When first encountering slope fields in calculus or elementary differential equations, students often ask "What is the purpose?"

A concise answer is that slope fields provide a way to graphically represent a first-order ODE of the form y' = f(x, y). Graphical representations can provide a range of insights regarding the overall behavior of solutions, at a glance. The same cannot often be said about symbolic representations.

This answer would be more compelling if it were possible to point to examples of the use of slope fields throughout history. However, I've had some difficulty charting this history.

The essential idea of a slope field, namely that y' = f(x, y) indicates the slope of the solution passing through any (x, y) that we choose, must have been clear since the beginning of the calculus. Supporting examples can be found in this Twitter exchange.

However, the earliest instance of (or reference to) an actual slope field that I've found so far was around the middle of the 20th century. See, for example, this 1944 note on the Teaching of Differential Equations from the American Mathematical Monthly.

Apart from the Twitter exchange above, I've searched the web, Google Scholar, and my personal library (including many textbooks and a couple of math history books).

Note: I did see the Stack Exchange question on the "Invention of the concept of vector field," but the concept of vector field is more general, and it's difficult to isolate any reference to slope fields from the answer to that question.


What is the earliest instance of a slope field as a graphical representation of the first-order differential equation y' = f(x, y)?

Here I refer not to the underlying idea of a slope field, but rather an actual picture, similar to what is now commonly found in introductory textbooks on elementary differential equations.

I would be happy to see any early examples, not necessarily the very earliest. I would also be happy to see early occurrences of the term "slope field" or "direction field," since these indicate the existence of actual pictures.


Progress! The question isn't settled yet, but I've discovered some new evidence. Starting from references in the paper by Tournès posted by Blåsjö below, I was able to find the following two sources that add to the narrative.

First, consider the book Graphical Methods by Carl Runge (based on lectures given in 1909-1910, published in 1912). Starting on p. 134 (numbered as p. 120 in the physical copy) he discusses the graphical solution of first-order ODEs.

In figure 88, he shows the same method as Bernoulli pictured in Blåsjö's answer, with one addition: he draws a "pencil of rays," which is basically a set of vectors. Since the direction vector indicated by the differential equation is the same for every point on a given isocline, that isocline corresponds to a unique direction vector. The pencil of rays depicts exactly those vectors, and they are labeled to indicate the isoclines to which they correspond.

While Runge evidently has a reason for doing this (he uses the pencil of rays to effect a change of coordinates later on), it's more difficult to quickly sketch multiple solution curves by visual inspection using this graphical representation than it is with a modern slope field, in which the directions are drawn on top of the isoclines or arranged uniformly across a rectangular grid. I did not find any occurrence of a slope field in its modern form, and neither "slope field" nor "direction field" appears in the text.

Second, consider the book Guide to Graphical Computing by Rudolf Mehmke, published five years later in 1917. Mehmke was aware of Runge's book (he cites a 1914 edition on p. 98). He writes the following (translated from German using Google Translate, with a little editing - I do not read or write German).

p. 116 (I've substituted equations for equation numbers in the original text):

...differential equations $F(x, y, \frac{dy}{dx}) = 0$ and $\frac{dy}{dx} = f(x, y)$ can be inferred from the geometric sense that they (generally speaking) assign a specific direction to each point of the plane, which must have the tangent of the integral curve going through the point there, or the presented differential equation requires, as one can say, a "directional field".

p. 121 (footnote omitted here):

If the current direction for a sufficient number of points is indicated by short dashes on... a series of isoclines (see Fig. 107), then any number of integral curves can already be drawn in a reasonably correct way according to the eye.

This is the exactly the modern idea (with the aid of computers, we often space these "dashes" uniformly along a rectangular grid, but I suppose that drawing them along isoclines only can also be considered modern). Moreoever, modern terminology is employed (perhaps even coined).

The evidence presented above appears to indicate that the modern conception of a slope field emerged between 1912 and 1917. Of course, there is the (somewhat unlikely) possibility that Runge used the modern form but did not present it in his book on graphical methods. There is also the possibility that Runge didn't use slope fields in their modern form, but others did.

Updated Questions

  1. Can anyone find an instance earlier than 1917, or any other corroborating evidence that modern slope fields did in fact originate close to the year 1917? The evidence is suggestive but not yet convincing.

  2. For the sake of completeness, it would be nice to see early instances of slope fields with segments uniformly distributed across a rectangular grid, as is now common.

Thanks for everyone's help so far!

