Senior Thesis at St. John's College

Introduction

This paper will explore how graphical presentation shapes and reflects the understanding of motion and time. Presenting motion in a picture necessarily requires ingenuity. Paper is static; motion is not. A map can show a nation’s boundaries and a painting can depict a scene or a face, but a diagram can only communicate motion and time by finding some symbolic way to invoke what it cannot actually depict. Yet scientific works routinely use diagrams to portray events that unfold in time. The works of Galileo and Newton, Einstein and Minkowski rely on diagrams to convey the movement of objects and the passage of time and to formulate mathematical descriptions of these movements. 

Focusing on diagrams as diagrams requires a change in attitude toward the texts examined. Scientific diagrams frequently appear to be transparent presentations of the subject matter they treat, with no content independent of the text. Indeed, the ideal diagram will attempt to efface itself. Its author wants the reader/viewer’s attention to look through the diagram as if it were transparent and to gaze on the subject matter of the work as if it were visible without interference or bias. A diagram that obviously interposes itself, appearing to distort or alter the subject matter it conveys, is faulty. However, all diagrams shape the subject matter in presenting it and if they do so subtly rather than obviously it is all the more important to notice influences that might otherwise escape our attention. 

This paper’s project is to look at the diagrams as diagrams and not mere vehicles for the concepts they are communicating. The subject of this essay is graphical rhetoric as it applies to the presentation of time and motion. We may wish to think very abstractly about the nature of time as it “really” is, but in reading the work of any author, what we confront is that author’s expression of time. Words are one form of expression, and someone might explore the verbal rhetoric of presentation of time—whether it is likened to a stream, to a circle, to a number, or is represented in some other way. The choices that authors make among the words available to them—and possibly the limitations imposed upon them by limited vocabulary or available comparisons—contribute to shaping the way in which they think about the subject. So, for instance, Einstein imagines time as measured by countless synchronous clocks permeating space; Aristotle would never—could never—imagine time through such a comparison. Similarly, time is also presented graphically and different authors show time differently in pictures and diagrams. The choices that authors make, and the constraints placed upon them by the limitations of the medium in which they work, also contribute to shaping the way in which they think of the subject.

The fundamental issue is present in Newton’s ‘Scholium to the Definitions’ in Philosophiæ Naturalis Principia Mathematica: 

1. Absolute, true, and mathematical time, in itself and by its nature without relation to anything external, flows uniformly and by another name is called “duration”…

2. Absolute space, by its nature, without relation to anything external, always remains similar and motionless…

Newton structures these as parallel definitions. However, the crucial difference that distinguishes them—the dynamism of time, the stasis of space—points to exactly the problem we face with all static representations of time we have.

Visual presentations of time differ in a variety of ways. Thus, to describe this subject it will be necessary to develop a taxonomy of many kinds of features found in diagrams. A number of distinctions will be useful. Is the presentation of time direct or indirect? Since a direct presentation of time would require showing a moving image, they have not been available until recently. In the diagrams of the texts explored below, all graphical presentations of time are indirect because paper is incapable of showing a moving image except in very primitive ways, e.g., a flip-book. Is the presentation of time explicit or implicit? Some show time explicitly by presenting it as a graphical object, often a “time line.” Others omit such a device and leave the reader/viewer to imagine the progress of time. Is the presentation of time sequential or simultaneous? Some diagrams show successive states of a moving system in separate drawings; some present the trace of an object’s trajectory as if the temporally extended sequence could be seen all together. There are other variations as well, all discussed below.

This examination of the graphical rhetoric of time might be compared to a reading of a poem or other piece of literature. It is possible to read a work of literature in order to explore the subject matter discussed—to read the Iliad as an exploration of courage or mortality, or to read Pride and Prejudice as a guide to social structures and romantic rituals of 19th century England. However, to examine these works as pieces of literature often means considering not exclusively, and perhaps not principally, what they say but how they say it. Similarly here. The principal aim of this paper is not to reach a conclusion about the nature of time but to explore how time is presented through the differing graphical presentations of different authors. This paper does not look at Newton and Galileo to get at the notions concerning falling bodies. It looks at the graphical portions of their works and thinks about them as one can treat a poem, focusing on the manner of their presentation. This is the project.

