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ZEN 3: Illusory Visions and Sensations

zazen-2

These are Yasutani-Roshi’s introductory lectures on Zen training from THE THREE PILLARS OF ZEN by Philip Kapleau.

There is little to comment here. Any comment is to empasize a point. The comments are in color.

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Lecture 3—Illusory Visions and Sensations

This is the third lecture. Before I begin I will assign you a new way of concentration. Instead of counting your exhalations, as heretofore, count “one” on the first inhalation, “two” on the next inhalation, and so on, up to ten. This is more difficult than counting on the exhalation, because all mental and physical activity is performed on the exhaled breath. For instance, just before pouncing animals take a breath. This principle is well known in kendo fencing and judo fighting, where one is taught that by carefully observing his opponent’s breathing his attack can be anticipated. While this exercise is difficult, you must try to practice it as another means of concentrating your mind. Until you come before me again you are to concentrate on counting the inhalations of your breath, not audibly but in the mind only.

Note that the counting regimen is modified slightly once again.

Makyo are the phenomena-visions, hallucinations, fantasies, revelations, illusory sensations—which one practicing zazen is apt to experience at a particular stage in his sitting. Ma means “devil” and kyo “the objective world.” Hence makyo are the disturbing or “diabolical” phenomena which appear to one during his zazen. These phenomena are not inherently bad. They become a serious obstacle to practice only if one is ignorant of their true nature and is ensnared by them.

The word makyo is used in both a general and a specific sense. Broadly speaking, the entire life of the ordinary man is nothing but a makyo. Even such Bodhisattvas as Monju and Karman, highly developed though they are, still have about them traces of makyo; otherwise they would be supreme Buddhas, completely free of makyo. One who becomes attached to what he realizes through satori is also still lingering in the world of makyo. So, you see, there are makyo even after enlightenment, but we shall not enter into that aspect of the subject in these lectures.

Make sure you understand the true nature of a makyo.

In the specific sense the number of makyo which can appear are in fact unlimited, varying according to the personality and temperament of the sitter. In the Ryogon [Surangama] sutra the Buddha warns of fifty different kinds, but of course he is referring only to the commonest. If you attend a sesshin of from five to seven days’ duration and apply yourself assiduously, on the third day you are likely to experience makyo of varying degrees of intensity. Besides those which involve the vision there are numerous makyo which relate to the sense of touch, smell, or hearing, or which sometimes cause the body suddenly to move from side to side or forward and backward or to lean to one side or to appear to sink or rise. Not infrequently words burst forth uncontrollably or, more rarely, one imagines he is smelling a particularly fragrant perfume. There are even cases where without conscious awareness one writes down things which tum out to be prophetically true.

Makyos involve all five physical senses—sight, hearing, taste, smell, and touch.

Very common are visual hallucinations. You are doing zazen with your eyes open when suddenly the ridges of the straw matting in front of you seem to be heaving up and down like waves. Or without warning everything may go white before your eyes, or black. A knot in the wood of a door may suddenly appear as a beast or demon or angel. One disciple of mine often used to see visions of masks—demons’ masks or jesters’ masks. I asked him whether he had ever had any particular experience of masks, and it turned out that he had seen them at a festival in Kyushu1 when he was a child. Another man I knew was extremely troubled in his practice by visions of Buddha and his disciples walking around him reciting sutras, and was only able to dispel the hallucination by jumping into a tank of ice-cold water for two or three minutes.

1 The southernmost of Japan’s main islands.

Visual maykos are very common.

Many makyo involve the hearing. One may hear the sound of a piano or loud noises, such as an explosion (which is heard by no one else), and actually jump. One disciple of mine always used to hear the sound of a bamboo flute while doing zazen. He had learned to play the bamboo flute many years before, but had long since given it up; yet always the sound came to him when he was sitting.

Maykos are often related to past experiences.

In the Zazen Yojinki we find the following about makyo: “The body may feel hot or cold or glasslike or hard or heavy or light. This happens because the breath is not well harmonized [with the mind] and needs to be carefully regulated.” It then goes on to say: “One may experience the sensation of sinking or floating, or may alternately feel hazy and sharply alert. The disciple may develop the faculty of seeing through solid objects as though they were transparent, or he may experience his own body as a translucent substance. He may see Buddhas and Bodhisattvas. Penetrating insights may suddenly come to him, or passages of sutras which were particularly difficult to understand may suddenly become luminously clear to him. All these abnormal visions and sensations are merely the symptoms of an impairment arising from a maladjustment of the mind with the breath.”

Maykos arise from maladjustment of the mind with the breath.

Other religions and sects place great store by experiences which involve visions of God or hearing heavenly voices, performing miracles, receiving divine messages, or becoming purified through various rites. In the Nichiren sect, for example, the devotee loudly and repeatedly invokes the name of the Lotus sutra, to the accompaniment of vigorous body movements, and feels he has thereby purged himself of his defilements. In varying degree these practices induce a feeling of well-being, yet from the Zen point of view all are morbid states devoid of true religious significance and hence only makyo.

Maykos are hindrances even when they induce a feeling of well-being.

What is the essential nature of these disturbing phenomena we call makyo? They are temporary mental states which arise during zazen when our ability to concentrate has developed to a certain point and our practice is beginning to ripen. When the thought-waves which wax and wane on the surface of the sixth class of consciousness are partially calmed, residual elements of past experiences ‘lodged” in the seventh and eighth classes of consciousness bob up sporadically to the surface of the mind, conveying the feeling of a greater or expanded reality. Makyo, accordingly, are a mixture of the real and the unreal, not unlike ordinary dreams. Just as dreams do not appear to a person in deep sleep but only when he is half-asleep and half-awake, so makyo do not come to those in deep concentration or samadhi. Never be tempted into thinking that these phenomena are real or that the visions themselves have any meaning. To see a beautiful vision of a Bodhisattva does not mean that you are any nearer becoming one yourself, any more than a dream of being a millionaire means that you are any richer when you awake. Hence there is no reason to feel elated about such makyo. And similarly, whatever horrible monsters may appear to you, there is no cause whatever for alarm. Above all, do not allow yourself to be enticed by visions of the Buddha or of gods blessing you or communicating a divine message, or by makyo involving prophecies which turn out to be true. This is to squander your energies in the foolish pursuit of the inconsequential.

Makyos are temporary mental states that arise when zazen practice is beginning to ripen. They are a mixture of the real and the unreal, not unlike ordinary dreams. They are inconsequential and should not be pursued.

But such visions are certainly a sign that you are at a crucial point in your sitting, and that if you exert yourself to the utmost, you can surely experience kensho. Tradition states that even Shakyamuni Buddha just before his own awakening experienced innumerable makyo, which he termed “obstructing devils.” Whenever makyo appear, simply ignore them and continue sitting with all your might.

