Length

“ A surface is defined to be length and breadth without depth; a line to be length without breadth or depth; a point to be what has neither length, breadth, nor depth. 'Tis evident that all this is perfectly unintelligible upon any other supposition than that of the composition of extension by indivisible points or atoms. How else could any thing exist without length, without breadth, or without depth? ”

“ No, you must not define length by means of a rigid scale, and define a rigid scale by means of length.
Phys. I admit I am hazy about strict definitions. There is not time for everything; and there are so many interesting things to find out in physics, which take up my attention. Are you so sure that you are prepared with a logical definition of all the terms you use? ”

“ We may now gain further familiarity with the operational attitude toward a concept and some of its implications by examining from this point of view the concept of length. Our task is to find the operations by which we measure the length of any concrete physical object. We begin with objects of our commonest experience, such as a house or a house lot. ”

“ A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line ”

“ To measure the length of a body instead of applying to it a yard-wand, one might listen to its sound; for the pitch of the sound given by two cords allows us to deduce their difference of length, and even the absolute length of each. ”

“ But it is not so in extension, where, whatsoever is more than just a foot or an inch, is not distinguishable from the standard of a foot or an inch; and in lines which appear of an equal length, one may be longer than the other by innumerable parts: nor can any one assign an angle, which shall be the next biggest to a right one. ”

“ For the length of the human body, from the crown of the head to the sole of the foot, is six times its breadth from side to side, and ten times its depth or thickness, measuring from back to front: that is to say, if you measure a man as he lies on his back or on his face, he is six times as long from head to foot as he is broad from side to side, and ten times as long as he is high from the ground. And therefore the ark was made 300 cubits in length, 50 in breadth, and 30 in [Pg 99] height. ”

“ Now, the body, being required to have—whether in length, reckoning from the point of the arm to the prominence of the buttock, or in height, reckoning from the summit of the withers to the ground—two heads and a half; whenever the head, by its geometrical length, shall give, in length or in height, to the body measured more than two and a half times its own length, it will be too short; and if it gives less, it will be too long.
‘In the case in which one of these faults exists there would be no further question of establishing by its geometrical length the proportions of the other parts. ”

“ There is only one way of measuring its real length. We must refer to the clock, to the sun, to some change in the physical world. We seem to live years in a dream; was the dream really a long one? The real length can only be determined, if at all, by a physical reference. ”

“ But now observe this other measurement. The animal has been moving, and we have the length of its stride. In each case it is only about three inches. You have an indication, you see, of a long body with very short legs attached to it. ”

“ If we first examine, for example, the process of measuring a length, it consists in our applying to the distance to be measured a length conceived as unchangeable, the unit of measure, until we have passed over the distance. ”

“ As in the sphere all the dimensions are not only equal, the length being equal to the depth and breadth, but are also identical, seeing that what in a sphere is called deep, may also be called long and wide. Likewise is it, as to height in divine wisdom, which is the same as the depth of power and the breadth of goodness. All these perfections are equal, because they are infinite. ”

“ Wherefore the geodesy or measuring of right lines is thus in length, heighth, and breadth, from whence the Painter, the Architect, and Cosmographer, may view and gather of many famous place the windowes, the statues or imagery, pyramides, signes, and lastly, the length and heighth, either by a single or double: the breadth by a double dimension onely, that is, they may thus behold and take of all places the nature and symmetry ”

“ The times of descent along planes of different length, slope and height bear to one another a ratio which is equal to the product of the ratio of the lengths by the square root of the inverse ratio of their heights. ”

“ Before their eyes in sudden view appear The secrets of the hoary Deep—a dark Illimitable ocean, without bound, Without dimension; where length, breadth, and height, And time, and place, are lost; ”

“ I suppose 33 must mean something in Geometry; what does it mean?” “Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.” And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 32. ”

“ Why, the height of a man, in nine cases out of ten, can be told from the length of his stride. It is a simple calculation enough, though there is no use my boring you with figures. I had this fellow's stride both on the clay outside and on the dust within. Then I had a way of checking my calculation. When a man writes on a wall, his instinct leads him to write about the level of his own eyes. Now that writing was just over six feet from the ground. ”

“ Similarly, the distance a boy can run may be considered as depending on the time. Or, we may say that the area of a square is a function of the length of a side, and the volume of a sphere is a function of its diameter. ”

“ Whether the unit of length adopted be an inch, a foot, a yard, a mile, the distance of the earth from the sun, or the distances of the fixed stars, the quantities thus measured are symbols admitting of one invariable interpretation. The exactness of the mathematical sciences is just this precision and invariability of their symbols, and is not, as commonly supposed, the source of any superior certainty as to the facts. ”

“ To us, who conceive three dimensions, a surface is that which hath length and breadth, excluding thickness: and a surface may be either plain in this third dimension, or it may be incurvated: so that the notion of a third dimension enters into our conception of a surface ”

“ At present not much attention is given to the definition of "point," since the term is not used as the basis of a proof, but every effort is made to have the concept clear. It is the custom to start from a small solid, conceive it to decrease in size, and think of the point as the limit to which it is approaching, using these terms in their usual sense without further explanation. [Pg 137]
2. Line. A line is breadthless length. ”

