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Every time you find where a line cuts a circle by solving two equations, you are using an idea Rene Descartes published in 1637. Before him, geometry and algebra were largely separate crafts. This post tells his story from reliable sources, explains what La Geometrie actually did, and works through a problem in the way he made possible, with the mistakes students most often make.
Who was Rene Descartes?
The MacTutor biography records that Descartes was born on 31 March 1596 in La Haye, Touraine (the town is now called Descartes), and died in Stockholm on 11 February 1650. His health was poor as a child. At the Jesuit college of La Fleche he was allowed to stay in bed until late morning, a habit he kept almost all his life. He took a law degree at Poitiers in 1616 to please his father, then joined the army and met the Dutch scientist Isaac Beeckman, who turned him towards mathematics and mechanics.
In 1628 he settled in Holland to work in quiet, keeping his address secret from all but a few friends such as Marin Mersenne. He held back his book on physics, Le Monde, after hearing that Galileo had been condemned.
What was La Geometrie?
In 1637 Descartes published his Discourse on Method at Leiden, with three appendices: one on optics, one on meteorology and La Geometrie. MacTutor calls La Geometrie by far the most important part, and the work that gave us Cartesian geometry. Descartes himself wrote to Mersenne that the other two appendices showed his method was better, and La Geometrie demonstrated it.
The Stanford Encyclopedia of Philosophy entry on Descartes' mathematics explains how it worked. Faced with a hard ancient problem, the Pappus problem, Descartes chose two unknown line segments and called them x and y. He then expressed every other length in the figure in terms of x and y, using similar triangles, and reduced the problem to an equation. The solution became a curve described by that equation.
Why did turning geometry into algebra matter?
MacTutor quotes a summary of what the method gave: algebra can recognise typical problems in geometry and bring together problems that look unrelated, and it can decide whether a construction is possible at all. A circle and a parabola stop being two separate pictures; they become a second-degree equation of two kinds. Fermat developed a similar approach at about the same time, as our post on Pierre de Fermat explains, and the two men clashed bitterly over method.
Worked example: where does a line cut a circle?
Find where the line y = x + 1 meets the circle x² + y² = 25, and the length of the chord between those points.
Substitute. Put y = x + 1 into the circle: x² + (x + 1)² = 25.
Expand. x² + x² + 2x + 1 = 25, so 2x² + 2x − 24 = 0, or x² + x − 12 = 0.
Solve. (x + 4)(x − 3) = 0, so x = 3 or x = −4.
Find y. For x = 3, y = 4. For x = −4, y = −3. The points are (3, 4) and (−4, −3).
Check. 3² + 4² = 25 and (−4)² + (−3)² = 25. Both lie on the circle.
Chord length. Use the distance formula: √((3 − (−4))² + (4 − (−3))²) = √(49 + 49) = 7√2 ≈ 9.90.
A ruler-and-compass drawing would only show roughly where they cross. The algebra gives exact points, and the discriminant of the quadratic tells you in advance whether there are two, one or no meeting points.
Common mistakes:
Forgetting the middle term when expanding (x + 1)², writing x² + 1.
Stopping at x. Each x needs its own y from the line equation, not from the circle, which would give extra wrong points.
Sign slips in the distance formula, such as 3 − (−4) taken as −1.
For more practice see our guides to coordinate geometry in Class 9 and to circles in coordinate geometry.
How did Descartes die?
MacTutor records that in 1649 Queen Christina of Sweden persuaded him to come to Stockholm. She wanted lessons at 5 a.m., so after a lifetime of late mornings he walked to the palace before dawn in a northern winter. Within a few months he died of pneumonia.
What can a student take from Descartes' way of working?
Name the unknowns first. His method starts by giving letters to the lengths you do not know.
Translate, then solve. A hard picture often becomes a routine equation.
Check against the original problem. Substituting back, as in step 5, is how you catch errors.
If a step such as completing the square or the distance formula keeps tripping you up, the Fermi Concept Library has every concept in mathematics with its own lesson and its place in the map.
Find where the line y = 2x − 5 meets the circle x² + y² = 10, then check both points.
