use a series of approximations to close in on an answer. In order to solve such an equation, physicists were forced to use a theory calledPerturbation. Which basically works in a third body by a series of successive approximations. Each approximation is smaller than the one before it, and by addingup a potentially infinite amount of these numbers, theoretical physicists hoped to arrive a working equation. Poincare knew that the approximation theory appeared to work well for the first couple of approximations, but what about further down the line, what effect wouldthe infinity of smaller approximations have? The multi-bodied equation Poincare was attempting was essentially a Non-linear equation. As opposed to adifferential or linear equation. For science, a phenomenon is orderly if its movements can be explained in the kind of cause-and-effect scheme represented by a differential equation. Newton firstintroduced the differential idea throughout his famous laws of motion, which related rates of change to various forces. Quickly scientists came to rely on lineardifferential equations. Phenomena as diverse as the flight of a cannonball, the growth of a plant, the burning of coal, and the performance of a machine can bedescribed by such equations. In which small changes produce small effects and large effects are obtained by summing up many small changes.4 A non-linear equation is quite different. In a non-linear equation a small change in one variable can have a disproportional, even catastrophic impact on othervariables.5 Behaviors can drastically change at any time. In linear equations the solution of one equation allows the solver to generalize to other solutions; innon-linear equations solutions tend to be consistently individual and unrelated to the same equation with different variables. In Poincares multi-bodied equation, he added a term that added nonlinear complexity to the system (feedback) that corresponded to the ...
