# „As a Line, I say, is looked upon to be the Trace of a Point moving forward, being in some sort divisible by a Point, and may be divided by Motion one Way, viz. as to Length; so Time may be conceiv'd as the Trace of a Moment continually flowing, having some Kind of Divisibility from an Instant, and from a successive Flux, inasmuch as it can be divided some how or other. And like as the Quantity of a Line consists of but one Length following the Motion; so the Quantity of Time pursues but one Succession stretched out as it were in Length, which the Length of the Space moved over shews and determines. We therefore shall always express Time by a right Line; first, indeed, taken or laid down at Pleasure, but whose Parts will exactly answer to the proportionable Parts of Time, as its Points do to the respective Instants of Time, and will aptly serve to represent them. Thus much for Time.“

p, 125
Geometrical Lectures (1735)

Obtido da Wikiquote. Última atualização 3 de Junho de 2021. História
1630 - 1677

### „Let the letters a b c denote the three angular points of a rectilineal triangle. If the point did move continuously over the lines ab, bc, ca, that is, over the perimeter of the figure, it would be necessary for it to move at the point b in the direction ab, and also at the same point b in the direction bc. These motions being diverse, they cannot be simultaneous. There-fore, the moment of presence of the movable point at vertex b, considered as moving in the direction ab, is different from the moment of presence of the movable point at the same vertex b, considered as moving in the same direction bc. But between two moments there is time; therefore, the movable point is present at point b for some time, that is, it rests. Therefore it does not move continuously, which is contrary to the assumption. The same demonstration is valid for motion over any right lines including an assignable angle. Hence a body does not change its direction in continuous motion except by following a line no part of which is straight, that is, a curve, as Leibnitz maintained.“

—  Immanuel Kant German philosopher 1724 - 1804

Section III On The Principles Of The Form Of The Sensible World

### „Every measurable thing except numbers is imagined in the manner of a continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them… measure or ratio is initially found… Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e. g., a quality… And since the quantity or ratio of lines is better known and is more readily conceived by us—nay the line is in the first species of continua, therefore such intensity ought to be imagined by lines… Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally.“

—  Nicole Oresme French philosopher 1323 - 1382

Tractatus de Configurationibus et Qualitatibus et Motuum (c. 1350)

### „For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow.“

—  Proclus Greek philosopher 412 - 485

Book III. Concerning Petitions and Axioms.

### „I shall speak twice over. As upon a time One came to be alone out of many, so at another time it divided to be many out of One: fire and water and earth and the limitless vault of air, and wretched Strife apart from these, in equal measure to everything, and Love among them, equal in length and breadth.“

—  Empedocles, livro On Nature

from fr. 17
Variant translations:
But come! but hear my words! For knowledge gained/Makes strong thy soul. For as before I spake/Naming the utter goal of these my words/I will report a twofold truth. Now grows/The One from Many into being, now/Even from one disparting come the Many--/Fire, Water, Earth, and awful heights of Air;/And shut from them apart, the deadly Strife/In equipoise, and Love within their midst/In all her being in length and breadth the same/Behold her now with mind, and sit not there/With eyes astonished, for 'tis she inborn/Abides established in the limbs of men/Through her they cherish thoughts of love, through her/Perfect the works of concord, calling her/By name Delight, or Aphrodite clear.
tr. William E. Leonard
On Nature
Original: (el) ἀλλ’ ἄγε μύθων κλῦθι· μάθη γάρ τοι φρένας αὔξει· ὡς γὰρ καὶ πρὶν ἔειπα πιφαύσκων πείρατα μύθων, δίπλ’ ἐρέω· τοτὲ μὲν γὰρ ἕν ηὐξήθη μόνον ῏ειναι ἐκ πλεόνων, τοτὲ δ’ αὖ διέφυ πλέον’ ἐξ ἑνὸς εἶναι, πῦρ καὶ ὕδωρ καὶ γαῖα καὶ ἠέρος ἄπλετον ὕψος, Νεῖκος τ’ οὐλόμενον δίχα τῶν, ἀτάλαντον ἁπάντηι. καὶ Φιλότης ἐν τοῖσιν, ἴση μῆκός τε πλάτος τε· τὴν σὺ νόωι δέρκευ, μηδ’ ὄμμασιν ἧσο τεθηπώς· ἥτις καὶ θνητοῖσι νομίζεται ἔμφυτος ἄρθροις, τῆι τε φίλα φρονέουσι καὶ ἄρθμια ἔργα τελοῦσι, Γηθοσύνην καλέοντες ἐπώνυμον ἠδ’ Ἀφροδίτην·
Contexto: But come, hear my words, since indeed learning improves the spirit. Now as I said before, setting out the bounds of my words, I shall speak twice over. As upon a time One came to be alone out of many, so at another time it divided to be many out of One: fire and water and earth and the limitless vault of air, and wretched Strife apart from these, in equal measure to everything, and Love among them, equal in length and breadth. Consider [Love] in mind, you, and don't sit there with eyes glazing over. It is a thing considered inborn in mortals, to their very bones; through it they form affections and accomplish peaceful acts, calling it Joy or Aphrodite by name.

