K. W. Pledge

"In this work more than any other it is rewarding to keep on looking at questions, which one considers solved, from another quarter, as if they were unsolved."     Wittgenstein

Systematics Vol 3 No 4 1966


In this paper the consequences of applying a certain type of symbolic method of investigation are demonstrated by its application to the study of particular scientific experiments. It is demonstrated that the scientific situations which are called experiments are bound by the same qualitative laws, structural principles, modes of patterning—however they may he called—as any other situations in which the presence of completing processes can be discerned. There enter into scientific work, as into every activity of whatever kind, certain qualitative and structural considerations which have direct bearing upon what can be realized by means of action within situations. Using Gurdjieff's generalized action symbol of the enneagram, which symbolizes the various patternings to which completing processes conform, it is demonstrated that both Newton's prism deviation experiment and the corresponding spectrometer experiment unmistakably exemplify these patternings. The results of the enquiry are consistent with an interpretation of scientific experiment as the study of specific completing processes by gradual removal of contingent factors from situations in which they are exemplified.





I. STUDY AND EXEMPLIFICATION OF STRUCTURE                                                          







1. Situation

2. Main Completing Process 

3. Structure of the event

4. Inner Harmony



1. Measurement Space

2. Structured Process

3. Setting up Procedure

4. Three fold  Structure







In this paper an attempt is made to show how certain extremely general ideas about structure are actually exemplified in the real situations of specific physical experiments.

I do not expect to have achieved more than an indication of how this can be done. It is now little more than two years since I first began to be able to see how precisely the octave structure of a completing process, with its necessary interventions from outside the process, is exemplified in the simple experiment of Isaac Newton to disperse white light through a prism.

What I first saw was communicated in a very brief internal paper delivered to the Integral Science Research Group of the Institute in 1963,* but the time was not yet ripe to develop the ideas it contained and I left the matter there. For a long time I ceased even to be able to see what I saw then, for my attention came to be occupied with other matters.

* I.S.E.R.G. The Group has now itself been dispersed, owing to the relentless pressure of the current education research programme of the Institute.


When I recently took up the task of making a full-length article out of that early paper I discovered that my power to see structure had moved on even in that short while. I could now see more in the simple physical situation of Newton's experiment than I had previously even dreamed could be there. Other work which had occupied me during the intervening time began to show itself as also possessing a rightful place in the structure that was unfolding before my eyes.

I must apologize in advance for the unfair demand which is made upon the reader who is not already familiar with Gurdjieff's extraordinary hook All and Everything or Ouspensky's faithful record, in his book In Search of the Miraculous, of Gurdjieff's early teaching about general structures and the principles according to which they are held together. It was Ouspensky's book which first gave me a convincing glimpse of the way to see into the general structure of situations. But my rigorous scientific training was equally compelling with its evidence that science can and does discover, but in a different way, valid knowledge about the way the world really is.

It was for me a most happy moment when I saw that both my painfully-acquired empirical science and the no less painfully-acquired structural ideas of Gurdjieff—which he himself gained by direct contact with ancient traditional sources—could be seen to have their own place. The one illuminated and completed the other. I began to see that Gurdjieff’s ideas and symbolism could begin to explain why the practice of science can be beautifully exciting and rewarding for theorist and experimentalist alike.

I saw that my training as a scientist, which refused to let me accept ideas without seeing for myself sonic kind of correspondence between them and what for me was the unshakable concrete reality of experimental fact, had not been wasted.

I knew I was seeing new things and looking at old things in new ways that made them meaningful and worthwhile and that was enough.

It became clear to me that the very notion of proof in general turns upon the possibility of establishing correspondences between correspond­ing elements in a structure. Where no correspondence can be established there is no possibility of proving anything whatsoever. The ability to "follow" a proof therefore becomes the power to perceive the corres­pondences involved.

The ability to follow general proofs of the exemplification of general structural principles turns then upon a power of the mind to perceive in what we may call a symbolic or figurative way.


Symbolism refers to pervasive structure. Situations may be more or less structured, more or less intelligible. It is common experience that structural features of one situation can often be transferred to another. The use of electrical analogues in treating acoustical problems is but one example. There are mathematical formulae such as the wave equation, mathematical techniques such as the differential and integral calculus, the very operations of arithmetic and number generally, which provide symbolisms and refer to pervasive structure in countless situations. In this paper we shall be concerned with the application of a special kind of structural symbolism which is non-mathematical but pervasive neverthe­less. It is a symbolism specifically concerned to elucidate the action structure of situations. In this paper, specifically those concerned in scientific experimentation.

Science is what scientists do. In studying what scientists do we have to study structured action-situations which involve arrangements and operations with extended material objects. We have to examine closely the adjustments that scientists make when they set up and operate their apparatus. We have to study the processes which the apparatus is specifically set up to study. We have to have some means of eliciting the significance of all these things and we have to find a means of doing it simply, clearly and with satisfying generality.

Experiments give data which arc factual. But the performance of any actual experiment involves considerations of intention, value, judg­ment, intelligent action towards the achievement of purposes and the like, which go beyond the kind of facts yielded by the experiment. They are involved in arriving at the facts, but by the time the facts have been produced they have disappeared. I his leads to certain naive misunder­standings concerning the nature of scientific facts in which the experi­mentalist himself, who knows just how much time, effort and difficulty are involved in producing the results, is not likely to share. Nevertheless, many men still talk as if the results of experiment are somehow objectively true and independent of the scientist who provided them. There is some truth in this notion, but like many another popular belief it requires to be taken with a grain of salt and hedged about with provisos before it becomes of real significance.

Generalized symbols cannot be used without importing into the situation just those kinds of considerations which are left out in deriving the results of science. Because of this, the manner of their use is somewhat different from, say, mathematics in scientific work. With them, one studies whole structures. If a part is studied using such a symbol, it is by reference to the whole situation from which it derives and to which it refers. In mathematics it is possible, and indeed necessary, to study parts in isolation and treat them as if they are independent from the wholes to which they refer. In generalized symbolism this is not possible. The connectedness is primary.

The use of such symbols as instruments of study requires a type of perception of the structural similarity common to diverse situations which is not markedly different from the kind of perception by which one, for example, learns through study and practice eventually to see what type of mathematical form corresponds to a definite physical situation. The difference is that one is working more from the general form to the situation than seeking to fit one of a variety of forms to the particular physical conditions. There is in both cases an aesthetic faculty involved in the recognition of the correspondence involved which is akin to the response to a work of art.

