# Must perpendicular (resp. orthogonal) lines meet?

In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space. Two lines are called perpendicular if they meet at a right angle. Two lines are called orthogonal if they are parallel to lines that meet at a right angle. Thus orthogonal lines could be skew (i.e., they need not meet), whereas perpendicular lines always intersect. [Edit: Evidence shows that this distinction most likely arose around the turn of the twentieth century. See below for details.]

Looking around on Quora, Answers.com and here, I have found numerous assertions that, in English, there is no difference whatsoever between 'orthogonal' and 'perpendicular.' However, given the situation in French, I have a gut feeling that the same distinction must once have been observed in English as well, but since there is now a greater focus on vectors (for which the concepts coincide) than on lines, it has gradually been lost. I would like confirmation of this, if possible.

My question, then, is as follows:

How have the two concepts referred to above as 'orthogonal' and 'perpendicular' lines historically been denoted in English and other major languages?

The best answers will include references to authoritative sources.

Edit. Zyx has provided an answer referring to Rouché and Comberousse's geometry text from 1900, where the word perpendiculaire is used for what we have called orthogonal here. This strongly suggests that, contrary to what I had assumed, even French usage has not been unchanging over time.

So Zyx may be correct in questioning my premise, and I am beginning to suspect that even in France, the use of orthogonal in the sense discussed here may have been introduced in the twentieth century. Let me give an example taken from 1952 geometry text that illustrates this usage (Géométrie dans l'espace: Classes de Première C et Moderne, 1952, Dollon and Gilet):

Deux droites sont orthogonales, si leurs angles sont droits.

Deux droites coplanaires formant quatre angles droits ont été appelées droites perpendiculaires [presumably in a lower-level book in the series]; on peut dire aussi qu'elles sont orthogonales.

Dans ce qui suit, nous réserverons en général l'expression droites orthogonales, pour deux droites non coplanaires et dont les angles sont droits.

Nearly identical conventions are found in Géométrie: Classe de Seconde C, 1964, by Hémery and Lebossé, except that they allow "orthogonal" lines to meet (thus perpendicular implies orthogonal, but not conversely):

Nous conviendrons d'appeler droites perpendiculaires deux droites à la fois concourantes et orthogonales.

However, Hadamard's Leçons de géométrie élémentaire, 1901, uses the word perpendiculaire to include both cases:

On dit que deux droites, situées ou non dans un même plan, sont perpendiculaires si leur angle, défini comme il vient d'être dit, est droit.

And Géométrie Élémentaire, 1903, by Vacquant and Macé de Lépinay agrees with Hadamard.

My conclusion is that I was much too quick in my question to call the distinction "traditional." It is most likely to have appeared in France sometime in the early to mid-twentieth century. (To pinpoint the date better, it would be best to check what was done in textbooks in the 1925-1940 period, such as those of P. Chenevier and H. Commissaire, but I don't have access to these. Vectors evidently first appeared in French school curricula in 1905. However, the scalar product was not taught systematically until 1947, so that would seem a possible time for the expression "orthogonal lines" to have been introduced.)

The examples given by Zyx show that usage in English in fact mirrors the earlier French usage, i.e. "perpendicular" is used everywhere. And I presume that the terms "skew perpendicular" and "intersecting perpendicular" would only be used where an author felt the distinction was needed. (In many cases, it will be clear from context whether two lines meet.)

Edit. The "new" French terminology dates at least from the turn of the century. Here is an excerpt from Cours de Géométrie élémentaire: à l'usage des élèves de mathématiques élémentaires, de mathématiques spéciales; des candidats aux écoles du Gouvernement et des candidats à l'Agrégation (1899) by Niewenglowski and Gérard, which was intended for both high-school and university-level students. This book is in fact referred to by Lebesgue in his Leçons sur l'intégration.

Considérons deux droites AB, CD, non situées dans un même plan; menons par un point quelconque O, des parallèles X'X et Y'Y à ces deux droites. [...]

Si les deux droites X'X et Y'Y sont perpendiculaires, nous dirons que les droites AB et CD sont orthogonales. Nous dirons aussi quelquefois qu'elles sont perpendiculaires, même si elles ne se rencontrent pas.

Thus these authors, unlike Hadamard, Rouché and Vacquant, appear to have a preference for droites orthogonales when the lines are not coplanar. However, this was not a hard-and-fast rule, and they allow that perpendiculaires can also "sometimes" be used in this case. The distinction only seems to have become settled later on.

• Since I have received a downvote, I would welcome any comments on how this question can be improved, or why it might be inappropriate. – David Dec 10 '15 at 7:35
• In all texts I encountered so far, this distinction is not made. Usually it is mentioned whether two (orthogonal, i.e. having orthogonal vectors) lines intersect or not. I never read a textbook that explicitly mentiones the distinction by using different words for it. I guess the same use of terminology as in French would be appropriate. – Mathematician 42 Dec 10 '15 at 8:35
• So the distinction is presumably useful in some geometries where lines parallel to lines that meet at a right angle do not themselves meet at a right angle? – rschwieb Dec 20 '15 at 3:29
• @rschwieb No, it's that lines parallel to ones meeting at a right angle may not meet at all. This question is about lines in space. – David Dec 20 '15 at 12:35
• @David hm, I see. Not many terms apply to skew lines, like that. – rschwieb Dec 20 '15 at 13:01

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.