Whenever makyo appear, simply ignore them and continue sitting with all your might.

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ZEN 2: Precautions to Observe in Zazen

zazen-1

These are Yasutani-Roshi’s introductory lectures on Zen training from THE THREE PILLARS OF ZEN by Philip Kapleau.

There is little to comment here. Any comment is to empasize a point. The comments are in color.

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Lecture 2—Precautions to Observe in Zazen

This is the second lecture. Now I want you to change your breathing exercise slightly. This morning I told you to count “one” as you inhaled and “two” as you exhaled. Hereafter I want you to count “one” only on the exhalation, so that one full breath [inhalation and exhalation] will be “one.” Don’t bother counting the inhalations; just count “one,” ” two,” “three,” and so forth , on the exhalation.

Note that the counting regimen is slightly modified in later sessions.

It is advisable to do zazen facing a wall, a curtain, or the like. Don’t sit too far from the wall nor with your nose up against it; the ideal distance is from two to three feet. Likewise, don’t sit where you have a sweeping view, for it is distracting, or where you look out on a pleasant landscape, which will tempt you to leave off zazen in order to admire it. In this connection it is important to remember that although your eyes are open you are not actually trying to see. For all these reasons it is wisest to sit facing a wall. However, if you happen to be doing zazen formally in a Rinzai temple, you will have no choice but to sit facing others, as this is the established custom in that sect.

Keep visual distractions to a minimum in the beginning.

In the beginning, if possible, select a room that is quiet as well as clean and tidy, one which you can regard as sacred. It may be asked whether it is satisfactory to do zazen on a bed so long as the room is clean and free from noise. For the ordinary healthy person the answer is no; there are any number of reasons why it is difficult to keep the mind in proper tension on a bed. A bedridden person, of course, has no choice.

A quiet, clean and tidy room provides less distractions.

You will probably find that natural sounds, like those of insects or birds or running water, will not disturb you, neither will the rhythmic ticking of a clock nor the purring of a motor. Sudden noises, however, like the roar of a jet, are jarring. But rhythmic sounds you can make use of. One student of mine actually attained enlightenment by utilizing the sound of the steady threshing of rice while he was doing zazen. The most objectionable sounds are those of human voices, either heard directly or over the radio or television. When you start zazen, therefore, find a room which is distant from such sounds. When your sitting has ripened, however, no noises will disturb you.

Keep sudden noises and sound of human voices to a minimum in the beginning.

Besides keeping your room clean and orderly you should decorate it with flowers and burn incense, since these, by conveying a sense of the pure and the holy, make it easier for you to relate yourself to zazen and thus to calm and unify your mind more quickly. Wear simple, comfortable clothing that will give you a feeling of dignity and purity. In the evening it is better not to wear night clothes, but if it is hot and a question of either doing zazen in pajamas or not doing it at all, by all means wear the pajamas. But make yourself clean and tidy.

Keep the room orderly, Decorate it with flowers. Burn incense. Wear simple, comfortable and dignified clothing.

The room ought not to be too light or too dark. You can put up a dark curtain if it is too light, or you can use a small electric bulb if it is night. The effect of a dark room is the same as closing your eyes: it dulls everything. The best condition is a sort of twilight. Remember, Buddhist zazen does not aim at rendering the mind inactive but at quieting and unifying it in the midst of activity.

Have comfortable lighting in the room. Aim at quieting and unifying the mind but not rendering it inactive.

A room that is neither too hot in summer nor too cold in winter is ideal. Punishing the body is not the purpose of zazen, so it is unnecessary to struggle with extremes of heat or cold. Experience has shown, however, that one can do better zazen when he feels slightly cool; too hot a room tends to make one sleepy. As your ardor for zazen deepens you will naturally become unconcerned about cold or heat. Nevertheless, it is wise to take care of your health.

Keep the room comfortably cool.

Next let us discuss the best time for zazen. For the eager and determined any time of day and all seasons of the year are equally good. But for those who have jobs or professions the best time is either morning or evening, or better still, both. Try to sit every morning, preferably before breakfast, and just before going to bed at night. But if you can sit only once-and you should sit at least once a day-you will have to consider the relative merits of morning and evening. Each has its advantages and disadvantages. If you find that either morning or evening is equally good and you ask which I recommend (because you can sit only once a day), I would say the morning, for the following reasons. No visitors come early in the morning, whereas in the evening you are likely to be interrupted. Also, morning—at any rate, in the city—is much quieter than evening since fewer cars are on the streets. Furthermore, because in the morning you are rested and somewhat hungry, you are in good condition for zazen, whereas in the evening, when you are tired and have had your meal, you are likely to be duller. Since it is difficult to do zazen on a full stomach, it is better not to sit immediately after a meal when you are a beginner. Before a meal, however, zazen can be practiced to good advantage. As your zeal grows it won’t matter when you sit, before, after, or during a meal.

If you are working, try to sit every morning, preferably before breakfast, and just before going to bed at night. 

How long should you do zazen at one sitting? There is no general rule, for it varies according to the degree of one’s eagerness as well as the maturity of one’s practice. For novices a shorter time is better. If you sit devotedly five minutes a day for a month or two, you will want to increase your sitting to ten or more minutes as your ardor grows. When you are able to sit with your mind taut for, say, thirty minutes without pain or discomfort, you will come to appreciate the feeling of tranquility and well-being induced by zazen and will want to practice regularly. For these reasons I recommend that beginners sit for shorter periods of time. On the other hand, should you force yourself from the beginning to sit for longer periods, the pain in your legs may well become unbearable before you acquire a calm mind. Thus you will quickly tire of zazen, feeling it to be a waste of time, or you will always be watching the clock. In the end you will come to dislike zazen and stop sitting altogether. This is what frequently happens. Now, even though you sit for only ten minutes or so each day, you can compensate for this briefness by concentrating intensely on the counting of each breath, thus increasing its effectiveness. You must not count absent-mindedly or mechanically, as though it were a duty.

You may do zazen anywhere from five to thirty minutes at one sitting. For novices a shorter time is better. Count your breaths with full awareness.

In spite of your being able to sit for an hour or more with a feeling of exquisite serenity, it is wise to limit your sitting to periods of about thirty or forty minutes each. Ordinarily it is not advisable to do zazen longer than this at one sitting, since the mind cannot sustain its vigor and tautness and the value of the sitting decreases. Whether one realizes it or not, a gradual diminution of the mind’s concentrative intensity takes place. For this reason it is better to alternate a thirty or forty-minute period of sitting with a round of walking zazen. Following this pattern, one can do zazen for a full day or even a week with good results. The longer zazen continues, however, the more time should be spent in walking zazen. In fact, one might advantageously add periods of manual labor to this routine, as has been done in the Zen temple since olden times. Needless to say, you must keep your mind in a state of clear awareness during such manual labor and not allow it to become lax or dull.

