So, this proof is worth looking at. One of you, Issac challenged it last time, how it's even legitimate. The idea notice what it says. The space described by a body urged by any centripetal force at the beginning of it's motion is the square of the time. So, what we've got here is times are being represented on the abscissa at the top.
Let's the times be represented by ab, ad. The spaces described under uniform centripetal motion by the areas abf and adh. Okay? So, this is, he actually refers to Galileo here. This is coming off of the standard Galilean proof of the mean speed here of distances, the area. Time is on the abscissa and that means that, at any point here, the instantaneous velocity is being represented by the ordinate dropping down.
Draw a straight line AGE. Yeah. Okay. Draw straight line AGE with the areas under it. That is ABG, representing the spaces under a uniform centripetal force. Greater than that corresponding to AAFH and extend the line ABD all the way out. That's the inner of the two straight lines.
Okay. Now what we're gonna look at here. I skipped this. And the space is described under a non uniform centripetal force by ABC and ADE where Is a tangent to the curve ACE. So that curve is going to represent non-uniform acceleration, and AD is tangential to the curve at A.
Now we're gonna draw a second line that also starts at a, and intersects up at, let's see where we want, at point g. Straight. Actually. Yeah that's right. Straight line a g e through those points. And it's only the point e by the way, that is actually intersecting the curve.
That's important. So what we've got is a straight line accelerating slower than the variable acceleration. And we have another straight line that represents uniform acceleration at the rate that is the average between a and e. Actually it's not even the average. It has a rate corresponding to where you would go at a uniform speed that would get you the velocity d e at the point where the curve gets that velocity.
I said this poorly. What we're doing here, is bounding the curved case. One of them is going to give you less distance cover. The other is going to give you more distance cover. Now what we're going to do is extend these so that we've got a ratio out here, little b, little d, etc.
That's the same ratio a capital B to a capital D is the same as the ratio of a little b to a little d. We're gonna use those second two ratios, we're gonna hold them fixed. And now we're going to move b and d holding that ratio fixed toward a, so that we're moving toward the very beginning at the time.
What's going to happen to those two straight lines? They're going to merge to the tangent at point a of the curve line. Now we already know from Galileo if it's uniform acceleration. Then the distance covered goes, it's the square of the time, not just the beginning, over an extended time.
So we've got two cases where we've got the distance covered over an extended time going into the time squared. One greater than the curve, the other less than the curve. And we go back now to the very beginning. And those two come together. Therefore at the very beginning.
Even though in the variable case distance has to be proportional to times squared. That's the argument. He elaborates this in the Principia the by dividing it into two parts. The first is purely mathematical talking about the areas. And the second part then says, the area represents distance, the absence of time, etc.
So all he's really doing here is teaching us a fairly fundamental principle of calculus. For the curve at the very beginning, the area described is going to be the square if the ratios are all held the same. And it's a ratios being held the same that let's him get the result.