Qualities of Figures

The next sections will enumerate and explore a number of features of diagrams that affect what is communicated and consequently affect the conception. They shall also examine the role of imagination in the process of observation and understanding.

Dynamism and Stasis

Physicists try to explain a world that is in motion, but the drawings and illustrations that they use are often (and until recently were always) motionless. The distinction between dynamism and stasis, however, is about more than the difference between moving realities and rigid illustrations. Illustrations can be made to move, given adequate technology. Even without computers and video equipment, a demonstration—like Newton’s bucket—can act as a dynamic illustration: if the reader were to fill a bucket with water, hang it from a tree, and spin it, he would see something in motion that demonstrates the content of the text. But the static nature of diagrams is not just a failure or an unfortunate limitation of the medium that prevents the capturing of reality. Static illustrations have advantages that dynamic pictures and demonstrations lack and vice versa. The distinction between stasis and dynamism in representation requires careful exploration. 

There is often a discrepancy between the systems described in lemmas and propositions and the figures depicting those systems. For instance, Newton’s Proposition 1 in the Principia describes a system in which a body (not shown in the figure) moves from point A to point B. The system contains motion. The figure of the system, however, contains no such motion. Since paper and ink are static, Newton cannot show the motion taking place in the system. The medium limits what can be directly shown.

Paper, as a static medium, is well suited to represent static systems. However, for dynamic systems, information is inevitably lost when represented statically. When one observes motion in reality, the moving body’s motion can be seen, and further, the magnitude of speed is observable, as is any acceleration. When this motion is depicted in the figure for Proposition 1, all that is seen is the overall trajectory. Merely looking at the figure is not enough to receive any aspect of the motion other than the set of locations of the body.

However, the presentation of a concept on paper provides advantages that reality cannot provide. When we observe motion in reality, the trajectory of the body is not seen in and of itself; we approximate ‘seeing’ it with our memory and imagination, but it is not ‘present’ before us in the same way the body is. The moving body is not seen simultaneously in two places. Newton, forced by the medium into creating purely static depictions, uses a line to show the locus of points making up the trajectory of the body. 

There is another species of motion present in this proposition: motion through propositional action. To complete the proposition, Newton says “let the number of the triangles [SAB, SBC, etc.] be increased and their breadth decreased in infinitum, and their ultimate perimeter ADF will be a curved line.” The changing of breadth and introduction of new elements is a spatial (and inescapably visual) change in the figure from one moment to the next; thus, motion is taking place in this imagined operation. Interestingly, this motion is not a simple moving-through-time, as was the previous example. Each ‘frame’ of such an ‘increase-in-number’ is a distinct and independent setup of the proposition. In each frame, the body moves along its specific trajectory, but each frame’s time interval between force-impulses is different, which produces the difference between each frame. This propositional motion is not a singular moving-through-time. It traverses a spectrum of similar and related ‘timelines’. This moving-through-multiple-timelines is an abstraction. Rather than having the body traverse each trajectory sequentially, the proposition has compressed each distinct timeline into its own ‘frame’ [The diagram for the proposition only shows a single timeline, with the time between the body’s movement from A to B still a non-infinitesimal magnitude.], and the imagined motion through the sequence of ‘frames’ is the operative motion. 

Abstraction

Abstraction occurs when a set of information is taken and, isolating a particular subset or aspect, only the subset is viewed. For example, if an algebraic function is plotted, it has a slope at any valid x-value. Plotting the slope-values (the subset) as y-heights creates an abstraction from the original function. The only part of the original set of information that remains to be observed is the slope, and other aspects (like the height of the original function) are lost. 

In the example from Newton, the isolated aspect is the entire flattened trajectory of a single ‘reality’ with a certain interval between each impulse. Phasing through each snapshot and allowing the triangles to increase in number and decrease in width is a motion that is not present in any given ‘reality’, yet is available in the proposition.

