# Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on.

I am aware that even Gantmacher 1959 has this terminology however I don't know where it originates from. I am also curious what he uses in the original Russian version in place for that word (though I don't know any Russian, I can handle a literal translation ala Körper etc.).

EDIT Since there are answers given towards the meaning of the word "pencil" which is really good to know, I would appreciate if the context is also taken into account. It is from the definition of the pencil forms that some sort of bundling or parameterization is involved. However the definition itself of the word pencil does not introduce the context.

Compare it with the word affine which comes from the similar meaning (Latin affinis) "adjacent,connected" but this is not preferred for some reason although the structure of matrix pencils resembles $a\lambda - b$ more an affine transformation in my opinion. Obviously, it might be a nonlinear function of $\lambda$ but that context looks like long forgotten (until recently the computational tools for quadratic and nonlinear eigenvalue problems started to emerge).

Thus, I would speculate that some circles deliberately avoided either pencil or the affine word at some point. That's what I would like to understand.

• It is a old terminology, already Cayley used it. See for instance jstor.org/stable/2369333?seq=1#page_scan_tab_contents – Francesco Polizzi Feb 8 '15 at 23:28
• suspect it is older than Cayley, used in projective geometry, a pencil of lines, a pencil of points. – Will Jagy Feb 9 '15 at 0:34
• A scan of the second Russian edition of Gantmacher's book (parts 1 and 2) is mburyakov.ru/phtf/lib/Gantmacher.pdf. In the table of contents of the English translation of the first edition at maths.ed.ac.uk/~aar/paper/gantmacher1.pdf (see pp. 6 and 197 of the file), section 6 of chapter 10 is called "pencils of quadratic forms" and chapter 12 is called "singular pencils of matrices". The corresponding names in the Russian version are пучок квадратичных форм and сингулярные пучки матриц. So the Russian word for pencil in math is пучок, which also means sheaf. – KConrad Feb 9 '15 at 0:34
• "The genus name is derived from the Latin root penicillum, meaning "painter's brush", and refers to the chains of conidia that resemble a broom." en.wikipedia.org/wiki/Penicillium – Will Jagy Feb 9 '15 at 2:07
• In italian we use the same term ("fascio") for both the mathematical objects "pencil" and "sheaf". – Francesco Polizzi Feb 9 '15 at 7:15

The Oxford English Dictionary has an example from 1665 of "pencil" in the sense of "A group of rays or a beam of radiation converging to or diverging from a point." And one from 1840 in the geometric sense of "A set of lines meeting in a point"

After Speaking about French, English, Italian and Russian I checked the Greek, my language. Then, I found the greek verb δέσμη which means "to bind".

My interpretation of the etymology is that pencil represents a group bound by a property. In the projective geometry (older in Greek) the term a pencil of lines, δέσμη ευθειών, means lines passing through a common point. It is like the common handle of the brush and the filaments - wire or bristles or other. It works with parallel lines also and then it can mean the light beam etc. Hesiod uses the verb as in "to tie together, as corn in the sheaf".http://www.perseus.tufts.edu/hopper/morph?l=de%2Fsmh&la=greek#lexicon . It is that old. Linearity follows the geometry notion e.g. of lines through a point.

The pencil, used for writing, in Greek as in Russian, is a different word and not a metaphore. I do not know why they coincide in English. Probably, because it means a paintbrush, but it is now obsolete.

• Stefanakis, I tried to fix a lot of the English, but I'm not entirely sure what you meant in some of the places. Please read the edited version and see I've not changed the meanins. Also, I'm not entirely sure about why you added the second linke from "Perseus.tutfts". – Amir Sagiv Aug 29 '16 at 15:48
• This answer seems only speculation, unfortunately. – Federico Poloni Aug 30 '16 at 11:53
• Thanks for the correction Mr. Sagiv. Just note that δέσμη is the noun not the verb. I tried to shorten my response unsuccessfuly I suppose. – A.S.Stefanakis Sep 1 '16 at 19:52
• Mr. Poloni it is not speculation. Starting from the Italian fascio, I found the exact same Greek meaning in Hesiod about 800bc. Then I noticed in wikipedia that the pencil meaning of paintbrush is considered obsolete, which explains why the original question was posed. Then I find that Cayley used it and also there are 1665 ac examples for pencil as a paintbrush. All that were (unsuccessfully shortened) and given with references for you to check. Then the Russian word has exactly the same meaning. – A.S.Stefanakis Sep 1 '16 at 19:53
• My response was not meant to be an answer but a hint for you to consider as also for me this english term was a surprise. I also gave some metafors (corn etc.) for you to see clearly the inspiration of the projective geometry. I have no doubts whatsoever. – A.S.Stefanakis Sep 1 '16 at 20:03