JMtfceUanea Curiofa. 169 



right Line EF, which from the Nature of 

 the Cycloid is parallel to BC } whence BM 

 is m EF, and EMsBFs the Arch VF 

 from the Nature of the Cycloid } and con- 

 sequently AM is S3 the Arch EHVF. 



By Propofition 25. Part II. Horolog. Ofcillatl 

 Hugert. the Time in which a Body at reft 

 in A defcribes the Cycloidal Arch AV, is 

 to the Time of Defcent thro' EV, as the 

 half Circumference to the Diameter. 



And (by the laft Propfition of the fore- 

 mention'd Part) the Time of Defcent thro* 

 VB, after the Defcent thro 5 AV (which is 

 equal to the Time of Defcent thro' KV, 

 after the Defcent thro' AK) is to the Time 

 of Defcent thro' AV, as the Arch VF, to 

 the Semicircumference ; and confequently to 

 the Time of Defcent thro' EV, as the Arch 

 FV, to the Diameter. Wherefore the Time 

 of defcribing the Curve AVB, is to the 

 Time of Defcent thro' EV, as the Arch 

 EHVF, to the Diameter EV. But the Time 

 of Defcent thro' EV, is to the Time of 

 Defcent thro' LB or EG, as EV to EF* 

 Therefore (by Equality ) the Time of de- 

 fcribing AVB, is to the Time of Defcent 

 thro' LB, as the Arch EHVF, to the Sub- 

 tenfe EF, that is, as AM to MB. Again, 

 the Time of Defcent thro' LB, is to the 

 Time of Defcent thro' AB, as LB to AB. 

 Therefore the Time of defcribing AVB, 

 is to the Time of Defcent thro' AB, in the 

 Ratio compounded of AM to BM, and LB 

 to BA, and confequently is equal to the 

 Ratio of AM x LB to MB x BA. 



But 



