„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)

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Isaac Barrow
1630 - 1677

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„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.

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„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><math>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}}}</math></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><math>x' = x - vt ; \qquad t' = t</math></center>.
The Structure of the Universe: An Introduction to Cosmology (1949)

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„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)

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