# if motion in infinitely small time interval is linear then why a tangent to a circle has only one common point with the circle

I cant understand the infinitely small linear relation in calculus. I don't yet know calculus. But my physics book use the conclusions from it without explanation.

Edit as asked by someone:

Instant is an infinitely small interval of time curve can change direction during that interval. Since point is smaller than displacement at an instant. I am getting even more confused.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Jan 24 at 4:25
• I'm not sure what you think calculus says, but it doesn't say what you think. Calculus tells you that you move a tiny bit along the curve "by following the tangent line" (what exactly that means is a bit technical, but ignore that for now), and then the new point on the curve where you are has its own tangent line, which generally has a different slope than the first one did.
– Ian
Commented Jan 24 at 4:42
• You're taking intuitive descriptions more literally than you should.
– Ian
Commented Jan 24 at 4:50
• "Infinitely small interval" is a physics way of talking that has no precise mathematical counterpart. Don't expect your physics texts to make mathematical sense. Commented Jan 24 at 5:42
• @Michael, if you want to teach nonstandard analysis to a student who hasn't even had Calculus, be my guest. (And do you think the authors of the physics textbook know about ultrafilters?) Commented Jan 24 at 8:55

Although physics and mathematics are closely related, you have discovered a difference between the things physicists care about and the things mathematicians care about.

Mathematicians define numbers in such a way that no matter how small a distance you can measure with one of those numbers, there is a much smaller number that measures a much smaller distance. You can keep choosing smaller and smaller numbers to measure smaller and smaller distances, far smaller than any distance that is physically impossible to observe.

Physicists are much more concerned about the real world and what they can observe in the real world. Since every actual physical measurement is a kind of approximation, physicists use approximate notions all the time. They will be very happy if some mathematician can make their approximation notions into an exact mathematical theory, but the primary concern of a physicist is not whether what they say is completely consistent mathematically; their concern is to be consistent with reality.

You are interpreting the physics text the way a mathematician would. Mathematically, you can have two points $$10^{-100}$$ meters apart on a circle of $$1$$ meter radius, and in that distance the direction changes. A line through those two points is a secant, not a tangent. But to the physicist, the distinction between a secant of this kind and a tangent is irrelevant.

I am oversimplifying even this point somewhat; physicists tend to know the standard model of calculus well enough to explain how the tangent line actually relates (through limits) to a secant line through two points that are at a "practically infinitesimal" distance from each other. But they mostly don't seem to care about this distinction as deeply as mathematicians do. That may explain why an author of your physics textbook would consider their description of motion along a curve adequate.

Long story short: there is no such thing as "infinitely small". Either you are a finite nonzero number, or you are zero, but there is no such thing as an "infinitely small" nonzero number.

If you represent the distance you travel as a function of time, then your speed during an interval of time is equal to the ratio of distance over time: $$\text{speed} = \frac{\text{distance traveled}}{\text{time of travel}}.$$

The derivative of "distance as a function of time" is sometimes referred to, informally, as the "instantaneous speed". However, if you take the name "instantaneous speed" literally and try to calculate it with the previous formula, you inevitably get:

$$\text{instantaneous speed} = \frac{\text{distance traveled in no time}}{\text{no time}} = \frac{0}{0} = \text{nonsense!}$$

There is no such thing as instantaneous speed, and there is no such thing as an infinitely small time, and no such thing as an infinitely small distance.

However, if you measure the ratio distance/time for smaller and smaller and smaller times, you'll notice that your measures converge to a number. This number is the derivative, informally referred to as instantaneous speed:

$$\frac{\text{distance traveled in 1 min}}{\text{1 min}} \to \frac{\text{distance traveled in 30 s}}{\text{30 s}} \to \frac{\text{distance traveled in 1 s}}{\text{1 s}} \to \frac{\text{distance traveled in 1 ms}}{\text{1 ms}} \to ... \to \text{instantaneous speed.}$$

At no point in this sequence have I dealt with "infinitely small time". I've only dealt with finite nonzero times. But I used smaller and smaller times and I noticed that the ratio converged to a number.

