„The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.“
— Archimedes, buch On the Equilibrium of Planes
Book 1, Proposition 9.
On the Equilibrium of Planes
„In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.“
— Archimedes, buch On the Equilibrium of Planes
Book 1, Proposition 13.
On the Equilibrium of Planes
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Archimedes4
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„It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.“
— Archimedes, buch On the Equilibrium of Planes
Book 1, Proposition 14.
On the Equilibrium of Planes
„If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity.“
— Archimedes, buch On the Equilibrium of Planes
Book 1, Proposition 4.
On the Equilibrium of Planes
„The centre of gravity of any cylinder is the point of bisection of the axis.“
— Archimedes, buch The Method of Mechanical Theorems
Proposition presumed from previous work.
The Method of Mechanical Theorems
„The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3.“
— Archimedes, buch The Method of Mechanical Theorems
Proposition 6.
The Method of Mechanical Theorems
„Proposition 14. The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth's shadow a ratio greater than that which 675 has to 1.“
— Aristarchus of Samos ancient Greek astronomer and mathematician
p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)
„The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple“
— Archimedes, buch The Method of Mechanical Theorems
of the portion adjacent to the base
Proposition presumed from previous work.
The Method of Mechanical Theorems
„The centre of gravity of a parallelogram is the point of intersection of its diagonals.“
— Archimedes, buch On the Equilibrium of Planes
Book 1, Proposition 10.
On the Equilibrium of Planes
„In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground“
— Thomas Little Heath British civil servant and academic 1861 - 1940
this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)
„If the earth were not round, heavy bodies would not tend from every side in a straight line towards the center of the earth, but to different points from different sides.“
— Johannes Kepler, buch Astronomia nova
As quoted by Bryant, ibid.
Astronomia nova (1609)
„A circle is a round straight line with a hole in the middle.“
— Mark Twain American author and humorist 1835 - 1910
Quoting a schoolchild in "English as She Is Taught"
„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)
„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
„Opinion is like a pendulum and obeys the same law. If it goes past the centre of gravity on one side, it must go a like distance on the other; and it is only after a certain time that it finds the true point at which it can remain at rest.“
— Arthur Schopenhauer, buch Parerga und Paralipomena
Vol. 2 "Further Psychological Observations" as translated in Essays and Aphorisms (1970), as translated by R. J. Hollingdale
Parerga and Paralipomena (1851), Counsels and Maxims
„We hang on to every line,
And walk straight down the middle of it.“
— Kate Bush British recording artist; singer, songwriter, musician and record producer 1958
Song lyrics, The Sensual World (1989)
„To avoid any assertion about the infinitude of the straight line, Euclid says a line segment“
— Morris Kline American mathematician 1908 - 1992
Quelle: Mathematical Thought from Ancient to Modern Times (1972), p. 175
Kontext: To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point.
„Take two opposites, connect the dots, and you have a straight line.“
— Yahia Lababidi 1973
Signposts to Elsewhere (2008)
„The classic instrument to measure drawn angles and to draw angles of a given measure is the protractor — essentially half a circular ring, subdivided by ray segments into 180 degrees. For reasons I was unable to find out, this instrument has recently been superseded by an isosceles right triangle — called geo-triangle, solid, transparant, made of plastic — with an angular division radiating from the midpoint of the hypotenuse to the other sides. Well, inside the triangle half a circle with the midpoint of the hypotenuse as its centre is indicated, and from the position of the degree numbers it becomes clear that it is the semicircle that really matters. One is inclined to say "an outrageously misleading instrument"…“
— Hans Freudenthal Dutch mathematician 1905 - 1990
Quelle: Mathematics as an Educational Task (1973), p. 363
„Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from each other… Euclid does not know zero angles, nor straight and bigger than straight angles…Euclid takes the liberty of adding angles beyond two and even four right angles; the result cannot be angles according to the original definitions…Nevertheless one feels that Euclid’s angle concept is consistent.“
— Hans Freudenthal Dutch mathematician 1905 - 1990
Quelle: Mathematics as an Educational Task (1973), p. 476-477