartmouth meeting of the American Mathe- - j received a B.S. degree in matical Society in 1940 [56g, pp. 229-2301, both i 1964 from Northwestern of which appear in his memorandum to Vannevar L j University, a M.S. in Bush. These are the need to use the binary num- 3 h 1965 from Colorado State ber system and the special advantage of elec- I University, and a Ph.D. in tronic parts over mechanical parts. Apparently, ,’ I* 1967 from Cornell only Konrad Zuse (2) in Germany had come to :\ University, all in electrical these same realizations (working in isolation in j i engineering. He is Berlin). If it was an idea whose time had come, h-k presently professor and only Wiener appears to have grasped these points chairman of the in the United States. department of Electrical

and Computer Engineering at Arizona State

3. Hardware University, and has previously served on the faculties of the University of Notre Dame and

Wiener’s estimate of the speed of his proposed Texas Tech University. He is involved in teaching partial difference analyzer [%b, pp. 130-1311 is and research in the areas of fault analysis, large- somewhat optimistic but not unreasonable. He scale systems, and mathematical system theory.

Dr. Saeks is a fellow of the IEEE and a member *See Editor’s Note at the end of this comment. of AMS, SIAM, ACM, ASEE, and Sigma Xi.



estimate is commensurate with the speed of the first generation of transistorized mainframe computers introduced in the late fifties (IBM 7090, 7094, etc.). But Wiener wrote his proposal before the invention of the transistor (1947) and its ap- plication to computers in the 1950s.

Wiener clearly recognized [85b, p. 1341 that a high-speed digital electronic device was required to cope with the extra computational load associated with the solution of a PDE vis-a-vis the solution of an ODE for which the analog differential analyzer had been developed. Interestingly, however, the memorandum makes little reference to the details of the required electronic central processing unit, possibly deferring to Bush’s greater expertise in this area, cf. (5). Moreover he alludes to the solution of nonlinear problems aris- ing in hydrodynamics, turbulence, analysis of shock waves and explosions, and the like but gives few details. Were the nonlinearities to be stored on the tape, implemented via some type of electronic multiplier, and the like?

Possibly the most outstanding and farsighted contribution of the memorandum is Wiener’s proposal for a magnetic tape. Although magnetic recording goes back to the turn of the century, and wire recorders were used in the broadcast industry by the early 1930s. Wiener’s realization of Turing’s “tape” as a multitrack magnetic tape in which the “bits” of a binary number are recorded simultaneously on parallel tracks came at least a quarter century ahead of the industry. Indeed, “bit- level” parallel processing did not again appear until the second generation of postwar computers (ILLIAC, WHIRLWIND, MANIAC) in which storage tubes and/or magnetic drums were employed

Association of America {7}, but it was Wiener who was consultant on computation. It is hard to estimate how much cross-fertilization occurred between these two men. Yet, had Bush circulated the Wiener memorandum, we might today be talking about the limitations of the Wiener-von Neumann, if not the Wiener machine, instead of the von Neumann machine.

References

1. P. G. Ciarlet, Numerical Analysis of the Finite Ele- ment Method, University of Montreal Press, Mon- treal, 1976.

2. R. Courant, K. Friedrichs, and H. Lewry, Uber die partiellen Differenzengleicheingen der mathema- tischen Physik, Math. Ann. 100 (1928) 32-74.

3. H. H. Goldstine, The Computer from Pascal to Von Neumann, Princeton University Press, Princeton, N.J., 1972.

4. S. J. Heims, John von Neumann and Norbert Wie-

in the late 1940s.

ner: From Mathematics to the Technologies of Life and Death, The MIT Press, Cambridge, Mass., 1980.

5. B. Randell, From analytical engine to electronic digital computer: the contributions of Ludgate, Torres, and Bush, Ann. Hist. Computing 4 (October 1982).

6. A. M. Turing, On computable numbers, Proc. Lon- don Math. Sot. series II, 42 (1936), 230, 266.

7. War Preparedness Committee of the AMS and MAA. Amer. Math. Monthly 47. (August-September 1940), 500-502.

Editor’s Note: On the question of the authorship of the stored program concept, Annals reported on the discussion between J. Presper Eckert, Herman H. Goldstine, Maurice V. Wilkes and Richard Clip- pinger in Vol. 4, No. 4. While not providing a de-

4. Conclusions

Although Wiener most certainly drew on the work of Turing and Stibitz, among others, his proposed

finitive answer to the question, some discussants suggested that the original concept developed during the period of the construction of the ENIAC, at a time when its authors were not free to publish

digital partial difference analyzer represents a unique synthesis of these ideas into a special-pur-

their proposal. Instead, John von Neumann was cast

pose computing machine. Indeed, most of the ele- in the light of a reporter instead of an originator of the idea. Readers are referred to this report for fur-

ments of the von Neumann machine, save the stored program, are present in the memorandum.

