§ 11. TEMPORAL IDEAS.



1. All our ideas are at once spatial and temporal. But just as the conditions for the spatial arrangement of impressions belong originally to the tactual and visual senses, and just as spatial relations are only secondarily carried over from these to all other sensations, so there are only two kinds of sensations, namely, the inner tactual sensations from movements and the auditory sensations which are primary sources of temporal ideas. Still, there is a characteristic difference between spatial and temporal ideas in the fact that in the case of spatial ideas the two senses mentioned are the only ones which can develop an independent spatial order, while in the case of temporal ideas the two most important kinds of sensation are merely those in which the conditions are most favorable for the rise of temporal ideas. These conditions are not entirely wanting in any sensations. This indicates that the psychological basis of temporal ideas is more general, and that it is not determined by the special structures of particular sense organs. In agreement with this view is the fact that we attribute to subjective processes, such as feelings and emotions, the same temporal attributes as we attribute to ideas. It is to be noted, however, that no justification for the conclusion that time perception is in itself a more universal form of perception, is to be found in this fact that the conditions of time perception are more general than are those of space perception. In the same way that we carry over spatial attributes from the two senses which give us space perception to other kinds of sensations, so also we give spatial attributes secondarily to feelings and affective processes, through the sensations and ideas inseparably connected with them. It may with equal right be doubted whether affective processes in themselves, without their related ideas, would have temporal attributes, for among the conditions of a temporal order are certain attributes of the sensation elements of ideas. The real facts in the case are that all psychical contents, are at once spatial and temporal. The spatial order arises from certain particular sensation elements; in normal cases where vision is present it rises chiefly from visual impressions, in blindness, from tactual impressions. Time ideas, on the other hand, can arise from all possible sensations.

2. Temporal compounds like spatial, and in contrast to intensive ideas, are characterized by the definite, unchangeable order of their component elements. If this order is changed, the given compound becomes another, even though the quality of its components remains the same. In spatial ideas, this unchangeableness of the order refers only to the relation of the elements to one another, not to the relation of the elements to the ideating subject. In temporal compounds, on the other hand, when the relation of one element is changed with respect to other elements, it is at the same time changed with respect to the ideating subject. There is no change of position in time analogous to that possible in the case of space compounds.

2a. This property of the absolute, unchangeable relation with respect to the ideating subject which belongs to every temporal compound, and every time element however short, is what we call the flow of time. Every moment in time filled by any content whatever, has, on account of this flow, such a relation to the ideating subject that no other moment can be substituted for it. With space the case is reversed; the very possibility of substituting any spatial element in its relation to the subject for any other element whatever, is what gives rise to the percept of constancy, or absolute duration as we express it by applying a time idea to a space idea. The idea of absolute duration, that is, of time in which no change takes place, is strictly speaking impossible in time perception itself. The relation of impressions to the subject changes continually. We speak of an impression as lasting, when its single periods in time are exactly alike so far as their sensation contents and affective contents are concerned, and differ only in the relation of these contents to the subject. The concept of duration when applied to time is, therefore, a merely relative concept. One time idea may be more lasting than another, but no time idea can have absolute duration. Even an unusually long unchanging sensation can not be retained. We interrupt it continually with other sensation and affective contents.

We may, however, separate the two temporal relations always united in actual experience, namely, that of the elements to one another, and that of the elements to the ideating subject, since each relation is connected with certain particular attributes of time ideas. In fact, this separation of the two relations found its expression in special terms for certain forms of occurrence in time, even prior to an exact psychological analysis of time ideas. If the relation of the elements to one another is alone attended to, without regard to their relation to the subject, temporal modes come to be discriminated, such, for example, as brief, long, regularly repeating, irregularly changing, etc. If, on the other hand, the relation of the subject is attended to, and the objective forms of occurrence neglected, we have as the chief forms of this relation the temporal stages, past, present, and future.



A. TEMPORAL TOUCH IDEAS.

3. The original development of temporal ideas belongs to touch. Tactual sensations, accordingly, furnish the general substratum for the rise of both the spatial and temporal arrangements of ideational elements (p. 114, 3). The spatial functions of touch, however, come from the outer tactual sensations, while the inner touch sensations which accompany movements are the primary contents of the earliest temporal ideas.