Newton does not present the final case, when the figure contains infinite triangles—nor could he. He could draw a curved trajectory, but he would then lose the depiction of finite polygons that he uses to make his argument. In order to more fully grasp the proposition, the reader must hold in his mind the (given) finite-triangle figure and an imagined infinite-triangle figure, juxtaposing the two.

The relationship between dynamism and stasis is central to concerns about the effect of a static medium on a dynamic concept. Representations can either directly or indirectly display motion, as is determined by the properties of the medium. This is explored in the next section, Direct and Indirect.

Direct and Indirect

The figure provided for Proposition 1 of Principia also illustrates another feature of diagrams that deserves attention, the distinction between direct and indirect representation. A feature in an illustration is represented “directly” when there is a simple one-to-one correspondence between what is shown and what is meant. The length of a line on a diagram directly presents the length of a feature of an object; the area on a map may similarly gives a direct, if scaled, presentation of an area on the ground.

Some features of what is to be shown, however, may not be capable of direct presentation and so must be given indirectly. Within a static medium like paper, any motion represented is represented indirectly, since the ink on the paper cannot move. To represent the motion directly would require a dynamic medium where the motion is visible. If the content has greater dynamism than what the medium can support, any dynamism must then be represented indirectly.

Because the static diagram can only show motion indirectly, the reader/viewer must move beyond what is actually presented to envision the motion the diagram is trying to convey. The reader must take on what we might call “depictive responsibility.” Since the reader cannot move the ink on the paper either, he must fall back upon his imagination.

Falling back upon the imagination is not necessarily a failure because in the end, often a concept is only understood when the reader holds some part of it in the mind, in the imagination. However, many systems can be challenging to imagine because of their sheer complexity. If the medium and figure can carry the “depictive responsibility” then that frees the reader to focus on matters other than depiction.

It’s quite possible that requiring the reader to carry the depictive responsibility helps develop the faculty of imagination. If that is the case, then a more limited medium would be beneficial to the larger-scale accrual of depictive and imaginative skill.

Imagination

Imagination, by its nature, is more dynamic than paper. It is capable of conceiving the dynamism of reality. However, this is a power and a danger. Since what is imagined need not obey the laws of physics, what is imagined can go beyond reality. Imagination allows the hypothesizing of alternate events, just as how diagrams can present alternate events (as explored in Adjacency and Comparison, below). However, there is little rigor in this hypothesizing. The one who imagines cannot assure himself that his mind’s imaginative capacity is accurately stepping through how the system would actually behave if the system was operating within physical laws.

Though the paper-bound diagram does not move, it is rigorous in what it does show: a snapshot, a particular state of the system. This rigor is not necessarily present in the imagination. Since there are no limits in place, imagination can misleadingly hypothesize, and ideas can come about that are not actually coincident with the capabilities of the system. Imagined systems may have inconsistencies that come to light only when one attempts to draw or write them in an unforgiving medium like ink-on-paper that ruthlessly juxtaposes earlier-and-later visions, exposing their differences.

Also pertinent is the relationship between the imagination and external mediums. Before an idea or system can be put down onto a physical, external medium, it must first be imagined to some degree. It is not a requirement that what is imagined is identical to what is put down; in fact, that’s almost certainly not the case. This is especially true for dynamic systems. The imagination formulates for itself a conception of the dynamic system that is distinct from what is translated to the static medium. This conception includes an intuitive understanding of the properties of the structural elements of the system. The medium cannot ‘show’ the intuition of the meaning for a given element. The meaning must be translated in large part into the text of the proposition, or the representation must explicitly show the meaning with elements of notation (as explored in Aspect and Dimensionality, below).

The imagination is present at both ends of the communicative process. The initial conceiver’s imagination has an idea. In order to communicate it to another, he must transfer it to an external medium (paper, speech, gesticulation, etc.). The receiver must then take his senses of the medium and from that develop the concept in his own imagination, such that if the representation on the medium was taken away, the receiver’s imagination can hold something roughly similar to what the initial conceiver held.