### „It cannot be justly inferr'd… We do not perceive the Thing, therefore there is no such Thing, that is a false Illusion, a deceitful Dream, that wou'd cause us to join together two remote Instants of Time. But nevertheless this is very True… That is, for as much Motion as there was, so much Time seems to have been elapsed; nor, when we mention such a Quantity of Time, do we merely mean any Thing else, than the Performance of so much Motion, to the continued successive Extension of which we imagine the Permanency as Things is co-extended.“

—  Isaac Barrow English Christian theologian, and mathematician 1630 - 1677

p, 125
Geometrical Lectures (1735)

### „Consider an event, for example the outburst if a nova… Suppose this event is observed from two stars in line with the nova, and suppose further that the two stars are moving uniformly with respect to each other in this line. Let the epoch at which these stars passed by each other be taken as the zero of time measurement, and let an observer A on one of the stars estimate the distance and epoch of the nova outburst to be x units of length and t units of time, respectively. Suppose the other star is moving toward the nova with velocity v relative to A.“

—  Gerald James Whitrow British mathematician 1912 - 2000

Let an observer B on the star estimate the distance and epoch of the nova outburst to be x<nowiki>'</nowiki> units of length and t<nowiki>'</nowiki> units of time, respectively. Then the Lorentz formulae, relating x<nowiki>'</nowiki> to t<nowiki>'</nowiki>, are<center>$x' = \frac {x-vt}{\sqrt{1-\frac{v^2}{c^2}}} ; \qquad t' = \frac {t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}$</center>
These formulae are... quite general, applying to any event in line with two uniformly moving observers. If we let c become infinite then the ratio of v to c tends to zero and the formulae become<center>$x' = x - vt ; \qquad t' = t$</center>.
The Structure of the Universe: An Introduction to Cosmology (1949)

### „Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. …the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order… an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,…), (x1, x2, x3,…), (dx1, dx2, dx3,…). This quantity retains the same value so long as… the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i. e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. …The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e. g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them…“

—  Bernhard Riemann German mathematician 1826 - 1866

On the Hypotheses which lie at the Bases of Geometry (1873)

### „Time is composed of time-atoms, i. e., of many parts, which on account of their short duration, cannot be divided. The Mutakallemim undoubtedly saw how Aristotle proved that time, space, and locomotion are of the same nature. …They, therefore, knew that if time were continuous and divisible ad infinitum, their assumed atom of space would of necessity likewise be divisible. Similarly, if it were supposed that space were continuous… the time-element… could also be divided. This has been shown by Aristotle in… Acroasis [Aristotelis stagyritae acroases physicae]. …An hour is, e. g., divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subjected to division, and in fact are indivisible.“

—  Maimónides, livro The Guide for the Perplexed

Guide for the Perplexed (c. 1190), Part I

### „All we are doing is looking at the time line, from the moment the customer gives us an order to the point when we collect the cash. And we are reducing the time line by reducing the non-value adding wastes.“

—  Taiichi Ohno Japanese businessman and engineer 1912 - 1990

### „To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished, always by a like proportional part.“

—  John Napier Scottish mathematician 1550 - 1617

The Construction of the Wonderful Canon of Logarithms (1889)

### „These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane… for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.“

—  Howard P. Robertson American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „I knew that the library was also the materialization of a certain level of the psyche, even as our world is. Only there, time is laid out like space is here. The windows of the library coincide with definite places in our space-time. In our world, these points of intersection may appear as natural objects, and these correlate with coordination points in your psyche. Moving toward these coordination points in your mind automatically lines up your consciousness to some extent with this other reality, and stabilizes conditions enough to allow for more or less conscious entry and return.“