The difference from working with mathematical forms lies in the way in which the perception penetrates into the investigation. When one has discovered the appropriate mathematics for dealing with a particular physical problem, there often remains only to apply its associated proce­dure of calculation and the problem is solved. In other words, the solution can be arrived at automatically without more ado. In studying the structure with the use of generalized symbolism the attention cannot leave the problem lest the substance of the structure vanish before one's eyes. The practical use of the symbol is rewarding only when it proceeds hand in hand with deliberate confrontation of the symbolic structure and the "irreducible and stubborn facts" of the situation under study.

 Gurdjieff himself put this in a nutshell by his reported remark: "Only what a man is able to put into the enneagram does he actually know, that is, understand. What he cannot put into the enneagram he does not understand."*

* Quoted by his pupil, P. D. Ouspensky in In Search of the Miraculous, Routledge & Kegan Paul, 1950, p. 294


The emphasis is on thorough grasp of the situation. In this there is no difference in degree between the college professor and the skilled garage mechanic. The practical test is, as always, that of effective action.

Human situations only comparatively rarely exemplify the full structuring of a completely general symbol. In particular, most artificial situations fall short of full exemplification, though elements of it may be perceptible. What one usually finds is that a situation contains features which correspond closely to one or another characteristic of the structur­ing a symbol describes and lacks others to an equally noticeable degree. Very many situations are found which exemplify the structure of a completing process. Others, rather more rarely, clearly show the pattern of three interacting processes necessary for the attainment of some desired end. Even more rarely do we find that inner recurrent pattern strongly established by which they are marked out as realized events.



We shall in this paper use the action symbol called the enneagram as an instrument of study and interpretation. It will be convenient at this stage to recapitulate some of the more striking characteristics of the structuring characterized in the enneagram before attempting to demon­strate how the whole patterning is exemplified in the examples that follow:

The symbol can first be looked upon as the symbol of a completing process of development. This is symbolized by the circle which contains the "figure of nine lines" from which the name of the symbol is derived. The process is to be imagined as originating from the uppermost point and developing along the circumference generated by clockwise rotation about the centre point.

The completion of the development is symbolized by the termination of the curve in its meeting with the starting-point and so forming a completed closed figure. Thus the bounding circle symbolizes the notion of a continually modified developing process in some way under restraint by an intentional act of will which enables it to come to its completion.

The end of the circle returns into the beginning; and this symbolizes the manner in which a co-ordinated sequence of actions sets out with the end already in view, already present though not yet in existence.



Fig. 1    The Generalized Action Symbol of the Enneagram showing its Three Interacting Processes


The circle once completed has no beginning and no end. This symbolizes the power of completed processes to perpetuate themselves repeatedly. Once made they alter the course of things. As an historical experiment, once made, cannot be unmade but must be taken into account in future science. The recurrence of the circle also symbolizes that processes can grow and develop in force and significance. As the simple process of Newton's experiment gave rise, with no change in the fundamental completing process involved, to the spectrometer.

The enclosing circle also symbolizes the requirement for isolation of a region within which the completing process may proceed. In order for an event to come about there must be some place at which it can be situated. An event is a really existing situation within the existing world which has "found its place" and holds on to it with a force of its own independent of the extraneous background of ever-changing contingencies.

Here we begin to touch the kernel of the matter and find a connection with science. Science is concerned with the study of order within the existing world. It carries on this study by producing and examining artificial situations from which contingency is, as far as is humanly possible, removed. These situations are called experiments and are the source of restricted but non-contingent information about completing processes within the world.

The setting-up of an experiment is the establishment of an event within a completing process. The power to remove contingencies is the condition of establishment of such events. More exactly, it is not the power to remove but to circumvent contingency which is the hallmark of the great experimentalist. Aston, Faraday, Michelson, Newton, were all brilliant men with a gift for performing just the right structuring action in an experimental situation which would circumvent the contingencies involved. There is even a saying about Michelson to the effect that "Michelson's interferometer was a marvellous instrument—in the hands of Michelson", There is plenty of evidence to show that the great advances in science are marked by an awareness of the presence of uncertainty in theoretical steps forward—contingency in experimental master strokes.*

* Cf. Brighter Than a Thousand Suns, by R. Jungk, Penguin Books, I960, and /. /. Thomson and the Cavcndish Laboratory, by Sir G. P. Thomson, Nelson, 1964; Reason and Chance in Scientific Discovery, by R. Taton, Sceince Editions, New York, 1962; The Art of Scientific Investigation, by W. I. B. Beveridge, Heinemann, 1950.  


The difficulty with the application of generalized action symbolism which does not arise with mathematics is the multiplicity of meaning. Whereas in the more familiar symbolisms there is a more or less one-to-one correspondence between symbol and meaning, in structural action symbolism the correspondences are one-to-many and correspondingly complex. Thus the inner triangle of the figure symbolizes the require­ment that not one but three independently derived and mutually inter­acting processes of development are necessary to ensure that one single such process shall be enabled to reach completion.

The triangle also serves to symbolize that the three processes must knit together according to the relationship of affirmation, denial and reconciliation specified by the three-term system or triad. The first com­pleting process transmits the affirmation in the relationship as the main process of the three. The second transmits the denying impulses to which the first is subject in consequence of hazard and uncertainty and con­tributed perforce by the environmental conditions through which it is required to proceed. The third process is concerned with bringing the development successfully to its intended conclusion through a reconcilia­tion of these two oppositely-acting impulses.

Another aspect which the triangle symbolizes concerns the qualitative and quantitative aspect of the interaction between the processes. The three impulses must be matched—must be of the appropriate kind and degree for correct matching and consequent fineness of quality in the final product. A striking example occurs in the spectrometer where the inner triad of collimator, prism and telescope allows of astonishingly fine matching in the whole process and an accuracy of measurement to fractions of a per cent in use.

The triple demarcation of the circle brought about by the inner triangle serves to indicate three main regions of the developing process. The first region is concerned with a stage of outgoing or expansion, the second with interaction or mediation, the third with return or concentrative receptivity. The third region concerns the finalizing process which intervenes at point 6 of the figure and ensures proper completion of the process at point 9. The second region is concerned with enabling the main process to continue and involves the entry of the second process at point 3. The first region is the domain within which the original process becomes established.

There are countless further interpretations of the inner triangle of which we may mention only one. This is connected with the location or fixation of a structure within a situation. There is a threefold action which, once accomplished, ensures that the parts of a process are harmoni­ously disposed with respect to it. In scientific work this is directly con­cerned with the location of apparatus in space. There are three kinds of spatial operations possible with apparatus: it can be placed in position, its elements can be aligned with respect to directions defined amongst themselves, and finally there can be rotations of directions thus defined, These three possibilities exhaust the instrumental possibilities allowed by space.

