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

y dx dy + z

There was therefore an error of

defect in making dy the divisor; which error was equal to z, i.e. NL the line comprehended between the curve and the tangent. Now by the nature of the curve yy=px, supposing to be the parameter, whence by the rule of differ

pdx ences 2y dy = p dx and dy:

2y

But if you multiply y + dy

by itself, and retain the whole product without rejecting the square of the difference, it will then come out, by substituting the augmented quantities in the equation of the curve, that p dx dy dy

dy=

2y

2y excess in making dy

truly. There was therefore an error of

pdx

which followed from the erroneous

2y

rule of differences.

dy dy

And the measure of this second error is

= 2. Therefore the two errors being equal and con2y trary destroy each other; the first error of defect being corrected by a second error of excess.

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

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

my

2x.

from similar triangles, as 2x to y so is m to n+z= Likewise from the nature of the parabola yy+2yn+nn=xp +mp, and 2yn+nn=mp: wherefore

yy

zyn +nn

=m: and be

cause yy=px, 22 will be equal to x. Therefore substituting

these values instead of m and x we shall have

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 ever 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. 24. For the fuller illustration of this point, I shall con

sider it in another light, and proceeding in finite quantities to the conclusion, I shall only then make use of one infinitesimal. Suppose the straight line MQ cuts the curve

AT in the points R and S. Suppose LR a tangent at the point R, AN the abscissa, NR and OS ordinates. Let AN be produced to O, and RP be drawn parallel to NO. Suppose AN=x, NR=y, NO=v, PS=2, the subsecant MN =s. Let the equation y=xx express the nature of the curve: and supposing y and x increased by their finite increments we get y+2=xx + 2xv+vv : whence the former equation being subducted, there remains z = 2xv+vv. And by reason of similar triangles

PS: PR:: NR : NM, i.e. z :v :: y:s=

vy

wherein if for y and z we substitute their values, we get

And supposing NO to be infinitely diminished, the subsecant NM will in that case coincide with the subtangent NL, and v as an infinitesimal may be rejected, whence it follows that

which is the true value of the subtangent. And, since this was obtained by one only error, i.e. by once ejecting one only infinitesimal, it should seem, contrary to what hath been said, that an infinitesimal quantity or difference may be neglected or thrown away, and the conclusion nevertheless be accurately true, although there was no double mistake or rectifying of one error by another, as in the first case. But, if this point be thoroughly considered, we shall find there is even here a double mistake, and that one compensates or rectifies the other. For, in the first place, it was supposed that when NO is infinitely diminished or becomes an infinitesimal then the subsecant NM becomes equal to the subtangent NL. But this is a plain mistake; for it is evident that as a secant cannot be a tangent, so a subsecant cannot be a subtangent. Be the difference ever so small, yet still there is a difference. And, if NO be infinitely small, there will even then be an infinitely small difference between NM and NL. Therefore FM or S was too little for your supposition (when you supposed it equal to NL); and this error was compensated by a

second error in throwing out v, which last error made s bigger than its true value, and in lieu thereof gave the value of the subtangent. This is the true state of the case, however it may be disguised. And to this in reality it amounts, and is at bottom the same thing, if we should pretend to find the subtangent by having first found, from the equation of the curve and similar triangles, a general expression for all subsecants, and then reducing the subtangent under this general rule, by considering it as the subsecant when v vanishes or becomes nothing.

25. Upon the whole I observe, First, that v can never be nothing, so long as there is a secant. Secondly, that the same line cannot be both tangent and secant. Thirdly, that when v or NO1 vanisheth, PS and SR do also vanish, and with them the proportionality of the similar triangles. Consequently the whole expression, which was obtained by means thereof and grounded thereupon, vanisheth when v vanisheth. Fourthly, that the method for finding secants or the expression of secants, be it ever so general, cannot in common sense extend any farther than to all secants whatsoever and, as it necessarily supposed similar triangles, it cannot be supposed to take place where there are not similar triangles. Fifthly, that the subsecant will always be less than the subtangent, and can never coincide with it; which coincidence to suppose would be absurd; for it would be supposing the same line at the same time to cut and not to cut another given line; which is a manifest contradiction, such as subverts the hypothesis and gives a demonstration of its falsehood. Sixthly, if this be not admitted, I demand a reason why any other apagogical demonstration, or demonstration ad absurdum should be admitted in geometry rather than this or that some real difference be assigned between this and others as such. Seventhly, I observed that it is sophistical to suppose NO or RP, PS, and SR to be finite real lines in order to form the triangle, RPS, in order to obtain proportions by similar triangles; and afterwards to suppose there are no such lines, nor consequently similar triangles, and nevertheless to retain the consequence of the first supposition, after such supposition hath been destroyed

:

1 See the foregoing figure.