Now let us discover the other diminution of velocity, which likewise may proceed to infinity, by the diminution of the gravity of the moveable, and this shall be represented by drawing other lines from the point A, which contein angles lesse than the angle BAE, which would be this line Ad, the which intersecting the parallels KL, Hi, FG, in the factors M, N, and O, symbolize unto us the levels FO, HN, KM, acquired within the instances AF, AH, AK, lesse than the opposite levels FG, Hi, KL, acquired in the identical occasions; however these latter by a moveable more ponderous, and those different by a moveable extra mild. AH being divided in two equal parts in F, the parallel Hi shall be double to FG, and sub-dividing FA, in two equal parts, the parallel produced from the purpose of the division shall be the half of FG; and persevering with the sub-division in infinitum, the following parallels shall be alwayes half of the next preceding; but it doth not so fall out within the lines intercepted between the tangent and the circumference of the circle: For if the same sub-division be made in FA; and supposing for instance, that the parallel which cometh from the point H, had been double unto that which commeth from F, this shall be more then double to the next following, and regularly the neerer we come in the direction of the contact A, we shall discover the precedent strains contein the following following three, four, ten, an hundred, a thousand, an hundred thousand, an hundred tens of millions of times, and more in infinitum.

Simpl.I will say for instance, that if that movement by the tangent have been one million of instances swifter than this by the secant, the pen, yea, and the stone also would come to be extruded. AC, and to it at right angles let there be drawn the Horizontal line AB, upon which the movement of the projection ought to be made; now the undertaking would proceed to move alongside the identical with a fair motion, if so be its gravity did not incline it downwards. Salv.Go to; tell me whether or not you believe that amongst the many lines that could be drawn between two factors, there may be multiple proper line onely. Tell me due to this fact, how much do you assume sufficeth to make that movement swifter than this? And this retreat in direction of A, representeth the first propension to the movement of descent, diminished in infinitum by the method of the moveable to the primary state of relaxation, which approximation is augmentable in infinitum. And couching the whole argument in its shortest phrases, we’ll say, that by contracting the angle EAB, the levels of velocity LK, IH, GF, are diminished; and moreover by the retreat of the parallels KL, Hi, FG, towards the angle A, the same levels are again diminished; and each these diminutions extend to infinity: Therefore the velocity of the movement of descent could very effectively diminish so much, (it admitting of a twofold diminution in infinitum) as that it may not suffice to restore the moveable to the circumference of the wheel, and thereupon might occasion the projection to be hindered and wholly obviated.

Salv.I have assumed for a truth, that the velocities of moveables descending naturally, will comply with the proportion of their gravities, with the favour of Simplicius, and of Aristotle, who doth in many locations affirm the same, as a proposition manifest: You, in favour of my adversary, deliver the identical into query, and say that its doable that the velocity increaseth with larger proportion, yea and better in infinitum than that of the gravity; so that all that hath been stated falleth to the ground: For sustaining whereof, I say, that the proportion of the velocities is far lesse than that of the gravities; and thereby I do not onely assist but confirme the premises. Page 180 however that the diminution of the same velocity, dependent on the diminution of the gravity of the moveable (which vvas the second trigger) doth also observe the same proportion, doth not so plainly appear, And vvho shall guarantee us that it doth not proceed in response to the proportion of the traces intercepted between the secant, and the circumference; or vvhether vvith a larger proportion? Sagr.I very nicely comprehend the entire discourse, and upon what it layeth all its stresse, yet neverthelesse methinks that he that would take pains to pursue it, might but start some additional questions, by saying, that of these two causes which render the descent of the moveable slower and slower in infinitum, it is manifest, that that which dependeth on the neighborhood to the first time period of the descent, increaseth alwayes in the identical proportion, like because the parallels alwayes retain the identical proportion to each other, &c.

And since C is the middle proportion between BA and AI; and GH is a center term between FG and GE; due to this fact, as BA is to C, so shall FG be to GH; that’s HG to GE, which was to be demonstrated. The unknown conclusion which you want to show, is it not, that the curved line ACB, is longer than the fitting line AB; the center term which is taken for identified, is that the curve-line ACB, is larger than the 2 strains AC and CB, the that are known to be better than AB; And if it be unknown whether the curve-line be larger than the one right-line AB, shall it not be rather more unknown whether or not or not it’s better than the 2 right strains AC & CB, that are identified to be better than the only line AB, & but you assume it as recognized? Now when the middle term is less known than the conclusion, it is called a proving ignotum per ignotius. It is true that the areas between the tangent and the circumference do progressively diminish in infinitum towards the contact; but additionally it is true on the contrary, that the propension of the moveable to descending groweth less & less in it, the nearer it’s to the primary term of its descent; that’s, to the state of rest; as is manifest from that which you declare unto us, demonstrating that the descending grave physique departing from relaxation, must passe thorow all of the levels of tardity comprehended between the said rest, & any assigned degree of velocity, the which develop much less and in infinitum.