The mechanical properties of the limbs are important physiological bases for the rise of these ideas. The arms and legs can be moved in the shoulder-joints and hip-joints by their muscles, and are at the same time subject to the action of gravitation drawing them downward. As a result there are two kinds of movements possible for these extremities. First, we have movements which are continually regulated by voluntary activity of the muscles and may, therefore, be indefinitely varied and accommodated at every moment to existing needs — we will call these the arrhythmical movements. Second, we have movements in which the voluntary energy of the muscles is operative only so far as it is required to set the limbs oscillating in their joints and to maintain this movement — rhythmical movements. We may neglect for our present consideration the arrhythmical movements exhibited in the various uses of the limbs, for there are always rhythmical movements combined with the arrhythmical, and the temporal attributes of the arrhythmical components are in all probability determined throughout by the dominating influence of the rhythmical components.

4. The significance of rhythmical tactual movements for the psychological development of time ideas is due to the same principle as that which gives them their importance as physiological processes, namely, the principle of the isochronism of oscillations of like amplitude. In walking, the fairly regular oscillations of our legs in the hip-joints not only reduce the amount of the muscular energy expended, but also reduce to a minimum the continual voluntary control of the movements. Furthermore, in natural walking the arms are supplementary aids. Their oscillation is not interrupted at every step as is that of the legs by the placing of the foot on the ground, so that they furnish, because of the continuity of their movement, a means for the more uniform regulation of the whole action.

Every single period of oscillation in such a movement is made up of a continuous succession of sensations which are repeated in the following period in exactly the same order. The two limits of the period are marked by a complex of outer tactual sensations, the beginning by the impression accompanying the removal of the foot from the ground, the end by the impression accompanying the return of the foot to the ground. Between these there is a continuous series of weak inner tactual sensations from the joints and muscles. The beginning and end of this series of inner sensations coincide in time with the appearance of outer sensations, and are more intense than the intermediate internal sensations. These more intense internal sensations accompany the impulse of movement coming to the muscles and joints and the sudden inhibition of these impulses, and they assist much in marking off the successive periods.

Connected with this regular succession of sensations is a regular and exactly parallel series of feelings. If we consider a single period in a series of rhythmical movements, there is always at its beginning and end a feeling of fulfilled expectation. Between the two limits of the period there is, beginning with the first movement, a gradually growing feeling of strained expectation, which suddenly sinks at the last moment from its maximum to zero, and gives place to the rapidly rising and sinking feeling of fulfillment. From this point on the same series is again repeated. Thus, the whole process of a rhythmical touch movement consists, on its affective side, of a succession of two qualitatively antagonistic feelings. In their general character these feelings belong to the series of straining and relaxing feelings (p. 91). One of these feelings is very rapid in its course, the other gradually reaches a maximum and then suddenly disappears. As a result, the most intense affective processes are crowded together at the extremities of the periods, and are made all the more intense through the contrast between the feeling of satisfaction and the preceding feeling of expectation. Just as this sharply marked limit between the different periods has its sensation substratum in the strong outer and inner tactual impressions arising at this instant, as above pointed out, so there is also a complete series of feelings of expectation corresponding to the continuous series of weaker inner tactual sensations accompanying the oscillatory movements of the limbs.

5. The simplest temporal ideas of touch are made up of the rhythmically arranged sensations which follow one another with perfect uniformity in the manner described, whenever we make a series of walking movements. But even in ordinary walking a slight tendency towards a somewhat greater complication arises. The beginning of the first of two successive periods is emphasized, both in the sensation and in the accompanying feeling, more than the beginning of the second period. In this case the rhythm of movement begins to be metrical. A simple regular succession of accented and unaccented ideas corresponds to the simplest measure, 2/8-time. It arises easily in ordinary walking because of the physiological superiority of the right side, and appears very regularly when several persons are walking together in marching. In the latter case even more than two periods may be united into one rhythmical unit. The same is true of the complicated rhythmical movements of the dance. But in such composite tactual rhythms the auditory temporal ideas have a decided influence.

B. TEMPORAL AUDITORY IDEAS.

6. The attribute of the auditory sense which most of all adapts it to the more accurate perception of the temporal relations in external processes, is the exceedingly short persistence of its sensations after the cessation of the external stimulus, as a result of which any temporal succession of sounds is reproduced with almost perfect fidelity in the corresponding succession of sensations. Connected with this fact are certain psychological properties of temporal auditory ideas. In the first place, temporal auditory ideas differ from temporal ideas of touch in that often only the extremities of the single intervals which go to make up the total idea, are marked by sensations. In such a case the relations of such intervals to one another are estimated by means of the apparently empty or heterogeneously filled intervals which lie between the limiting sensations.

