Pentatonic scale

“ Even the finest scale, sensitive to a millionth of a gram, remains stationary when only a tenth of a millionth is placed upon it, although we can see a body of such minute weight under the microscope. In the same way it requires a definite expenditure of energy in order to bring the sense organs, or the central organ, into action, and all stimuli less than this limit or threshold produce no experience of their presence. ”

“ The ratio of any element of area in the volume-pressure or the entropy-temperature diagram, or in any other in which the scale of work and heat is unity, to the corresponding element in the volume-entropy diagram is represented by or The cases in which this ratio is 0, or changes its sign, demand especial attention, as in such cases the diagrams of constant scale fail to give a satisfactory representation of the properties of the body, while no difficulty or inconvenience arises in the use of the volume-entropy diagram. ”

“ In the best Latin poets, the first foot of each part of the pentameter is a dactyl. The pentameter scarcely exists except in conjunction with the hexameter, to which it always succeeds in elegaic verse. ”

“ A favoured economist like Von Thünen—with whose aims Petty's thought exhibits much affinity though he lacks Von Thünen's conspicuous patience—may make similar experiments upon a small scale. Most of us, however, must get on as best we may without any economic laboratory whatever. In this respect Petty was no exception. Experiment being impossible, he substituted what he called Political Arithmetick, a beginning of what is now called statistics. ”

“ Some theorists maintain that our diatonic major scale is alone a true scale, and that any other regular succession of tones in which the two semitones of the diatonic scale occur upon other intervals than 3-4 and 7-8 is, properly speaking, a Mode. According to this doctrine, which was evidently suggested by the ecclesiastical Modes, our minor scale must be called a Mode, and the scales with steps exceeding a whole-tone, of which some examples will presently be given, are Imperfect Modes. It is unnecessary to refute such pedantic definitions; suffice it to remember that they exist. ”

“ The body of the Pentacrinite, between the summit of the column and the base of the arms, is small, and composed of the pelvis, and the costal, and scapular plates, (See Pl. 51. Pl. 52. Fig. 1. 3. and Pl. 53. Fig. 2. 6. E. F. H.) The arms and fingers are long and spreading, and have numerous joints, or tentacula; each joint is armed at its margin with a small tubercle, or hook, (Pl. 53. Fig. l7.,) the form of which varies in every joint, to act as an organ of prehension; these, arms and fingers, when expanded, must have formed a net of greater capacity than that of the Encrinites. ”

“ The inner surface of each scale is triangular, with the basal margin protuberant. The scales continue to grow or be added to, only in about the ten upper whorls, which form but a small part of the whole peduncle; in the lower part, the scales become further and further separated from each other. The surface of attachment, in full-grown specimens, is broad; but in two very young specimens, which I removed with great care after the action of potash, I found the peduncle ending in a filiform prolongation, such as often occurs in Scalpellum vulgare and in Lepas fascicularis. ”

“ It was more than a chance that produced the decimal system, for the universe is pentalogic, as all of the fundamental series discovered in nature are pentalogic by reason of the five concomitant properties. The origin of the decimal system was the recognition by primitive man of the reciprocal pentalogic systems involved in the two hands of the human body, and the pentalogic properties are always in pairs. ”

“ The unit of the scale must constantly be the same, although realized in the metals, or the whole operation fails in the most essential part. This realizing the unit is like adjusting a pair of compasses to a geometrical scale, where the smallest deviation from the exact opening once given must occasion an incorrect measure. The metals, therefore, are to money what a pair of compasses is to a geometrical scale. ”

“ But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above; and therefore does the same thing in such a non-resisting medium as the absolute weight does in a vacuum. ”

“ The outer rows of the great scales are irregular pentagons. The row that runs down the middle between these are regular hexagons, and round the whole circumference the large scales are inclosed by a kind of quadrangular frame firmly united; the broadest and largest of these scales being nearest the tail. The lowest of all, as it were in the centre of the lowest part of the figure, is notched, the centre of this division answering to a line drawn through the middle of the oval, and the head or occiput. ”

“ The artist did not allow himself the licence of placing Pentheus in the tree, for this had been too grotesque a sight for the fourth-century painter. The frequent references to the thicket [176] and the protection it was or the inconvenience it caused, is happily brought out in the picture, but the poet has not been followed in details. Pentheus does not appear with the thyrsos, talaric chiton, and dishevelled hair, for the simple reason that he would have been indistinguishable from the maenads. As he appears in the painting the contrast is striking and the eye at once grasps the situation. ”

“ For our part, when we realize that a pentacle is a synthetic figure which expresses in concrete form a profound truth of nature, we can see nothing more ridiculous in it than in the figures of Euclid, and nothing half so comical as the symbols in a modern work on chemistry. ”

“ But psychology is passing into a less simple phase. Within a few years what one may call a microscopic psychology has arisen in Germany, carried on by experimental methods, asking of course every moment for introspective data, but eliminating their uncertainty by operating on a large scale and taking statistical means. This method taxes patience to the utmost, and could hardly have arisen in a country whose natives could be bored. Such Germans as Weber, Fechner, Vierordt, and Wundt obviously cannot; ”

