Thousands of years later, the solution finally arrived. We need - somehow - to store fewer results. That’s space efficient!īut this still doesn't scale - to multiply numbers in the 1000s, you’d need to pre-compute thousands of results. Pre-computing just 24 results now lets us multiply any two numbers between 1 and 12. On the face of it, this looks a lot more complicated than just multiplying the damn numbers!īut with the right cheat-sheet, it’s actually simpler: the right-hand-side is always a subtraction of square numbers divided by 4, so by pre-computing these quarter-squares, we can multiply any two numbers by just doing a subtraction. A classic is the 'quarter-square multiplication', which reduces multiplication to subtraction. The Babylonians came up with many of these. The next hack came through mathematical identities - 'cheat codes' to express one calculation in terms of another. But when your cheat-sheet is more of a cheat-stone-tablet, every result matters. But this wasn’t space-efficient - there are 66 ways to multiply two numbers just between 1 and 12 (12 choose 2), so the corresponding cheat-sheet would need 66 pre-computed results. They realized they could do multiplications once, and write the results on a 'cheat-sheet' to avoid having to do them again. This is the core insight underpinning most number-crunching tricks and, as we'll see, the slide rule.Ĥ,000 years ago, the Babylonians came up with one of the first ways to “hack” multiplication. But it’s also remarkable how quickly it’s been lost to obscurity in the three decades since. It’s remarkable how little the slide rule has changed in the three centuries spanning these events. Newton used it to derive the equations of motion, and NASA used it to solve those very equations to take us to the moon. Millennia of their work culminated in the greatest mechanical calculating device there is: the slide rule. A great deal of people, throughout history, have spent a great deal of time coming up with tricks to do it faster. With a little less patience, he wouldn’t have proved his theory, and elliptical orbits would have eluded us for longer.įor most of history, basic calculations were a serious bottleneck to science. But it was remarkable that Kepler managed it - his calculations were riddled with mistakes! Only after repeating his procedures 70 times did he offset his computational errors. 4 years and 900 pages of calculations later, he confirmed his theory by determining Mars’s true orbit. Kepler first formulated his hypothesis: planets had elliptical orbits with two foci, and then set out to prove it mathematically. At the turn of the 16th century, Johannes Kepler set out to disprove Copernicus, who believed planets orbited in perfect, concentric circles from the sun.

Slide ruler