Interval

“ Rather, which of all the intervals, infinite in number as they are, shall time be? If time be considered the interval of the regular movement, it will not be the particular interval of every regular movement; otherwise, as there are several regular movements, there would be several kinds of time. If time be defined as the interval of movement of the universe, that is, the interval contained within this movement, it will be nothing else than this movement itself. ”

“ The difference of interval between lives means merely that they take their bliss in a much more concentrated form, and therefore work out the result of an equal expenditure of force in much less time. ”

“ For we measure the time interval itself from some beginning point to some end. This is why a voice that has not yet ended cannot be measured, so that one could say how long or how briefly it will continue. ”

“ We observe the tract between a sunrise and its return and, as the movement is uniform, we thus obtain a Time-interval upon which to measure ourselves, and we use this as a standard. We have thus a measure of Time. Time itself is not a measure. How would it set to work? And what kind of thing is there of which it could say, "I find the extent of this equal to such and such a stretch of my own extent?" ”

“ The Lat. intervallum, from which the English word has come through the French, originally meant a space between the palisades on a rampart (vallum) , or between the rampart and the tents of the legionaries. In medical language “interval” is used of the intervening periods between attacks or paroxysms of a disease, particularly of the periods of a rational or normal condition of mind sometimes experienced by an insane person, a “lucid interval”; this phrase frequently occurs in legal documents from the 13th to the 15th centuries, non compos mentis sed gaudet lucidis intervallis. ”

“ What shall I do with all the days and hours
That must be counted ere I see thy face?
How shall I charm the interval that lowers
Between this time and that sweet time of grace? ”

“ All that is requisite is that we should be immediately conscious through an infinitesimal interval of time. In this infinitesimal interval we directly perceive the temporal sequence of beginning, middle, and end. ”

“ They will not arrive at the same estimate for the distance in space or the lapse of time between the two events, but the interval is a genuine physical fact, the same for all. If a body can travel freely from one event to the other, the interval between the two events will be the same as the time between them as measured by a clock travelling with the body. If such a journey is physically impossible, the interval will be the same as the distance as estimated by an observer to whom the two events are simultaneous. ”

“ As we measure intervals as straight lines, it costs more muscular effort to measure from right to left than the other way, and an interval lying to the left seems to us consequently longer than it really is. Now I have been a reader for more years than Herr Münsterberg ”

“ We commonly think that there are two separate kinds of interval between two events, an interval in space and an interval in time: between your [Pg 53] departure from London and your arrival in Edinburgh, there are 400 miles and ten hours. We have already seen that another observer will judge the time differently; it is even more obvious that he will judge the distance differently. An observer in the sun will think the motion of the train quite trivial, and will judge that you have traveled the distance traveled by the earth in its orbit and its diurnal rotation. ”

“ It is only [Pg 610] as parts of this duration-block that the relation of succession of one end to the other is perceived. We do not first feel one end and then feel the other after it, and from the perception of the succession infer an interval of time between, but we seem to feel the interval of time as a whole, with its two ends embedded in it. The experience is from the outset a synthetic datum, not a simple one; and to sensible perception its elements are inseparable, although attention looking back may easily decompose the experience, and distinguish its beginning from its end. ”

“ The first measure of exercise 2 contains the major second which is an easily recognized interval. The last measure but two of exercise 2 contains the major seventh. This interval is ordinarily a difficult interval to intonate but coming as it does in exercise 2 it is easy to intonate because of the ascending scale on the second half of the measures. For the present, it is not advisable to practice the major seventh except in some such sequence as exercise 2. All other normal intervals may be practiced separately and in combinations. ”

“ There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval.
5. If the interval is unlimited towards the right (or towards the left) , the function has a limit at ∞ (or at −∞) .
10. Discontinuity of Functions.—The discontinuities of a function of one variable, defined in an interval with the possible exception of isolated points, may be classified as follows:
(1) The function may become infinite, or tend to become infinite, at a point. ”

“ However different the distance and the interval may appear to observers in movement or to observers in different systems of movement in relation to ourselves and to one another, in themselves they are the same distance and the same interval for all. ”

“ When medical men are called upon to attend a commission of lunacy, they are always asked, whether the patient has had a lucid interval? A term of such latitude as interval requires to be explained in the most perspicuous and accurate manner. ”

“ That which has weighed most with us, however, is, that if there were any considerable interval of time between the reality and the appearance, the images would often be interrupted and confused by clouds formed in the meantime, and similar disturbances of the medium. Let this suffice with regard to the simple measures of time. ”

“ Resultant relative duration—that is, an interval of flowing duration—admits of the same division into real and imaginary. It is real when a real continuous flowing connects the before with the after; in all other suppositions it will be imaginary. ”

“ The longer an interval the less we feel able to estimate it accurately, and, as a consequence, the more frequently it is judged as the same as the time-unit or a simple fraction of that unit. The females are less exact in their estimates than the males, and less exact for long than for short intervals, and as an accompaniment of their inexactitude we find the frequent occurrence of multiples of 15, 30, and 60. ”

“ The width of the interval depends, as is natural, upon the intensity of the thought or emotion of which the concrete is intended to be an expression. For example, suppose the statement, "You are the culprit," be answered by the surprised and indignant interrogation, "I?" The emphatic words here used may be appropriately uttered with intervals of a tone, a third, a fifth, or an octave, according to the emphasis supposed necessary. ”

