Grids before graphs

"Where exactly is it?" To answer that with numbers, we use a system of coordinates: a grid, like the lines on a map or on graph paper. Think of it as giving every point an address.

India used grids long ago. In the cities of the Sindhu-Sarasvatī Civilisation, streets ran North–South and East–West, about 10 metres apart. A merchant could find a shop by counting street-gaps East or West, then North or South, from the city centre. Two counts, one address: that is a coordinate system at work.

Coordinates on the Earth and in the sky:

• Ujjayinī was used as the central line of longitude, from which other places were measured.
• Āryabhaṭa replaced the Greek 'chords' with 'sines', which made the coordinates of a star or a city much easier to work out. He also mapped the sky in coordinates.

Why we can have negative coordinates. Brahmagupta treated zero and negative numbers as numbers you can calculate with. The origin of our plane is zero, and half of each axis holds negative values. Without his work, the four-part plane of this chapter would not be possible.

The ideas travelled.

• Brahmagupta's work was translated into Arabic, and the Ujjayinī line entered Arabic maps as 'Arin', their zero of longitude.
• Al-Bīrūnī studied in India and perfected the astrolabe, which let sailors find their position from the stars.
• Omar Khayyām solved algebra problems by turning them into coordinates in a plane.

Finally, after the work of Fermat, René Descartes (1637) set out the big idea: any point on a flat surface is fixed by just two numbers, its distances from two lines at right angles. That joined geometry (shapes) and algebra (numbers).