We can now come to consider the recurrent figure within the circle. This repeats itself according to the recurring number sequence 142857 . . . .* The pattern signifies an interwoven connectedness between the structure of the three processes of the situation which, when attained, organizes the whole into a significant event. Correspondences are necessary between the points in the three processes connected by this figure which are made to hold by certain actions performed between them. The per­formance of these actions is the work of synchronization between the process which makes a patterned and properly working whole.

* The repeated decimal common to all non-integral fractions with 7 as denomi­nator. It demonstrates the incommensurability of sevenfoldness with unity and serves to express in number symbolism one of the necessary incompatibilities of structure. A derivation of the figure is contained in In Search of the Miraculous, p. 289


We come now to the symbolic notation of the musical octave with its seven intervals between the eight notes from do to its first harmonic do'.

do re mi / fa sol la si / do'

The gaps marked with oblique strokes indicate the two semitone-intervals in the octave.

This is an ancient means of indicating the characteristics of a developing process in a convenient and economical way. The transition from note to note successfully conveys the character of a transformation. The note remains a note, that is, it remains a sound, but changes in pitch. Thus, by analogy, a transformation involves a change in quality of some material vehicle which retains its own nature during the change. The transition to a finer quality is conveyed by the rise in pitch involved.

The semitones between the notes mi-fa and si-do' symbolize two kinds of discontinuity implicit in the development of completing processes by which they require to be reinforced by the intervention of other processes in order to reach completion. This has already been treated above by reference to the enneagram and the three interacting processes it requires.

There is a special condition in the transformation process associated with the transition following the note sol which is described by Gurdjieff .* This lies between the two steps of the transformation involving the semi­tones referred to above, at a stage of the process where interaction with disturbing factors has already occurred. That point is just being leached at which correcting effects which will bring the process to finality are at last able to intervene. It is, in other words, a stage of particular hazard for the completion of the process to which it refers.

*Cf.  All and Everything   by G. Gurdjieff, Routledge  &  Kegan  Paul,   1950, p. 754-5. "Harnel-Aoot"


Generally speaking, it is sometimes possible, in highly sophisticated and well-organized situations, to trace strong exemplification of all the component structuring of the enneagram pattern. Such situations are invariably those which have evolved painfully over the years to meet more and more completely and appropriately some fairly well-defined need or purpose. They arise in science in those experimental arrange­ments which have become evolved to deal experimentally with one particu­lar phenomenon in one particularly specialized way. A striking example will be studied in the present paper when we come to examine the structure involved in the operation of an optical spectrometer and elicit its exemplification of the enneagram structure. Nevertheless, the ennea­gram can be to some extent studied piecemeal by picking out, say, the three-process structure of a petrol-engine where the main completing process concerns the air, the secondary deals with fuel injection, and the third ignition. Another example occurs in the thermionic diode with a strongly exemplified main completing process from cathode to anode.

We have now barely outlined the weapons of attack upon the structural problems with which we are concerned. As described above the striking power of a generalized symbol is probably only dimly evident. It will become clearer only when we have succeeded in demonstrating a real correspondence between the patterns it manifests and the structure of the concrete world of scientific experiment.




It will be convenient to approach the study of situations which exemplify the pattern of structure depicted by the enneagram in a certain order. We adopt the following procedure:

1        First we shall seek for the isolated region within which the process proceeds. Strictly speaking, the three do's of the three processes are generated from outside this region. This serves to define its boundaries.

2        Next we shall consider the fundamental completing process. Often we can recognize the similarity between the two notes do and do' which initiate and terminate the process.

3   We shall then look for the manifestation of the inner recurrent figure of the symbol. Very often it is this only which gives the clue to the real character of the event which is realized upon its closure.

4.  Finally we shall examine the situation according to the various meanings of the inner triangle symbol: the three interacting processes; the fixation of the whole within its situation; the qualitative and quantitative aspects of the relationship; the three regions, etc.

We now continue, without more ado, to conduct our studies through the consideration of particular physical experiments. The experiments chosen are Newton's famous experiment of 1660 to show that "the light of the Sun consists of rays differently refrangible"; and the corresponding, though more refined, experiments performed today with the direct descendent of Newton's apparatus which is the optical prism spectrometer.



 I  Situation

Newton was a brilliant and painstaking experimentalist. To such a man there is granted eventually an intuitive grasp of structure. Such men become so expert in the field of their absorbing interest that they come to be able instinctively to do the right thing, make the right adjust­ment, find the right way around some difficulty. All this comes with the feeling for structure. There comes, too, the knowledge that the complexity of the world is virtually limitless, hence the need for generalized symbol­isms to express its multiplicity.

Newton it was who remarked, shortly before the end of his brilliant life:

"I do not know what I may appear to the world, but to myself I seem lo have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

This same man wrote the awe-inspiring Principia and the famous Optics. When we turn in the Optics to the pages at which Newton records his own description of the experiment by which he demonstrated the formation of the coloured spectrum from white light, we read the following: *

"In a very dark chamber, at a round hole, about one-third part of an inch broad, made in the shut of a window, I placed a glass prism, whereby the beam of the Sun's light, which came in at that hole, might be refracted upwards towards the opposite wall of the chamber, and there form a coloured image of the Sun."

* Optics, by Sir Isaac Newton. London, 1704, Bk. 1 Pt. 1. Prop. 2. Th. 2


       Newton was not only a brilliant experimentalist; he was a brilliant formulator and expositor as well. In this single condensed sentence he has managed to convey with precision the first two of our set of four groupings of structural aspects. He has given, in fair exactness, the eight notes in the octave of the main completing process, and assigned the region in which it proceeds.

We now apply the symbolism according to the procedure outlined above.

The circle, taken as providing the place within which the completing process and potential event is to become established is, of course, the "very dark chamber". It is dark because the event is to be an experiment with light, so it is specially prepared. It is a chamber which will contain the experiment.

2.   Main Completing Process

The seven steps and eight notes of the main completing process with reference to which the experiment is made are quite well enough described for the purposes of exposition. We can recognize the following eight necessary stages of the process from his descriptions:


    do1         The Sun as light source

    re1        The hole

    mi1       The entering beam

    fa1        The refraction

    sol1      The prism

    la1        The upwards deviation of the refracted beam

    si1        The opposite wall

    do’1      The Sun as coloured image


Table 1. Main Completing Process of Newton’s Experiment


  The correspondence between the Sun as original radiating source •Hid the image as re-radiating source unmistakably points them out as do and do' of the main completing process. The interval between the notes mi-fa is filled by the prism face and so the second process in the experiment must be concerned with the effects of the prism upon the light. The remaining si-do' semitone interval is filled and the process completed by the atoms of the wall surface as it absorbs and re-emits the deviated beam incident upon it.

