Although I treat "drawing" here in its artistic sense --as the act or product of the act of making marks, designs or tracings on a surface--I am glad to be reminded that the word "draw," as in the sense of a horse drawing a cart, means to drag, pull, or bear. The word's etymology cannot be broken down any further. The OED traces the word "draw" to a Teutonic word meaning "to pull." The Webster's New World Dictionary traces "draw" through Middle English "drawen," to Old English "dragan" which is akin to Old Norse, "draga," meaning to drag, and the German word "tragen," meaning to bear or carry. All this is then traced to a hypothetical Indo-European base, "dherach-," meaning to pull or draw along. Out of this, Webster's traces the Latin word for pull, drag, or draw, "trahere," from which we get the word "traction." In its most general sense, then, to draw means to act upon something and make it move. Movement--of both agent and object--is essential to the idea of drawing. And ironically, one finds that the word "drawing," with which we often associate the freedom of self-expression, is fundamentally related to the idea of bearing a burden, echoing Merleau-Ponty's phrase "the labor of vision." 
Drawing functions as a medium both to and through other forms of art, but especially through painting, which is often conflated with drawing. Drawing is traditionally a learning medium for the "bigger" practices of sculpture, painting and architecture; and Ingres says, "drawing is the probity of art." The Grove Dictionary of Art treats drawing sometimes as the servant, sometimes as the law-giver of painting; and one runs into problems distinguishing the two. The Grove's description of drawing can equally apply to painting: "the language of drawing is used to record, outline, or document images that are observed, imagined, recalled from memory or copied."  Jansen's History or Art and Gardner's Art through the Ages distinguish among sculpture, architecture, and painting; but when they treat it at all, they consider drawing a subgenre of painting. The Grove lists "line, form, value, and texture" rather than color, as the prerogatives of drawing; but it subsequently includes such "colored" media as pastel, guache, tempera, watercolor and ink as appropriate to drawing. Nor does the Grove make a clear distinction in instruments, the "added-on organs"  as Merleau-Ponty says, proper to drawing and painting: one may draw with a brush as well as with a pencil. As a practitioner of both drawing and painting, I do not readily distinguish them, and I agree with Merleau-Ponty's claim, "that the whole of painting is present in each of its modes of expression; there is a kind of drawing, even a single line, that can embrace all of painting's bold potential."  I therefore feel justified in discussing some works, such as the Lascaux Cave paintings, as drawings, or in applying to drawing various observations that have been made about painting.
As part of the larger process of painting, drawing is treated by the Grove as a medium through which artists work out plans and ideas. Judged on their own terms, such drawings serve as records of invention, original thought and primary observation, "as thoughts and images were first rendered into graphic form."  In terms of time and thought, drawing is here seen as a less mediated activity--more personal, spontaneous and intimate than painting, which traditionally takes planning and deliberation. Drawings done preliminary to painting carry "the aura of fiery inspiration" and become repositories of the artist's style. Since many preliminary drawings are unsigned, people make identifications of author or school by style alone.
Drawing as a teaching tool, as "the exact study of appearances"  serves science as well as art. It is a learning medium for vision itself, but not always in a way that "invents its own ends,"  as Merleau-Ponty describes in painting. Students of biology or medical anatomy are commonly called upon to draw what they see in a microscope or on the laboratory table. They cannot see it the way they are required to see it until they draw it; but the ends of scientific and medical illustration are specific and quite different from the experience of artistic drawing. In a section of Sally Price's essay The Mystique of Connoisseurship she confuses artistic and scientific drawing. At one point Price describes a biology student who "rashly bucked the system by depicting what she (and her partner) saw, and was reprimanded by the teacher for having 'failed to see correctly."  From this, Price concludes that "the rhetoric of respect for each person's perceptual and judgmental integrity can run into awkward moments when its principles are tested in the context of an established hierarchy of authority." Price in this particular instance is confusing art and illustration. In Victor Shklovsky's words, Price does not "distinguish properly between the laws of practical language and the laws of poetic language." 