—  Jane Roberts American Writer 1929 - 1984

Fonte: Psychic Politics: An Aspect Psychology Book (1976), p. 41

### „As Magnitudes themselves are absolute Quantums Independent on all Kinds of Measure, tho' indeed we cannot tell what their Quantify is, unless we measure them; so Time is likewise a Quantum in itself, tho' in Order to find the Quantity of it, we are obliged to call in Motion to our Assistance as a Measure… and thus Time as measurable signifies Motion; for if all Things were to continue at Rest, it would be impossible to find out by any Method whatsoever how much Time has elaps'd; and the several Ages wou'd roll on imperceptibly and undistinguish'd. Do I say we shou'd not perceive how Time flows? No indeed, nor any Thing else, but remain like Stocks or Stones in a continual Insensibility. We perceive nothing, unless so far as we may be instigated by some Change affecting the Senses, or that our Souls are mov'd and excited by the internal Operation of the Mind. We esteem the Quantities and different Degrees of Things according to the Extension or Intension of Motions striking upon us either interiorly or exteriorly. So that the Quantity of Time so far as we can observe; depends upon the Extension of Motion.“

—  Isaac Barrow English Christian theologian, and mathematician 1630 - 1677

p, 125
Geometrical Lectures (1735)

### „…the famous assertion by Einstein that the length of a rod depends on its velocity and on the chosen definition of simultaneity. …is based on the fact that we do not measure the length of the rod, but its projection on a system at rest. How the length of the projection depends on the choice of simultaneity can be illustrated by reference to a photograph taken through a focal-plane shutter. Such a shutter… consists of a wide band with a horizontal slit, which slides down vertically. Different bands are photographed successively on the film. Moving objects are therefore strangely distorted; the wheels of a rapidly moving car for instance, appear to be slanted. The shape of the objects in the picture will evidently depend on the speed of the shutter. Similarly, the length of the moving segment depends on the definition of simultaneity. One definition of simultaneity differs from another because events that are simultaneous for one definition occur successively for another. What may be a simultaneity projection of a moving segment for one definition is a "focal-plane shutter photograph" for another.“

—  Hans Reichenbach American philosopher 1891 - 1953

The Philosophy of Space and Time (1928, tr. 1957)

### „Minkowski… supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'…. The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle…. Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity…. It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found through the medium of his 'Principle of Equivalence'.“

—  James Jeans British mathematician and astronomer 1877 - 1946

The Growth of Physical Science (1947)

### „Suppose a surface to be part red and part blue; so that every point on it is either red or blue, and of course, no part can be both red and blue. What then, is the color of the surface in the immediate neighborhood of the point. …it follows that the boundary is half red and half blue. In like manner, we find it necessary to hold that consciousness essentially occupies time… Thus, the present is half past and half time to come. …Take another case: the velocity of a particle at any instant of time is its mean velocity during an infinitesimal instant in which that time is consumed. Just so, my immediate feeling is my feeling through an infinitesimal duration containing the present instant.“

—  Charles Sanders Peirce American philosopher, logician, mathematician, and scientist 1839 - 1914

The Law of Mind (1892)

### „Transnational corporate networks, and their resulting spatial patterns, are always in a continuous state of flux. At any one time, some parts may be growing rapidly, others may be stagnating, others may be in steep decline.“

—  Peter Dicken British geographer 1938

Fonte: Global Shift (2003) (Fourth Edition), Chapter 8, Transnational Production Networks, p. 250

### „But you may perhaps wonder why I explain Time without Motion, and will say, does not Time imply Motion? I answer no, as to its absolute and intrinsic Nature; any more than it does Rest. The Quantity of Time, in itself, depends not on either of them; for whether Things move on, or stand still; whether we sleep or wake, Time flows perpetually with an equal Tenor.“

—  Isaac Barrow English Christian theologian, and mathematician 1630 - 1677

p, 125
Geometrical Lectures (1735)

### „By August, 1913, all links in the chain of moving assembly lines were complete except the last and most spectacular one - the one we had first experimented with one Sunday morning just five years before. Again a towrope was hitched to a chassis, this time pulled by a capstan. Each part was attached to the moving chassis in order, from axles at the beginning to bodies at the end of the line. Some parts took longer to attach than others; so, to keep an even pull on the towrope, there must be differently spaced intervals between delivery of the parts along the line. This called for patient timing and rearrangement until the flow of parts and the speed and intervals along the assembly line meshed into a perfectly synchronized operation throughout all stages of production. Before the end of the year a power-driven assembly line was in operation, and New Year's saw three more installed. Ford mass production and a new era in industrial history had begun.“

—  Charles E. Sorensen American businessman 1881 - 1968

Fonte: My Forty Years with Ford, 1956, p. 130-131 ; As cited in: EyeWitness to History (2005)

### „Some of these quantities refer directly to a point of time. That is true of "capital value" as also of such quantities as demand and supply prices. Other terms – as e. g. "income", "revenue", "return", "expenses", "savings", "investments" – imply, however, a time period for which they are reckoned. But in order to be unambiguous they must also refer to a point of time at which they are calculated.“

—  Gunnar Myrdal Swedish economist 1898 - 1987

Fonte: Monetary Equilibrium (1939), p. 46-7