Alternate a thirty or forty-minute period of sitting with a round of walking zazen.

A word about food. It is better to eat no more than eighty percent of your capacity. A Japanese proverb has it that eight parts of a full stomach sustain the man; the other two sustain the doctor. The Zazen Yojinki (Precautions to Observe in Zazen), compiled about 650 years ago, says you should eat two-thirds of your capacity. It further says that you should choose nourishing vegetables (of course meat-eating is not in the tradition of Buddhism and it was taboo when the Yojinki was written) such as mountain potatoes, sesame, sour plums, black beans, mushrooms, and the root of the lotus; and it also recommends various kinds of seaweed, which are highly nutritious and leave an alkaline residue in the body. Now, I am no authority on vitamins and minerals and calories, but it is a fact that most people today eat a diet which creates too much acid in the blood, and a great offender in this respect is meat. Eat more vegetables of the kind mentioned, which are alkalinic in their effect. In ancient days there was a yang-yin diet. The yang was the alkaline and the yin the acid, and the old books cautioned that a diet ought not be either too yang or too yin. This is substantially what I have just told you.

Eat nutritious, mostly vegetarian food. Be somewhat hungry.

When sitting it is a good idea to have a notebook and pencil before you, because a variety of insights will flash into your mind and you will think: “I must write this down before I forget it.” Relationships which previously were incomprehensible will suddenly be clarified and difficult problems will be abruptly solved. If you do not jot these things down, they will worry you and thus interfere with your concentration.

Note down any insight that occurs during the zazen session, if you plan to access it later.

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ZEN 1: Theory and Practice of Zazen

Zazen

These are Yasutani-Roshi’s introductory lectures on Zen training from THE THREE PILLARS OF ZEN by Philip Kapleau.

There is little to comment here. Any comment is to empasize a point. The comments are in color.

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Lecture 1—Theory and Practice of Zazen

What I am about to tell you is based upon the teachings of my revered teacher, Daiun1 Harada-roshi. Although he himself was of the Soto sect, he was unable to find a truly accomplished master in that sect and so went to train first at Shogen-ji and then Nanzen-ji, two Rinzai monasteries. At Nanzen-ji he eventually grasped the inmost secret of Zen under the guidance of Dokutan-roshi, an outstanding master.

1 A Zen name meaning “Great Cloud.” See “clouds and water” in section x. His other name is Sogaku.

While it is undeniably true that one must undergo Zen training himself in order to comprehend the truth of Zen, Harada-roshi felt that the modem mind is so much more aware that for beginners lectures of this type could be meaningful as a preliminary to practice. He combined the best of each sect and established a unique method of teaching Zen. Nowhere in Japan will you find Zen teaching set forth so thoroughly and succinctly, so well suited to the temper of the modem mind, as at his monastery. Having been his disciple for some twenty years, I was enabled, thanks to his grace, to open my Mind’s eye in some measure.

These lectures can be very useful in starting a grass-root movement for general spiritual advancement.

Before commencing his lectures Harada-roshi would preface them with advice on listening. His first point was that everyone should listen with his eyes open and upon him—in other words, with his whole being—because an impression received only through the hearing is rather shallow, akin to listening to the radio. His second point was that each person should listen to these lectures as though they were being given to him alone, as ideally they should be. Human nature is such that if two people listen, each feels only half-responsible for understanding, and if ten people are listening each feels his responsibility to be but one-tenth. However, since there are so many of you and what I have to say is exactly the same for everybody, I have asked you to come as a group. You must nonetheless listen as though you were entirely alone and hold yourselves accountable for everything that is said.

This is very good advice on listening.

This discourse is divided into eleven parts, which will be covered in some eight lecture sessions. The first involves the rationale of zazen and direct methods of practice; the next, special precautions; and the following lectures, the particular problems arising from zazen, to­gether with their solution.

In point of fact, a knowledge of the theory or principles of zazen is not a prerequisite to practice. One who trains under an accomplished teacher will inevitably grasp this theory by degrees as his practice ripens. Modem students, however, being intellectually more sophisticated than their predecessors in Zen, will not follow instructions unreservedly; they must first know the reasons behind them. Hence I feel obliged to deal with theoretical matters. The difficulty with theory, however, is that it is endless. Buddhist scriptures, Buddhist doctrine, and Buddhist philosophy are no more than intellectual formulations of zazen, and zazen itself is their practical demonstration. From this vast field I will now abstract what is most essential for your practice.

Do not delay practice. Start practicing while supporting it with theory.

We start with the Buddha Shakyamuni.1 As I think you all know, he began with the path of asceticism, undergoing tortures and austerities which others before him had never attempted, including prolonged fasting. But he failed to attain enlightenment by these means and, half-dead from hunger and exhaustion, came to realize the futility of pursuing a course which could only terminate in death. So he drank the milk which was offered him, gradually regained his health, and resolved to steer a middle course between self-torture and self-indulgence. Thereafter he devoted himself exclusively to zazen for six years2 and eventually, on the morning of the eighth of December, at the very instant when he glanced at the planet Venus gleaming in the eastern sky, he attained perfect enlightenment. All this we believe as historical truth.

1 The traditional Japanese term is O-Shaka-sama. It is both respectful and intimate. The O and sama are honorifics, and rather than attempt an arbitrary translation of them, I have followed the usual English rendering of this title. (See “Buddha” in section X.)
2 Other accounts say six years elapsed from the time he left his home until his supreme enlightenment.

This practice is about the middle-way (no extremes).

The words the Buddha uttered involuntarily at this time are recorded variously in the Buddhist scriptures. According to the Kegon sutra, at the moment of enlightenment he spontaneously cried out: “Wonder of wonders! Intrinsically all living beings are Buddhas, endowed with wisdom and virtue, but because men’s minds have become inverted through delusive thinking they fail to perceive this.” The first pronouncement of the Buddha upon his enlightenment seems to have been one of awe and astonishment. Yes, how truly marvelous that all human beings, whether clever or stupid, male or female, ugly or beautiful, are whole and complete just as they are. That is to say, the nature of every being is inherently without a flaw, perfect, no different from that of Amida or any other Buddha. This first declaration of Shakyamuni Buddha is also the ultimate conclusion of Buddhism. Yet man, restless and anxious, lives a half-crazed existence because his mind, heavily encrusted with delusion, is turned topsy-turvy. We need therefore to return to our original perfection, to see through the false image of ourselves as incomplete and sinful, and to wake up to our inherent purity and wholeness.

We need to return to our original perfection.

The most effective means by which to accomplish this is through zazen. Not only Shakyamuni Buddha himself but many of his disciples attained enlightenment through zazen. Moreover, during the 2,500 years since the Buddha’s death innumerable devotees in India, China, and Japan have, by grasping this selfsame key, resolved for themselves the most fundamental question, What are life and death? Even in this day there are many who have been able to cast off worry and anxiety and emancipate themselves through zazen.

