Must perpendicular (resp. orthogonal) lines meet?
From "Earliest Known Uses of Some of the Words of Mathematics (O)" ...
ORTHOGONAL is found in English in 1571 in A geometrical practise named Pantometria by Thomas Digges (1546?-1595): "Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle." (In Billingsley's 1570 translation of Euclid, an orthogon (spelled in Latin orthogonium or orthogonion) is a right triangle.) (OED2).
also
ORTHOGONAL VECTORS. The term perpendicular was used in the Gibbsian version of vector analysis. Thus E. B. Wilson, Vector Analysis (1901, p. 56) writes "the condition for the perpendicularity of two vectors neither of which vanishes is A·B = 0." When the analogy with functions was recognised the term "orthogonal" was adopted. It appears, e.g., in Courant and Hilbert's Methoden der Mathematischen Physik (1924).
There are also notes on orthogonal matrix and orthogonal function, and orthocenter (the last of which includes an anecdote about the coining of the term in 1865).
The site's "P" page has less to say about the other term:
PERPENDICULAR was used in English by Chaucer about 1391 in A Treatise on the Astrolabe. The term is used as a geometry term in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.
Note that Billingsley also appears in the "orthogonal" entry above. A deeper dive into his translation of Elements may be in order, to see if he explains his own thinking about the distinction between "perpendicular" and "ortho[gonal]".
Anecdotally, I (an American) was formally introduced to "orthogonal" in the context of vectors in Pre-Calculus. (The term may have been mentioned in passing when we learned about orthocenters in Geometry.) So, the term to me has always connoted a directional relationship independent of position. I've also seen the term "perpendicularly skew" for lines in space. Be that as it may ... I don't appear to be alone in using "orthogonal" and "perpendicular" interchangeably ---"perpendicular" just seems friendlier to use with students--- but in formal circumstances, I would probably be inclined to follow the French convention of sharp distinction. (That said, I'd feel obliged to explicitly acknowledge the convention, to avoid confusing my audience.)
In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space.
I suspect that the premise of a traditional distinction between intersecting and nonintersecting orthogonal pairs of lines may be incorrect. The references below have examples from 1900-1921 in textbooks written in English, French and German. Today such a distinction is probably limited to dimension $3$ as presented in some pre-university books, or courses for school teachers.
Problems with the distinction include
- it does not work well for parametric families of (pairs of) lines.
- in higher dimension there is a clear notion of orthogonality between linear subspaces but it would be complicated to have to judge whether there is an intersection in order to choose the mot juste.
- there are too many words like orthocenter, orthologic, orthopole in 2-3 dimensional Euclidean geometry that are incongruous with the idea of orthogonal lines not intersecting. If the orthocenter is the intersection of some orthogonal lines (altitudes) they must be orthogonal to the sides of the triangle. To then alter the language from 2 to 3 dimensions would be strange.
Search results:
1903 UK translation of Franz Hocevar's Solid Geometry book into English has examples of "perpendicular" lines in 3-d being used to include the case of skew lines. Page 10 : "prove that if a straight line be perpendicular to two intersecting lines but does not meet them, it is normal to the plane containing them" and other similar uses on the same page. This was the first old text listed at https://www.google.com/#q=solid+geometry&tbm=bks .
1921 (in USA) Charles Austin Hobbs, Solid Geometry, p271: "in solid geometry, two skew lines are either perpendicular to each other or are oblique to each other".
1900 Eugene Rouche et Ch. de Comberousse, Traite de Geometrie, V, Geometrie dans l'Espace, p.10 "on dit que deux droites non situees dans le meme plan sont perpendiculaires l'un a l'autre lorsque leur angle entre est droit". https://books.google.com/books?id=w8Q0AQAAMAAJ&pg=PA10
Rouche's book was in its 7th edition in 1900 and looks like it was a standard text of its time.
The references added to the question support the idea that a distinction between orthogonal and perpendicular is made only in introductory school-books. I guess the rationale is to avoid the possible language confusion for students, between "perpendicular" as a relation between two objects and "the perpendicular" drawn from a point to a line or plane. The perpendicular sounds unique, and is unique. However, if skew lines can be perpendicular, then there are many lines through a point that are perpendicular to a given line, but are not "the" perpendicular to that line.
They are tantamount to the same. "Orthogonal" is a term used for more general objects, like planes, whereas "perpendicular" began with, and sticks with lines. As geometry expanded in dimension, so did the definition change. "Orthogonal" would include "Perpendicular" in particular, however, the terms are used synonymously now with no loss of meaning.