Newton's Principia : the mathematical principles of natural philosophy by Newton Isaac Sir 1642-1727 & Chittenden N. W. Life of Sir Isaac Newton & Adee Daniel ca. 1819-1892. (1846) bkp CU-BANC & Motte Andrew d. 1730 & Hill Theodore Preston. Early American mathematics books. CU-BANC

Author:Newton, Isaac, Sir, 1642-1727 & Chittenden, N. W. Life of Sir Isaac Newton & Adee, Daniel, ca. 1819-1892. (1846) bkp CU-BANC & Motte, Andrew, d. 1730 & Hill, Theodore Preston. Early American mathematics books. CU-BANC
Language: eng
Format: epub
Tags: Newton, Isaac, Sir, 1642-1727, Mechanics -- Early works to 1800, Celestial mechanics -- Early works to 1800
Publisher: New-York : Published by Daniel Adee

K

/IK

O S P /~R Q M

and the right line RG falls upon the right line Q.E ; and at the end of the motion where the arc CD is equal to the arc Ca, and RG falls upon

the right line ST. And the area* Y or — IEF — IGH begins and ends

also where the resistance is nothing, and therefore where IEF and

IGH are equal ; that is (by the construction), where the right line RG falls successively upon the right lines Q,E and ST. Therefore those areas begin and vanish together, and are therefore always equal. Therefore the area OR

IEF — IGH is equal to the area Z, by which the resistance is ex

pressed, and therefore is to the area PINM, by which the gravity is ex pressed, as the resistance to the gravity. Q.E.D.

THE MATHEMATICAL PRINCIPLES

[BOOK 11.

COR. 1. Therefore the resistance in the lowest place C is to the force

OP

of gravity as the area ^ ~ IEF to the area PINM.

COR. 2. But it becomes greatest where the area PIHR is to the area IEF as OR to OQ. For in that case its moment (that is, PIGR — Y) becomes nothing.

COR. 3. Hence also may be known the velocity in each place, as being in the subduplicate ratio of the resistance, and at the beginning of the mo tion equal to the velocity of the body oscillating in the same cycloid with out any resistance.

However, by reason of the difficulty of the calculation by which the re sistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition following.

PROPOSITION XXX. THEOREM XXIV.

If a right line aB be equal to the arc of a cycloid which an oscillating body describes, and at each of its points D the perpendiculars DK be erected, which shall be to the length of the pendulum as the resistance of the body in the corresponding points of the arc to the force of grav ity ; I say, that the difference between the arc described in the whole descent and the arc described in the whole subsequent ascent drawn into half the sum of the same arcs will be equal to the area BKa which all those perpendiculars take up.

Let the arc of the cycloid, de scribed in one entire oscillation, be expressed by the right line aB, equal to it, and the arc which would have been described in vaciw by the length AB. Bisect AB in C, and the point C will represent the lowest point of the cycloid, and CD Mill be as the force arising from gravity, with which the body in D i,s urged in the direction of the tangent of the cycloid, and will have the same ratio to the length of the pendulum as the force in D has to the force of gravity. Let that force, therefore, be expressed by that length CD, and the force of gravity by the length of the pendulum; and if in DE you