“ Given in any species, the proportions found to exist among the bones, and we may, and zoologists do, predict from any one, the dimensions of the rest; just as, when knowing the proportions subsisting among the parts of a geometrical figure, we may, from the length of one, calculate the others. ”

“ As a measure of length, it is unknown to the mass of our citizens; and if retained for the purpose of superficial measure, the foot will supply its place, and fix the acre as in the fourth resolution. ”

“ Most striking, perhaps, in their general appearance was the disproportion between the legs of these creatures and the length of their bodies; and yet—so relative is our idea of grace—my eye became habituated to their forms, and at last I even fell in with their persuasion that my own long thighs were ungainly. ”

“ But these speculations lead us out of science into the shadowy land of metaphysics, of which we long to know something, but are condemned to know so little. Area is got by multiplying length by breadth. Cubic content is got by multiplying length by breadth and by height. Of all the conceptions respecting space, that of a line is the simplest. ”

“ the greater the length, the more beautiful will the piece be by reason of its size, provided that the whole be perspicuous. ”

“ Taking into consideration the mean of observations made at divers times—rejecting the timid estimate of those who assigned to this object a length of two hundred feet, equally with the exaggerated opinions which set it down as a mile in width and three in length—we might fairly conclude that this mysterious being surpassed greatly all dimensions admitted by the learned ones of the day, if it existed at all. ”

“ We have measured, by means of sixty-three triangles, an arc of more than eight degrees in length; and when we have rigidly corrected our results, we shall know the exact value of the degree, and consequently of the mètre, in this part of the globe.”
A cheer of satisfaction could not be repressed amongst the others. ”

“ The child should be taught a definite length of step for the purpose of measurement. In proportion to his size he can learn to step off two feet or a yard. He should also know the length in inches of his shoe for the purpose of checking shorter measurements. ”

“ But, when a man is arrived at the period of his full growth, every one of these dimensions becomes double in length, except the face, which, with the top of the head, undergoes but very little alteration in length. A well-proportioned and full-grown man, therefore, is ten times the length of his face; the breadth of his shoulders will be two faces, and in like manner all the above lengths will be double. ”

“ The length of a man's outspread arms is equal to his height. ”

“ A cord, whether laid on itself or extended in only one direction, and though conceived of the utmost conceivable thinness, can not be conceived with less than three dimensions. Nor can a line or a point. When we think of a mathematical plane or line or point, we do nothing more than fix our attention on length and breadth, regardless of thickness ”

“ Everyone seems to over-estimate the proper height.
Another favourite illusion is to ask a person to mark on the wall a height from the floor which would represent the length of a horse’s head: here the majority guess far too little—for a horse’s head is much longer than most people imagine, ranging from 25 to 34 inches. In a recent experiment 5 persons out of 6 under-estimated the proper height. ”

“ and so I penned
It down: until it came at last to be,
For length and breadth, the bigness which you see. ”

“ Measure how many times the diameter of the sun will go into its course in twenty-four hours. And thus we can see whether Epicurus was correct in saying the sun was only as large as it appeared to be; for as the apparent diameter of the sun is about a foot, and as the sun would go a thousand times into its course in twenty-four hours, it would have travelled a thousand feet, that is, three hundred arms' length, which is the sixth of a mile. ”

“ The length of its body is always adapted to the depth of the hole or crevice in which the animal lives, and may vary from half an inch to two or three inches, the diameter of the columns being greatest where the length is least. ”

“ The Diameter of the Field it discovers, or the subtense of the visual angle, measured upon the Object-plate, in the first length is of 8 inches and 7 lines: in the second, of 12 inches and 4 lines: in the third, of 13 inches: an in the fourth, of a little more than 16 inches. ”

“ Or if it be only meant that we think of a particular height, which must be a precise, positive, determinate idea, even this supposition may in the same way be shewn to be exceedingly fallacious, and an inversion of the natural order of our ideas. For take any given height of a man, whether tall, short, or middle-sized, and let that height be as visible as you please, yet the actual height to which it amounts must be made up of the length of the different parts, the head, the face, the neck, the body, limbs, &c. ”

“ In figuring, multiply the length by the width by the thickness, by the number of pieces. If any piece is less than 1′′ thick figure it as 1′′. Combine all results that are the same in price per foot. Reduce to square feet by dividing by 144. ”

“ It is sometimes found three feet in length, and weighing eighteen pounds. According to the accounts in the Elements of Natural History, it is met with from two to fifteen feet long, grows quickly, is very tenacious of life, and lives to a great age. ”

“ The size generally in use in this country is about twenty-seven feet in length, a little over seven feet broad, three feet eight inches deep, carrying eight oars, double-banked, and weighing when empty 4,000 pounds. ”

“ There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it. But some foolish people have got hold of the wrong side of that idea. You have all heard what they have to say about this Fourth Dimension?"
"I have not," said the Provincial Mayor.
"It is simply this, That space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to these planes, each at right angle to the others. ”