The Chesterfield projectFitting one number to Stephenson's experiments
Turn two sentences into data, build a model with one unknown, and measure how wrong it is.
First the learner checks Smiles's units. A gradient of 1 in 260 means rising one foot for every 260 feet travelled, and a mile is 5,280 feet, so that is 5,280 divided by 260, or 20.3 feet in the mile: "about 20", as Smiles says. Then comes the model. On level track an engine only has to overcome friction, which the learner writes as a fraction r of the train's weight. On a slope it also has to lift the train, which costs the gradient G. So the share of power spent on the climb is G divided by G plus r. There is one unknown, r, and two sentences to learn it from.
Share of an engine's power spent climbing, Smiles's statements against one-number models, our Python run, 28 September 2026| Model | At 1 in 260 (Smiles: half) | At 1 in 100 (Smiles: at least three quarters) |
|---|
| r learned from the first sentence (1 in 260) | 0.500 | 0.722 |
| r learned from the second sentence (1 in 300) | 0.536 | 0.750 |
| r fitted to both by least squares (about 1 in 274) | 0.513 | 0.733 |
| Unit check | 20.3 feet per mile | 52.8 feet per mile |
Using only the first sentence, r comes out at exactly 1 in 260, but then the model predicts that a 1 in 100 climb uses 72.2 percent of the power, less than the "not less than three fourths" Smiles reports. Using only the second sentence gives r of 1 in 300, and then the model says the gentler slope takes 53.6 percent, not half. The two statements cannot both be exactly true in this model. So the learner does what machine learning does: defines a loss, the sum of squared errors across both statements, and searches for the r that makes it smallest. The answer is about 1 in 274, which predicts 51.3 percent and 73.3 percent.
The interesting part is interpreting the result. "Not less than three fourths" is not a measurement but a floor, so strictly the fitted model breaks it slightly. Either Stephenson's figures were rounded, or the simple model leaves something out, such as air resistance, which grows with speed. A good data scientist reports both the fitted number and the doubt, rather than hiding the conflict. Finally the learner uses the model to predict an untested case, a 1 in 50 climb, and gets about 85 percent, then writes down why that prediction deserves less trust than the fitted points.
Ages 8 to 11
Roll a toy car down ramps of different steepness and chart how far it goes.
Ages 11 to 15
Code the one-number model in Python and try values of r by hand.
Ages 15 and up
Define a loss function, search for the r that minimises it and explain why the data disagree.
Smiles's figures, our model
The quotations come from Project Gutenberg's edition of Samuel Smiles's Life of George Stephenson and of his Son Robert Stephenson. The model, the fitting and the predictions are our own work.