This derivative number turns out to be extremely useful, and the whole motivation of calculus is to formalise this concept of convergence and limits and get tons of applications of derivatives.

It's the same thing for a tangent. Derivatives are very intimately linked with tangents.

If you have a point $$A$$ on a circle, and a point $$B_0$$ on the curve at a finite nonzero distance from $$A$$, then you can draw the secant $$(D_0)$$ that goes through $$A$$ and $$B_0$$. And then if you take a point $$B_1$$ on the curve, slightly closer to $$A$$, so that the distance between $$A$$ and $$B_1$$ is smaller, you get a new secant $$(D_1)$$. And then if you take $$B_2, B_3, B_4, ...$$ closer and closer to $$A$$ on the curve, you'll notice the successive secants $$(D_2), (D_3), (D_4), ...$$ appear to converge towards a line $$(T)$$, and we call this line $$(T)$$ the tangent to the circle at $$A$$. The tangent at $$A$$ only touches the circle at $$A$$, but it is the limit of the secants that touched the circle at $$A$$ and $$B_n$$, with $$B_n$$ coming closer and closer to $$A$$. For this reason some people sometimes, very informally, say that the tangent touches the circle "at two infinitely-close points", even though that is a very clumsy wording that doesn't make any literal sense.

• -1 for "there is no such thing as 'infinitely small'." Commented Jan 25 at 9:30
• Making such comments in an answer to a question specifically about infinitesimals, and listing the infinitesimals tag, is surprising. Commented Jan 25 at 12:35
• With your logic, ∞ doesn't exist either. Every number, no matter how large, is still finite. Commented Jan 25 at 12:47
• I was just pointing the non sequitur in your answer. It's the same as claiming that ∞ doesn't exist because no number is infinite. Commented Jan 25 at 13:22
• There is in fact a meaningful theory of infinitesimals, which gives just the behaviour that physicists, engineers etc. use informally (without invoking hyperreals as Mikhail Katz's answer does). The cost is that to use it we must discard the law of excluded middle. For an introduction, see math.andrej.com/2008/08/13/… Commented Jan 25 at 14:45

What you may be referring to is the historical idea of a curve as an infinite-sided polygon (with infinitesimal sides), and the idea of the tangent line as the line through two infinitely close points (such as the adjacent sides of the infinite-sided polygon). Such ideas were common in the 17th century and specifically in Leibniz. Starting with a smooth curve, one would obtain in this way the tangent line, with equation of type $$y=mx+b$$ where $$m$$ is the slope.

You seem to have heard in a physics class that the tangent direction is the direction through two points separated by an infinitesimal time interval. Such a description is in fact correct up to an infinitesimal correction in direction (which is good enough for the physicists).

In more detail: Today we have a theory of infinitesimals that vindicates Leibniz's ideas, but it turns out that one needs to introduce some modifications to 17th century procedures. Thus, the tangent line would not be $$y=mx+b$$ but rather $$y=st(m)\,x+st(b)$$ where "st" is the standard part which rounds off a finite hyperreal number to the nearest real number. So the tangent line is not literally the line through two infinitely close points on the curve, but rather the standard part thereof.

In this way, there is no contradiction with the fact that the tangent line only touches a (say, strictly convex) curve in only one point.

• Got that, sach? standard part? hyperreal? Commented Jan 24 at 8:58
• OP accepts that OP is "new one bear" , @GerryMyerson , though this technically correct answer might be useful to most users , it might not be useful to OP who might not even know what this answer is trying to convey. It is my observation & it is not a fault of the answerer here.
– Prem
Commented Jan 24 at 15:09
• @Prem They were saying "I am a new one, bear with me please" and just forgot the comma, not calling themselves a "new one bear" :P Commented Jan 24 at 15:36
• I am well aware of that , @Idran , which was why I "quoted" it ! I used it to show the level of the OP & this technically correct answer is not at that level.
– Prem
Commented Jan 24 at 15:41
• @Henry, the point is that if you have two infinitely close but distinct points, you can draw a unique line through them, and then take standard part of its slope. By contrast, in the non-infinitesimal approaches, it is impossible to determine the tangent line from a single secant line. Commented Jan 25 at 13:06

Start with simply 'small time interval' without 'infinitely'. You can approximate your movement with linear shift from start point to end; instead of tangent you then have secant through those points.

