140 PHILOSOPHICAL TKANSACTIONS. [anNO 16q6-7. 



want it likewise, for the most part, for they creep only at the bottom of the 

 water, but there are many fish that have them double. 



On the Ratio of the Time, in which a heavy Body passes over a Right Line, join- 

 ing tiuo given Points, to the shortest Time ivhich, hij the Force of Gravity, it 

 passes from the one of these Points to tlie other by the Arc of a Cycloid.* 

 N° 225, p. 424. Translated from the Latin. 



Theorem. In the cycloid avd, (fig. 8, p. 3) whose base ad is parallel to the 

 horizon, and vertex v is downwards, if from a there be any- how drawn the 

 right line ab meeting the cycloid in b, from whence draw the right line bc per- 

 pendicular to the cycloid at b, to which from a there is drawn the perpendicular 

 AC. I say that the time in which a body from rest falling from a, by the force 

 of its gravity, describes the right line ab, is to the time in which it runs along 

 the curve avb, as the right line ab is to the right line ac. 



Through b draw bl parallel to the axis ve of the cycloid, and bk parallel to 

 the base ad, meeting the axis in g, and a circle described on the diameter ev in 

 F and H, and lastly meeting the cycloid in k. Draw the right line ep, which 

 from the nature of the cycloid, will be parallel to the riglit line bc. Whence 

 BM is equal to ef, and em to bf, which by the cycloid, is equal to the arc vf ; 

 and therefore am is equal to the arc ehvf. 



By prop, 25, part 2, of Huygens's Horologium Oscillatorium, the time in 

 which a body falling from rest passes over av, is to the time falling through 

 ev, as a semicircumference is to its diameter, and by the last prop, of the same, 

 the time in which a body runs over vb, after having passed over av, (viz. 

 equal to the time in which a body passes over kv after running over ak) is to 

 the time of sliding along av, as the arc vf is to the semicircumference; and 

 therefore to the time of the fall through ev as the arc fv to the diameter. 

 Therefore the time in which a body runs over the curve avb is to the time of 

 the fall through ev, as the arc ehvf to the diameter ev. But the time of the 

 fall through ev is to the time of tiie fall through lb, or eg, as ev to ep; 

 therefore, by equality, tlie time of a body's running along avb is to the time 

 of the fall through lb, as the arc ehvf to the subtense ef : that is, as the right 

 ine AM to the right line mb. Again, the time of the fall through lb, is to 

 the time of sliding along ab, as lb to ab ; therefore the ratio of the time in 

 which the body passes along avb, to the time of running over ab, is composed 

 of the ratio of am to mb, and of the ratio of lb to ba ; and therefore is equal 

 to the ratio of am X lb to mb X ba. But am X lb is equal to mb X ac, 

 because each is equal to double the triangle abm : and therelore the time in 



* This anonymous pnper has very much the character ola proJuclion of Sir Isaac Newton'*. 



