Scales (balance)

“ Caroline. But if the scales contained different weights, the centre of gravity would no longer be in the fulcrum of the lever, but remove towards that scale which contained the heaviest weight; and since that point would no longer be supported, the heavy scale would descend, and out-weigh the other.
Mrs. B. True; but tell me, can you imagine any mode by which bodies of different weights can be made to balance each other, either in a pair of scales, or simply suspended to the extremities of the lever? for the scales are not an essential part of the machine ”

“ If two weights A and B be placed in the scales of the best balance we cannot be sure that the equilibrium of the beam indicates exact equality, because the arms of the beam may be unequal or unbalanced. But if we take B out and put another weight C in, and equilibrium still exists, it is apparent that the same causes of erroneous weighing exist in both cases, supposing that the balance has not been disarranged ”

“ Whenever cheating is practiced by the "false balance," it is of course done in a small way, to avoid any observation by the eye of the inequality of the two arms of the scale-beam, and the weight of the scale hanging from the shorter arm is made a little greater than that of the other, so that they may balance. Scales may be rendered very accurate by making the fulcrum or pivot of hardened steel, and of a wedge shape, with a sharp edge, in order to avoid friction as much as possible. ”

“ When you go a journey and your luggage is weighed, it is not put on scales, but on a spring; the weight depresses the spring a certain amount, and the result is indicated by a needle on a dial. The same principle is used in automatic machines for finding your weight. The spring balance shows weight, but scales show mass. So long as you stay in one part of the world, the difference does not matter ”

“ Of mechanical ones the most familiar is that of the scales. If accurately made and not clogged by dirt or rust, a pair of scales cannot be perfectly balanced: eventually one scale will descend and the other ascend—they will assume a heterogeneous relation. ”

“ After careful thought and a comparison of different scales, I have come to the conclusion that the scale for standards of size given in the accompanying Table is the one I would recommend to be adopted. I have endeavoured to arrange one which would be easily intelligible to the ordinary workman, and that would in every possible case coincide with the old system, so as not to cause more expense in altering the present sizes than may be absolutely necessary. ”

“ Should the scales contain too large or too small units, they may be multiplied or divided, and a new unit found by doubling or halving the units, or by dividing or multiplying them with any number, to gain the desired result. ”

“ The median point of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller. When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls. ”

“ If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place. ”

“ It is upon this principle, as we shall see, that all accurate processes of measurement depend. If for a weight in a scale of a balance we substitute another weight, and the equilibrium remains entirely unchanged, then the weights must be exactly equal. The general test of equality is substitution. Objects are equally bright when on replacing one by the other the eye perceives no difference. Objects are equal in dimensions when tested by the same gauge they fit in the same manner. ”

“ Weights imply scales, of which we have also mention; and scales involve the primary theorem of mechanics in its least complicated form—involve not a qualitative but a quantitative prevision of mechanical effects. And here we may notice how mechanics, in common with the other exact sciences, took its rise from the simplest application of the idea of equality. For the mechanical proposition which the scales involve, is, that if a lever with equal arms, have equal weights suspended from them, the weights will remain at equal altitudes. ”

“ When the masses are unequal the point is at unequal distances from the centers of the masses. These unequal distances have the same ratio as the masses themselves, but the larger mass is always the shorter distance from the point. If 1 pound is to balance 4 [Pg 26] pounds it is obvious that the 1-pound mass must be 4 times as far from the point of balance as the 4-pound mass.
Fig. 13. Mass of 4 units balanced by 1 unit.
Hence, to balance two masses in a rectangle, the point of balance will be found by proportion, placing it [Pg 27] on a line which divides the rectangle into parts of 2 to 3. ”

“ Why are we so interested in the type of scale that measures a dependent variable? The crux of the matter is the relationship between the variable's level of measurement and the statistics that can be meaningfully computed with that variable. ”

“ When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150 : 200 is not the same as 100 : 150. The mean ratio is as unity to √1·00 × 2·00 or 1 to 1·41. ”

“ Arbitrary Scales.—An arbitrary scale can be constructed by selecting any physical property of a substance which varies regularly with the temperature, such as the volume of a liquid, or the pressure or density of a gas, or the electrical resistance of a metal. ”

“ After the scale is thus leveled, if from any cause whatever, such as shifting the scale on a table, or shifting the table itself, the scale support is thrown out of level, the center of gravity of the poise and beam is shifted from the vertical line above the support, and its moment immediately becomes greater than the torsional resistance, and the beam tips out of balance, and cannot be used as a correct scale until the support is again leveled. ”

