Measured by Shadows

What this book is about

At school, trigonometry starts with a table. sin 30° = 0.5, cos 60° = 0.5 — learn it. Nobody says where those numbers came from, why they do not depend on the size of the triangle, or what you can actually do with them. So for most people the whole subject settles into a heap of meaningless letters.

Not one number in this book falls from the sky. Wherever one is needed it is built on that page — usually with a drawing and a single division. What you need before starting is this much: how to divide, what a ratio is, and that the angles of a triangle add to 180°. Pythagoras helps but is not assumed; chapter 16 arrives at it anyway.

Read it with a tape measure or a scale beside you. At least five times the book asks you to get up and measure something. Those measurements are the lesson; the rest is explanation. And stop at page 37 — the same question is answered there by two completely different roads and both give the same number. That page is the spine of the book.

What is inside

• Shadows: why the same sun casts two different ones, and why a bigger triangle keeps the

same ratio

• The angle's own numbers — the angles you can work out by hand, half an equilateral

triangle, a square cut across, and how the table is actually built

• Measuring for real: make a clinometer, get the height of a tree, the far bank of a river

by two routes, and what a surveyor does

• The ratios as a subject of their own — tan is sin over cos, the unit circle, and a circle

unrolled into a wave

• Where there is no right angle: the sine rule, the cosine rule, a pier of the Padma bridge,

and triangulating a whole country

• Who built it — Thales at the pyramids, Hipparchus and the first table, Āryabhaṭa and

*ardha-jyā*, a word's fifteen-hundred-year journey, Mādhava and the Kerala school, and Radhanath Sikdar measuring Peak XV

• Two experiments, every formula on one sheet, worked exercises, a bilingual glossary and a quiz