This is especially noticeable in the case of rhythmical auditory ideas. There are in general two possible forms of such ideas; continuous, or only rarely interrupted successions of relatively lasting sensations, and discontinuous successions of strokes, in which only the extremities of the rhythmical periods are marked by external sounds. For a discontinuous succession of entirely uniform sounds the temporal attributes of the ideas are in general more apparent than for lasting impressions, since in the former case any effects from the tonal qualities as such are entirely obviated. We may confine our consideration to discontinuous series, because the principles which apply here hold for continuous successions also. In fact, the rhythmical division in the latter case, is made by means of certain single accents which are either given in the external impression or voluntarily applied to it.

7. A series of regular strokes made in this way as the simplest form of temporal auditory ideas, as for example a series of ticks of a clock or of a metronome, is distinguished from the simplest form of temporal touch ideas, described above (p. 162), mainly by the absence of all objective sensation content in the intervals. The external impressions here do nothing but divide the separate intervals from one another. Still, the intervals of such a series are not entirely empty, they are filled by subjective affective and sensational contents which correspond fully to those observed in tactual ideas. Most emphatic of all are the affective contents of the intervals which consist of successive periods of expectation. This expectation gradually rises in each period and is at the end of such a period suddenly fulfilled. Even the sensation substratum for this feeling is not entirely absent; it is merely more variable. Sometimes it is nothing but the sensations of tension of the tympanic membrane, sometimes it includes also accompanying sensations from other parts of the body. Finally, in those cases in which an involuntary rhythmical movement is connected with the auditory series, it is a series of some other kind of inner tactual sensations. The ordinary course of sensations and feelings which are aroused by a regular succession of taps, can be represented in some such curve as that shown in Fig. 20, where the dotted line shows the succession of sensations which come with the taps. 1, 2 and 3 correspond to the points of greatest intensity of these taps. The full-drawn line represents the feelings. The parts of the line lying above the horizontal line represent feelings of tension and expectation. The parts of the curve below the base line correspond to the feelings of relief which come with each tap.

The influence of the subjective elements on the character of time ideas shows itself most clearly in the case of the rhythmical auditory impressions in the effect produced by different rates of succession of the sensations. A certain medium rate of about 0.2 sec. is found to be most favorable for the union of a number of successive auditory impressions, and it is easy to observe that this is the rate at which the above mentioned subjective sensations and feelings are most pronounced in their alternation. If the rate is made much slower, the strain of expectation is too great and passes into an unpleasurable feeling which becomes more and more unendurable. If, on the contrary, the rate is accelerated, the rise of the feeling of expectation is interrupted so soon that the feeling is barely noticeable. Thus, in both directions, limits are approached at which the synthesis of the impressions into a rhythmical time idea is no longer possible. The upper limit is about one second, the lower about 0.1 sec.

8. Then again, this influence of the course of our sensations and feelings upon our perception of temporal intervals, shows itself just as clearly in the changes which our ideas of such an interval undergo when the conditions of perception are varied without changing the objective length of the interval. Thus, it has been observed that in general a period divided into intervals is estimated as longer than one not so divided. We have here a phenomenon analogous to that observed in the illusion with interrupted lines (p. 139). The overestimation may be very much greater for temporal intervals than for spatial intervals. This is obviously due to the fact that the repeated alternations of sensations and feelings in an interval of time have a greater influence on perception than does the interruption of the movement through points of division in the case of the similar space-illusion. On the other hand, a limit can be reached in the division of time intervals, provided long enough periods are taken, at which the illusion disappears or is reversed. In the latter case the divided interval may appear shorter than the undivided interval. This reversed illusion is obviously due to the strong feeling of tension which increases during the empty interval up to the moment of the arrival of the final impression. Furthermore, if in a series of regular beats, single strokes are emphasized by their greater intensity or by some qualitative peculiarity, the result is that the interval preceding and usually also the interval following the emphasized stroke are overestimated in comparison with the other intervals of the same series. Finally, if a certain rhythm is produced successively with weak and strong beats, the rate appears slower in the first case than in the second. These phenomena are also explicable from the influence of the sensation and affective changes. An impression different from the rest, produces a change in the course of the sensations, and especially in the course of the feelings which precede its apprehension, for there must be a more intense strain of expectation and a correspondingly stronger feeling of relief or satisfaction. The feeling of expectation lengthens the interval preceding the impression, the feeling of relief lengthens the interval following. The case is different when the whole series is made up at one time of weak impressions, and at another of strong ones. In order to perceive a weak impression we must concentrate our attention upon it more. The sensations and feelings of tension are, accordingly, more intense, as may be easily observed, for weaker beats than for stronger ones. Here too, the different intensities of the subjective elements which give rise to the temporal ideas are reflected in the differences between these ideas. The effect is, therefore, not only lost, but even reversed, when we compare, not weak beats with strong, but strong beats with still stronger beats.

