On this page
Every JEE and CBSE physics student is told that the area under a velocity-time graph gives the distance travelled. Few are told that the idea is about 650 years old, or that it began as a picture drawn by a French scholar who never saw a stopwatch. This post covers who Nicole Oresme was, what his graphs actually showed, a worked example in his style, and the graph mistakes that still cost students marks.
Who was Nicole Oresme?
According to the Stanford Encyclopedia of Philosophy entry on Oresme, he was born around 1320 in Normandy, took his master of arts at the University of Paris by 1341/42 and later taught theology there. From 1362 he served the future King Charles V of France, became dean of Rouen Cathedral in 1364 and was made bishop of Lisieux in 1377. He died on 11 July 1382.
He wrote on economics and astronomy as well as mathematics. The same entry notes that he argued the Earth could rotate daily on its axis, yet in the end affirmed his belief in a stationary Earth. Oresme separated "this can be argued" from "this is true".
What did Oresme's graphs show?
His key work for physics is the Tractatus de configurationibus qualitatum et motuum (On the Configurations of Qualities and Motions). The SEP entry explains the set-up. A horizontal base line (longitudo) stands for time. Vertical lines raised on it (latitudines) stand for the speed at each instant. Join the tops of those lines and you get a figure. Constant speed gives a rectangle. Uniform acceleration from rest gives a right triangle.
The MacTutor biography of Oresme describes this as finding the logical equivalence between tabulating values and graphing them, and proposing a graph for a variable magnitude whose value depends on another variable. That is exactly what you do when you plot v against t.
MacTutor also quotes the historian Marshall Clagett's warning that Oresme has been credited with inventing analytic geometry before Descartes and the law of free fall before Galileo, and that none of these claims is, in fact, true. What he did was still remarkable: he turned a change you cannot see into a shape you can measure.
How did he prove the mean speed rule?
Scholars at Oxford in the 1330s had stated what is now called the Merton rule: a body moving with uniform acceleration covers the same distance, in a given time, as a body moving steadily at the speed it has at the midpoint of that time. Both MacTutor and the SEP credit Oresme with the first proof, and he did it with his figures.
The argument is pure geometry. Draw the triangle for uniform acceleration. Draw a rectangle on the same base whose height is the speed at the middle instant. The small triangle the rectangle cuts off at the top right exactly fills the gap at the top left. So the two figures have equal area, and equal area means equal distance.
A worked example in Oresme's style
A car speeds up uniformly from 4 m/s to 16 m/s in 6 s. How far does it travel?
Draw the figure. The v-t graph is a straight line from (0, 4) to (6, 16). The shape under it is a trapezium: a rectangle 6 s wide and 4 m/s tall, with a triangle on top whose height is 12 m/s.
Area by pieces. Rectangle: 6 × 4 = 24 m. Triangle: ½ × 6 × 12 = 36 m. Total 60 m.
Oresme's mean speed. The midpoint speed is (4 + 16)/2 = 10 m/s. A car moving steadily at 10 m/s for 6 s covers 60 m. Same answer.
Check with the equations. The acceleration is (16 − 4)/6 = 2 m/s². Then s = ut + ½at² = 4 × 6 + ½ × 2 × 36 = 24 + 36 = 60 m.
The NCERT Class 11 chapter Motion in a Straight Line states the modern version in its summary: the area under the velocity-time curve between two times equals the displacement in that interval.
Which graph mistakes does this help you avoid?
Reading slope when you need area. On a v-t graph, slope is acceleration and area is displacement. Say which one the question wants before you touch the graph.
Averaging speeds blindly. (u + v)/2 is the average speed only when acceleration is uniform. For a journey at 20 m/s for 1 hour and 60 m/s for 1 hour the average is 40 m/s, but for 20 m/s over one kilometre and 60 m/s over the next it is not.
Ignoring area below the axis. When velocity goes negative, that area counts as negative displacement. Distance adds the sizes; displacement adds the signs.
Mixing up x-t and v-t graphs. Check the label on the vertical axis first. Our guide to reading physics graphs has more practice on this.
The SEP also credits him with the first proof that the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... grows without limit, by grouping terms: 1/3 + 1/4 and 1/5 + 1/6 + 1/7 + 1/8 each exceed 1/2, and so on forever.
What can a student take from Oresme's way of working?
Turn a rule into a picture. Before you use an equation of motion, sketch the v-t graph and see the answer as an area.
Prove it two ways. Oresme had geometry; you have geometry and algebra. When both give 60 m, you can trust the answer.
Separate what is possible from what is true. He argued for a rotating Earth and still did not claim it was proved. In your own work, mark which steps you have checked and which you assumed.
The SEP entry notes that some scholars believe Oresme's graphs influenced later work on motion, including Galileo's; you can read about Galileo's own experiments in our post on Galileo's inclined plane. If you want to watch the area under a v-t graph build up as a body accelerates, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Try it today: take three kinematics problems you solved with equations and solve each again by drawing the v-t graph and finding its area.