Now decrease your small interval - you will see that your secant will be getting closer to tangent. If you decrease your interval to 'infinitely small', then starting and ending points will be merged and you see first that your movement approximation (from point to point) becomes precise and second your secant line becomes tangent.

I’m not sure what exactly your question is. You should learn calculus with rigor before doing physics, and that is what I recommend you do.

However, if you insist on relying on intuition, here’s the best I can do.

Imagine swinging a ball on a string in a circle above your head. If you let go at any point, what does the ball do? It travels in the direction of the tangent line. You can try this physically and you’ll see a rough version of it, but you should see it. That’s what the derivative of a curve tells you, what direction the curve is going at an instant.

• If physically it is working then mathematically it means tangent can have more than one point of contacts with the circle please read all the comments to the question Commented Jan 24 at 6:14
• @Malady Many American universities and high schools have "non Calculus based" physics courses intended for non-STEM majors who will haven't taken or may never take Calculus. Commented Jan 24 at 10:14
• Yeah I don’t think those classes are good. Calculus was created to do physics. You really should learn it. There’s no way to understand things like uniform circular motion without it. I don’t know though. Commented Jan 24 at 13:03
• Calculus was created to do physics, but the infinitesimal version was good enough tor a couple of centuries of physics research, and it has mostly been mathematicians, not physicists, who worried about it not being rigorous enough. Today, it can be formulated rigorously without resorting to the less intuitive epsilons and deltas. I do agree that the effort of doing that is worthwhile, as it clarifies questions like this one. Commented Jan 25 at 15:06

If you haven't taken Calculus then I'm guessing Michael's answer is a bit too technical to make full sense to you. Unfortunately, a fully coherent, solid understanding to your question requires not only Calculus but a university senior course called Real Analysis.

By infinitesimal, I guess that you are referring to $$dx$$ or $$dy$$. Consider an intuitive, less formal explanation to what is $$dx$$? Imagine $$dx$$, as the smallest possible positive real number. See ** below. In practice, $$dx$$ is primarily considered in context of a ratio, $$\frac{dy}{dx}$$. This ratio is in many ways a slope.

Regarding, "Ïf motion in an infinitely small time interval is linear then why does a tangent to a circle have only one common point with the circle?" It is possible for a line to touch a circle at exactly one point. This has more details.

By definition, to be tangent means to contact at only one point. If a line contacts a circle at two points, it would be called a secant line.

Motion in an infinitely small time interval doesn't have to be linear. However, the real analysis course concludes that if we include some other concepts like limits, assuming $$dx$$ is linear works out the same even for an arc. However, in practice $$dx$$ is always linear.

So small that the length of $$dx$$ and a related arc on the circle in some contexts are

Regarding learning calculus by rigor before attempting physics, many students have found learning physics alongside calculus helps to understand the other topic better. I'm guessing you are in a university physics course and do not have an option to add Calculus to your courseload this semester. Possibly your degree program does not even require Calculus.

** In reality there is no smallest possible positive real number. For any "$$dx$$" we can take "$$dx\div 10$$" to get a smaller number.

• Obviously my answer is not rigorous or entirely mathematically sound. Unfortunately, any rigorous answer I know of is likely to create more confusion. With all due respect, and no offense intended, Mikhail answers are likely to confuse a student who hasn't started Calculus. Commented Jan 24 at 10:11

I think so. You are considering circle and tangent line which both have domain of real numbers. Thus tangent to a circle has only one common point with the circle.

For any curved graph, such as a circle, there is never a point where the graph is perfectly linear, no matter how deep you zoom in. You probably understand that a horizontal interval of a graph is enclosed by $$2$$ endpoints, one on the left and one on the right. Now, an "infinitely small interval" is so small that both the left and right endpoints of the interval merge together and become one. This means that any infinitely small interval has only 1 point in it. Therefore, the tangent line only intersects one unique point of the graph.