It is clear that seven steps can be roughly distinguished in this way but, unlike other ways of approaching problems, we have to face an unavoidable seeming vagueness in attempting to make precise ascriptions. The generalized symbolic method studies situations as connected wholes; and this holds good throughout. We cannot make precise and separate from one another elements which are in reality always connected. All we can do is point to certain nodal regions within which a given character is more or less exemplified and then pass on to the next such nodal region by a kind of withdrawal of the attention from one character and transfer to the next. The technique is exactly in correspondence with the procedure by which we ascribe seven colours to the spectrum. We cannot say where one colour terminates and the next succeeds, nevertheless we can distinguish the seven—no more and no less.

3.  Structure of the Event

We have enough material now to begin to construct the enneagram of the experiment. To make the whole study more clear, we first put down a schematic diagram of Newton's apparatus, showing the steps labelled with the octave notes which they characterize. The diagram is Newton's own in all essentials. We then set down the circle with the eight notes of the main octave distributed around it in the usual manner.













Fig. 2   Schematic of Newton's Prism Refraction Experiment















Fig. 3   Enneagram Indicating Octave of Main Completing Process in Newton's Experiment


The inner triangle is dotted-in for completeness, but for the moment we shall concentrate on elucidating the significance of the inner recurrent figure joining the numbers 142857 ... on the circumference of the circle.

For this purpose we simply have to turn again to Newton's own account of his procedure in the Optics.

(1 -4) Connection

From the first quotation above we extract the following statement .which describes his first action in setting-up the experiment. It corres­ponds to the establishment of the line 14 of the recurrent figure. It is simply that: ". . . at a round hole ... I placed a glass prism. . .." This is an action which places hole and prism in a certain state of connectedness and meaning for each other. The action is simply per­formed but its significance takes some little time to comprehend. It sets the stage, as it were, for the synchronizing actions which will follow in the inner recurrent pattern.

(4-2) Connection

      The next sentence immediately succeeds that previously quoted: "The axis of the prism (that is, the line passing through the middle of the prism from one end of it to the other and parallel to the edge of the refracting angle) was in this and the following experiments perpendicular to the incident rays."

This selling of the prism axis perpendicular to the incident beam is of immense significance. It is an act of standardization which immediately eradicates one of the contingencies in the situation, an act of limita­tion which removes certain possibilities and retains only a single ordered set. For each one of the set of possibilities which is retained there is one and only one condition of connectedness holding between prism face and incident beam.

(2-8) Connection

This connection is only weakly exemplified in this experiment, as will become clear when we discover it strongly made in the spectrometer. At present we need merely note that a connection between beam and opposite wall is assured by the construction of the room. The incident beam will always provide some result at the wall as long as the actions so far described which connect the points 142 are not revoked.

(8-5-7) Connection

For elucidation of these connecting acts we again appeal to Newton's account, which continues: "... About this axis I turned the prism slowly, and saw the refracted light on the wall, or coloured image of the Sun, first to descend, and then to ascend."

     Turning to the enneagram figure, we plot the action as follows: With his attention directed to the wall at point 8, Newton rotates the prism at point 5, thereby varying the deviation of the transmitted beam at 7.

(7-1) Connection

Now we come to the act which makes the whole experiment into an event. Newton describes it by saying: "Between the descent and ascent, when the image seemed stationary, / stopped the prism, and fixed it in that posture, that it should be moved no more."

 To see the connectedness here in its dependence upon deviated beam and hole, we have to reflect that the perfection with which this last act is completed depends in the last instance upon the precision with which the point of minimum deviation is able to be determined. This is set directly by the hole—too small a hole will provide too dim a final image, leaving out of account diffraction effects—too large a hole will just give a blur. Hence the adjustable slit of the modern spectrometer. The problem only becomes properly resolved when monochromatic beams are used, which correspond to modification in do-i and hence fall outside the domain of the experiment as such. Newton found it a considerable problem, since he was using white light and hence overlapping coloured images. Nevertheless he recognized the commitment involved in reaching the point of closure of the recurrent figure, for he goes on to tell us that he   afterwards   standardised this position for his prismsin other experiments.

4,   Inner Harmony

The wealth of symbolic meaning in the inner triangle of the figure precludes any attempt to give other than a few examples of its application to the experiment. We shall consider its establishment by degrees, as follows.

Point 9 is established even before the prism is placed in position and even before the experiment exists as the one we are studying. It is established when the initiating and completing notes sound as one. In other .words, when the Sun simply casts its own image upon the wall opposite the hole in the window. There is no triangle because there is no inter­action, no outgoing and no returning.

Points 3 and 6 are established, and with them the base of the triangle and its two sides 93 and 69, when the prism is interposed between hole and wall.

With the prism placed in position there comes at once a threefold separation of the form described above. A completing process comes into being which comprises regions of outgoing and returning mediated by interaction.

In considering the further significance of the triangle we may visualize it as a skew, or asymmetrical triangle so long as the outgoing and returning processes are lacking in mutual adjustment. When they become balanced (here enters a harmony into the whole situation. In Newton's experiment precisely this harmonization occurs at the moment of achievement of the minimum deviation position. This is the point, already mentioned: ". . . between the ascent and the descent, when the image seemed stationary."

Newton gives his reason, really an aesthetic reason, for choosing this particular position as the termination of his setting-up operations. He says: "For in that posture the refractions of the light at the two sides of the refracting angle, that is, at the entrance of the rays into the prism, and at their going out of it, were equal to one another."

The very experiment itself becomes symbolic of harmony at this point. The structure of harmony cuts through all levels and shows itself in all acts of harmony wherever and whenever they may be performed.

The act of harmony signifies a realization of value within a factual situation. Anyone who has actually performed this very experiment will instantly recall the aesthetic satisfaction of achieving the minimum devia­tion position at the end of the setting-up sequence. There is an unmistak­able awareness that something has been realized which comes with the final adjustment. It. is the reward of the true experimentalist.



We now part company with the writings of Newton, who knew nothing of the spectrometer as we know it today. One wonders what he would have achieved if he had.

We pursue our chosen line of attack by first considering the spectro­meter in its capacity as the place of experiment.

