Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/204

 of rest, are alwayes determinate, and answer in proportion to the parallels comprehended between two right lines that concur in an angle, like to the angle BAE, or BAD, or any other infinitely more acute, alwayes provided it be rectilineall. But the diminution of the spaces thorow which the moveable is to be conducted along the circumference of the wheel, is proportionate to another kind of diminution, comprehended between lines that contain an angle infinitely more narrow and acute, than any rectilineal angle, how acute soever, which is that in our present case. Let any point be taken in the perpendicular AC, and making it the centre, describe at the distance CA, an arch AMP, the which shall intersect the parallels that determine the degrees of velocity, though they be very minute, and comprehended within a most acute rectilineal angle; of which parallels the parts that lie between the arch and the tangent AB, are the quantities of the spaces, and of the returns upon the wheel, alwayes lesser (and with greater proportion lesser, by how much neerer they approach to the contact) than the said parallels of which they are parts. The parallels comprehended between the right lines in retiring towards the angle diminish alwayes at the same rate, as v. g. 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 point of the division shall be the half of FG; and continuing the sub-division in infinitum, the subsequent parallels shall be alwayes half of the next preceding; but it doth not so fall out in 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 example, that the parallel which cometh from the point H, were double unto that which commeth from F, this shall be more then double to the next following, and continually the neerer we come towards the contact A, we shall find the precedent lines contein the next following three, four, ten, an hundred, a thousand, an hundred thousand, an hundred millions of times, and more in infinitum. The brevity therefore of such lines is so reduced, that it far exceeds what is requisite to make the project, though never so light, return, nay more, continue unremoveable upon the circumference.

I very well comprehend the whole discourse, and upon what it layeth all its stresse, yet neverthelesse methinks that he that would take pains to pursue it, might yet start some further questions, by saying, that of those two causes which render the descent of the moveable slower and slower in infinitum, it is manifest, that that which dependeth on the vicinity to the first term of the descent, increaseth alwayes in the same proportion, like as the parallels alwayes retain the same proportion to each other, &c.