Scaling

“ It is not always possible to get a view against a face or side of the object, but, with irregularly shaped bodies or under special conditions, it may be necessary to take a view of corners, sloping planes, curved or irregularly shaped surfaces.
Fig. 231 shows a hexagonal nut in the three normal projections; from the top view it is readily seen that the front view shows the side of the hexagon in a contracted scale, and that therefore the scaling and dimensioning for the horizontal direction have all to be done in the top or plan view ”

“ 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. ”

“ This effect of scale is simply due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to the area of surface; and others, such as gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume, of the body.
The strength of an iron girder obviously varies with the cross-section of its members, and each cross-section varies as the square of a linear dimension; but the weight of the whole structure varies as the cube of its linear dimensions. ”

“ 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. ”

“ Suppose that, instead of pressing down the empty scale, you put something that has weight into it; then, as soon as this weight is equal to that in the other scale, the beam will become horizontal. In fact, one scale has just as much tendency to move towards the centre of the earth as the other has, and as neither can go down without pulling the other up, they neutralise one another. ”

“ Scales and Steelyards.—In the common scale-beam we have a lever, the two arms of which are equal, and therefore equal weights suspended at the ends balance. If they be not exactly equal, a heavier weight will be necessary on the shorter arm than on the longer. The inequality will injure the buyer if the prop be too near the scale in which the weights are placed, and the seller if it be too near that which holds the article to be sold. ”

“ Yes; everything on which there is a duty when it is taken off the boats has to be weighed, and to do this there are big scales, and for each one a scale house. When a scale is being used there are two men in the scale house. One of these is the Government weigher, who sets the scale to a balance and notes down the weight in a book. The other man, who is an employee of the company, writes the weight also in a book of his own; and he is called the company’s checker. But though there are half a dozen scales, almost everything, when it is possible, is unloaded in front of Scale No. ”

“ 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. ”

“ In working to regular scales, such as 1⁄2, 1⁄8, or 1⁄16 size, a good plan is to use a common rule, instead of a graduated scale. There is nothing more convenient for a mechanical draughtsman than to be able to readily resolve dimensions into various scales, and the use of a common rule for fractional scales trains the mind, so that computations come naturally, and after a time almost without effort.
The protractor shown in fig. 193 is an instrument for laying down and measuring angles on paper; it is [129] used in connecting with a scale to define the inclination of one line to another. ”

“ It will be seen that if a horizontal line is drawn from the foot of one of these figures, say G, to point O on the edge of the incline, then dropped vertically to o·, then again carried on to o·· where the other figure g is, we find it is the same height and also that the other vanishing scale is the same width at that distance, so that we can work from either one or the other. In the event of the rising ground being uneven we can make use of the scale on the level plane. ”

“ Square A to the left of the vertical is 21⁄2 feet wide, 1 foot from the vertical and 2 feet from the base, and is worked out in the same way.
Note.—It is necessary to know how to work to scale, especially in architectural drawing, where it is indispensable, but in working 51 out our propositions and figures it is not always desirable. A given length indicated by a line is generally sufficient for our requirements. To work out every problem to scale is not only tedious and mechanical, but wastes time, and also takes the mind of the student away from the reasoning out of the subject. ”

“ Scales are proportioned rules or mathematical instruments of wood, metal, etc., on which are marked lines and figures for the purpose of measuring sizes and distances. It is usual to make scales in the proportion of parts of an inch equalling a foot; the most generally adopted scale for machine drawing is one and a half inches, equalling one foot ”

“ Def.—A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either unchanged or reversed according as the scalar is positive or negative. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multiplication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector. ”

“ Draw to scale such parts of the sketch as can be easily measured, and put in the remainder in proportion, and make these sketches at the scale at which you are used to working in the office. They will be of immense advantage in giving you a sense of absolute scale.
"There is such a thing as 'absolute scale,' and scale is not simply proportion. A drawing might be made in good proportion, and the building look well if executed a thousand feet long, and yet lose all its effectiveness if executed but one hundred feet in length, the relative proportions of the parts remaining the same. ”

