I answer, the true reason hereof is plainly this: because q being unit, qo is equal to o : and therefore 2 x + o - qo = y = 2 x , the equal quantities qo and o being destroyed by contrary signs.
27. As, on the one hand, it were absurd to get rid of o by saying, Let me contradict myself; let me subvert my own hypothesis; let me take it for granted that there is no increment, at the same time that I retain a quantity which I could never have got at but by assuming an increment: so, on the other hand, it would be equally wrong to imagine that in a geometrical demonstration we may be allowed to admit any error, though ever so small, or that it is possible, in the nature of things, an accurate conclusion should be derived from inaccurate principles. Therefore o cannot be thrown out as an infinitesimal, or upon the principle that infinitesimals may be safely neglected; but only because it is destroyed by an equal quantity with a negative sign, whence o - po is equal to nothing. And as it is illegitimate to reduce an equation, by subducting from one side a quantity when it is not to be destroyed, or when an equal quantity is not subducted from the other side of the equation: so it must be allowed a very logical and just method of arguing to conclude that if from equals either nothing or equal quantities are subducted they shall still remain equal. And this is a true reason why no error is at last produced by the rejecting of o . Which therefore must not be ascribed to the doctrine of differences, or infinitesimals, or evanescent quantities, or momentums, or fluxions.
28. Suppose the case to be general, and that is equal to the area ABC whence by the method of fluxions the ordinate is found , which we admit for true, and shall inquire how it is arrived at. Now if we are content to come at the conclusion in a summary way, by supposing that the ratio of the fluxions of x and is found [Sect. 13.] to be 1 and , and that the ordinate of the area is considered as its fluxion, we shall not so clearly see our way, or perceive how the truth comes out, that method as we have shewed before being obscure and illogical. But if we fairly delineate the area and its increment, and divide the latter into two parts BCFD and CFH , [See the figure in sect. 26.] and proceed regularly by equations between the algebraical and geometrical quantities, the reason of the thing will plainly appear. For as is equal to the area ABC , so is the increment of equal to the increment of the area, i.e. to BDHC ; that is to say And only the first members on each side of the equation being retained, = BDFC : and dividing both sides by o or BD , we shall get = BC . Admitting therefore that the curvilinear space CFH is equal to the rejectaneous quantity and that when this is rejected on one side, that is rejected on the other, the reasoning becomes just and the conclusion true. And it is all one whatever magnitude you allow to BD , whether that of an infinitesimal difference or a finite increment ever so great. It is therefore plain that the supposing the rejectaneous algebraical quantity to be an infinitely small or evanescent quantity, and therefore to be neglected, must have produced an error, had it not been for the curvilinear spaces being equal thereto, and at the same time subducted from the other part or side of the equation, agreeably to the axiom, If from equals you subduct equals, the remainders will be equal . For those quantities which by the analysts are said to be neglected, or made to vanish, are in reality subducted. If therefore the conclusion be true, it is absolutely necessary that the finite space CFH be equal to the remainder of the increment expressed by equal, I say, to the finite remainder of a finite increment.
29. Therefore, be the power what you please, there will arise on one side an algebraical expression, on the other a geometrical quantity, each of which naturally divides itself into three members. The algebraical or fluxionary expression, into one which includes neither the expression of the increment of the abscissa nor of any power thereof; another which includes the expression of the increment itself; and the third including the expression of the powers of the increment. The geometrical quantity also or whole increased area consists of three parts or members, the first of which is the given area; the second a rectangle under the ordinate and the increment of the abscissa; the third a curvilinear space. And, comparing the homologous or correspondent members on both sides, we find that as the first member of the expression is the expression of the given area, so the second member of the expression will express the rectangle or second member of the geometrical quantity, and the third, containing the powers of the increment, will express the curvilinear space, or third member of the geometrical quantity. This hint may perhaps be further extended, and applied to good purpose, by those who have leisure and curiosity for such matters. The use I make of it is to shew, that the analysis cannot obtain in augments or differences, but it must also obtain in finite quantities, be they ever so great, as was before observed.