At this point we are compelled by reason of limited space to refer the reader to the usual textbooks and laboratory manuals for details of the appearance, construction and operation of the instrument.* In this paper we shall assume the reader to be familiar enough with its working to be able to follow the points we shall select to illustrate our purposes. A schematic diagram of the optics involved in the instrument is given below.


* Textbook of Light, by G. R. Noakes, Maemillan, 1946, contains much of the relevant information. Another valuable source is Experimental Optics, by Wagner, John Wiley, 1929, but any textbook of Physics to Intermediate Standard is suitable. In England manipulation of the prism spectrometer is now an established part of the syllabus in Physics at Advanced Level.



Fig 4 Schematic of Optical Prism Spectrometer

1.  Measurement Space


The spectrometer as the place within which the experiment will be situated shows an unmistakable evolution from Newton's dark room with a hole in the window shutter. The whole has become an organized mechanism which contains within itself everything necessary for the existence,** which means the performance, of the prism deviation experi­ment. The hole has evolved into a complete sub-instrument capable of independent adjustment on its own account. We could devote an enneagram to this instrument alone, the collimator, and we would find a setting-up sequence corresponding to the recurrent inner lines of Newton's experiment. The harmony achieved through the act of closing the pattern would be the production of the parallel beam which the instrument is designed to achieve. The telescope likewise is a sub-instrument which has evolved out of the wall that plays its part in the structure of Newton's experimental set-up.

** Cf. In.Search of the Miraculous. loc. sit. p. 288

Both of these sub-instruments are within the place at which the same experiment as before will become situated within the existing world. But the place itself has moved on from Newton's room and in doing so a step forward of the first significance has been made.

The step forward consists in the previous setting-up, at the place where the experiment will become situated, of another space which is so structured that certain kinds of events taking place within it will inevitably become measured.

This step marks the completed transformation of the situation of Newton's experiment from that of a qualitative experiment which pro­duces a phenomenon, exhibits a physical effect, into an instrument of measurement. The measurement space is brought into being by means of another enneagram which treats the instruments involved in the experiment as material objects capable of undergoing only operations of positional arrangement, directional displacement and rotational configuration according to the three kinds of space.

When the spectrometer is first taken out of the laboratory cupboard it has to be checked to ensure that the principal axes of the telescope and collimator move in the same plane perpendicular to the main axis of the instrument, about which both telescope and prism table rotate. When this condition is attained, the harmony of the spectrometer as a measuring-instrument has become realized.

It can then be used in combination with a gauge* to provide traces in the measurement space of the displacements in the space which occur when the experiment is performed and the actions which realize its harmony have all been made.

* Cf, Towards an Objectively Complete Language, by J. G. Bennett. H. H Bortoft and K W. Pledge, Systematics, Vol. 3, No. 3. Dec. 1965. Sect. IV, Pt 1. p. 220


The whole structured situation is very well symbolized for our purposes by a schematic diagram of the actual arrangement of the apparatus elements as a measurement instrument shown below. The central point represents the central axis of the instrument. The collimator is rigidly connected perpendicular to it. The first circle represents the prism table which is free to rotate about the central axis. The second circle represents the freedom of rotation of the telescope about the same axis. The outer most circle signifies the divided circle which is the scale of the gauge by reference to which displacements in the measurement space are traced.


Fig. 5   Schematic of Measurement Space of Spectrometer

With these observations we leave our consideration of the first aspect of patterning symbolized by the enneagram and exemplified in the optical prism spectrometer. We have not the space to pursue the study further in this paper, but it should be evident that the considerations outlined above are of far-reaching importance for the understanding of measurement as, in general, a contrived technique impressed upon the structure of the experimental situation.

2.   Structured Process

When we compare the simple schematic of Newton's experiment in Fig. 2 with the corresponding diagram of Fig. 4 for the spectrometer, we certainly discover refinements and modifications in the latter; but there can be no doubt in our minds that the main completing process is in both cases one and the same.

So we find ourselves in the fortunate position of having two different exemplifications of the same completing process. Hence we stand a good chance of eliciting, from consideration of both, their common main octave structure.

We are helped in this by yet another consideration. Generally speaking we find that the effect of all the additional complications of apparatus in the spectrometer is in fact to simplify the experimental situa­tion. The parallel beams involved behave effectively in the same way us the "idealized rays" according to which the refraction phenomenon is usually explained in elementary textbooks. The corresponding "idealized rays" are drawn in in heavy lines on the spectrometer schematic of Fig. 4. This supplies us with yet a third "idealized" exemplification which is particularly clear in showing the features of the second process.

The Octave Structure

When we consider these three together we are led to make the following statements about the notes of their completing processes:

First Process

do1   The sounding of the first note of the main octave consists in the provision of a radiating light source.

re1    The second note concerns the selection out from this radiation of light which is well-defined in direction and solid angle.

mi1   The transmitted light thus defined enters the experimental domain and travels onward by expansion.

Second Process

Interval and do2 A plane surface is interposed in its path. The normal to its surface defining some angle with the beam now incident upon it. In the spectrometer a lens at this point ensures that light from the beam is uniformly incident. The glass surface is the second source in the experi­ment.

fat and re2 Light in the incident beam enters the prism surface and interacts selectively there with the glass (hence the occurrence of re2 here in the enneagram).

sol1 and mi2 The light is dispersed (sol1) and its path deviated (mi2). These conditions persist during its transmission through the medium.

Third Process

Interval and do3 The second surface of the prism becomes the exit surface through which the transmitted beam emerges, eventually to mani­fest the effects of interactions which it has undergone. The angle of this face to the first is a determining factor. The objective lens of the telescope performs certain finalizing functions and, in particular, gathers together as one whole emergent beams of corresponding colour and deviation.

lat, fa2 and re3 In the spectrometer, the sounding and blending together of these three notes signifies the finalizing action of "bringing to a focus".

la1 is the condition of the emergent light as carrying with it the results of all its previous interactions in a form able later to be manifested.

fa2 is "the deviation of the already deviated" at the second prism face and, in the spectrometer, at the telescope lens as well. It is another interaction analogous to fa1 where the incident beam interacts with the first prism face.

re3 we may take as the definition of directions which takes place at exeunt from the second prism face. The directions are directed towards some final or ultimate place at which each and every colour will become displayed correspondingly.

si1, sol2 and mi3 These three notes likewise correspond to the condition of the light "coming to a focus" in the focal plane of the spectrometer telescope objective lens, or arriving at the final displaying screen of Newton's wall.

