An introduction to astronomy: designed as a text-book for the use of students in college. By Denison Olmsted ...

CENTRAL FORCES3 91
Tllese wele atlls sccrtaillntd as fcits, many ye-ars before
New'cton delonlstratedl, by matlhematical recasoni'. t:hat tlhey
tMare neccssnaritly involved in the laws of inertia andl gravitation..T1he funldamental principles of all mecllanical action, pertaiini:g alike to terrestrial bodlies and to the worlds scattered
t)hrouthoutt space, are tile following.":1. M;atter, until acted on by ext aneous firce, w vii remain
perpe)tually in its present conditiotn, whlther of 1cst or straight
11uniform motion.
2. All motions colulmmnicated to a body coexist inll the m1otion of thle body.
3. To every action there is an equal and opposite reactioln.
4. All mas-ses tend toward each other, with at force vary'ing
directly as the quantity of matter, and ilvcrsely asa thle square
of the dlistance.'lhe three first, thle laws of inerttia, of coexisten! mot0tiot9, and
of equal actio)n2 (tm!d reactiown, were seen to be the true first
principles in the Mchltalics of terrestrial t bodics. itult thtey a re
equally essetlltlial inl Asttronomll. e..........t celtestial Mechanllics; and
not on-lty 0does no1 fact in this science militate against thelmc, b)ut,
onl tfil ctontrary, thley fb)rml thle basis of all correct reasoning o-n
the motionls of thie lheavenly bodies. T.'hie fourth, usually c(taled
tile ladt)t otE ig:it1, is fialr more prominlent in A.strononmy thlan
in Mecllhanics, but harmonizes withl all the fitcts of bothl. WeVe
proceed to showr tthat Keplter's laws and other laws of tenltral
forces, are' the necessary consequences of the above-named inecianlical principles.
1 7 2. IWhatever path a body describes under the influence
of:t pro(jectile alld a centripetal force, the rad-iuts vector of thal
pat/h passes overt equal spaef s' in., equttal timns.
[Let 8 (Fi';g 34) be the center of attraction, anmid suppose the
proje-tile force hill tlhe lie tY to be suchlt (s to causs to i body
to pass over the cqual spaces PQ, QR, &e., eacht in a certtain
unit of time. When the body reaclhes Q, let the action toward
S be sutlfficient to move it; over QV' inl lhe same timte in whlichl
by the original illpunlse it would (describe Q1T,'Thlien itt will
in thte satie time describe thle diagonal QO of the parallelograin. Join -S and CS. Thelie triangles QSC ald QSl:t are
equal; blut QSttI:;QSP;.' Q8(. (.:QSi.. That is, the >areas dc