Though the imagination can support mild comparisons of temporally- or spatially-distinct things, direct visual comparisons are best effected through visual adjacency, discussed in the next section.

Adjacency and Comparison

An illustration can adjacently show things that are, in reality, only sequential, or entirely separate. It can place side-by-side things that, in reality, never appear together. These adjacencies can be tremendously useful, making ink-and-paper illustrations preferable to unmediated contemplation of things as they appear in reality. The author/illustrator should be aware of how best to make illustrations that take advantage of this capacity.

A moving object is never seen at both its start and end points simultaneously. A moving object’s trajectory is never directly visible. (Though an indirect method, a cloud chamber captures the trajectory of a particle with tiny droplets of alcohol vapor; in other demonstrations, one might try to capture a path by rolling balls on carbon paper or by some other means.) A static visual representation of a trajectory, however, is eminently visible. Further, since multiple, temporally-distinct moments can be presented simultaneously in a visual representation, their position relative to each other is available to be seen. This ease of comparison—possible in a static medium—is impossible in everyday reality. Parts of Proposition 1’s visible trajectory ABCDEF can be compared to other parts of itself specifically because it is static, and multiple moments in the body’s movement can be seen simultaneously. 

Since the points A and B are visible simultaneously, their relative locations can be compared. This comparison is something that we approximate with our memory and imagination, but it is indirect (in that we must hold a previous state in memory, which may itself be faulty, in order to compare it to the current state). For complicated trajectories, this process becomes arduous indeed, and ultimately impossible to imagine accurately. Paper, with enduring consistency, holds ‘prior’ and ‘current’ states present for the mind to peruse at ease.

Distinct from singular trajectories (complicated or otherwise), paper allows a further removal from reality by enabling the simultaneous depiction of alternate trajectories. Consider the diagram for Newton’s Proposition 1 in the Principia. If the body passes point B without interference, it will continue along to point c unabated. If the body is struck by a sudden impulse at point B, it will go off at an angled trajectory BC. These two trajectories are mutually exclusive (in that a body cannot be both impulsed and unimpulsed). Only one of these trajectories would occur (and therefore be visible) in reality. Since depictions on paper are not bound by the laws of reality (that is, they can show things that are physically impossible or that reality is incapable of showing), both trajectories can be shown simultaneously. This adjacency allows comparisons to be observed that would otherwise be impossible in reality. 

In order to better differentiate and compare the adjacent elements, their aspect must be communicated by the figure and/or the text.

Aspect and Dimensionality

Each object in an illustration has a variety of features and characteristics, each of which can be used to convey information. A skillful author/illustrator can manipulate each of these “dimensions” of every object in a picture to show different parts of what he or she wants to say.

Points and lines on paper have definite spatial arrangements; that is, their position is quantifiable within the dimensions of paper (the x and y axes). Information about their place and orientation is embedded in their place and orientation. [Alternately, information about place and orientation could be embedded in a non-spatial property like color or pattern]. If there are other spatial qualities that are to be communicated, they can be shown in many ways. Often this spatial information is not explicitly derivable solely from the diagram; it takes the form of descriptions in the text of the proposition:

“This line is double the length of this line”,
“this line is perpendicular to this line”,
“this point is in the center of the circle”, etc. 

Each line can have further properties than mere spatial arrangement. These “extraspatial” properties can be communicated in the text of the proposition:

“this line correlates to a magnitude of speed”,
“this line correlates to a magnitude of time”,
“this line is movable” / “this line is static”,
“this line is a hypothetical trajectory” / “this line is the real trajectory”,  etc.

Additionally, the extraspatial properties can be communicated visually within the figure itself. For example, in Proposition 1, lines Bc, Cd, De, etc. possess a dotted pattern that enables the dotted lines to be distinguished from the solid lines. A difference in patterning allows immediate differentiation. Though not used in the Principia, color is another aspect that can be correlated to spatial and extraspatial information.