“ If two successive actions or sets of circumstances to which the attention of the spectator is specially directed follow one another by a very narrow interval of time, they easily appear continuous, so that there seems absolutely no time for the introduction of an intermediate step. ”

“ In a second class of cases, time will enter as a principal ground of similarity. When we hear a clock pendulum beat time after time, at equal intervals, and with a uniform sound, we confidently expect that the stroke will continue to be repeated uniformly. A comet having appeared several times at nearly equal intervals, we infer that it will probably appear again at the end of another like interval. ”

“ However often such real points of position are interpolated, by real points of matter being interposed, their number will always be finite, the number of intervals intercepted on the first interval, & at the same time constituting that interval, will be finite ”

“ The interval between one moment and the next would have to be infinitesimal, since, if we take two moments with a finite interval between them, there are always other moments in the interval. Thus if there are to be no infinitesimals, no two moments are quite consecutive, but there are always other moments between any two. ”

“ From five o’clock to eight is on certain occasions a little eternity; but on such an occasion as this the interval could be only an eternity of pleasure. ”

“ We may measure duration by the succession of thoughts in the mind, as we measure length by inches or feet, but the notion or idea of duration must be antecedent to the mensuration of it, as the notion of length is antecedent to its being measured....Reason, from the contemplation of finite extended things, leads us necessarily to the belief of an immensity that contains them. In like manner, memory gives us the conception and belief of finite intervals of duration. ”

“ I am now saying this sentence. means that for an interval starting a short time in the past and extending to a short time in the future, I am expressing the utterance which is Example 5.4. Of course, “short” is relative, as always in tenses. Even a long sentence takes up only a short part of a whole day; in a geological context, the era of Homo sapiens would only be a “ze'i” interval. ”

“ A continuous line, or a continuous time is understood to be generated, not by repetition of points or instants, but by a continuous progressive motion, in which some intervals are parts of other intervals; the points themselves, or the instants, which are continually progressing, are not parts of the intervals. ”

“ But for us, conscious beings, it is the units that matter, for we do not count extremities of intervals, we feel and live the intervals themselves. Now, we are conscious of these intervals as of definite intervals. ”

“ Time may be considered either as a relation or as a quantity. In fact, intervals of successive duration are, like distances, real relations; but when we think of the greater or less extent of space which can be measured with a given velocity between two correlated terms of time, these same intervals exhibit themselves under the form of continuous quantities. ”

“ We have, however, seen reasons, which, applied to time, would lead us to declare that an absolute insight would view the temporal order as a discrete series of facts ordered as any succession of facts expressing one purpose would be ordered, viz. like the whole numbers. On the other hand, we have no reason to suppose that our human consciousness distinctly observes intervals of time that in brevity anywhere nearly approach to the final truth about the temporal order. Within what is for us the least observable happening, a larger insight may indeed discriminate multitudes of events. ”

“ The smaller intervals are congenial with indifference, monotony, doubt, melancholy, and sadness; the group of moderate intervals affirms occupation, pleasure, and desire, which grow more ardent as we approach the extreme intervals ”

“ When a singer is accompanying himself on the piano, keeping good time in spite of the fact that the notes are uneven in length, and meanwhile using his feet on the pedals, what has he got left to beat time with? No one has located the stimulus to which accurate time perception responds, though, in a general way, we are pretty sure that change of one sort or another is the datum. With longer intervals, from a minute to several hours, the sign of duration is probably the amount happening in the interval, or else such progressive bodily changes as hunger and fatigue. ”

“ I refer to them in passing only because they are eminently suited to show the very relative and precarious character of our concrete notions of duration.
Through all divergencies, the formula enunciated by Vierordt, the principal initiator of these researches, [163] remains stable; our consciousness of time comes, not from a sensation, but from a judgment, and in our retrospective appreciation of duration, we diminish intervals that are long, and increase those that are short. ”

“ The odd thing about these figures is the recurrence they show in so many men of about three fourths of a second, [Pg 618] as the interval of time most easy to catch and reproduce, Odder still, both Estel and Mehner found that multiples of this time were more accurately reproduced than the time-intervals of intermediary length; [537] and Glass found a certain periodicity, with the constant increment of 1.25 sec., in his observations. ”

“ For, since we measure all duration by time or by successive intervals, we can scarcely conceive that there may be duration without succession. Even the word “permanence” which we employ in the definition of duration, and which seems to exclude all notion of change, is always associated in our thought with succession and time. The difficulty we experience in forming a concept of standing duration is as great at least as that which we find in conceiving absolute space without formal extension and parts. ”

“ But a moment of time, we repeat, cannot persist in order to be added to others. If the sounds are separated, they must leave empty intervals bet ween them. If we count them, the intervals must remain though the sounds disappear: how could these intervals remain, if they were pure duration and not space? It is in space, therefore, that the operation takes place. It becomes, indeed, more and more difficult as we penetrate further into the depths of consciousness. Here we find ourselves confronted by a confused multiplicity of sensations and feelings which analysis alone can distinguish. ”

“ It has been shown, for example, that a person looking at a small light which is flashed on and off at intervals has a tendency to make rhythmic groupings of those flashes, by overestimating or underestimating the lengths of the intervals. ”

“ for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal. ”