Si1 is the condition of the light as being concentrated into seven more-or-less specific definite colours.

sol2 is the fixation and final dispersion of the effect of the prism refraction by the formation of the whole spectrum in consequence.

mi3 manifests as the persistence within one well-defined spatial region of the image forming the displayed effect which is the spectrum.

do'1 signifies the image as being itself a re-radiating light source. As such, it forms the first note of another consequent octave by which it comes eventually to be perceived. This second octave is taken for granted in the whole previous treatment. Nevertheless, it is evident that the experi­ment begins from do\ just as much as it does from do’1. The whole is designed to provide material for observation via the display at point 8.






















Fig. 6   Enneagram indicating main octave of Spectrometer


The ascriptions of the notes in the octave here derived are quite precise enough for our present purposes. For the moment we need only note that the two lenses are placed at points 2 and 7 respectively in Fig. 6. We might have expected them to fill the two intervals at points 3 and 6, as indeed they do. But it should be clear from Newton's experiments that these two intervals are already filled by the two prism faces and that therefore the lenses may quite plausibly play other roles in the whole,

It is characteristic of structuring considerations that one and the same clement may play different, though perhaps closely similar, roles when considered as entering into different kinds of structure. In much the same way a piece of paper may in different situations serve as a bookmark, a place to record a telephone number, or a dollar bill. Which role is appro­priate depends upon the total context under consideration.


3.   Setting-Up Procedure

We continue our study of the action-structure of the spectrometer by examining how the inner recurrent figure is manifested in its operation. It will be evident from the previous discussion of the same figure with reference to Newton's experiment that the event which comes into being at the closing of the figure refers to the setting-up of the spectrometer for minimum deviation.

Reference to the usual textbooks of experimental physics soon dis­closes that a standard procedure has evolved for this over the years. This procedure is always followed if one wants to line-up a spectrometer accurately with the greatest economy of action. A typical textbook of this kind* reads:

"Before proceeding to make measurement with the spectrometer it is necessary to see that certain conditions hold true, as follows:

1. The principal axes of the telescope and collimator must be perpendicular to the main axis of the instrument, i.e., the axis    about which the telescope and prism table rotate.**

2. The telescope must be focused for parallel incident rays, that is, for infinity.

3. The collimator must be focused for parallel emergent rays.

4. The prism must be adjusted so that the faces which include the angle to be measured are both parallel to the axis about which the telescope and table turn.

Wagner adds the two further conditions:**

                      (a)   The axes of telescope and collimator must pass through the main axis of the instrument.

(b)      It must be possible to make the axes of the telescope and collimator coincident."

* Wagner, loc. cit. p. 24, This book contains the practical work in optics given to officers at a U.S. Navy Postgraduate School, some of whom later "will have duty as inspector* of military telescopic instruments to be purchased for use in  the  Navy".

** We note that that this refers to and establishes the measurement space as described above in part I of this section.


Wagner then notes that these five conditions "are to be established in the order given"; and goes on to describe the procedure of doing this in the usual way.

We may begin at this point to make correspondences with the points and connecting recurrent inner lines of the enneagram, as follows:

Point 8 refers to the first action in the lining-up sequence, which is the removal of the telescope from the spectrometer and its adjustment to focus parallel incident rays in the focal plane. This is usually done by the simple expedient of focusing on any convenient distant object. This operation connects a certain place—the focal plane at point 8 with the condition of various parallel beams entering the instrument: which, we notice, refers to point 5 where the deviated and dispersed beams are first present in the prism.

Point 2 is the scene of the next operation. This is the lining-up of the collimator to make it provide parallel beams. The telescope and collimator literally are "lined-up" on opposite sides of the prism table while the operation is performed. This establishes the line 2-8 in the figure. The entering beam for the experiment in any case becomes defined at point 2, so there can be little doubt of the correctness of our ascription here. Point 4. The first prism face is then lined-up so as to become parallel to the axis about which telescope and table turn. This is done by a rather complex set of adjustment operations which take longer to describe than to do. In these operations, both telescope and collimator are used in con­junction with the prism face, itself acting as a reflector. We note that it could not be done if telescope and collimator were not already aligned. Completion of the operation really establishes the line 4-2 by which the collimator beam stands in a well-defined perpendicular relationship to the first prism face.

Point 1 is the collimator slit itself, which is adjustable. Up until this point in the sequence the experimental operation which produces the phenomenon has been in the background. It now begins to make itself felt and the collimator slit is made as narrow as is required. In these earlier setting-up operations plenty of light has been an advantage; now it has served its purpose. In making this observation we distinguish between the inner lines 1-4 and 7-1, and find how naturally the sequence moves along the zig-zag path of 4-1-7.

Point 7. We may regard this point as the point or place in the experiment at which the experimenter enters into or intervenes in the workings of Nature. It is at tins point that the subtle and delicate operation is per­formed by which, it) this experiment, the phenomenon of minimum deviation is actualized. We have already referred to the beauty of this achieve­ment. It is a synchronizing operation which, once again, is better and easier to do than describe. But it very clearly illustrates the action at this point 7 by which all three interacting processes are brought into a single harmony of synchronous adjustment.

The collimator is fixed in position, thus ensuring the establishment of the first process. The prism and telescope which are the instruments of the other two processes are then used together in one single operation with respect to the image of the slit in the focal plane. This operation consists in coming to precisely the same condition as did Newton, for the image moves first towards and then away from one uniquely defined situation which occurs at the position of minimum deviation. With all its immense accuracy and precision, the setting-up procedure culminates in the same simple harmonizing act that Newton described by:

"Between the descent and ascent, when the image seemed stationary, I stopped the prism, and fixed it in that posture, that it should be moved no more."

We see this operation as the establishment of the zig-zag line 8-5-7 in this figure.

Point 5 is the place where the material of the prism and the prism angle determine the dispersion and deviation of the refracted beam transmitted through it. In coming to the position of minimum deviation through the operation at point 7, the phenomenon is brought into a condition at which a particularly simple relationship holds between the nett deviation of the coloured beam for which it is established, the prism angle, and the index of refraction of its material. All these three have reference to point 5 and determine the final setting position at minimum deviation. Hence the setting-up sequence terminates at this point.

We ask ourselves why the sequence should begin at point 8 and work backwards along the sequence against the direction of the arrows given by the recurrent decimal 142857. . . . An answer would seem to be that, unlike Newton's experiment, which is a pure manipulative experi­ment to produce the phenomenon, the spectrometer is a measuring instrument and the inner lines have already been traversed in lining-up the instrument to produce the calibrated measurement space. This is one possible explanation worth exploring.

