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I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and Linear Algebra were. By real-world application, I mean a device, instrument or technology which made lives better and would have been simply impossible without Calculus or Linear Algebra.

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    $\begingroup$ You might get more replies if you move this question to hsm.stackexchange.com, history of science and mathematics se. I would recommend that. (I love the question,but it would fit better there.) $\endgroup$
    – Sue VanHattum
    Commented Apr 21 at 15:27
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    $\begingroup$ I think that "simply impossible without Calculus or Linear Algebra" is a very strong assertion. For example, who in the world would say that somthing like the Antikythera mechanism could exist without the "Calculus and Linear Algebra" that we know today? $\endgroup$
    – Pedro
    Commented Apr 22 at 3:59
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    $\begingroup$ Cross-posted: hsm.stackexchange.com/questions/17436/… $\endgroup$
    – user12357
    Commented Apr 22 at 14:49
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    $\begingroup$ The answers raise the question in my mind, what do you mean by "calculus"? Do you mean the formal system of calculation that began (publicly) with Leibniz? Or do you include problems now solved by calculus that were originally solved by other forms of analysis (often by clever geometrical arguments)? Or perhaps you mean something else? $\endgroup$
    – user1815
    Commented Apr 22 at 16:20
  • $\begingroup$ Or you can change the context and use the example of Riemannian geometry, which was used by Einstein to formulate General Relativity which was used to give us exact timekeeping and location determination via GPS satellites. $\endgroup$ Commented Apr 23 at 10:56

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I want to find what the earliest real-world applications of Calculus were. By real-world application, I mean a device, instrument or technology which made lives better.

Let's assume that:

  • Earliest = Close to the date of the "beginning" of Calculus.
  • Beginning of Calculus = date of the first publication on the subject (1684, [1]).
  • Technology = practical use of scientific discoveries.

In this sense, I would mention the following work by Daniel Bernoulli (who attempted to answer whether inoculation against smallpox should be encouraged even though it can cause death):

In the first, which gives the idea, we read

I am tempted to believe that by abandoning in England the usage where one was to inoculate the newborn infants, one is less complied to the general good of humanity, than to the fear to decree this method next to the vulgar, who would impute to it without examination, the ordinary accidents at that age. One day will come perhaps where one will not be forced to these fatal variations. We will be able to enjoy then all the advantages that inoculation offers us; & one will be astonished to have them so long neglected.

In the second, which gives the math, we read

... ce qui fait $\frac{mn}{mq-1}dq=dx$, dont l'intégrale est $n\ln(mq-1)=x+C$.

Further information can be found in [2], where we read

For him there was no doubt: inoculation had to to be promoted by the State.


[1] Gottfried Wilhelm Leibniz. Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus. Acta Eruditorum (Oct. 1684), p. 467-473, 1684.

[2] Bacaër, N. (2011). Daniel Bernoulli, d’Alembert and the inoculation of smallpox (1760). In: A Short History of Mathematical Population Dynamics. Springer, London. https://doi.org/10.1007/978-0-85729-115-8_4


Of course, whether this would be “simply impossible without Calculus” cannot be answered.

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The Mercator projection (1569).
This map projection revolutionized naval navigation.

The vertical coordinate $y$ depends on the latitude $\varphi$. In modern language, it is the integral of the secant function. But 1569 was before the notion of "integral" from Newton & Leibniz. Nevertheless, those guys back then did come up with the formula $$ y(\varphi) = R \ln\left[\tan\left(\frac{\pi}{4} + \frac{\varphi}{2}\right)\right] . $$


There is a nice COMAP module on this. Indeed, many of the COMAP modules sound exactly like what you are looking for.