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    $\begingroup$ I looked at a few older (before 1910s) ODE books on my shelves and found this from 1897 (see also Fig. 4 on p. 48 of the 1902 edition). Older math books, especially before 1900, had very few diagrams except those necessary to follow the text discussion. If it helps, old differential equations books often have diagrams where envelopes and singular solutions are discussed. I looked in Boole's well known (in 2nd half of 1800s) ODE treatise and didn't see anything. $\endgroup$ Jan 29, 2019 at 12:29
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    $\begingroup$ @DaveLRenfro Thanks! Sounds like usage was uncommon at least until 1900. In section 5 of your reference titled "Geometrical meaning...", it's stated that "Through every point on the plane, there will pass a particular curve, for every point of which x, y, dy/dx will satisfy the equation." This is close to an explicit description of a slope/direction field, but the term is not used. I just checked Boole; didn't see any such discussion. We seem to be narrowing it down! I'm starting to suspect that pictures of slope fields, and perhaps the term itself, is quite recent. $\endgroup$ Jan 30, 2019 at 10:19
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    $\begingroup$ I think we should distinguish the idea of slope fields from the modern looking pictures of them. Arguably, the idea appears at least in Poincare's seminal 1880-s papers on qualitative theory of ODE (I did not have time to check for pictures). I would not expect it to appear much earlier. "Vector field" only facilitates things if it is a known entity with understood behavior, and that did not happen until Gibbs and Heaviside built a home for it in 1880-s. This is probably why D'Alambert, Euler, Poisson et al., do not have them even in fluid mechanics, which seems (to us) to cry out for them. $\endgroup$
    – Conifold
    Jan 31, 2019 at 10:19
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    $\begingroup$ @Conifold It's possible to draw small segments with slopes given by y' = f(x, y) without having a fully developed notion of a vector field (or even of a vector). For example, Mehmke (see link in question) talks about indicating direction with "short dashes." Not sure about Poincaré; I think you're referring to this work. $\endgroup$ Jan 31, 2019 at 10:54
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    $\begingroup$ Right, thank you. Which is why I was so struck by the absence of segments or arrows before 1900-s where they would seem to us so natural. In addition to fluid mechanics, there is a long tradition, going back to Huygens, of drawing characteristics for (we would say) first order PDE, which is, of course, equivalent to ODE slope fields. Hamilton and Jacobi perfected it, and Hamilton invented "vector", yet I am not aware of the ODE/vector field/flow line connection, either in the text or pictorially, there either. $\endgroup$
    – Conifold
    Jan 31, 2019 at 19:04

1 Answer 1


Johann Bernoulli explains the idea of a direction field quite explicitly (Modus generalis construendi omnes aequationes differentiales primi gradus, Acta Eruditorum, November 1694). He focusses on drawing isoclines rather than slope segments. There is no figure in that work but Bernoulli drew an example in his correspondence:

enter image description here

Corresponding to:

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Dominique Tournès, Résolution graphique des équations différentielles, discusses this and many other historical examples.

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    $\begingroup$ This is certainly a graphical representation of an ODE and a useful answer, but personally I'd shy away from calling Bernoullis "directrices" a slope field. But I have not read Bernoullis article you cite, so maybe he does indeed explain the slope field in words there. $\endgroup$ Jan 30, 2019 at 12:13
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    $\begingroup$ @MichaelBächtold Yes. Very relevant - the article indicates that what we now call isoclines were introduced by Johann Bernoulli in 1694 and were applied by Van der Pol in his famous equation. However, it does not appear to treat slope fields directly. I still wonder when slope fields with segments originated or were popularized. If anyone can access the 1944 reference in my original post to put an upper bound on the introduction of slope fields, that would help (that article refers to a "direction field," but I haven't been able to access a full copy yet to see the figures). $\endgroup$ Jan 30, 2019 at 13:02
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    $\begingroup$ The curves in Bernoulli's figure are constructed by drawing line segments from one isocline circle to the next, the slopes of the segments being determined by the slope value associated with the isocline in question. So they function very much like a slope field and this is Bernoulli's reason for introducing them. The difference with a modern slope field is cosmetic in my opinion. $\endgroup$ Jan 30, 2019 at 13:19
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    $\begingroup$ From a modern perspective, once one has understood things, many differences seem cosmetic. I do think there is a genuine nontrivial step involved in passing from the directrices to slope fields. To make an analogy: the difference between visualising a function of two variables by a surface in 3d or representing it by isoclines in 2d might seem cosmetic, but I doubt if it was so obvious the first time people came up with these visualisations. $\endgroup$ Jan 30, 2019 at 14:13
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    $\begingroup$ What strikes from browsing Tournes' paper, as well as other other early examples of "vector fields" (in Euler, D'Alambert, Poisson, etc.) is the absence of said fields. What we would call flow lines are prominent in the drawings, as well as the components in the text, but they are not bundled into wholes, nor are there arrows in the drawings to link those wholes to. The earliest such thing in Tournes appears to be in Schreiber from... 1922. I wanted to check Poincare's 1880-s papers, but didn't get around to it. Are there earlier examples? $\endgroup$
    – Conifold
    Jan 31, 2019 at 0:02

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