“ Pennant had a male badger of that weight; the eyes are very small, the ears short and rounded; the neck thick, and the whole shape of the body clumsy, and which being covered with long coarse hairs, like bristles, adds to its awkward appearance; each hair next the root is of a dirty, yellowish white, the middle is black, and the extremity grey ”

“ An investigator may fancy his problem solved when it is merely restated in a new form. The new and neater form may be a step toward eventual solution. Achieved solution it generally is not.
The influence of the statistical method, as exemplified in Graunt's Observations and in Petty's writings, can be traced in two directions. One springs primarily from Graunt, flows through Petty's Essays, and leads, as has already been said, to modern vital statistics: the other proceeds from Petty's Political Arithmetick through Davenant and Gregory King to Arthur Young and Chalmers. ”

“ Suppose there is a known source of carbonic acid: the determining the amount of it in a room can also be used for measuring another element, which would otherwise defy calculation—I mean the amount of ventilation of a closed space of definite construction. Imagine to yourselves a room with its walls, windows, and doors, its numberless penetrable places through which the air holds ingress and egress. It is impossible to measure the velocity of the air at each crack, to measure each little hole, the diameter of each pore, even if one had the means of measuring such minute velocities and sections ”

“ But when the numbers are varied, it is no longer possible to use the index for specific phases. For the principle that the entropy of any body has an arbitrary additive constant is subject to limitation, when different quantities of the same substance are concerned. ”

“ And the heat in this latter experiment, though less than in the former, was yet greater than the rod of a pendulum can ordinarily [304] contract in the hottest country; for metals receive a great heat when exposed to the open sun, certainly much greater than that of a human body. But pendulums are not usually so exposed, and without doubt in these experiments were kept cool enough to appear so to the touch; which they would do in the hottest place, if lodged in the shade. ”

“ Oftentimes also we constitute Pentacles, by making the revolution of some kinde of name, in a square Table, and by drawing about it a single or double Circle, and by writing therein some holy Versicle competent and befitting this name, or from which that name is extracted. And this is the way of making the Pentacles, according to their several distinct forms and fashions, which we may as we please either multiply or commix together by course among themselves, to work the greater efficacie, and extension and enlargement of force and vertue. ”

“ In 1324, under Edward II, it was enacted that "the inch shall have length of three barley corns, round and dry, laid end to end; twelve inches shall make one foot, and three feet one yard." While this standard for measurement was the accepted one, scientists were at work on a plan to establish a standard for length that could be recovered if lost, and Huygens, a noted philosopher and scientist of his day, suggested that the pendulum, which beats according to its length, should be used to establish the units of measurement. ”

“ Whibble sat, luckily for himself, at a safe distance, regarding him. The four malefactors, feigning a profound absorption in their work, watched him furtively with the keenest interest. For even within the limits of the Three Gases, Plattner's practical chemistry was, I understand, temerarious.
They are practically unanimous in their account of Planner's proceedings. He poured a little of the green powder into a test-tube, and tried the substance with water, hydrochloric acid, nitric acid, and sulphuric acid in succession. ”

“ In 1774 the Royal Society offered a reward of a hundred guineas for a method that would obtain an invariable standard, and Halton proposed a pendulum with a moving weight upon it, so that by counting the beats when the weight was in one position and again when in another, and then measuring the distance between the two positions, a distance could be defined that could at any time be duplicated. ”

“ Moreover, if at the end of any finite amount of stirring we should take our elements of volume in any ordinary form but sufficiently small, the average square of the density of the coloring matter, as determined by such element of volume, would approximate to any required degree to its value before the stirring. But if we take any element of space of fixed position and dimensions, we may continue the stirring so long that the densities of the colored liquid estimated for these fixed elements will approach a uniform limit, viz., that of perfect mixture. ”

“ since, again, they saw that the attributes and the ratios of the musical scales were expressible in numbers; since, then, all other things seemed in their whole nature to be modelled after numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. ”

“ Similarly with the measurements of the ancient buildings. In this case if all our various ‘errors’ were thrown together into one group of statistics we should find that the resultant curve of facility displayed, not two peaks only, but a succession of them; and these of various magnitudes, corresponding to the frequency of occurrence of each particular measurement. We might take an average of the whole, but hardly any rational purpose could be subserved in so doing; whereas each separate point of maximum frequency of occurrence has something significant to teach us. ”

“ Thus it is found that the pendulum standard cannot compete in accuracy and certainty with the simple bar standard, and the method would only be useful as an accessory mode of restoring the bar standard if at any time again destroyed.
Unit of Density.
Before we can measure the phenomena of nature, we require a third independent unit, which shall enable us to define the quantity of matter occupying any given space. All the changes of nature, as we shall see, are probably so many manifestations of energy ”

“ If we project the point-rows to any point in the plane, we obtain two projective pencils having the same center. At most two pairs of self-corresponding rays may present themselves. Such a ray is clearly a line joining two corresponding points in the two point-rows. The result may be stated as follows: The system of rays joining corresponding points in two protective point-rows has at most two rays in common with any pencil in the plane. For that reason the system of rays is called a pencil of rays of the second order. ”

“ If two pencils with centres S1 and S2 are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space. ”