A limitation of the figure as shown in Newton and Galileo is that two lines may have identical visual characteristics (save for locationand orientation: otherwise they would not be distinguishable as two lines) yet represent distinct things. For example, in Proposition 1, lines AB, BC, etc. are solid lines that represent the trajectory of the hypothetical body. Lines SA, SB, etc. are solid lines (indistinguishable in kind from AB, BC, etc.) that display the distance between the body and the center of force at equal intervals of time. These lines are of wholly separate kind, and themselves give no account of what they are representing. The figure only makes sense when examined in conjunction with the text. The text gives meaning to the lines, and the reader/viewer must hold that meaning in memory to sustain the understanding, rather than discovering the understanding from the figure independently.

Even more cluttered are the dotted lines. Bc, Cd, De, etc. are alternate trajectories, Sc, Sd, Se, etc. are distances, and the lines Cc, Dd, Ee, etc. are most convoluted indeed. They measure the deflection of the hypothetical trajectory of the body (as it would have been absent the “sudden impulse”) compared with the impulsed, actually-traversed trajectory. They bridge a chasm separating the hypothetical and the actual. They link two independent realities. If one imagines a real body orbiting and being affected by a center of force, there is no line or magnitude or entity or thing operative or present in reality that would correlate to Cc. It is merely a propositional artifice. These three types of dotted lines are similarly indistinguishable in kind, and again require memory to hold them distinct. They are not self-evident.

Aspect communicates a meaning embedded in an element. Concepts such as time can be articulated explicitly with certain species of aspect, or can be implicitly present in a figure without a dedicated element.

Explicit and Implicit

Another distinction, similar but not quite identical to that between direct and indirect presentation, separates explicit from implicit features of a diagram. Even an idea that can only be indirectly presented may be given explicitly if there is some feature of the diagram that is devoted deliberately to showing it. In other cases, the idea may be merely implicit and left entirely to the reader/viewer’s imagination. 

This distinction can be seen in the diagram for Galileo’s Proposition 1 Theorem 1 in his Discorsi e dimostrazioni matematiche, intorno à due nuove scienze. The proposition sets out to show that falling bodies traverse spaces proportional to the squares of the time elapsed. “Time” is an essential part of the proposition, but as an abstract concept it cannot be shown directly in the picture. Nevertheless, the diagram gives us an an explicit representation of time: “line AB represent[s] the time in which the space CD is traversed by a moveable”. In this case, time is again shown indirectly, in that no part of the figure moves, but time is explicitly represented in that a part of the structure is designated to correspond to time.  

Illustrations of other propositions, by contrast, have no explicit presentation of time but rely on the reader to fill in the idea which is implicitly required for the demonstration. For instance, in Galileo’s ‘experiment’, the presentation of time is quite similar to that of Newton’s propositions, in that time is implicitly included. Multiple, separate moments of the pendulum’s path are shown simultaneously. The weight is at C before it reaches D, yet C and D are shown simultaneously. Motion is described in this experiment, yet the motion must be imagined by the reader, as none is shown.

In the first example, Galileo wants a quantified conclusion about time and the traversal of space, but is only comparing motions in the second. Perhaps that is the reason for choosing explicit or implicit presentations of time. When the end goal of a proposition is to have time in a ratio or proportion, it is then necessary to use an explicit representation. It becomes more like a graph than a picture.

Since these two pairs of qualities—direct versus indirect, explicit versus implicit—are independent of one another, there are four permutations of how they can occur:

Indirect and explicit is present in Galileo’s Prop 1 Theorem 1.

Indirect and implicit is present in the illustration of Galileo’s pendulum experiment.

Direct and explicit would require a dynamic medium wherein objects can move, and time is represented in a visible structure. An animated illustration with a clock or a ‘progress bar’ showing the passage of time would be a direct, explicit presentation of time.

Direct and implicit would require a dynamic medium wherein objects can move, but time is only visible in that motion is visible.

