ἐξ ἴσου → evenly, equally, in like measure, on a level with, on equal terms

[Okrad]

Hi. In definition 4 and 7, Euclid uses:

"ἐξ ἴσου" :

δʹ. Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ ̓ ἑαυτῆς σημείοις κεῖται.
ζʹ.  ̓Επίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ ̓ 6. And the extremities of a surface are lines.
ἑαυτῆς εὐθείαις κεῖται.

Here is what one commentator says, "A (straight) line is a curve that lies symmetrically with the points on itself.
[The commonly quoted Heath's translation says "...lies evenly with the points...", but in his notes he says "we can safely say that the sort of idea which Euclid wished to express was that of a line ... without any irregular or unsymmetrical feature distinguishing one part or side of it from another."] "

Can anyone hep me understand this phrase? It is important so I can understand what Euclid is seeing and saying. Is the line moving out of "ἴσου" as if ἴσου were a phenomenon or is is he saying because of ἴσου, it is lying on itself?

Perplexing...

[spiros]

ἐξ ἴσου = equally, in like measure, on a level with, on equal terms
ἐξ ἴσου - Ancient Greek (LSJ)

εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται → A straight line is a line which lies evenly with the points on itself
UGARIT - Translation Alignment Editor ::: Alexander von Humboldt Lehrstuhl für Digital Humanities - Univerity Of Leipzig

εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς σημείοις κεῖται → línea recta es la que está en situación de igualdad respecto a sus puntos
ἐκ - Ancient Greek (LSJ)

Euclid has said “A straight line is a line which lies evenly with the points on itself.” IMHO, this means that a straight line is the line which the points lie evenly pointing to the same slope, they aren't going down at first then going up again like a curve, they are evenly lying there, the points are even.
https://digital.aakash.ac.in/aakashassist/#/student/answer/294838-explain-the-meaning-of-the-following-fourth-definition-given-by-euclid-a-straight-line-is-a-line-which-lies-evenly-with-the-points-on-itself

French analysis here
Mathématiques et philosophie : de l'Antiquité à l'âge classique - Roshdi Rashed - Βιβλία Google