Miscellanea Curiosa, Vol. 1 / Containing a collection of some of the principal phaenomena in nature, accounted for by the greatest philosophers of this age

by Unknown

16. And because they are in Number infinite (though equal to a finite Magnitude) the Duration is infinite: And consequently the impressed Force, and Motion thence arising, never to be wholly extinguished (without some further Impediment) but perpetually approaching to A, in the Nature of Asymptotes.

17. The Spaces Fl, Fm, Fn, &c. are therefore as Logarithms (in Arithmetical Progression increasing) answering to the Lines AF, AL, AM, &c. or to FL, LM, MN, &c. in Geometrical Progression decreasing.

18. Because FL, LM, MN, &c. are as ¹⁄_m, ¹⁄_mm, ¹⁄_m3, &c. (infinitely) terminated at A; therefore (by ¶ 10) their Aggregate FA or dh, is to DH, (so much Length as would have been dispatched, in the same time, by such impressed Force undiminished) as 1 to m - 1 = n.

19. If therefore we take, as 1 to n, so AF to DH; this will represent the Length to be dispatched, in the same time, by such undiminished Force.

20. And if such DH be supposed to be divided into equal Parts innumerable (and therefore infinitely small;) these answer to those (as many) Parts unequal in FA, or hd.

21. But, what is the Proportion of r to 1, or (which depends on it) of 1 - r to 1, or 1 to m; remains to be inquired by Experiment?

22. If the Progression be not infinitely continued; but end (suppose) at N, and its least Term be A = MN; then, out of V / R - 1 = ¹⁄_m + ¹⁄_mm + ¹⁄_m3, &c. is to be subducted A / R - 1 (as at ¶ 10.) that is (as by Division will appear) ^A⁄_R + ^A⁄_R2 + ^A⁄_R3 &c. That is (in our present Case) ^a⁄_m + ^a⁄_mm + ^a⁄_m3 &c. And so the Aggregate will be 1-a / m + 1-a / mm + 1-a / mmm &c. = 1-a / n.

And thus as to the Line of Projection, in which (secluding the Resistance) the Motion is reputed uniform; dispatching equal Lengths in equal Times. Consider we next the Line of Descent.

23. In the Descent of Heavy Bodies, it is supposed that to each Moment of Time, there is superadded a new Impulse of Gravity to what was before: And each of these, secluding the Consideration of the Air's Resistance, to proceed equally (from their several beginnings) through the succeeding Moments. As (in the erect Lines) 1 1 1 1, &c. 1 1 1, &c. 1 1, &c. 1, &c. and so continually, as in the Line of Projection.[16]

24. Hence ariseth (in the transverse Lines) for the first Moment 1, for the second 1 + 1, for the third 1 + 1 + 1, and so forth, in Arithmetical Progression: As are the Ordinates in a Triangle, at equal distance.