Explicit and Implicit Representations of Time in Einstein and Minkowski

The distinction between explicit and implicit representations of time appears interestingly in a comparison of diagrams used by Einstein and Minkowski to illustrate fundamental ideas in Special Relativity.

Einstein, in Über die spezielle und die allgemeine Relativitätstheorie, presents a diagram that shows a ‘snapshot’ of a system where a train moves with a velocity v along the side of a stationary embankment.  In order to explain the idea of Relativity of Simultaneity, he describes a situation in which light from A and B will simultaneously reach a stationary midpoint M. If M' is moving at the rate v (denoted by the moving point M'), however, the observer at M' will see the light from B before the light from A. The light flashes at A and B, events that a the “stationary” observer M judges to be simultaneous, are judged not to be simultaneous by the moving observer M'. Einstein’s thesis of the relativity of time and space from different frames of reference entails that there can be no ‘snapshot’ of multiple frames of reference. 

Einstein rails against a simplistic understanding of time, yet the diagram he provides does not serve him well. Since it does not represent time explicitly, the diagram is ambiguous about which frame of reference the diagram is depicting. He equates the points A and B for the train and the embankment, but the flashes at A and B are only simultaneous for observer M; if considered from the standpoint of M', the diagram bizarrely shows two events that occur at different times as if they happened together. A and B for M are not the same as for M'. Einstein’s textual explanation is clear, but his diagram misleadingly invites the viewer/reader to imagine a single, unambiguous ‘snapshot’ of events that shows ‘what is really happening’. In Einstein’s picture, the reader has to keep tellinghimself ‘this is only from one perspective’ to remain cognizant of the truth of the system.

In Raum und Zeit, Minkowski introduced diagrams that give an explicit representation of time. If he were to present the situation Einstein is describing, it would look somewhat like the illustration given on the right. By representing time explicitly, Minkowski can clearly show both frames of reference, for they are delineated by their angle. By notating time and space in such a fashion, Minkowski makes the system far less ambiguous than it is in Einstein’s diagram. It is elementary to see that the beams of light from A and B reach the stationary observer M simultaneously, for the point MAB represents a single event, a point specified in both space and time. However, concerning the moving observer, the light from B reaches the observer M' moving at velocity v at M'B. The light from A later reaches the observer M' at M'A ,thus the light beams do not appear simultaneous to the moving observer. 

By making time explicit (using the trick that Galileo used), Minkowski’s diagram more clearly demonstrates Einstein’s insight and the behavior of his system.

Presentations of systems can have behaviour of sorts, if they are imagined to or defined to move in specified arrangements and relationships. This mechanical behaviour can be articulated even if the figure cannot move. In principle all four permutations of direct, indirect, explicit, and implicit can be produced with this method of construction.

Definite Mechanism versus Symbolic Structure

Species of diagrams can be distinguished by their “behaviour”. The level of interaction between parts of a diagram determines how it behaves, if at all.
A figure can be said to be mechanistic if it has parts that are held in arrangement with other parts according to a set of rules governing the dependencies and independencies of each part. For instance, line X, if defined as perpendicular to another line Y, is dependent upon Y. If line Y were oriented differently, line X would have a consequently different angle. This arrangement of lines can be considered a mechanism of sorts. Though this mechanism cannot move on paper, its movement can be conceived and imagined, and this “behaviour” is a consequence of its construction.
A figure can be said to have a symbolic structure if its elements, while potentially relating to each other, are exclusively symbolic elements. These figures are fully static, not only in their paper-and-ink representation, but also in their hypothetical, imagined action. 

Newton’s Lemma 7 functions similarly to a mechanism. b slides along rd, the arcs ACB and Acb are held similar, RD and rd are held parallel, etc.  As B moves towards A along ACB, other elements of the construction move in tandem, articulating, through their motion, the logic of the proof. With flexible enough materials, aspects of the lemma could be constructed physically as a true mechanism, and would operate (approximately) as the figure is only imagined to.