9. The tendency found in the case of rhythmical touch impressions for at least two like periods to unite and form a regular metrical unit, shows itself in auditory ideas also, only in a much more marked degree. In tactual movements, where the sensations which limit the single periods are under the influence of the will, this tendency to form a rhythmical series shows itself in the actual alternation of weaker and stronger impressions. With auditory sensations, on the other hand, where the impressions are produced altogether by external sounds, this tendency may lead to the following characteristic illusion. In a series of beats which are exactly alike in intensity and are separated by equal periods of time, certain single beats, occurring at regular intervals, are always heard as stronger than the others. The rhythm that most frequently arises when there is nothing to determine it, is that known as 2/8-time. This tendency to mark time can be overcome only by an effort of the will, and then usually only for very fast or very slow rates, where, from the very nature of the series, the limits of rhythmical perception are nearly reached. If the effort is made to unite as many impressions as possible in a unitary time idea, the phenomena become more complicated. We have accents of different degrees which alternate in regular succession with unaccented members of the series and thus, through the resulting divisions of the whole into groups, the number of impressions which may be comprehended in a single idea is considerably increased. The presence of two different grades of accent gives 3/4-time and 5/8-time, the presence of three grades gives 4/4-times and 6/4-time, and as forms with three feet there are 9/8-time and 12/8-time. More than three grades of accentuation or, when the unaccented note is counted, more than four grades of intensity, are not to be found in either musical or poetical rhythms, nor can we produce more by voluntary formation of rhythmical ideas. Obviously, these three grades of accentuation mark the limits of the possible complexity of temporal ideas, in a way analogous to that in which the maximal number of included impressions (§ 15, 6) marks the limits of the length of temporal ideas.

The phenomena of subjective accentuation and the influence of this accentuation on the sensation which go to make up rhythms, show clearly that temporal ideas, like spatial ideas, are not derived from objective impressions alone, but that there are always connected with these objective factors subjective elements which help by their character to determine the mode of apprehending the objective stimulus. The primary cause of the accentuation of a particular impression is always to be found in the increase in the intensity of the preceding and concomitant feelings and inner tactual sensations of movements. This increase in the intensity of the subjective elements is then carried over to the objective impression, and makes the latter also seem more intense. The strengthening of the subjective elements may be voluntary, when the tension of the muscles which produce inner tactual sensations is voluntarily intensified, thus producing a corresponding intensification in the feeling of expectation. Or the strengthening of the subjective elements may be involuntary, as when a strong impulse toward a grouping of the elements of the temporal idea is brought about as an immediate consequence of the fluctuations in sensation and feeling which take place during the effort to include as many factors as possible in the percept.

C. GENERAL CONDITIONS FOR TEMPORAL IDEAS.

10. If we seek to account for the rise of temporal ideas on the basis of the phenomena just discussed, we must start with the fact that a sensation thought of by itself can no more have temporal than it could have spatial attributes. Position in time can be developed only when single psychical elements enter into certain characteristic relations with other such elements. This condition holds for temporal ideas just as much as for spatial ideas. The nature of the union is, however, characteristic and essentially different for the two kinds of ideas.