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    $\begingroup$ How a thing done without Calculus illustrates something that "would have been simply impossible without Calculus"? $\endgroup$
    – Pedro
    Commented Apr 22 at 4:24
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    $\begingroup$ @Pedro one might argue that this was an early discovery in the field we now call Calculus, and one with immediate application. $\endgroup$ Commented Apr 24 at 19:55
  • $\begingroup$ @StevenGubkin Under the criterion "a thing that we now call Calculus", 1569 does not seem to be "early". $\endgroup$
    – Pedro
    Commented Apr 25 at 11:04
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By real-world application, I mean a device, instrument or technology which made lives better and would have been simply impossible without Calculus or Linear Algebra.

In the case of calculus, I don't think you're going to find anything that concrete for hundreds of years. Newton and Leibniz didn't have that type of application in mind. They were interested in abstract and theoretical problems in physics and astronomy. And in fact when Newton wrote up the Principia for publication, he translated everything into the language of geometry, because his audience didn't even know algebra. People weren't building widgets that required calculus in the 17th, 18th, and 19th centuries. Slide rules didn't even become common until the 20th century. People probably made use of Taylor's theorem as far back as the 18th century to do things like calculate trig tables and tables of logarithms, but those aren't widgets like you're talking about, they're just more math that would only later indirectly be used to build a bridge or a steam engine.

In the case of linear algebra, nobody actually formulated it as a separate subject until very late. Hermite died in 1901. Banach formalized the definition of a vector space in 1920. Before then, they had systematic methods for solving linear equations, and they knew about determinants, but there was nothing like the subject the way we describe it now.

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  • $\begingroup$ Newton and Leibniz were not interested in applications? On the contrary basically every mathematician before 1900 was very much motivated by applications. $\endgroup$
    – Dirk
    Commented Apr 27 at 8:19
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Maxwell's equations (1861) are a set of coupled partial differential equations. They led to Heinrich Hertz demonstrating radio waves in 1887. Marconi sent messages to British battleships in 1899.

I suppose someone could have discovered radio transmission without knowing Maxwell's equations. Calculus is used extensively in airplane design but birds and the Wright brothers didn't use it.

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In the late 18th century, ships leaving Europe for other continents routinely brought with them books such as trigonometrical tables and astronomical almanacs.

Those almanacs included ephemerides, a.k.a predicted positions of various astronomical bodies (sun, moon, planets, stars), which would have been possible but certainly much less accurate and reliable without calculus. These predicted positions greatly improved the ability of contemporary navigation officers to accurately compute the current positions of their ships. This certainly predates any sort of electromagnetic/radiocommunication applications of calculus.

It is possible to claim that the improved ephemerides made the lives of sailors better (or at least longer...), as they increased the probability of not getting lost at sea and thus surviving the trip. The fact that naval powers of the time such as England and France were willing to fund costly and sophisticated astronomical observatories had a lot to do with the seafaring applications of astronomy.

Side note: More broadly, it is also possible to claim that all modern technology was developed between 1700 and now, which suspiciously coincides with the development of Newton/Leibniz calculus. A sample exhibit for such a claim being the Scilly naval disaster of 1707 with over 1,400 casualties. The lost ships belonged to one of best navies of the time and had professionally trained navigation officers on board. And still, they struck the rocks while almost in their home waters.

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  • $\begingroup$ A lot of great answers here, but I think this one is the best. $\endgroup$
    – BonsaiOak
    Commented Apr 23 at 16:01
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Maybe Archimedes' figuring out that the volume of a sphere is 2/3 the volume of the smallest cylinder surrounding that sphere? This work was performed some time in the third century BCE.

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    $\begingroup$ How did this discovery make lives better? $\endgroup$
    – Pedro
    Commented Apr 22 at 3:12
  • $\begingroup$ @Pedro: I was thinking in terms of how long it took to come up with the math to explain why it is so, and not just the answer itself. $\endgroup$ Commented Apr 24 at 1:22
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I tried researching it (online, not exhaustively). The earliest I could find for a real technological (economic) benefit of calculus was structural (civil) engineering as early as the 1700s with beam loading calculations.