Galileo’s Proposition 1 Theorem 1, however, is decidedly different from a mechanism. Its elements do not move relative to each other, and they are entirely symbolic. [Though, this isn’t to say that symbolic elements and mechanical structure are mutually exclusive]. No ‘action’ happens with the figure. A physical mockup of the figure would ‘operate’ identically to the printed figure—neither would have any operation at all. A physical mockup would provide essentially identical information to the viewer.

Some mechanical systems depicted have incredibly intricate behavior. This behavior is not directly available to the reader, for the medium cannot display it. The text of the proposition and the given spatial arrangement in the figure provide a set of information often sufficient for understanding. However, one of the most important aspects of observing and interacting with physical systems is that continued interaction builds a larger understanding than what is possible through only observation. There are aspects of behavior of movable systems that are obscured when the system is shown statically.

Application and Discussion

Diagrams are a key element of the authors’ presentations. The text alone can provide a full understanding, given the investment of sufficient thought and imagination on the part of the reader. However, the authors discussed above provide more information than the text alone. They provide diagrams that can act as aids to the imagination. By scrutinizing the aspects of the visual form chosen by the author, one can gain a sense of how the author conceived of his concepts. The visual form is indicative of the visuospatial ‘language’, the graphical rhetoric, the set of tools that the author was fluent in, in order to create the diagram in the first place.

Further, the visual form chosen by the author continues to resonate within the reader. The form of the figure can influence the reader’s conception of the system and the concept.

If the medium was incapable of allowing the representation of time as a line, time would not be represented as a line. The visual language, the graphical rhetoric used by the author to articulate his conception of time would be consequently affected. That the medium can allow such a representation informs how the author thinks about the concept’s representation. 

This is true for all concepts. One’s ability to represent a given concept is deeply tied to what the medium can support. Limitations of a medium require a thinker to go out of his way to represent a concept, if an aspect of the concept is of the same type as the medium’s limitation.

Though Newton and Galileo, Einstein and Minkowski all tackle motion and time as concepts and represent the concepts in their diagrams, each utilizes a different graphical rhetoric. Some elements are common among them, like the use of distance to represent distance. Other elements are unique to each author. By nature of each having a different graphical rhetoric, each author conceives of his concepts uniquely.

The medium’s fundamental effect on possible graphical rhetorics determines in large part how an author represents his concepts. Paper, a static medium, can only support indirect representations of motion, and thus requires that the graphical rhetorics used utilize indirect representations of motion. 

Graphical representations reveal and shape authors’ understanding of time. Compare the presentation of time in Galileo’s Proposition 1, Theorem 1 in the Fourth Day of Two New Sciences with that of Newton’s Proposition 1 in the Principia. Galileo describes the parabolic trajectory of a falling object as it moves forward. As discussed above, Galileo in this demonstration provides an explicit representation of time. In fact, his postulate that a body traveling without interference will go with forward motion in a straight-line and with uniform in speed allows him to use the same line both to represent time explicitly and to show the path of the body as it would have proceeded were it not drawn downwards. Galileo also imagines that the downward motion of the falling body is always parallel with itself. This is an acknowledged fudge: Galileo was aware that “down” means “toward the center of the earth,” and that downward motion is not ever really parallel in two different locations. However, overlooking small divergences in direction over short distances allows Galileo in a single diagram to marry point-for-point the equable distances and the equable times. These artifices allow him to equate the parabola of equable-time abscissa and accelerated-motion ordinates with the parabola of equable-distance abscissa and accelerated-motion ordinates. The smooth, continuous infinitude of the horizontal time-line matches the smooth, continuously accelerating downward motion.

Newton, thinking more generally, explores the question of falling bodies with ‘down’ as non-parallel. In the first step of the proof, one might think that there is an explicit representation of time in the same manner as in Galileo’s proof: the body is imagined to move uniformly from A to B. This line AB , by beingtraced out with equable motion, equates equal distances with equal times. However, at this point, the demonstration takes a turn. At B the body is hypothesized as continuing equably to c. However, the body receives an instantaneous impulse from S, which instead results in it going to C. It is important to note that, while it is stated that the time intervals from A to B to C and so on are equal, the proposition does not specify that each inertial line AB, BC, CD, etc. has equal length. Therefore, since the line ABCDEF does not have an equal correspondence of distance and time, it is not an explicit presentation of time.

