It's this moment where now both sides realize that no quarter will be asked, and no quarter will be given. Surrender is no longer even an option. You're in this to the end. You either die or you triumph, and there is no other alternative. Amid a spiral of brutality and low morale, Mehmed makes Giustiniani an enticing offer. The grand vizier urges Mehmed to seek a truce with his rival.

This is the story of a book that could have changed the history of the World. To the untrained eye, it is nothing more than a small and unassuming Byzantine prayer book. For faintly visible beneath the prayers on its pages are other, unique, writings - words that have been lost for nearly two thousand years. The text is the only record of work by one of the world's greatest minds - the ancient Greek, Archimedes - a mathematical genius centuries ahead of his time. Hidden for a millennium in a middle eastern library, it has been written over, broken up, painted on, cut up and re-glued. But in the nick of time scientists have saved the precious, fragile document, and for the first time it is revealing just how revolutionary Archimedes' ideas were. If it had been available to scholars during the Renaissance, we might have reached the Moon over a hundred years ago.

Hours before Mehmed II launches his final assault, an ominous sight shakes both sides. Rumors of a 40-ship fleet's imminent arrival have swirled for weeks, but it's exact whereabouts remained unknown. Ottoman cannons reduced the city walls to rubble, and Venetian reinforcements arrived too late. The conquest of Constantinople ushers in a new era for the Ottoman Empire, being a dominant force in world politics for 300 years. Mehmed II, in many senses, changed the nature of world history.

Mehmed II sends his elite troops into the teeth of Emperor Constantine's defenses along the city's ancient stone walls. The defenders, led by Italian soldier-of-fortune Giovanni Giustiniani, repel every attack. Mehmed's men dig underground tunnels in an attempt to shatter city walls. The tides turn against the Ottomans when a naval blockade founders.

Sultan Mehmed II's siege of Constantinople enters its third week. Mehmed's artillery teams continue their relentless bombardment of Constantinople's walls. After his navy suffers a devastating defeat, Mehmed moves his ships overland to the Golden Horn in a daring, visionary feat. In the shadow of betrayal, Giustiniani attacks the Ottoman fleet.

Andrew Graham-Dixon travels to Istanbul to immerse himself in the tumultuous world of the Byzantine Empire. He decodes the iconography of the art of the period and explains its continuing relevance.