Another is that from the point 8 the enneagram shows the main completing process of the experiment as connecting-up to the second enneagram which involves observation via do1. If a photographic plate is placed in the focal plane of the telescope the instrument becomes a spectro graph. When the plate is exposed and later developed and fixed and the spectrum ii depicts is studied, (he studying will begin the new enneagram and will necessarily involve (he experimenter in a much more cognitive role. So, however the experiment carries on from point 8, it has to be tailored to fit this final end of producing the displayed spectrum. Hence the various steps of adjustment begin from point 8 and work back around the zig-zag figure.

At this point we leave the setting-up sequence and carry on to the next and final section of this part of our study. It will be evident that the attempt to demonstrate the role of the recurrent figure in the experi­ment could be carried to almost any depth of detail and precision. The beauty of the general symbolic method comes largely from its use as an instrument for studying situations in depth. There is no end-point at which we can terminate our enquiries with the assurance that there is no more to be discovered. There is always more to be seen, always more to be grasped and understood. The only limit is set by our own will to seek for what is there to be found.

4.   Three-Fold Structure

We come finally to study the spectrometer in some of its three-fold aspects. Since the spectrometer is designed as a three-fold instrument this is an almost inexhaustible task. Since, in addition, the inner triangle of the enneagram has an unlimited wealth of meaning, we shall again be compelled to select from the variety of exemplifications available a few of particular interest for our present study.


The triad set up by the three sub-instruments of the spectrometer is an obvious first choice. Very much can be learned from it of the workings of the triad, the structure of relatedness. It will already have become apparent that whole experiment is symbolic of the structure of relatedness, down even to very triangular shape of the prism which makes the experiment what it is, and was for Newton.

The triad of collimator-prism-telescope is not established all at once at the beginning of the experiment. It becomes established only when the position of minimum deviation has become achieved—exactly as in Newton's experiment. However, there are certain refinements in the manner by which it comes to be which are of interest. So we can profitably examine its establishment by degrees, following the way in which we studied it in that first experiment.

Referring to the enneagram, point 9 sounds its two notes do1 and do’1 in the spectrometer when the collimator and telescope have been given such a perfection of adjustment within the measurement space that they are exactly matched. This occurs for the two sub-instruments in two roles.

1 As extended material objects playing roles in the triad of the measurement-space, they have previously to be lined-up in such a way that:   “It must be possible to make the axes of the telescope and colli­mator coincident."*

*  Wagner, loc. cit   p. 25. Spectrometer adjustments.


In this condition they coincide as potential angular measurement-objects and express the do, and do\ of the measurement enneagram.

2. The initiating and completing notes of the experiment sound as one when the image of the radiating light source formed with light transmitted through the collimator slit is brought into focus in the focal plane of the telescope.

Once established, this is a condition which recurs throughout the experiment in all the setting-up operations. It is fixed once and for all, as far as the experiment is concerned, only when that inner harmony becomes realized by which the recurrent figure is synchronized into one whole and the triad is established.

Three Impulses**

When the prism is placed upon the prism table the third role in the relationship enters the scene in embodied form and we are able to speak of the three relational impulses which the three elements transmit:

1. The collimator transmits the affirmation of the existence of the source of the first process in the experiment. The fiat lux.

2. The prism transmits a denial through setting-up conditions by which the transmission of the light from the first process becomes opposed, limited and subject to fragmentation.

3. The telescope, when used as a third instrument in relationship with the other two, becomes able to transmit the means of reconciling them which is, in the last analysis, the free and independent reconciling will of the scientist as experimenter—by virtue of which he can perform intentional actions involving progressive approximation.

** For further elucidation of the three impulses and the manner by which they enter into different modes of combination the reader is referred to The Dramatic Universe, by. J. G. Bennett, Hodder & Stoughton, 1961, Vol. IIPart II, p. 69 ff. The Triad - Will.


Perfect Reconciliation

The prior condition at which the telescope and collimator are per­fectly matched corresponds to the condition of their perfect recon­ciliation—complete receptivity combined with total donation. It is possible only when the two are not committed in any way to participation in the interaction for which the prism acts as a denying source. The moment this denying source enters the situation, the point 9 becomes the triangle 3-6-9 in the symbol and the perfection of the reconciliation becomes compromised by denying elements.*

* We note that when the reconciling source enters fully into the situation there is the completion of the inner recurrent figure and the triangle 369 becomes the pattern 142857 . . . endlessly recurring. We have not the space to pursue this observation further here. It signifies the establishment of a permanent hold upon existence for the event concerned, through perpetual renewal.


This corresponds to the establish­ment of three distinct kinds of process:

Three Kinds of Process

                       1.         The function performed by the collimator so clearly typifies the outgoing process that it deserves the name of paradigm in this role.

2.          The prism likewise typifies the second process which provides the field of interaction, and sets up the condition of denial towards the first process.

3.          The connection of the telescope with a finalizing role has been already sufficiently stressed earlier in this paper. It is likewise a paradigm in typifying the returning concentrative process by which a final image is formed.

Power of Reconciliation

It is a mark of the fineness of quality of reconciliation expressed by the instrument as a whole that it can perform measurements to an extreme degree of precision. If we look to see how this quality enters the machine, we see that it all turns upon the provision and manner of use of the parallel beam.

The parallel beam presents both collimator and telescope with a common mediating power, or free energy, which is at their disposal. The collimated beam is an already reconciled entity. It therefore has the power to engender harmony into the situations in which it participates. In the spectrometer this shows itself in the smoother manner in which the light passes through its intervals in completing the main process of the experi­ment. It is helped into the prism. The deviated light is afterwards eased into position at the final image. There is more harmony in the spectro­meter as the place where Newton's experiment takes place and the experi­ment is more perfectly performed than Newton's as a consequence.




The process of qualitative transformation by which the steps of the completing processes transform one into another is triadic. It is expressed by Gurdjieff in a deceptively simple generalized formula as follows:

"The higher blends with the lower in order to actualize the middle and thus becomes either higher for the preceding lower or lower for the succeeding higher."*

* Cf. All and Everything, p. 751: "A new arising from the previously arisen through the 'Harnel-Miatznel'." The higher stands to the lower in the relation of a greater activity for the situation. The lower is always more passive than the higher.

    We can express the action of transformation by reference to some of the steps we have already elicited in the main completing process common to Newton's experiment and the spectrometer. Thus:

Entry of Light into the Experimental Domain

The light radiated out from the presence of the source (doL) blends with the slit and its surrounds (re,) in order to actualize the beam transmitted through the slit and thus becomes an expanding cone of light (mi,) well-defined in direction . . . travelling towards the prism face.