We must cast off worry and anxiety about life and death.

Between a Nyorai (i.e., a supremely perfected Buddha) and us, who are ordinary, there is no difference as to substance. This “substance” can be likened to water. One of the salient characteristics of water is its conformability: when put into a round vessel it becomes round, when put into a square vessel it becomes square. We have this same adaptability, but as we live bound and fettered through ignorance of our true nature, we have forfeited this freedom. To pursue the metaphor, we can say that the mind of a Buddha is like water that is calm, deep, and crystal clear, and upon which the “moon of truth” reflects fully and perfectly. The mind of the ordinary man, on the other hand, is like murky water, constantly being churned by the gales of delusive thought and no longer able to reflect the moon of truth. The moon nonetheless shines steadily upon the waves, but as the waters are roiled we are unable to see its reflection. Thus we lead lives that are frustrating and meaningless.

The basic nature is to be adaptable.

How can we bring the moon of truth to illumine fully our life and personality? We need first to purify this water, to calm the surging waves by halting the winds of discursive thought. In other words, we must empty our minds of what the Kegon sutra calls the “conceptual thought of man.” Most people place a high value on abstract thought, but Buddhism has clearly demonstrated that discriminative thinking lies at the root of delusion. I once heard someone say: “Thought is the sickness of the human mind.” From the Buddhist point of view this is quite true. To be sure, abstract thinking is useful when wisely employed—which is to say, when its nature and limitations are properly understood—but so long as human beings remain slaves to their intellect, fettered and controlled by it, they can well be called sick.

Discursive thought is the sickness of the human mind.

All thoughts, whether ennobling or debasing, are mutable and impermanent; they have a beginning and an end even as they are fleetingly with us, and this is as true of the thought of an era as of an individual. In Buddhism thought is referred to as “the stream of life-and-death.” It is important in this connection to distinguish the role of transitory thoughts from that of fixed concepts. Random ideas are relatively innocuous, but ideologies, beliefs, opinions, and points of view, not to mention the factual knowledge accumulated since birth (to which we attach ourselves), are the shadows which obscure the light of truth.

The nature of thought is to be mutable and impermanent and not fixed. By becoming fixed thought obscures the light of truth.

So long as the winds of thought continue to disturb the water of our Self-nature, we cannot distinguish truth from untruth. It is imperative, therefore, that these winds be stilled. Once they abate, the waves subside, the muddiness clears, and we perceive directly that the moon of truth has never ceased shining. The moment of such realization is kensho, i.e., enlightenment, the apprehension of the true substance of our Self-nature. Unlike moral and philosophical concepts, which are variable, true Insight is imperishable. Now for the first time we can live with inner peace and dignity, free from perplexity and disquiet, and in harmony with our environment.

Enlightenment is the apprehension of the true substance of our Self-nature.

I have spoken to you briefly about these matters, but I hope I have succeeded in conveying to you the importance of zazen. Let us now talk about practice.

The practice of zazen now follows. 

The first step is to select a quiet room in which to sit. Lay out a fairly soft mat or pad some three feet square, and on top of this place a small circular cushion measuring about one foot in diameter to sit on, or use a square cushion folded in two. Preferably one should not wear trousers or socks, since these interfere with the crossing of the legs and the placing of the feet. For a number of reasons it is best to sit in the full-lotus posture. To sit full-lotus you place the foot of the right leg over the thigh of the left and the foot of the left leg over the thigh of the right. The main point of this particular method of sitting is that by establishing a wide, solid base, with the crossed legs and with both knees touching the mat, you achieve absolute stability. With the body thus immobile, thoughts are not stirred into activity by physical movements and the mind more easily becomes tranquil.

With the body immobile thoughts are not stirred into activity by physical movements and the mind more easily becomes tranquil.

If you have difficulty sitting in the full-lotus posture because of the pain, sit half-lotus, which is done by putting the foot of the left leg over the thigh of the right. For those of you who are not accustomed to sitting cross-legged, even this position may not be easy to maintain. You will probably find it difficult to keep the two knees resting on the mat and will have to push one or both of them down again and again until they remain there. In both the half- and the full-lotus posture the uppermost foot can be reversed when the legs become tired.

The posture should be comfortable and not painful.

For those who find both of these traditional zazen positions acutely uncomfortable, an alternative position is the traditional Japanese one of sitting on the heels and calves. This can be maintained for a longer time if a cushion is placed between the heels and the buttocks. One advantage of this posture is that the back can be kept erect easily. However, should all of these positions prove too painful, you may use a chair.1

1 See section IX for sketches of all these postures, including one widely used in the Southeast Asian Buddhist countries.

The important aspect of posture is keeping the back erect.

The next step is to rest the right hand in the lap, palm upward, and place the left hand, palm upward, on top of the right palm. Lightly touch the tips of the thumbs to each other so that a flattened circle is formed by the palms and thumbs. Now, the right side of the body is the active pole, the left the passive. Hence during practice we repress the active side by placing the left foot and left hand over the right members, as an aid in achieving the highest degree of tranquility. If you look at a figure of the Buddha, however, you will notice that the position of these members is just the reverse. The significance of this is that a Buddha, unlike the rest of us, is actively engaged in the task of saving.

The posture is an aid in achieving the highest degree of tranquility.

After you have crossed your legs, bend forward so as to thrust the buttocks out, then slowly bring the trunk to an erect posture. The head should be straight; if looked at from the side, your ears should be in line with your shoulders and the tip of your nose in line with your navel. The body from the waist up should be weightless, free from pressure or strain. Keep the eyes open and the mouth closed. The tip of the tongue should lightly touch the back of the upper teeth. If you close your eyes you will fall into a dull and dreamy state. The gaze should be lowered without focusing on anything in particular. Experience has shown that the mind is quietest, with the least fatigue or strain, when the eyes are in this lowered position.

In zazen, eyes are open without focusing on anything in particular, and in lowered position.

The spinal column must be erect at all times. This admonition is important. When the body slumps, not only is undue pressure placed on the internal organs, interfering with their free functioning, but the vertebrae by impinging upon nerves may cause strains of one kind or another. Since the body and mind are one, any impairment of the physiological functions inevitably involves the mind and thus diminishes its clarity and one-pointedness, which are essential for effective concentration. From a purely psychological point of view, a ramrod erectness is as undesirable as a slouching position, for the one springs from unconscious pride and the other from abjectness, and since both are grounded in ego they are equally a hindrance to enlightenment.

Be careful to hold the head erect; if it inclines forward or backward or sideward, remaining there for an appreciable length of time, a crick in the neck may result.

The posture should be well balanced so as not to cause strain on any body part.