“ After elaborating this last point and explaining the decimal subdivisions on the scales of the movable circle, he says that “the numbers and divisions on the fixed Circle, are the very same that the moveable are, . .” There is no drawing of the slide rule in this publication. ”

“ According to the first law, the sum of the weights of the products of a chemical process is always equal to the sum of the weights of the factors. This law must now be illustrated by experiments, and approximate quantitative determinations should be introduced thus early into the course of study. All that is required for this purpose is a common pair of scales, capable of weighing two or three hundred grammes, and turning with a decigramme. We use in our laboratory some platform-scales, made by the Fairbanks Company, which are inexpensive, and serve a very useful purpose. ”

“ When reducing by scales, it is obviously not essential to use the same scale as that to which the original is made; the dimensions on one scale may be readily transferred to any other, and the student will do well to make himself familiar with the operation. ”

“ I saw, methought, a pair of golden scales hanging by a chain in the same metal over the table that stood before me; when, on a sudden, there were great heaps of weights thrown down on each side of them. I found upon examining these weights, they showed the value of everything that is in esteem among men. I made an essay of them, by putting the weight of wisdom in one scale, and that of riches in another, upon which the latter, to show its comparative lightness, immediately "flew up and kicked the beam." ”

“ II.30 We all know that, when a balance is in equilibrium, the tendency to turn in one direction equals the tendency to turn in the other. Each weight produces on its side a tendency to turn, measured by the product of the weight by its arm. The weight on the left produces, on that side, a tendency measured by 5,000,000 × 20; while the weights on the right make a combined opposite tendency measured by 200,000,000×.10+200,000,000×.25+90,000,000×.33 1/3. The equality of these opposite tendencies represents the equation of exchange. ”

“ Their merits, when every consideration is taken into view, are nearly in the inverse ratio of the sums denoting their aggregate dissonance. That of Mr. Hawkes is the best, and, in many respects, has a remarkable analogy to the one derived from the preceding investigations.
Cor. 2. As the aggregate dissonance of the changeable scale is calculated on the same principles, in Prop. VIII., as that of the Douzeave in this, a comparison of the results in Table VIII. with those in Table XIII., will furnish us with the relative dissonance of different systems for these different scales. ”

“ If, however, the form of distribution remains uniform, it would first be necessary for those advocating the use of any of the other quantitative definitions to show that the units of their scales are equal under some reasonable hypothesis. A ratio or a deviation statable only in scale units which are not demonstrably equal is a hazard, with the chances badly weighted against its reliability. So far as both the Binet and the Point scales are concerned we have found that the units are not equal. ”

“ The items in this scale are ordered, ranging from least to most satisfied. This is what distinguishes ordinal from nominal scales. Unlike nominal scales, ordinal scales allow comparisons of the degree to which two subjects possess the dependent variable. For example, our satisfaction ordering makes it meaningful to assert that one person is more satisfied than another with their microwave ovens. Such an assertion reflects the first person's use of a verbal label that comes later in the list than the label chosen by the second person. ”

“ It will be found much more advantageous in dealing with educational statistics to consider only the median point, and to use only the n/2 formula given in a previous paragraph, for practically all educational scales are or may be thought of as continuous scales rather than scales composed of discrete steps. ”

“ The interest of the ensemble which has been described lies in the fact that it may be in statistical equilbrium, both in respect to exchange of energy and exchange of particles, with other grand ensembles canonically distributed and having the same values of and of the coefficients , , etc., when the circumstances are such that exchange of energy and of particles are possible, and when equilibrium would not subsist, were it not for equal values of these constants in the two ensembles. ”

“ 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. ”

“ Ordinary measurement by a foot rule, a surveyor’s chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars, are sufficient instances of this procedure.
In the second case, where p y/x = q, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ Lines balance when in opposite attitudes. We get Balance in all inversions, whether single or double.
Fig. 129
Here similar lines are drawn in opposite attitudes and we get Measure and Shape-Balance. In the above case the axis of balance is vertical. The balance is, therefore, symmetrical. Symmetrical Balance is obtained by the single inversion of any line or lines on a vertical axis. Double inversion gives a Balance of Measures and Shapes on a center. We have no Symmetry in double inversions. All this has been explained.
Fig. 130
We have Measure and Shape-Balance on a center in this case. ”

“ Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another.
First, let A and B have the same ratio to C: A shall be equal to B.
For, if A is not equal to B, one of them must be greater than the other; let A be the greater. ”