It is interesting then to wonder how much von

ther amplifications of the controversy.

Neumann and Wiener influenced each other. Their paths had crossed on several occasions (3, pp. 20- 21, 51-52, 176) prior to 1940, at which time both von Neumann and Wiener were consultants to the joint War Preparedness Committee of the Amer- ican Mathematical Society and the Mathematical

References

Burks, Arthur W., and Alice R. Burks. 1981. “The ENIAC: First General-Purpose Electronic Com- puter,” (and commentary). Annals Hist. Comput. 3:l: 310-399.

196 l Annals of the History of Computing, Volume 9, Number 2, 1987 \


Metropolis, N., and J. Worlton. 1980. “Errors in Com- puting History.” Annals Hist. Comp 2:l: 53-57.

“Minutes of 1947 Patent Conference, Moore School of Electrical Engineering, Univ. Pa.” Annals Hist. Comp. 7:2: 100-116.

Randell, B. 1982. “From Analytical Engine to Elec- tronic Digital Computer: Contributions of Ludgate, Torres, and Bush.” Annals Hist. Comp. 4:327-341.

J.A.N. Lee

Wiener Bibliography

[23b] Wiener, Norbert (1923). “Nets and the Dirichlet problem” (with H. B. Phillips), J. Math. Physics 2, pp. 105-124. (Reprinted in Norbert Wiener: Collected Works, Vol. I, The MIT Press, Cambridge, Mass., 1976, pp. 333-352.

126~1 Wiener, Norbert (1926). “The operational calculus,” Math. Ann. 95, pp. 557-584. (Reprinted in Nor- bert Wiener: Collected Works, Vol. II, pp. 397-424.)

[48f,61c] Wiener, Norbert (1961). Cybernetics, Second Edition, The MIT Press, Cambridge, Mass,, and John Wiley and Sons, New York.

[53d] Wiener, Norbert (1953). “Les Machines a calculer et la forme (Gestalt). Les machines a calculer et la pen&e humaine,” Colloques Internationaux du Centre National de la Recherche Scientifique, Paris, pp. 461- 463. (Reprinted in Norbert Wiener: Collected Works, Vol. IV, pp. 422-424.)

[53h] Wiener, Norbert (1953). Ex-Prodigy: My Child-

hood and Youth, Simon and Schuster, New York, The MIT Press, Cambridge, Mass., 1965.

[56g] Wiener, Norbert (1956). I Am a Mathematician: The Later Life of a Prodigy, Doubleday, Garden City, New York, The MIT Press, Cambridge, Mass. 1964.

[58f] Wiener, Norbert (1958). “My connection with cybernetics: Its origins and its future,” Cybernetica (Belgium) I, 1-14. (Reprinted in Norbert Weiner: Collected Works, Vol. IV, MIT Press, Cambridge, Mass., 1985, pp. 107-120.)

[60b] Wiener, Norbert (1960). “The brain and the machine” (summary of an address), in Dimensions of Mind, ed. S. Hook, Collier Books. (Proc. Third An- nual New York Univ. Institute of Philosophy held on May 15-16, 1959, 113-117. (Reprinted in Norbert Wiener: Collected Works, Vol. IV, pp. 684-688.)

[60d] Wiener, Norbert (1960). Some moral and tech- nical consequences of automation, Science 131 (1960), 1355-1358. (Reprinted in Norbert Wiener: Collected Works, Vol. IV, pp. 718-721.)

[62b] Wiener, Norbert (1962). “The mathematics of self- organizing systems,” in Recent Developments in In- formation and Decision Processes, Macmillan, New York, 1-2. (Reprinted in Norbert Wiener: Collected Works, Vol. IV, 260-280.)

164el Wiener, Norbert (1964). God and Golem, Inc.-A Comment on Certain Points Where Cybernetics Im- pinges on Religion, The MIT Press, Cambridge, Mass.

[85b] Wiener, Norbert (unpublished). “Memorandum on mechanical solution of partial differential equations,” in Norbert Weiner: Collected Works, Vol IV, The MIT Press, Cambridge, Mass., 1985, 125-134.


the wiener memorandum on the mechanical solution of partial differential equations

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