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

— Proclus

...
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)

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 "If two right lines cut one another, they will form the angles at the vertex equal." by Proclus?

Proclus 18

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)

“Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.”

Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)

“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)

“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)