“ You might, at first, imagine that, given what we may call our scale of solidity, or scale of depth, the diminution from nature would be in regular proportion, as, for instance, if the real depth of your subject be, suppose, a foot, and the depth of your bas-relief an inch, then the parts of the real subject which were six inches round the side of it would be carved, you might imagine, at the depth of half an inch, and so the whole thing mechanically reduced to scale. ”

“ Mark pulled some of the tools about, lifted the sledge which stood upright, and had left it’s mark on the iron “scale” which lay on the ground an inch deep. Scale consists of minute particles which fly off red-hot iron when it is hammered—the sparks, in fact, which, when they go out, fall, and are found to be metal; like the meteors in the sky, the scale shooting from Vulcan’s anvil, which go out and drop on the earth. ”

“ Scales of Construction. —Scales of construction are intended to afford means of measuring more minute quantities than scales of distances. Of the former there are two kinds, known respectively as the Diagonal and the Vernier scale. The diagonal is the more frequently employed. Its construction involves no peculiar difficulty, as it consists simply of an ordinary scale of distances, with the addition of a number of parallel lines crossed by other parallel lines drawn diagonally from the smaller points of division. An example will best show the construction and mode of using this scale. ”

“ Diag ́onal Scale, a scale which consists of a set of parallel lines drawn on a ruler, with lines crossing them at right angles and at equal distances. One of these equal divisions, namely, that at the extremity of the ruler, is subdivided into a number of equal parts, and lines are drawn through the points of division obliquely across the parallels. With the help of the compasses such a scale facilitates the laying down of lines of any required length to the 200th part of an inch. ”

“ 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. ”

“ 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 ”

“ XXXV How to Form a Vanishing Scale that shall give the Height, Depth, and Distance of any Object in the Picture
If we make our scale to vanish to the point of sight, as in Fig. 89, we can make SB, the lower line thereof, a measuring line for distances. Let us first of all divide the base AB into eight parts, each part representing 5 feet. From each division draw lines to 8th distance; by their intersections with SB we obtain 80 measurements of 40, 80, 120, 160, &c., feet. Now divide the side of the picture BE in the same manner as the base, which gives us the height of 40 feet. ”

“ I need not say that we may make scales in any proportion ad infinitum—one, two, or three hundredths of an inch to a foot, or to 10, 100, or 1,000 feet, as we do for drawing maps. In the same way we may give details on a scale of 6 inches to a foot, or half the actual size; 2 inches to a foot, or a sixth of the actual size, &c. Having chosen his scale, the architect is enabled to give to each part of the plan exact relative dimensions. If he has adopted the scale of one-eighth of an inch to a foot, and wishes to indicate a door 4 feet in width, he takes 4/8ths. ”

“ With a protractor, measure the angle A of the triangle whose sides were just measured, and compare the ratios of the sides or the functional values with those given in the Table, Appendix III, for the same angle. The larger the scale of the drawing, the greater the accuracy. By making use of the hundredths scale of the framing square together with a finely pointed pair of dividers, variation in values should not be great. ”

“ Scales.—There are sixty scales on the lateral line, and twenty-two in a vertical row between the first dorsal and centre of the belly. The scales are rather small, their bases truncated and furrowed to near the middle (striées en éventail) by six grooves corresponding to eight minute lobes of the margin. A narrow border of the outer rounded edge is very minutely streaked, producing teeth on the margin, visible under a lens. The length and breadth of a scale, taken from the side, are about equal, being two and a half lines. ”

“ When the mass is narrowed to the point where the vertical dimension exceeds the horizontal, it becomes a chair for one person. A low bowl may be intended for pansies, but as soon as the service changes and we design it for goldenrod, it becomes a vertical mass. The fable of the fox who, upon being invited to dine with the stork, found the tall vases unfitted for his use illustrates the change of mass with the change of service.
Figures 1 and 4, Plate 4, are examples of horizontal masses with the dark lines indicating the dominance of the horizontal lines and planes. ”

“ 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. ”

“ 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. ”

“ 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 ”