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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    $\begingroup$ So Newton’s work was not ‘real-world’? $\endgroup$
    – Jon Custer
    Commented Apr 21 at 16:38
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    $\begingroup$ I passed your question unchanged to chatGPT, which gave me an answer I agree with: computer graphics. $\endgroup$
    – M. Lonardi
    Commented Apr 21 at 19:22
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    $\begingroup$ If the point is that mathematics might take time to come to fruition then shouldn't you be interested in late applications rather than the earliest ones? Those that came long after the original discovery and were not a motivation when it was made (as in Newton's case). Linear error-correcting codes would be an example of that. $\endgroup$
    – Conifold
    Commented Apr 22 at 1:28
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    $\begingroup$ Regarding linear algebra: solving systems of linear equations dates back to at least the 2nd century CE, as in Chapter 8 of the Chinese text "The Nine Chapters of the Mathematical Art". Neither calculus nor linear algebra are good examples of mathematics initially developed only for its own sake; both were developed to solve concrete problems - cf. e.g. problem 8-1 of the above text. A better (and common) example is the number theory now used for the cryptography that underpins e-commerce. $\endgroup$
    – Alexis
    Commented Apr 22 at 2:11
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    $\begingroup$ Cross-posted: matheducators.stackexchange.com/questions/27741/… $\endgroup$
    – user12357
    Commented Apr 22 at 14:50

3 Answers 3


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

These almanacs included ephemerides, that is predicted positions of various astronomical bodies (sun, moon, planets, stars), which would have been available, 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.

On a broader perspective, astronomical navigation benefited from 3 major improvements during the 18th century:

  1. invention of the sextant
  2. greatly improved astronomical predictions (calculus with perturbation theory)
  3. invention of the marine chronometer

However, the availability of marine chronometers was, for decades after their invention, limited by their very high cost, and many ships persisted in using the lunar method instead.

It is possible to claim that improved ephemerides made the lives of sailors better (or at least longer...), as they increased the probability of not getting lost at sea and 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 seafaring applications of astronomy.

Side note: As an example of the dire consequences of being unable to compute an accurate position for a ship (or fleet for that matter), consider 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.


Applications of calculus actually predate its formal invention. For example, computation of volumes. One can argue whether Archimedes computations of volumes and areas have "real world applications", but Kepler's book on volumes certainly has, which is evident from its title: "Stereometry of wine barrels". In this book he considers a real life problem: how to find from few measurements of a barrel how much wine it can contain.

Speaking of linear algebra, this name appeared very late, but mathematicians used it very long before (without knowing that they are using linear algebra). One of the earliest results in "linear algebra" is that addition of 2-dimensional vectors is commutative. This is a famous theorem of Apollonius (2 century bc). Its real world application was to astronomy: "epicycle description is equivalent to eccentric description", this is how Ptolemy states it.


Maybe I will write the obvious answer:

Isaac Newton and Gottfried Leibniz are said to have invented calculus independently (even if there is a priority dispute). Both were working on the math but also on applications.

Isaac Newton used calculus to reformulate the concepts of motion and dynamics (position, velocity, acceleration, force) in terms of derivatives (or fluxions as he called it, motion in this form was already discussed in Newton's founding work De Methodis Serierum et Fluxionum published in 1763). He also applied calculus to describe the motion of astronomical bodies and in optics. These steps led to the development of what we now call classical physics (and all of its derivatives like planetary motion prediction and mechanical engineering). Newton's Principia may have avoided to use calculus, but there is some of it, for more details check answers here: Why is calculus missing from Newton's Principia?

As for Leibniz, I cannot provide a complete picture of everything he did but he certainly applied calculus to physics like: the brachistochrone problem (Newton also got a solution for it), optics, and rigid beams calculations (see here).


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