When a lay person first sees an open cadaver, she sees nothing but chopped meat. If she is an artist she may draw it as chopped meat and be perfectly right. But the medical student has to get over that phenomenological experience of the cadaver. The time-honored method for doing so is through drawing. In this respect, drawing is a medium through which the eye learns to differentiate (e.g., organs or microscopic structures), rather than to connect what it sees. The ends of scientific illustration are indeed the opposite of artistic drawing. The former enables the mind to divide the whole into its parts, so that those parts may be considered distinctly and separately. The latter requires the mind to bring all the parts before one's eye into a totality, to create a whole out of parts which may be spacially distant from each other. This is what Merleau-Ponty means when he says that "to see is to have at a distance ." 
Drawing is primitive in both historical and developmental senses. The Lascaux Cave paintings attest to it as one of the most ancient media in Western culture. It is also a universal practice among children. Mediation within drawing is readily apparent insofar as it is arguably the child's earliest mode of self-expression that requires tools. Humans use only their own bodies to sing and dance and talk, but they cannot draw without a tool to extend the eye and hand into a force that leaves a mark, nor without a support upon which to leave that mark.
Developmentally, drawing is a medium through which children work out their understanding of the world, and both their spacial and emotional placement in it. From toddlerhood to adolescence, children draw as part of play. However, they don't do this from direct observation, but rather, out of imagination and memory. Through drawing they are forming ideas of the world. At an early age, however briefly, they draw the sky as a blue streak at the top of the page, because the sky is "up." They draw lines radiating out of the sun, because light and heat come "out" of it. Later, drawing becomes a medium through which children work out their fears, hostilities, and ego ideals. They draw appealing animals and scary monsters. At a later age (among middle-class Americans, at least) boys draw fighters, and girls draw beauties. Psychotherapists commonly use drawing as a medium through which verbally inarticulate children can express their thoughts and emotions.
Drawing's fundamental and greatest importance (to me, anyway) is as a mediator of the hand and eye, and through that mediation "to impart the sensation of things as they are perceived and not as they are known."  For a very simple example, we "know" the sky is "up," but we perceive it, however unconsciously, as meeting the horizon. Through the physical activity of drawing, one really does physically connect with the world and see it as it really is. Shadows are no longer just black, and water is no longer just blue. Through this action, drawing is a medium to Plato's "divine madness" and to Merleau-Pony's "delirium which is vision itself."  One could see drawing, in the Peircean sense, as an index which connects and activates the eye and the hand, as well as the self and the world. The drawing itself, like the footprint, is an index of this action. We "draw" the pencil across the paper, and that movement is the index of the movement of our hand and eye. There is indeed a "system of equivalences."  among the modes of art production exactly because through drawing one has learned to see the entire world with the entire body.
Drawing is a medium through which we develop perception by slowing down the act of seeing. In its definition of "draw" the OED quotes Ophelia from Hamlet: "He falls to such perusal of my face As he would draw it."  Shakespeare understands that the act of drawing is an act of prolonged, intense, deliberate looking. I quibble with Walter Benjamin when he says, "Since the eye perceives more swiftly than the hand can draw, the process of pictorial reproduction was accelerated so enormously that it could keep pace with speech."  No one can argue with Benjamin's point that the clicking camera is faster than the drawing hand, particularly if one regards drawing as nothing other than a "process of pictorial reproduction." But Benjamin is not interested in the actual act of drawing, and therefore does not concern himself with the fact that the act of drawing slows down and expands the act of seeing. Merleau-Ponty understands that drawing--"this gift [that] is earned by exercise"  --expands perception by slowing down vision. He is fascinated with the length of time that Cezanne would devote to a single painting. Victor Shklovsky discusses this idea of art slowing down perception: "The technique of art is to make objects 'unfamiliar,' to make forms difficult, to increase the difficulty and length of perception because the process of perception is an aesthetic end in itself and must be prolonged." 
Recently I had a student of mine try to copy a drawing by Tiepolo. Within minutes I realized it was too difficult for her, but I had her keep at it anyway. We spent the entire class time on it, and came away with a renewed appreciation for the anti-intuitive nature of great drawing. The jutting head was especially hard to get. The brow is farther forward than the nose. This is not how we "know" a human head should be. One smudge and a thin line defines the lower eyelid, the bridge of the nose, and the concave cheekbone. This is what Merleau-Ponty calls seeing the invisible.  During the course of the lesson, the student said, "This guy was a genius. How can he see that?" How can Tiepolo see
as an eye, a nose and a mouth? They seem nowhere near our ideas of what these things should be; but in the context of the total drawing we perceive them for exactly what they are.