“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

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)

Adopted from Wikiquote. Last update Sept. 14, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right li…" by Proclus?

Proclus18

Greek philosopher 412–485

Proclus (412–485) Greek philosopher

Proposition XV. Thereom VIII.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“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 (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“If two right lines cut one another, they will form the angles at the vertex equal.”

Proclus (412–485) Greek philosopher

...
This... is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales...
Proposition XV. Thereom VIII.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“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 (1861–1940) British civil servant and academic

this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)

“All causes of natural effects have to be given through lines, angles and figures, for otherwise it is impossible for the reason why”

Robert Grosseteste (1175–1253) English bishop and philosopher

De Lineas, Anguilis et Figuris (On Lines, Angles and Figures) as quoted in Neil Lewis, "Robert Grosseteste" http://plato.stanford.edu/entries/grosseteste/ Stanford Encyclopedia of Philosophy (2007, 2013) citing Baur, Ludwig (ed.) Die Philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln (1912) pp.59–60 <br class="br">Context: The consideration of lines, angles and figures is of the greatest utility since it is impossible for natural philosophy to be known without them... All causes of natural effects have to be given through lines, angles and figures, for otherwise it is impossible for the reason why (propter quid) to be known in them.