The story of the British Library's Royal Manuscripts collection reaches its end with the last great flowering of illumination, in the magnificent courts of the Tudors. Dr Janina Ramirez investigates astrological texts created for Henry VII, and unwraps his will - still in its original, extravagantly-decorated velvet and gold cover. She hears music written for Henry VIII, which went unperformed for centuries; and reads love notes between the king and Anne Boleyn, written in the margins of a prayer book. Nina also visits Bruges, the source of many of the greatest manuscripts, where this medieval art form collided with the artistic innovations of the Renaissance.

Dr Hannah Fry travels down the fastest zip wire in the world to learn more about Newton's ideas on gravity. His discoveries revealed the movement of the planets was regular and predictable. James Clerk Maxwell unified the ideas of electricity and magnetism, and explained what light was. As if that wasn't enough, he also predicted the existence of radio waves. His tools of the trade were nothing more than pure mathematics. All strong evidence for maths being discovered.
But in the 19th century, maths is turned on its head when new types of geometry are invented. No longer is the kind of geometry we learned in school the final say on the subject. If maths is more like a game, albeit a complicated one, where we can change the rules, surely this points to maths being something we invent - a product of the human mind. To try and answer this question, Hannah travels to Halle in Germany on the trail of perhaps one of the greatest mathematicians of the 20th century, Georg Cantor. He showed that infinity, far from being infinitely big, actually comes in different sizes, some bigger than others. This increasingly weird world is feeling more and more like something we've invented. But if that's the case, why is maths so uncannily good at predicting the world around us? Invented or discovered, this question just got a lot harder to answer.

American troops launch widespread 'search and destroy' operations; body count, not territory, becomes the measure of success in Vietnam. Charles Brown fights for survival on the bloody slopes of Hill 875 and at home, the American public begins to question U.S. military strategy.

The enemy gains ground when the massive Tet Offensive catches the Americans by surprise. At Khe Sanh and Pleiku, U.S. troops are under siege. Americans mount a counteroffensive but the shock of the assault ignites anti-war fervour at home.

The turmoil of 1968 helps Nixon win the presidency. Troop strength in Vietnam peaks and the draft accelerates. Karl Marlantes endures bitter jungle fighting on Hill 484, and draftee Arthur Wiknik gets a grim introduction to combat on 'Hamburger Hill'. But shortly after winning both hills, US troops abandon them.

Troop withdrawals begin as the Americans train their ARVN allies to take over the war. Gery Benedetti patrols the hostile waters of the Mekong Delta. James Anderson leads a battalion into Cambodia after Nixon's controversial order. Don DeVore struggles to survive a fierce night attack, and Anne Purcell gets promising news about her husband.