Newton’s construction of a curved trajectory relies upon a summation of distinct, instantaneous moments whose smooth continuity has to be provided in the imagination of the reader. The graphic shows the step-by-step-ness of Newton’s thinking, in contrast to Galileo’s ‘equable matching’ of continuous magnitudes. Without access to Galileo’s artifice and without an explicit depiction of time, Newton uses stepwise, additive inertial trajectories of equal time to produce the final curved trajectory. This thinking requires more intricate graphics to communicate, but is more powerful due to its generality. 

Darwin’s depiction of time is similar, yet distinct from that of Newton’s or Galileo’s or Einstein’s.

Darwin’s vertical axis is a direct representation of time. That is, as one looks upward, the increasing values of the Roman numerals correspond to later moments in time. The horizontal axis is less clear. Species divergence occupies horizontal space, but it is not rigorously represented. That is, point m4, though occupying the same vertical slice as now-extinct species C, is not claimed to be now identical to C.

Darwin treats these points as breaking the line of succession “at regular intervals by small numbered letters marking the successive forms which have become sufficiently distinct to berecorded as varieties”. However, “these breaks are imaginary, and might have been inserted anywhere”. Having the breaks occur along each horizontal time delineation is a mere diagrammatic simplification. It would be impossible to depict the true line of succession of an actual species, for there would be far too many lines and breaks to remain visually coherent and legible.

Darwin allows for the intervals of time between each Roman numeral to be variable, from a thousand generations to millions. However, this is not intended to be a rigorous number. The variety/species/genus divergences u5 and x5 need not occur at precisely the same generation or time for the content of the diagram to be communicated: “But I must here remark that I do not suppose that the process ever goes on so regularly as is represented in the diagram, though in itself made somewhat irregular, nor that it goes on continuously”.

Most curiously, Darwin allows the vertical axis to not only represent time, but also to “represent a section of the successive strata of the earth’s crust including extinct remains”. Due to the fact that the geologic strata build up in a manner generally correlating with the forward march of time, the vertical axis can represent both. With such a representation, Darwin displays an conception of time that possesses unique correlations in his mind.

Einstein’s limited depiction of multiple frames of reference injures the representability of more complicated systems, while Minkowski’s visual notation allows and easily scales to such tasks. It is important to note that Einstein’s 1905 paper had no pictures whatsoever. Was he thinking algebraically to begin with, and only included figures for his popularization, Über die spezielle und die allgemeine Relativitätstheorie, as aids to the imagination? Minkowski’s graphical presentation expands the imaginative possibilities of Relativity theory; without it, it is hard to imagine that General Relativity would ever have been conceived. General Relativity is, essentially, non-Euclidean Minkowski space-time. It is a hard idea to picture, to depict graphically, but it’s hard to even conceive of if Space-Time were not grasped geometrically.

Conclusion

By exploring a range of graphical rhetorics concerning the depiction of motion and time in a static medium, it is possible to observe details of the presentation that affect the author’s and the reader’s conception of the concept. This exploration provides a valuable vantage point—one that is often overlooked—precisely because the rhetoric of diagrams so often is to efface themselves and point instead towards the concept.

Texts Referenced

Densmore, Dana. Newton’s Principia: The Central Argument. Santa Fe: Green Lion Press, 2010.

Galilei, Galileo. Two New Sciences. Trans. Stillman Drake. Wisconsin: The University of Wisconsin Press, 1974. 

Einstein, Albert. Über die spezielle und die allgemeine Relativitätstheorie. Trans. Robert W. Lawson. New York: Penguin Books, 2006. 

H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl. The Principle of Relativity. Trans. W. Perrett and G. B. Jeffery. Mineola: Dover Publications, 1952.

Darwin, Charles. On the Origin of Species. John Murray, 1859.