When you have established a correct posture, take a deep breath, hold it momentarily, then exhale slowly and quietly. Repeat this two or three times, always breathing through the nose. After that breathe naturally. When you have accustomed yourself to this routine, one deep breath at the beginning will suffice. Now bend the body first to the right as far as it will go, then to the left, about seven or eight times, in large arcs to begin with, then smaller ones until the trunk naturally comes to rest at center.

Start the zazen session with a couple of deep breaths and settling down in the correct posture.

You are now ready to concentrate your mind.1 There are many good methods of concentration bequeathed to us by our predecessors in Zen. The easiest for beginners is counting incoming and outgoing breaths. The value of this particular exercise lies in the fact that all reasoning is excluded and the discriminative mind put at rest. Thus the waves of thought are stilled and a gradual one-pointedness of mind achieved. To start with, count both inhalations and exhalations. When you inhale, concentrate on “one”; when you exhale, on “two”; and so on, up to ten. Then you return to “one” and once more count up to ten, continuing as before. It is as simple as that.

1 For additional information on concentrating the mind, see pp. 128-29.

Start to concentrate your mind by counting incoming and outgoing breaths.

As I have previously pointed out, fleeting ideas which naturally fluctuate in the mind are not in themselves an impediment. This unfortunately is not commonly recognized. Even among Japanese who have been studying and practicing Zen for five years or more there are many who misunderstand Zen practice to be a stopping of consciousness. There is indeed a kind of zazen that aims at doing just this,2 but it is not the traditional zazen of Zen Buddhism. You must realize that no matter how intently you count your breaths you will still perceive what is in your line of vision, since your eyes are open, and you will hear the normal sounds about you, as your ears are not plugged. And since your brain likewise is not asleep, various thought-forms will dart about in your mind. Now, they will not hamper or diminish the effectiveness of zazen unless, evaluating them as “good,” you cling to them or, deciding they are “bad,” you try to check or eliminate them. You must not regard any perceptions or sensations as an obstruction to zazen, nor should you pursue any of them. I emphasize this. “Pursuit” simply means that in the act of seeing, your gaze lingers on objects; in the course of hearing, your attention dwells on sounds; and in the process of thinking, your mind adheres to ideas. If you allow yourself to be distracted in such ways, your concentration on the counting of your breaths will be impeded. To recapitulate: let random thoughts arise and vanish as they will, do not dally with them and do not try to expel them, but merely concentrate all your energy on counting the inhalations and exhalations of your breath.

2 See p. 45.

Fleeting ideas which naturally fluctuate in the mind are not in themselves an impediment. The impediment comes from clinging to thought-forms or trying to eliminate them.

In terminating a period of sitting do not arise abruptly, but begin by rocking from side to side, first in small swings, then in large ones, for about half a dozen times. You will observe that your movements in this exercise are the reverse of those you engage in when you begin zazen. Rise slowly and quietly walk around with the others in what is called kinhin, a walking form of zazen.

End the zazen session with movements that are the reverse of those you engage in when you begin zazen. 

Kinhin is performed by placing the right fist, with thumb inside, on the chest and covering it with the left palm while holding both elbows at right angles. Keep the arms in a straight line and the body erect, with the eyes resting upon a point about two yards in front of the feet. At the same time continue to count inhalations and exhalations as you walk slowly around the room. Begin walking with the left foot and walk in such a way that the foot sinks into the floor, first the heel and then the toes. Walk calmly and steadily, with poise and dignity. The walking must not be done absent-mindedly, and the mind must be taut as you concentrate on the counting. It is advisable to practice walking this way for at least five minutes after each sitting period of twenty to thirty minutes.

Walk for at least five minutes after each sitting period of twenty to thirty minutes.

You are to think of this walking as zazen in motion. Rinzai and Soto differ considerably in their way of doing kinhin. In the Rinzai method the walking is brisk and energetic, while in the traditional Soto it is slow and leisurely; in fact, upon each breath you step forward only six inches or so. My own teacher, Harada-roshi, advocated a gait somewhere between these two and that is the method we have been practicing here. Further, the Rinzai sect cups the left hand on top of the right, whereas in the orthodox Soto the right hand is placed on top. Harada-roshi felt that the Rinzai method of putting the left hand uppermost was more desirable and so he adopted it into his own teaching. Now, even though this walking relieves the stiffness in your legs, such exercise is to be regarded as a mere by-product and not the main object of kinhin. Hence those of you who are counting your breaths should continue during kinhin, and those of you who are working on a koan should carry on with it.

Walk naturally and comfortably. Walking is zazen in motion.

This ends the first lecture. Continue to count your breaths as I have instructed until you come before me again.

Zen and Zazen

Zen

Definitions from THREE PILLARS OF ZEN By Philip Kapleau.

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ZEN:

An abbreviation of the Japanese word zenna, which is a transliteration of the Sanskrit dhyana (ch’an or ch’anna in Chinese), i.e., the process of concentration and absorption by which the mind is tranquilized and brought to one-pointedness.

As a Mahayana Buddhist sect, Zen is a religion whose teachings and disciplines are directed toward Self-realization, that is to say, to the attainment of satori, which Shakyamuni Buddha himself experienced under the Bo tree after strenuous self-discipline. The Zen sect embraces the Soto, Rinzai, and Obaku sects.

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ZAZEN (pronounced “zah-zen,” each syllable accented equally):

Zazen is the principle discipline of Zen. Zazen is not “meditation”. In the broad sense zazen embraces more than just correct sitting. To enter fully into every action with total attention and clear awareness is no less zazen.

The prescription for accomplishing this was given by the Buddha himself in an early sutra: “In what is seen there must be just the seen; in what is heard there must be just the heard; in what is sensed (as smell, taste or touch) there must be just what is sensed; in what is thought there must be just the thought.”

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SATORI:

The Japanese term for the experience of enlightenment, i.e., Self-realization, opening the Mind’s eye, awakening to one’s True-nature and hence of the nature of all existence. See also “kensho.”

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KENSHO (lit., “seeing into one’s own nature”):

Semantically, kensho and satori have virtually the same meaning, and they are often used interchangeably. In describing the enlightenment of the Buddha and the Patriarchs, however, it is customary to use the word satori rather than kensho, the term satori implying a deeper experience.

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Schrodinger: Nobel Lecture, 1933

Wave-Mechanics

This paper presents the Nobel Lecture delivered on December 12, 1933 by Erwin Schrodinger. A summary and comments are provided.

Summary & Comments:

Schrodinger starts with the Fermat principle of the shortest light time. Light propagates in different media with different velocities, following a path that takes least amount of time. Fermat considers light propagating as one-dimensional rays for its math. However, in reality, light propagates as a two-dimensional wave front. The curvature of the wave front takes part in the phenomena of reflection, refraction and diffraction.

