# Where was the word “pencil” first used in (projective) geometry and what is the reason behind this curious name?

The title is pretty self-explanatory: A pencil in projective (or algebraic) geometry is the family of all lines through a point. The above-linked website tells me that Cremona, on page x of Elements of Projective Geometry, asserts that Desargues was the first to make use of this geometric concept.

However, inspection of the page shows that Cremona does not claim that Desargues came up with the name (in fact, the page says: "The geometric forms [...] are found, names excepted, in Desargues [...]"). Therefore, I'm left wondering about, firstly, who did come up with this name and where. Secondly, I'm interested in the reason why this concept got the name "pencil". The name is surprising enough for me to suspect that whoever came up with it had a good reason for it. I tried to look the term up in Lo Bello's book on mathematical terminology, but there is no relevant information to be found under "pencil" in that book.

Edit:

Has anyone ascertained whether Desargues (or Pascal) actually wrote “pencil” (or “pinceau”) — or else, who introduced the word?

• Mactutor's biography on Desaurges mentions that his major work from 1639 (which can be found here , in French) "begins with the concept of pencils through lines". It is likely that Desaurges used the word "pinceau" which means "brush" but also sometimes "pencil" (usually "crayon") – Spencer Nov 25 '16 at 15:51
• see also mathoverflow.net/a/248618/454 – Gerald Edgar Nov 25 '16 at 16:37
• Using guesswork at the French on that page image, D. may be using ordonnance des lignes droićts to mean "pencil". – Spencer Nov 25 '16 at 16:42
• I don't know who coined the term pencil, but the why always seemed obvious to me when I first heard it: many depictions of the concept (including the example in your first link) look like the tip of a pencil seen side-on, especially if you remember sharpening pencils with a penknife rather than a pencil-sharpener, so the tip had flat-ish edges instead of being a circular cone. – TripeHound Nov 26 '16 at 5:45

From the Earliest Known Uses of Some of the Words of Mathematics site :

PENCIL OF LINES. Desargues coined the term ordonnance de lignes, which is translated an order of lines or a pencil of lines [James A. Landau].

Boyer's A History of Mathematics (1968, p. 396) also attributes this terminology to Desargues. From the section on Blaise Pascal, when discussing his Hexagrammum Mysticum Theorem :

... he followed the special language of Desargues, saying that if $A, B, C, D, E,$ and $F$ are successive vertices of a hexagon in a conic, and if $P$ is the intersection point of $AB$ and $DE$ and $Q$ is the intersection point of $BC$ and $EF$, then $PQ$ and $CD$ and $FA$ are lines "of the same order" (or, as we should say, the lines are members of a pencil, whether a pencil point or a parallel pencil).

The intuition behind the choice of pencil in the English translation is not entirely clear. Perhaps "order", as in the translation "an order of lines", was already too strongly associated with order. Boyer hints at this translated usage, writing :

[Desargues assumes] that the parabola has a focus "at infinity" and that parallel lines meet at "a point at infinity". The theory of perspective makes such ideas plausible, for light from the sun ordinarily is considered to be made up of rays that are parallel - comprising a cylinder or a parallel pencil of rays - whereas rays from a terrestrial source are treated as a cone or a pencil point. The cylinder is merely a cone the vertex of which is infinitely distant, and a parallel pencil of lines is simply a family of lines all of which go through the same point at infinity. Desargues similarly studied a sheaf or bundle of planes through a point, finite or infinite.

Reading further, it appears that Desargues was fond of colourful terminology. For example, he calls a conic section a "coup de rouleau" - i.e., incidence with a rolling pin.

EDIT

Also worth considering is that Desargues introduced these ideas and terms in a work not translated for at least two hundred years after its first appearance. Desargues' Brouillon Projet d'une atteinte aux événements des rencontres du Cône avec un Plan (trans: Rough Draft of an Attempt to Deal with the Outcome of a Meeting of a Cone with a Plane) first appeared in Paris, 1639. According to Boyer, Desargues published his works not to sell but to distribute to friends. The work was completely lost until in 1847 a hand written copy made by Phillipe de Lahire, one of Desargues' few admirers, was found in a Paris library. Brouillon Projet is described by Boyer as one of "the most unsuccessful great books ever produced".

This is a question about English terminology. As others on here have pointed out, the French terminology is different.

The original meaning of the English word “pencil” is a fine brush; this is also the meaning of French pinceau (as opposed to French "crayon" = English "pencil"). According to the Oxford English Dictionary the earliest English attestation of “pencil” in its mathematical sense is in a book from as late as 1840. The underlying image is evidently that of lines converging at a single point in the same way that the fine strands on a small pointed brush converge at the tip.