The members of a temporal series a b c d e f, can all be immediately presented as a single whole, when the series has reached f, just as well as if they were a series of points in space. In the case of a spatial idea, however, the elements would, on account of original ocular reflexes, be arranged in relation to the point of fixation, and this fixation point could, at different times, be any one of the impressions a to f. In time ideas, on the other hand, it is always the impression of the present moment in relation to which all the rest are arranged in time. When a new impression becomes, in a similar manner, the present impression, even though its sensation contents are exactly the same as that of the earlier idea, still, it will be perceived as subjectively different, for though the affective state accompanying a sensation may, indeed, be related to the feelings of another moment, the two can never be identical. Suppose, for example, that following the series a b c d, e f, there is a second series of impressions, a' b' c' d' e' f', in which a' = a, b' = b, c' = c, etc., so far as the sensation elements are concerned. Let us represent the accompanying feelings by a b g d e j and a' b ' g ' d ' e ' j '. Then a and a', band b ', g and g ', etc., will be similar feelings, because the sensations are the same; but they will not be identical, because every affective element depends, not only upon the sensation with which it is immediately connected, but also upon the state of the subject as determined by the totality of its experiences. The state of the subject is different for each of the members of the series a' b' c' d' . . ., than it was for the corresponding member of the series a b c d..., because when the impression a' arrives, a has already been present, and so a' can be associated with a, while no such thing was possible in the case of a. Analogous differences in the affective states show themselves in composite series when repeated. The successive states in such series are never identical, however much the subjective conditions of the momentarily present feelings may agree, for every one of them has its characteristic relation to the totality of psychical processes. If we assume, for example, that a number of beats follow each other, as represented in Fig. 20 (p. 166), with the sensations 1, 2 and 3 all alike, there will appear a difference in the feeling conditions connected with 3 as contrasted with 2, for 2 can be associated only with 1, whereas 3 can be associated with both 2 and I. Similar statements can be made of each point in the feeling curve and sensation curve which lies between two beats. For example, a" is related to corresponding points of the same order which have preceded it. This relation is strong for the point which immediately precedes it, such as a', and weaker for the corresponding points further back, such as a. Each point thus has its specific position in the succession of feelings.

11. Since every element of a temporal idea is, as above remarked, placed in some fixed relation to the impression immediately present, it follows that this present impression will have an attribute which makes it more prominent than any of the other elements of the same idea. This attribute is similar to that possessed by the point of fixation in the field of vision, or by the central points of the tactual surfaces, and consists in the fact that the present impression is the most clearly and distinctly perceived of all the elements of the idea. But there is a great difference in that this most distinct perception in the temporal idea is not connected with the physiological organization of the sense-organ, but is due entirely to the general attributes of the ideating subject, as expressed in the affective processes. The momentary feeling accompanying the immediately present impression is what helps to make it the impression most clearly perceived. We may, accordingly, call the part of a temporal idea which forms the immediate impression the fixation-point of the idea, or in general, since it does not depend on external structure, as does the fixation-point of spatial ideas, we may call it figuratively the inner fixation-point. The impressions which lie outside this point of fixation, that is, impressions that have preceded the present, are indirectly perceived. They are arranged in a regular gradation of diminishing degrees of clearness, from the fixation-point. A unitary temporal idea is possible only so long as the degree of clearness of each of its elements has some positive value. When the clearness of any element sinks to zero, the idea divides into its components.

12. The inner fixation-point of temporal percepts differs essentially from the outer fixation-points of spatial percepts in that its character is primarily determined, not by sensations, but by affective elements. Since these affective elements are continually changing, in consequence of the varying conditions of psychical life, the inner fixation-point is also always changing. This change of the inner fixation-point is called the continuous flow of time. By the phrase continuous flow we mean to express the fact that no moment of time is like any other, and that no such moment can return (cf. sup. p. 161, 2a). This fact is connected with the one-dimensional character of time, which is due to this very condition that the inner fixation-point of temporal ideas is continually moving forward, so that a single point can never recur. The arrangement of time in one dimension, with reference always to a changing point of fixation in which the subject represents himself, is what gives rise to the result that the elements of time ideas have a fixed relation, not only with respect to one another, but also with respect to the ideating subject (p. 160, 2).