We begin to see from this formulation just how down-to-earth these expressions and notions of generalized structure are. We are familiar in our everyday lives with the structuring of light falling upon slits and take it for granted. In fact, we make many more assumptions about the ubiquity and pervasiveness of generalized structuring than we usually realize. We are accustomed to assume as a matter of course that the world is "logical", "coherent" and "consistent"—and these are assump­tions about generalized qualitative structure.

It is noticeable, also, that this kind of picturing of situations and what is going on in them is precisely what we come to when we consider the world as a place for action. In this world we as scientists set up experi­ments, make adjustments, take readings, produce and interpret records of results and diagrams of apparatus. The world of practical science is a world of transformation conceived of in this kind of way.

We continue with the expression of the transformations involved in the experiment in these terms as follows:

Incidence Upon the Prism

The light present in the well-defined conical beam (mi1) travels towards and blends with the presence of the plane surface of the prism face (do2) in order to actualize a third beam (fa1) interacting with the glass.

We can also express in this way the effect on the octave of the inter­vention of the collimator lens into the completing process. Thus:

The intervention of the collimator lens into the first interval of the main completing process assists the blending of the incident beam with the prism face by previously itself blending with it in order to actualize a uniform parallel beam whose light may . . . etc.

The correspondence in form between Gurdjieff's generalized formula and the way in which we commonly understand these kinds of trans-formation is remarkable. It is clearly a means of expressing the structure of transforming situations whose depth is limited only by our ability to penetrate into and see what is going on. We can pursue the processes into the prism. Thus:

Refraction Into the Medium

1.  The light present in the incident beam (fat) enters the medium through the prism interface (do2) and blends with the material of the glass in order to actualize an interaction and thus becomes a dispersing cone of light (sol1). . . travelling towards the second prism face within the medium.

2.           The presence of the plane surface (do2) blends with the material properties of transparency and opacity of the glass (re2) in order to actualize an interface which shall evenly refract and uniformly deviate into its interior (mi2) the light paths of beams incident upon it.

Dispersion we ascribe to the light itself and therefore to the condition denoted by soli. Deviation being an effect of the prism material we ascribe to mi2. Snell's law, of course, comes in at this point, when the phenomenon is related to a measurement space.

Here we may leave the main completing process at a very appropriate place for the reader to begin to complete the final expressions for himself. Much of the preliminary ground has already been covered and sufficiently excavated in the second section of this last part.

The exposition given above in this paper of the correspondences which may be found between the various structurings expressed by the enneagram symbol and features of the structure exemplified by the experi­ments considered may seem somewhat arbitrary and unconvincing on a first reading. The determined and experimentally-minded reader is invited to consider for himself the correlations to be discovered between the symbol and the most economical practical procedure fol­lowed in actually performing that well-known elementary experiment by which the index of refraction of glass is determined from a sample in the form of a rectangular block—pins, paper, pencil, ruler, protractor and all. There is a stage in this procedure at which the experimenter finds it necessary to move his eye right round from one face of the block to the other in order to finish plotting the course of the "idealized ray", defined by the lined-up pins, after refraction through the block. The reader should find, after only a little difficulty occasioned by the unfamiliarity of the symbolism, that this can he easily and convincingly correlated with the (2-8) line in the zig-zag inner pattern. Consideration of the vital role played by the eye in that experiment, in conjunction with the use of his spatial freedom by the experimenter, is particularly illuminating. In that experiment, as with Newton's, it is the experimenter himself who contrives the setting-up of the measurement space. It is not automated into the apparatus as it is in the spectrometer.



I have endeavoured in this paper to demonstrate empirically the applicability of Gurdjieff's generalized enneagram symbolism to a piece of scientific work. I am sufficiently familiar with the work in question to be able to bring both it and the symbol into direct contact. To my mind this is almost certainly the only possible way to come to some understand­ing of what this generalized symbol really is about.

We talk about the power of mathematics, but more generally any symbol which refers to the structure pervading a real situation has power when used in relation to that situation. The structure or patterning to which the enneagram symbol refers is so extremely general that its power as a device for coming to understand structuring in all kinds of situations must be virtually unlimited. I have no doubt that this was one of the purposes for which it was originally created.

The enneagram is a device, in its use a method, for coming to under­stand general structural principles. The study of their exemplification in particular situations leads one to a new view-point which is anti-temporal.

Structure is timeless and refers to that which remains beyond the actualizations of temporal process. Processes actualize temporally in accordance with patterns—which are anti-temporal because they remain preserved and untouched, unmodified by the changes which take place successively in time.

The enneagram points to a world which is already present but virtual. A world which is ordered, structured, patterned - latent with forms of meaning already waiting to become realized in the actualization of existing situations. The symbol thus expresses the latent patterning of the present moment*

* Cf.   Towards an Objectively Complete Language,   by Bennett, Bortoft   and Pledge/ .Systematics. 1, No. 3, 1965.


The symbol is an obvious representation of the structure of a perfectly co-ordinated process actualizing within a present moment. This is symbolized by the structural figures confined within the circle. In an experiment the central point about which the circle is circumscribed is the will of the scientist S by reference to whom the whole experiment into existence, is set up, adjusted, measurements are performed. . . . All that is compatible with the performance of that particular experiment has its place within the circle; all else is excluded by the initiating decision of S. Thus the circle initially symbolizes the compatibility bracket set up by that act which separates the relevant from the irrelevant. The setting-up of the inner triad symbolized by the triangle corresponds to the establish­ment of a compresence with the basic apparatus—as the prism establishes the refraction effect by its compresence with light-source, slit and screen. The state of coalescence comes about when the inner recurrent pattern is closed. The three kinds of linkages refer to the points at the corners of the triangle. The three recurrent elements can be ascribed to the three kinds of processes involved and hence to the three sides of the triangle. The symbol which denotes within-ness has a two-fold meaning as the connection of circumferential points with the centre and also the area contained within the circle. But once again we must beware of trying to tie down meanings which are really many-to-one. The advantage of the enneagram symbol as compared with other means of representation lies in its ability to communicate the structural connectednesses immedi­ately and unambiguously.

I hope to have shown in this paper that the enneagram symbol is directly relevant to the scientific procedure. But science is only one of the fields in which human beings engage themselves with a view to achieving purposes. Wherever there is something to be done, there is, if it is worth doing, some value to be realized in the doing of it. Satisfaction comes when the situation within which our efforts are applied transforms from struggle into harmony. When that happens something has both made and found its proper place within the existing world.