Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/371

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Sect. XII.
of Natural Philopoſophy.
283

the force with which the corpuſcle attracted by the whole ſphere will be as the area ANB.

Cor. 4. And univerſally if the centripetal force tending to the ſeveral particles of the ſphere be ſuppoſed to be reciprocally as the quantity V; and DN be made as ; the force with which 5 corpuſcle is attracted by the whole ſphere will be as the area ANB.


Proposition LXXXI. Problem XLI.

The things remaining as above it is required to meaſure the area ANB. (Pl. 23. Fig. 1.)

Plate 23, Figure 1
Plate 23, Figure 1

From the point P let there be drawn the right line PH touching the ſphere in H; and to the axis PAB letting fall the perpendicular HI, biſect PI in L; and (by prop. 12. book 2. elem.) is equal to . But becauſe the triangles SPH, SHI are alike. or is equal to the rectangle PSI. Therefore is equal to the rectangle contained under PS and PS + SI + 2SD; that is under PS and 2LD. Moreover is equal to , or , that is, . For or (by prop. 6. book 2. elem.) is equal to the rectangle ALB. Therefore if inſtead of we write ; the