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You write 27.847 without thinking. Stevin's readers in the 1580s would have written 27 847/1000 and done the arithmetic with fractions. Simon Stevin argued that tenths, hundredths and thousandths could make all business arithmetic as easy as working with whole numbers. The next year he proved a law of mechanics with a drawing so simple it needs almost no algebra. This post covers both, with worked examples.
Who was Simon Stevin?
The MacTutor biography records that he was born in Bruges in 1548 and died in The Hague in February 1620. He worked as a bookkeeper and cashier in Antwerp, travelled in Poland, Prussia and Norway between 1571 and 1577, and was a clerk in the tax office at Bruges. He moved to Leiden in 1581 and entered the university there in 1583, aged 35.
At Leiden he became tutor and adviser to Maurits of Nassau, who led the Dutch army against Spain. Stevin advised on fortifications, windmills, locks and ports, and in 1600 set up an engineering school at Leiden where teaching was in Dutch. He was a strong believer in writing science in Dutch rather than Latin.
What did Stevin's booklet on decimals say?
In 1585 he published De Thiende (The Tenth), and a French version, La Disme, the same year. A 1921 article by Vera Sanford in The Mathematics Teacher, La Disme of Simon Stevin: the first book on decimals, gives an English translation. Stevin dedicates it to "Astrologers, Surveyors, Measurers of Tapestry, Gaugers, Stereometers in general, Mint-Masters, and to all Merchants", and promises to teach how all computations met in business can be done with integers alone, without fractions.
His notation put a small circled number after each digit to show its place: 0 for the whole number (the "commencement"), 1 for tenths ("primes"), 2 for hundredths ("seconds"), and so on. So 27⓪8①4②7③ meant 27 + 8/10 + 4/100 + 7/1000.
MacTutor notes that Stevin did not invent decimals, which had been used in the Arab world and in China long before. What he did was introduce their use in mathematics in Europe. He also predicted that decimal coins, weights and measures would come into general use. An English translation appeared in 1608, and his notation was taken up by Clavius and Napier and developed into the one we use today.
Worked example 1: Stevin's own addition
In La Disme, Proposition I adds three numbers: 27⓪8①4②7③, 37⓪6①7②5③ and 875⓪7①8②2③.
Line up the places, just as you line up units, tens and hundreds.
Add as whole numbers: 27847 + 37675 + 875782 = 941304.
Put the places back: the last digit is in the thousandths place, so the sum is 941.304.
Check with fractions: 27 847/1000 + 37 675/1000 + 875 782/1000 = 941 304/1000. Stevin's whole point is that step 2 needs no fractions at all.
How did Stevin prove the law of the inclined plane?
His 1586 book De Beghinselen der Weeghconst (The Principles of the Art of Weighing) is scanned on archive.org from the Koninklijke Bibliotheek. In it he takes a triangle ABC with a horizontal base, where side AB is twice as long as side BC. Around it he hangs a closed loop of 14 equal balls, evenly spaced on a string, so four balls rest on AB, two on BC, and the rest hang below.
Then he argues. If the four balls on AB outweighed the two on BC, the loop would roll. But after rolling, the balls would sit exactly as before, so the loop would roll again, and again, forever. Stevin calls such perpetual motion false. So the loop does not move. The hanging part below is symmetric and balances itself, so take it away: the four balls on AB balance the two on BC. Weights on the two slopes balance when they are in proportion to the lengths of the slopes.
MacTutor adds that the same book contains the triangle of forces, and that in 1586 Stevin dropped two lead balls, one ten times heavier than the other, about thirty feet from a church tower in Delft and found they fell in the same time.
Worked example 2: a ramp and a hanging weight
Today we write the force along a frictionless slope as mg sinθ. For a slope of length L and height h, sinθ = h/L.
A 10 kg block rests on a smooth ramp 5 m long and 3 m high. Take g = 9.8 m/s².
Force along the slope: mg × h/L = 10 × 9.8 × 3/5 = 58.8 N.
Stevin's view: a vertical side is just a very steep slope of length 3 m. Weights balance in proportion to slope lengths, so the hanging mass is 10 × 3/5 = 6 kg.
Check: 6 × 9.8 = 58.8 N. The two methods agree.
Mistakes to avoid:
Using cosθ for the force along the slope. mg cosθ presses into the slope; mg sinθ pulls along it.
Ignoring friction silently. Stevin's loop and this example assume smooth surfaces.
Measuring the wrong length. Use the slope's length, not its horizontal base.
Galileo later studied motion on slopes rather than balance, as our post on Galileo's inclined plane explains. For decimal skills in school maths, see our guide to number systems in Class 9. If a step in a slope problem still feels unclear, the Fermi Concept Library has every concept in physics and mathematics with its own lesson, so you can fill that one gap.
What can a student take from how he worked?
Write for the people who will use it. He addressed surveyors and merchants, not only scholars.
Find a proof you can see. His loop of balls replaced pages of argument with one picture.
Use an impossibility as a tool. Knowing perpetual motion cannot happen told him the answer.
Test claims by experiment. He dropped lead balls instead of trusting old authority.
Try it: on a smooth double ramp with slopes 4 m and 2 m long sharing the same top, what mass on the 2 m side balances 3 kg on the 4 m side?