13. If we try to give an account of the means through which this reciprocally interdependent order of the parts of an idea, and the determination of these parts with reference to the ideating subject, originate, it is obvious that these means can be nothing but certain of the elements connected with the idea itself. These elements, however, considered in themselves, have no temporal attributes, they gain such attributes through their union. We may call these elements temporal signs, after the analogy of local signs. The characteristic conditions for the development of temporal ideas indicate from the first that these temporal signs are, in the main, affective elements. In the course of any rhythmical series every impression is immediately characterized by the concomitant feeling of expectation, while the sensation is of influence only in so far as it arouses the feeling. This may be clearly perceived when a rhythmical series is suddenly interrupted. Furthermore, the only sensations which are never absent as components of all time ideas are the inner tactual sensations. In the case of tactual time ideas these inner tactual sensations fuse immediately with the tactual sensations which arise from the movements of the part of the body in action, while in auditory and other ideas which are brought into the time form, they stand out distinctly from the other outerimpressions as subjective accompanying phenomena. We may, accordingly, regard the feelings of expectation as the qualitative temporal signs, the inner tactual sensations described, as the intensive, temporal signs of a temporal idea. The idea itself must then be looked upon as a fusion of the two kinds of temporal signs with each other and with the objective sensations arranged in the temporal form. Thus, the inner tactual sensations, as a series of intensive sensations, give a uniform measure for the arrangement of the objective sensations; the accompanying feelings, on the other hand, furnish the qualitative characteristics of these impressions which are necessary for the temporal ideas.

13a. The inner tactual sensations play a similar part in the formation of both time ideas and space ideas. This common sensational substratum leads very naturally to a recognition of a relation between these two forms of perception, which finds its expression in the geometrical representation of time by a straight line. Still, there is an essential difference between the complex system of temporal signs and the systems of local signs in the fact that the former is based primarily, not on the qualitative attributes of sensations connected with certain special external sense-organs, but on feelings which may come in exactly the same way from the most widely differing kinds of sensation, for these feelings are not dependent on the objective content of the sensations, but on their subjective synthesis. The marked variations in the conditions which control the course of these feelings explain, furthermore, why it is that our time ideas are much less certain than our space ideas. The influence of the course of the feelings in any given case shows itself in the fact that the degree of certainty of any subjective estimation of a time interval depends primarily on the duration of the interval. Our comparison of temporal qualities, as, for example, in the case of successive rhythmical periods, is most accurate, other things being equal, for those intervals which are most favorable in point of length for rhythmical division. This favorable interval is, in the case of auditory sensations about 0.2 seconds (7). It may be observed when such an interval is given that the exactness of perception is conditioned by the favorable succession of feelings of expectation and fulfillment. Such a favorable succession makes it possible to recognize with greatest certainty when a new impression interrupts the feeling of expectation before it has risen to the same intensity as in a preceding case, or when, on the other hand, the new impression has, by its delay, allowed the feeling to reach a higher degree of intensity. When the succession of impressions is very slow the feelings of expectation become excessively intense. When, on the other hand, the succession is very rapid, it is almost possible to notice a feeling of surprise accompanying every impression. Even this feeling of surprise, however, can reach only a moderate intensity because of the relatively small degree of intensity attained by the preceding feelings of tension. For the facts of time memory compare § 16.

13b. Here again we have on the question of the psychological origin of time ideas the same opposed nativistic and genetic theories which we had in the case of spatial ideas (p. 124, 12a). In this case, however, nativism has never developed a theory in any proper sense. It usually either limits itself to the general assumption that time is a "connate form of perception", or else it falls back on a so-called sense of time which is arbitrarily identified with some one of the senses, as for example, with the kinesthetic sense of the muscles. In no case does nativism attempt to give any account of the influence of the elements and conditions of temporal ideas which can be actually demonstrated. The genetic theories of older psychology, as, for example, that of herbart, seek to deduce time perception from ideational elements only. This is, however, pure speculation and loses sight of the conditions given in actual experience.

References. Vierordt, Der Zeitsinn, 1868. mach, (English trans.) Analysis of Sensations. This is an attempt to develop a nativistic theory. Meumann, Phil. Stud., vols. 8 and 9. Schumann, Zeitschr. f. Psych., vols. 4, 14, 18. Nichols, Amer. Journal of Psychol., vol. 4. On Rhythm: Meumann, Phil. Stud., vol. 10. Bolton, Amer. Journal of Psychol., vol. 6. Bücher, Arbeit und Rhythmus, 3rd. ed., 1902. Smith, Phil. Stud., vol. 16. Wundt, Grundz, 5th ed., vol. Ill, Chap. 15; Lectures, lectures 17 and 18.