19. And yet it should seem that, whatever errors are admitted in the premises, proportional errors ought to be apprehended in the conclusion, be they finite or infinitesimal: and that therefore the of geometry requires nothing should be neglected or rejected. In answer to this you will perhaps say, that the conclusions are accurately true, and that therefore the principles and methods from whence they are derived must be so too. But this inverted way of demonstrating your principles by your conclusions, as it would be peculiar to you gentlemen, so it is contrary to the rules of logic. The truth of the conclusion will not prove either the form or the matter of a syllogism to be true; inasmuch as the illation might have been wrong or the premises false, and the conclusion nevertheless true, though not in virtue of such illation or of such premises.
I say that in every other science men prove their conclusions by their principles, and not their principles by the conclusions. But if in yours you should allow yourselves this unnatural way of proceeding, the consequence would be that you must take up with Induction, and bid adieu to Demonstration.
And if you submit to this, your authority will no longer lead the way in points of Reason and Science.
20. I have no controversy about your conclusions, but only about your logic and method: how you demonstrate? what objects you are conversant with, and whether you conceive them clearly? what principles you proceed upon; how sound they may be; and how you apply them? It must be remembered that I am not concerned about the truth of your theorems, but only about the way of coming at them; whether it be legitimate or illegitimate, clear or obscure, scientific or tentative. To prevent all possibility of your mistaking me, I beg leave to repeat and insist, that I consider the geometrical analyst as a logician, i.e. so far forth as he reasons and argues; and his mathematical conclusions, not in themselves, but in their premises; not as true or false, useful or insignificant, but as derived from such principles, and by such inferences. And, forasmuch as it may perhaps seem an unaccountable paradox that mathematicians should deduce true propositions from false principles, be right in the conclusion and yet err in the premises; I shall endeavour particularly to explain why this may come to pass, and show how error may bring forth truth, though it cannot bring forth science.
21. In order therefore to clear up this point, we will suppose for instance that a tangent is to be drawn to a parabola, and examine the progress of this affair as it is performed by infinitesimal differences. Let AB be a curve, the abscissa AP = x , the ordinate PB = y , the difference of the abscissa PM = dx , the difference of the ordinate RN = dy . Now, by supposing the curve to be a polygon, and consequently BN , the increment or difference of the curve to be a straight line coincident with the tangent, and the differential triangle BRN to be similar to the triangle TPB , the subtangent PT is found a fourth proportional to RN : RB : PB : that is, to dy : dx : y . Hence the subtangent will be But herein there is an error arising from the aforementioned false supposition, whence the value of PT comes out greater than the truth: for in reality it is not the triangle RNB but RLB which is similar to PBT , and therefore (instead of RN ) RL should have been the first term of the proportion, i.e. RN + NL , i.e. dy + z : whence the true expression for the subtangent should have been There was therefore an error of defect in making dy the divisor;1
22. If you had committed only one error, you would not have come at a true solution of the problem. But by virtue of a twofold mistake you arrive, though not at science, yet at truth. For science it cannot be called, when you proceed blindfold, and arrive at the truth not knowing how or by what means. To demonstrate that z is equal to let BR or dx be m and RN or dy be n .
By the thirty-third proposition of the first book of the Conics of Apollonius, and from similar triangles, as 2 x to y so is m to Likewise from the nature of the parabola yy + 2 yn + nn = xp + mp , and 2 yn + nn = mp : wherefore and because yy = px , will be equal to x . Therefore substituting these values instead of m and x we shall have i.e. which being reduced gives 23. Now, I observe, in the first place, that the conclusion comes out right, not because the rejected square of dy was infinitely small, but because this error was compensated by another contrary and equal error. I observe, in the second place, that whatever is rejected, be it every so small, if it be real, and consequently makes a real error in the premises, it will produce a proportional real error in the conclusion.
Your theorems therefore cannot be accurately true, nor your problems accurately solved, in virtue of premises which themselves are not accurate; it being a rule in logic that conclusio sequitur partem debiliorem . Therefore, I observe, in the third place, that when the conclusion is evident and the premises obscure, or the conclusion accurate and the premises inaccurate, we may safely pronounce that such conclusion is neither evident nor accurate, in virtue of those obscure inaccurate premises or principles; but in virtue of some other principles, which perhaps the demonstrator himself never knew or thought of. I observe, in the last place, that in case the differences are supposed finite quantities ever so great, the conclusion will nevertheless come out the same: inasmuch as the rejected quantities are legitimately thrown out, not for their smallness, but for another reason, to wit, because of contrary errors, which, destroying each other, do, upon the whole, cause that nothing is really, though something is, apparently, thrown out. And this reason holds equally with respect to quantities finite as well as infinitesimal, great as well as small, a foot or a yard long as well as the minutest increment.