The Hamilton’s principle states that the motion of a dynamical system in a given time interval is such as to maximize or minimize the action integral. (In practice, the action integral is almost always minimized.) Hamilton’s principle considers the movement of mass points in a field of forces.

Schrodinger saw the propagation of mass points as similar to the rays of light. So, he took the bold step of applying Hamilton’s principle to the dynamics within the atom addressing both particle and wave characteristics of electrons. To meet quantum requirements, Schrodinger postulated the minimum value given by Hamilton’s principle to be restricted to integral multiples of Planck’s quantum (h).

The new mechanics of Hamilton principle allowed better accommodation of the diffraction phenomena. Schrodinger replaced electrons by hypothetical waves, which when diffracted and captured by the nucleus generated a diffraction halo, which then appeared as the atom. The dynamics of the atom could then be solved in terms of a variable that could be determined in two different ways and matched, and then verified as an integer multiple of h to obtain an accurate value.

From the viewpoint of Disturbance theory, the atomic configuration is more like a galaxy. The electronic region is like a rotating whirlpool of field-substance that is increasing in quantization toward the center. At the center, the field-substance collapses into a nucleus. It is to be seen if this model can provide a modification of Schrodinger’s equation that is more general in usefulness.

The text of the lecture now follows. The heading below links to the original materials.

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The Fundamental Idea of Wave Mechanics

On passing through an optical instrument, such as a telescope or a camera lens, a ray of light is subjected to a change in direction at each refracting or reflecting surface. The path of the rays can be constructed if we know the two simple laws which govern the changes in direction: the law of refraction which was discovered by Snellius a few hundred years ago, and the law of reflection with which Archimedes was familiar more than 2,000 years ago. As a simple example, Fig. 1 shows a ray A-B which is subjected to refraction at each of the four boundary surfaces of two lenses in accordance with the law of Snellius.

SFig 1

Fermat defined the total path of a ray of light from a much more general point of view. In different media, light propagates with different velocities, and the radiation path gives the appearance as if the light must arrive at its destination as quickly as possible. (Incidentally, it is permissible here to consider any two points along the ray as the starting- and end-points.) The least deviation from the path actually taken would mean a delay. This is the famous Fermat principle of the shortest light time, which in a marvellous manner determines the entire fate of a ray of light by a single statement and also includes the more general case, when the nature of the medium varies not suddenly at individual surfaces, but gradually from place to place. The atmosphere of the earth provides an example. The more deeply a ray of light penetrates into it from outside, the more slowly it progresses in an increasingly denser air. Although the differences in the speed of propagation are infinitesimal, Fermat’s principle in these circumstances demands that the light ray should curve earthward (see Fig. 2), so that it remains a little longer in the higher “faster” layers and reaches its destination more quickly than by the shorter straight path (broken line in the figure; disregard the square, WWW1W1 for the time being). I think, hardly any of you will have failed to observe that the sun when it is deep on the horizon appears to be not circular but flattened: its vertical diameter looks to be shortened. This is a result of the curvature of the rays.

SFig 2

According to the wave theory of light, the light rays, strictly speaking, have only fictitious significance. They are not the physical paths of some particles of light, but are a mathematical device, the so-called orthogonal trajectories of wave surfaces, imaginary guide lines as it were, which point in the direction normal to the wave surface in which the latter advances (cf. Fig. 3 which shows the simplest case of concentric spherical wave surfaces and accordingly rectilinear rays, whereas Fig. 4 illustrates the case of curved rays).

SFig 3

It is surprising that a general principle as important as Fermat’s relates directly to these mathematical guide lines, and not to the wave surfaces, and one might be inclined for this reason to consider it a mere mathematical curiosity. Far from it. It becomes properly understandable only from the point of view of wave theory and ceases to be a divine miracle. From the wave point of view, the so-called curvature of the light ray is far more readily understandable as a swerving of the wave surface, which must obviously occur when neighbouring parts of a wave surface advance at different speeds; in exactly the same manner as a company of soldiers marching forward will carry out the order “right incline” by the men taking steps of varying lengths, the right-wing man the smallest, and the left-wing man the longest. In atmospheric refraction of radiation for example (Fig. 2) the section of wave surface WW must necessarily swerve to the right towards W1W1 because its left half is located in slightly higher, thinner air and thus advances more rapidly than the right part at lower point. (In passing, I wish to refer to one point at which the Snellius’ view fails. A horizontally emitted light ray should remain horizontal because the refraction index does not vary in the horizontal direction. In truth, a horizontal ray curves more strongly than any other, which is an obvious consequence of the theory of a swerving wave front.) On detailed examination the Fermat principle is found to be completely tantamount to the trivial and obvious statement that–given local distribution of light velocities–the wave front must swerve in the manner indicated. I cannot prove this here, but shall attempt to make it plausible. I would again ask you to visualize a rank of soldiers marching forward. To ensure that the line remains dressed, let the men be connected by a long rod which each holds firmly in his hand. No orders as to direction are given; the only order is: let each man march or run as fast as he can. If the nature of the ground varies slowly from place to place, it will be now the right wing, now the left that advances more quickly, and changes in direction will occur spontaneously. After some time has elapsed, it will be seen that the entire path travelled is not rectilinear, but somehow curved. That this curved path is exactly that by which the destination attained at any moment could be attained most rapidly according to the nature of the terrain, is at least quite plausible, since each of the men did his best. It will also be seen that the swerving also occurs invariably in the direction in which the terrain is worse, so that it will come to look in the end as if the men had intentionally “bypassed” a place where they would advance slowly.

The Fermat principle thus appears to be the trivial quintessence of the wave theory. It was therefore a memorable occasion when Hamilton made the discovery that the true movement of mass points in a field of forces (e.g. of a planet on its orbit around the sun or of a stone thrown in the gravitational field of the earth) is also governed by a very similar general principle, which carries and has made famous the name of its discoverer since then. Admittedly, the Hamilton principle does not say exactly that the mass point chooses the quickest way, but it does say something so similar – the analogy with the principle of the shortest travelling time of light is so close, that one was faced with a puzzle. It seemed as if Nature had realized one and the same law twice by entirely different means: first in the case of light, by means of a fairly obvious play of rays; and again in the case of the mass points, which was anything but obvious, unless somehow wave nature were to be attributed to them also. And this, it seemed impossible to do. Because the “mass points” on which the laws of mechanics had really been confirmed experimentally at that time were only the large, visible, sometimes very large bodies, the planets, for which a thing like “wave nature” appeared to be out of the question.