Certainly, the advent of electrical engineering (or E&M) in the late 1800s, showed significant usage of calculus and ODEs in technological applications. However, I think there were likely more (but am not strong on early engineering and science/technology). For example, I suspect ballistics and artillery used calculus in creation of ranging tables. But I was unable to verify that.

P.s. Comments: I agree that this is a fascinating question. And non-trivial to answer. Definitely would post it at HSM instead...those gu...er...people are awesome. Here, the community is smaller and not really as strong (on a topic like this). Also, you will tend to get answers that are not really question-responsive--for instance the Archimedes thing was, yes, an "application", but NO, not one of economic/technological impact. I would also post it as two separate questions (the math fields are very different, in timing, content, and applications).

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  • $\begingroup$ The first example might also connect quite well to linear algebra. Once you arrange several beams into a truss, analysing the respective forces between them becomes a big system of linear equations and questions like if it is an over- or under-determined system are quite important. $\endgroup$
    – mlk
    Commented Apr 23 at 9:52
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I'd like to offer you another idea, to look at the question from the Methodology POV. IMHO most of the mathematical/science solution have had some application before they were understood, and then explained, classified, etc. If we accept this as a historical fact, the answer of the question should be "unknown"/or "no solution:, like in a maths problem which is a valid answer when there is no other definitive answer and a solution can't be found from the given data/

P.S. I ask you to excuse me for not using the correct mathematical terms and lingo, slang. I haven't learned maths in English, yet...

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  • $\begingroup$ This is a good point, but would probably make more sense as a comment. Thanks for joining us. $\endgroup$
    – Sue VanHattum
    Commented Apr 23 at 16:12
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It's easier to say what is not touched by calculus or linear algebra. All physics after Newton involved Calculus.

Your touchscreen, for example requires an understanding of capacitance, and simple capacitors are governed by dv/dt=1/C i, since capacitance is defined as the charges stored per volt applied: C=q/v. Inductors, like the fuel injectors in your car, follow a similar law. Any modern engineering feat requires computer simulation, in which the physical forces (usually involving derivatives) are then approximated. This leads to large systems of equations to solve, whether you are using Finite Difference, Finite Volume, or Finite Element (Galerkin) methods.

This is also how meteorologists are able to make predictions about weather: they are solving a modified Navier-Stokes system.

In supervised machine learning, you tell the machine what is "good" and "bad" categorizations, and then it learns to build a model for prediction that minimizes the error with respect to your labels. This minimization process is then in the hands of calculus.

Audio analysis, filters are based on selectively diminishing certain frequencies of audio to shape the sound without changing the actual notes. In an analogue filter, this is done with resistors, capacitors, and inductors. These last two, as mentioned above, are governed using equations from calculus. In digital filters, the sound information is decomposed using Fourier Analysis into fundamental frequencies, and you can digitally modify what frequencies to highlight or attenuate freely. Applied Fourier Analysis is part calculus and part linear algebra.

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I nominate search engines. Algorithms like page rank are excellent applications of linear algebra.

I also nominate neural networks. Deep learning and neural networks are underneath generative AI, like large language models. Training a neural network involves gradient descent in order to minimize a loss function--so calculus. And then just like the air we breathe is 80% nitrogen, it seems like 99% of what happens in an LLM involves matrix multiplication, vector embeddings, etc.

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    $\begingroup$ How do these applications qualify as "earliest" as required by the question? $\endgroup$ Commented Apr 22 at 22:50
  • $\begingroup$ @JochenGlueck These are upper bounds. Since linear algebra was not formalized as a subject until about 1900, we have a lower bound. Back propagation with gradient descent in artificial neural networks emerged around 1974 (Werbos). But at that early time, it did not meet the OP requirement of "device, instrument or technology which made lives better." $\endgroup$
    – user52817
    Commented Apr 22 at 23:37
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Back in Greece long ago around 400 to 200 BC by Eudoxus and Archimedes according to Wikipedia calculating the area under a parabola for instance.

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