# What is the origin of the cut and weigh method of integration, is it Galileo's?

I recently heard a story of a clever method of approximating an integral which, instead of using numerical techniques, relied on physically drawing out the graph of a function, cutting it out, and weighing it.

I have not been able to find very much information about it (partially because I haven't been able to find if the technique has a specific name), but I have found an anecdote in which Galileo uses the method to find the area of a cycloid (See Cycloid from Wolfram Mathworld).

Do we know who came up with the cut and weigh technique, or if not, is Galileo's use the first to have been recorded?

• I know this as the chemists method. – Chris May 2 '15 at 22:35
• As always, it is difficult, almost impossible to prove that someone was the first, especially in such a simple thing. But I suppose this was not used before the wide spread of paper. (It is difficult for me to imagine a mathematician cutting a cycloid of a sheet metal:-) – Alexandre Eremenko May 2 '15 at 22:40

To work the weighing method also needs a standard of area, cut from the same sheet. Galileo was allegedly balancing the cycloid cutout with that of its generating circle, and found that they were as $$3:1$$, but did not believe it because he expected that they are incommensurable. The first association that comes to mind is Archimedes's Method, where he cuts cone, sphere and cylinder into thin slices, and then balances them on a lever. Of course, this was a purely mental exercise, but in the famous Eureka incident (if Vitruvius's tell tale is believed) Archimedes was presumably balancing the suspicious crown and a slab of gold on physical scales. Of course, that was to compare densities rather than areas or volumes, but the re-purposing is straightforward.
At a risk of giving material to conspiracy theorists I'll point out another quirk of history. Another "desciple" of Galileo's, Cavalieri, reinvented Archimedes's method of "indivisibles" in 1627, and in a streamlined form that did not require levers and balancing. In 1644, the same year that Torricelli published his cycloid story, Cavalieri used indivisibles to show that cycloid is to the generating circle as $$3:1$$. Torricelli lavished Cavalieri with praise for indivisibles, but wrote: