Quotes about parallelogram

A collection of quotes on the topic of parallelogram.

Quotes about parallelogram

“The centre of gravity of a parallelogram is the point of intersection of its diagonals.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 10.
On the Equilibrium of Planes

“The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 9.
On the Equilibrium of Planes

“While I was pursuing the concept of geometrical product, as this idea was established by my father… I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third… and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC. ]
From this harmony I came to see a whole new area of analysis was opening up which could lead to important results.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

Idea

“Grasping spatial gestalts as figures is mathematizing of space. Arranging the properties of a parallelogram such that a particular one pops up to base the others on it in order to arrive at a definition of parallelogram, that is mathematizing the conceptual field of the parallelogram.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 133

Mathematics

“To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle.
According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient.”

Proclus (412–485) Greek philosopher

And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application...
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

Men, Books, Problem, Parting