• I think it is reasonable to assume that until fairly recent times, the basic vocabulary used in solid geometry was unaffected by considerations related to vectors and $n$-dimensional geometry. I would guess until at least 1900 for mathematicians and until 1960 or so in schools. In any event, to the extent that the terminology is used with respect to two lines, the distinction is logical in dimension $n$. What is more likely is that anyone thinking about dimension $n$ is mostly thinking in terms of vectors anyway. – David Dec 19 '15 at 9:50
• Also, if you are uncertain whether there is an intersection, you are free to use "orthogonal," as this includes the "perpendicular" case (much as identical lines are usually called parallel, these days). – David Dec 19 '15 at 10:05
• Updated answer. There is no reason to limit orthogonality to 1-dimensional linear subspaces. All you need is the vector between any pair of points in one subspace to be perpendicular to the vector between any pair in the other subspace. I found some examples in a web search that illustrate why I am suspicious of the premise of the question. – zyx Dec 20 '15 at 8:25
• Thank you for this answer. I hadn't realized that the French usage I mentioned was actually relatively recent. Your example from Rouché's textbook is what alerted me to this. I've edited my question to include additional details. – David Dec 20 '15 at 12:33
• Perhaps you can determine what the exact terms used in English are when a distinction needs to be made. Do you say "skew perpendicular", "perpendicular skew", "intersecting perpendicular", etc.? – David Dec 20 '15 at 12:40

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.)

• Thank you for an informative and helpful answer. The part of the answer that draws on sources, however, doesn't answer the question directly. The question was not about the history of the words per se, but rather the history of how the two concepts I referred to in my question have been denoted, in space. The three citations are silent on this. Perpendicular skew may well be the answer for "orthogonal," but I'd like to see it sourced. And that leaves open whether the other kind should be called perpendicular or perpendicular intersecting, or something else. – David Dec 19 '15 at 9:56
• My wordy anecdote seems to over-power my stated interest in Billingsley's potential role in this. After all, Elements discusses the geometry of three-dimenional figures; I don't know off-hand how or whether (what I've called) "perpendicularly skew" lines are specially named, but they certainly appear, for instance, in the opposite edges of a regular tetrahedron. If Billingsley (or one of this contemporaries) has an opportunity to describe these lines using "his" term "perpendicular", but chooses to reserve the term for coplanar lines, then you have a key data point. – Blue Dec 19 '15 at 10:58
• [continued] ... But also, given what an astrolabe is for, Chaucer's Treatise likely describes lines in both the plane and space, so his use of "perpendicular" might be particularly insightful. In any case ... My point wasn't to provide mere definitional sources nor to offer a definitive answer, but to provide leads for further investigation, citing specific early places in the mathematical literature in which an author had probably faced the question at hand: What do we call these specially-directed lines in space? – Blue Dec 19 '15 at 11:16

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.

• What are you saying the definition of two "perpendicular" lines is? Do they need to meet? – David Dec 10 '15 at 8:43
• @David Ah yes sorry, should have mentioned that part. According to my sources they do not need to meet to be perpendicular/orthogonal. – BLAZE Dec 10 '15 at 8:46
• Could you give the sources (for both words)? Also, what would you then call intersecting perpendicular lines? – David Dec 10 '15 at 8:48
• @David I wish I could, but my sources are from what I was taught and not written down (anywhere I can find). Perhaps, it's best to wait for a user to give you a better answer with some evidence, as I can't seem to do that. – BLAZE Dec 10 '15 at 8:57

Orthogonal is from Ancient Greek - orthos means 'correct, straight, right' and gonos means 'angle': Wiktionary:Orthogonal.

Perpendicular is from early Latin from pendare, meaning 'to weigh carefully', as with a plumb line used to get a right-angle from a horizontal. Dictionary.com:Perpendicular.

They mean exactly the same thing, but have different origins.

• btw/afaik, Gonio (angle) derives from Sanskrit. Tri_Kona_Mathra/Mithi = trigonometry .. – Narasimham Dec 19 '15 at 6:07
• Note that the etymology of orthogonal mentions an angle, which assumes a point of intersection; by contrast the etymology of perpendicular only involves direction (any plumb line would be perpendicular to any horizontal line; no point of intersection is required). This is opposite to their supposed distinction in meaning! – Marc van Leeuwen Dec 19 '17 at 14:40

In both cases, vector dot product should vanish. If minimum distance ( along vector cross product) is zero, they are perpendicular (being in the same plane), else skew orthogonal ( there is a minimum skew distance along their common normal).

EDIT1:

We can distinguish between the two or disambiguate between them using a comprehensive 3D picture.

We could bring in 4 points $P(P_1,P_2), Q(Q_1,Q_2)$ on two lines $P,Q.$

If the volume of tetrahedron spanned by these 4 points ( evaluated by means of well known determinant formula) is zero, then $P$ is perpendicular to $Q$. Else, $P$ is skew orthogonal to $Q.$

• Could you provide citations for these words from (preferably older) geometry books? – David Dec 19 '15 at 9:51
• I have no citations per se, but replied more from what I thought would quickly conjure up a mental image of vector contact, based on what I came across commonly. With perpendicularity, the referenced plane is immediately clear; without contact skew situation we are laboring to think about direction of the ( minimum distance) common normal. – Narasimham Dec 19 '15 at 10:58