The smallest, elementary components of matter which we today, much more specifically, call “mass points”, were purely hypothetical at the time. It was only after the discovery of radioactivity that constant refinements of methods of measurement permitted the properties of these particles to be studied in detail, and now permit the paths of such particles to be photographed and to be measured very exactly (stereophotogrammetrically) by the brilliant method of C. T. R. Wilson. As far as the measurements extend they confirm that the same mechanical laws are valid for particles as for large bodies, planets, etc. However, it was found that neither the molecule nor the individual atom can be considered as the “ultimate component”: but even the atom is a system of highly complex structure. Images are formed in our minds of the structure of atoms consisting of particles, images which seem to have a certain similarity with the planetary system. It was only natural that the attempt should at first be made to consider as valid the same laws of motion that had proved themselves so amazingly satisfactory on a large scale. In other words, Hamilton’s mechanics, which, as I said above, culminates in the Hamilton principle, were applied also to the “inner life” of the atom. That there is a very close analogy between Hamilton’s principle and Fermat’s optical principle had meanwhile become all but forgotten. If it was remembered, it was considered to be nothing more than a curious trait of the mathematical theory.

Now, it is very difficult, without further going into details, to convey a proper conception of the success or failure of these classical-mechanical images of the atom. On the one hand, Hamilton’s principle in particular proved to be the most faithful and reliable guide, which was simply indispensable; on the other hand one had to suffer, to do justice to the facts, the rough interference of entirely new incomprehensible postulates, of the so-called quantum conditions and quantum postulates. Strident disharmony in the symphony of classical mechanics – yet strangely familiar – played as it were on the same instrument. In mathematical terms we can formulate this as follows: whereas the Hamilton principle merely postulates that a given integral must be a minimum, without the numerical value of the minimum being established by this postulate, it is now demanded that the numerical value of the minimum should be restricted to integral multiples of a universal natural constant, Planck’s quantum of action. This incidentally. The situation was fairly desperate. Had the old mechanics failed completely, it would not have been so bad. The way would then have been free to the development of a new system of mechanics. As it was, one was faced with the difficult task of saving the soul of the old system, whose inspiration clearly held sway in this microcosm, while at the same time flattering it as it were into accepting the quantum conditions not as gross interference but as issuing from its own innermost essence.

The way out lay just in the possibility, already indicated above, of attributing to the Hamilton principle, also, the operation of a wave mechanism on which the point-mechanical processes are essentially based, just as one had long become accustomed to doing in the case of phenomena relating to light and of the Fermat principle which governs them. Admittedly, the individual path of a mass point loses its proper physical significance and becomes as fictitious as the individual isolated ray of light. The essence of the theory, the minimum principle, however, remains not only intact, but reveals its true and simple meaning only under the wave-like aspect, as already explained. Strictly speaking, the new theory is in fact not new, it is a completely organic development, one might almost be tempted to say a more elaborate exposition, of the old theory.

How was it then that this new more “elaborate” exposition led to notably different results; what enabled it, when applied to the atom, to obviate difficulties which the old theory could not solve? What enabled it to render gross interference acceptable or even to make it its own?

Again, these matters can best be illustrated by analogy with optics. Quite properly, indeed, I previously called the Fermat principle the quintessence of the wave theory of light: nevertheless, it cannot render dispensible a more exact study of the wave process itself. The so-called refraction and interference phenomena of light can only be understood if we trace the wave process in detail because what matters is not only the eventual destination of the wave, but also whether at a given moment it arrives there with a wave peak or a wave trough. In the older, coarser experimental arrangements, these phenomena occurred as small details only and escaped observation. Once they were noticed and were interpreted correctly, by means of waves, it was easy to devise experiments in which the wave nature of light finds expression not only in small details, but on a very large scale in the entire character of the phenomenon.

SFig 5

Allow me to illustrate this by two examples, first, the example of an optical instrument, such as telescope, microscope, etc. The object is to obtain a sharp image, i.e. it is desired that all rays issuing from a point should be reunited in a point, the so-called focus (cf. Fig. 5 a). It was at first believed that it was only geometrical-optical difficulties which prevented this: they are indeed considerable. Later it was found that even in the best designed instruments focussing of the rays was considerably inferior than would be expected if each ray exactly obeyed the Fermat principle independently of the neighbouring rays. The light which issues from a point and is received by the instrument is reunited behind the instrument not in a single point any more, but is distributed over a small circular area, a so-called diffraction disc, which, otherwise, is in most cases a circle only because the apertures and lens contours are generally circular. For, the cause of the phenomenon which we call diffraction is that not all the spherical waves issuing from the object point can be accommodated by the instrument. The lens edges and any apertures merely cut out a part of the wave surfaces (cf. Fig. 5b) and – if you will permit me to use a more suggestive expression – the injured margins resist rigid unification in a point and produce the somewhat blurred or vague image. The degree of blurring is closely associated with the wavelength of the light and is completely inevitable because of this deep-seated theoretical relationship. Hardly noticed at first, it governs and restricts the performance of the modern microscope which has mastered all other errors of reproduction. The images obtained of structures not much coarser or even still finer than the wavelengths of light are only remotely or not at all similar to the original.

A second, even simpler example is the shadow of an opaque object cast on a screen by a small point light source. In order to construct the shape of the shadow, each light ray must be traced and it must be established whether or not the opaque object prevents it from reaching the screen. The margin of the shadow is formed by those light rays which only just brush past the edge of the body. Experience has shown that the shadow margin is not absolutely sharp even with a point-shaped light source and a sharply defined shadow-casting object. The reason for this is the same as in the first example. The wave front is as it were bisected by the body (cf. Fig. 6) and the traces of this injury result in blurring of the margin of the shadow which would be incomprehensible if the individual light rays were independent entities advancing independently of one another without reference to their neighbours.

SFig 6

This phenomenon – which is also called diffraction – is not as a rule very noticeable with large bodies. But if the shadow-casting body is very small at least in one dimension, diffraction finds expression firstly in that no proper shadow is formed at all, and secondly – much more strikingly – in that the small body itself becomes as it were its own source of light and radiates light in all directions (preferentially to be sure, at small angles relative to the inci dent light). All of you are undoubtedly familiar with the so-called “motes of dust” in a light beam falling into a dark room. Fine blades of grass and spiders’ webs on the crest of a hill with the sun behind it, or the errant locks of hair of a man standing with the sun behind often light up mysteriously by diffracted light, and the visibility of smoke and mist is based on it. It comes not really from the body itself, but from its immediate surroundings, an area in which it causes considerable interference with the incident wave fronts. It is interesting, and important for what follows, to observe that the area of interference always and in every direction has at least the extent of one or a few wavelengths, no matter how small the disturbing particle may be. Once again, therefore, we observe a close relationship between the phenomenon of diffraction and wavelength. This is perhaps best illustrated by reference to another wave process, i.e. sound. Because of the much greater wavelength, which is of the order of centimetres and metres, shadow formation recedes in the case of sound, and diffraction plays a major, and practically important, part: we can easily hear a man calling from behind a high wall or around the corner of a solid house, even if we cannot see him.

Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light. What is gained by changing from the old view to the new is that the diffraction phenomena can be accommodated or, better expressed, what is gained is something that is strictly analogous to the diffraction phenomena of light and which on the whole must be very unimportant, otherwise the old view of mechanics would not have given full satisfaction so long. It is, however, easy to surmise that the neglected phenomenon may in some circumstances make itself very much felt, will entirely dominate the mechanical process, and will face the old system with insoluble riddles, if the entire mechanical system is comparable in extent with the wavelengths of the “waves of matter” which play the same part in mechanical processes as that played by the light waves in optical processes.

This is the reason why in these minute systems, the atoms, the old view was bound to fail, which though remaining intact as a close approximation for gross mechanical processes, but is no longer adequate for the delicate interplay in areas of the order of magnitude of one or a few wavelengths. It was astounding to observe the manner in which all those strange additional requirements developed spontaneously from the new undulatory view, whereas they had to be forced upon the old view to adapt them to the inner life of the atom and to provide some explanation of the observed facts.

Thus, the salient point of the whole matter is that the diameters of the atoms and the wavelength of the hypothetical material waves are of approximately the same order of magnitude. And now you are bound to ask whether it must be considered mere chance that in our continued analysis of the structure of matter we should come upon the order of magnitude of the wavelength at this of all points, or whether this is to some extent comprehensible. Further, you may ask, how we know that this is so, since the material waves are an entirely new requirement of this theory, unknown anywhere else. Or is it simply that this is an assumption which had to be made?

The agreement between the orders of magnitude is no mere chance, nor is any special assumption about it necessary; it follows automatically from the theory in the following remarkable manner. That the heavy nucleus of the atom is very much smaller than the atom and may therefore be considered as a point centre of attraction in the argument which follows may be considered as experimentally established by the experiments on the scattering of alpha rays done by Rutherford and Chadwick. Instead of the electrons we introduce hypothetical waves, whose wavelengths are left entirely open, because we know nothing about them yet. This leaves a letter, say a, indicating a still unknown figure, in our calculation. We are, however, used to this in such calculations and it does not prevent us from calculating that the nucleus of the atom must produce a kind of diffraction phenomenon in these waves, similarly as a minute dust particle does in light waves. Analogously, it follows that there is a close relationship between the extent of the area of interference with which the nucleus surrounds itself and the wavelength, and that the two are of the same order of magnitude. What this is, we have had to leave open; but the most important step now follows: we identify the area of interference, the diffraction halo, with the atom; we assert that the atom in reality is merely the diffraction phenomenon of an electron wave captured us it were by the nucleus of the atom. It is no longer a matter of chance that the size of the atom and the wavelength are of the same order of magnitude: it is a matter of course. We know the numerical value of neither, because we still have in our calculation the one unknown constant, which we called a. There are two possible ways of determining it, which provide a mutual check on one another. First, we can so select it that the manifestations of life of the atom, above all the spectrum lines emitted, come out correctly quantitatively; these can after all be measured very accurately. Secondly, we can select a in a manner such that the diffraction halo acquires the size required for the atom. These two determinations of a (of which the second is admittedly far more imprecise because “size of the atom” is no clearly defined term) are in complete agreement with one another. Thirdly, and lastly, we can remark that the constant remaining unknown, physically speaking, does not in fact have the dimension of a length, but of an action, i.e. energy x time. It is then an obvious step to substitute for it the numerical value of Planck’s universal quantum of action, which is accurately known from the laws of heat radiation. It will be seen that we return, with the full, now considerable accuracy, to the first (most accurate) determination.

Quantitatively speaking, the theory therefore manages with a minimum of new assumptions. It contains a single available constant, to which a numerical value familiar from the older quantum theory must be given, first to attribute to the diffraction halos the right size so that they can be reasonably identified with the atoms, and secondly, to evaluate quantitatively and correctly all the manifestations of life of the atom, the light radiated by it, the ionization energy, etc.

SFig 7

I have tried to place before you the fundamental idea of the wave theory of matter in the simplest possible form. I must admit now that in my desire not to tangle the ideas from the very beginning, I have painted the lily. Not as regards the high degree to which all sufficiently, carefully drawn conclusions are confirmed by experience, but with regard to the conceptual ease and simplicity with which the conclusions are reached. I am not speaking here of the mathematical difficulties, which always turn out to be trivial in the end, but of the conceptual difficulties. It is, of course, easy to say that we turn from the concept of a curved path to a system of wave surfaces normal to it. The wave surfaces, however, even if we consider only small parts of them (see Fig. 7) include at least a narrow bundle of possible curved paths, to all of which they stand in the same relationship. According to the old view, but not according to the new, one of them in each concrete individual case is distinguished from all the others which are “only possible”, as that “really travelled”. We are faced here with the full force of the logical opposition between an

either – or (point mechanics)

and a

both – and (wave mechanics)

This would not matter much, if the old system were to be dropped entirely and to be replaced by the new. Unfortunately, this is not the case. From the point of view of wave mechanics, the infinite array of possible point paths would be merely fictitious, none of them would have the prerogative over the others of being that really travelled in an individual case. I have, however, already mentioned that we have yet really observed such individual particle paths in some cases. The wave theory can represent this, either not at all or only very imperfectly. We find it confoundedly difficult to interpret the traces we see as nothing more than narrow bundles of equally possible paths between which the wave surfaces establish cross-connections. Yet, these cross-connections are necessary for an understanding of the diffraction and interference phenomena which can be demonstrated for the same particle with the same plausibility – and that on a large scale, not just as a consequence of the theoretical ideas about the interior of the atom, which we mentioned earlier. Conditions are admittedly such that we can always manage to make do in each concrete individual case without the two different aspects leading to different expectations as to the result of certain experiments. We cannot, however, manage to make do with such old, familiar, and seemingly indispensible terms as “real” or “only possible”; we are never in a position to say what really is or what really happens, but we can only say what will be observed in any concrete individual case. Will we have to be permanently satisfied with this. . . ? On principle, yes. On principle, there is nothing new in the postulate that in the end exact science should aim at nothing more than the description of what can really be observed. The question is only whether from now on we shall have to refrain from tying description to a clear hypothesis about the real nature of the world. There are many who wish to pronounce such abdication even today. But I believe that this means making things a little too easy for oneself.

I would define the present state of our knowledge as follows. The ray or the particle path corresponds to a longitudinal relationship of the propagation process (i.e. in the direction of propagation), the wave surface on the other hand to a transversal relationship (i.e. normal to it). Both relationships are without doubt real; one is proved by photographed particle paths, the other by interference experiments. To combine both in a uniform system has proved impossible so far. Only in extreme cases does either the transversal, shell-shaped or the radial, longitudinal relationship predominate to such an extent that we think we can make do with the wave theory alone or with the particle theory alone.

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