174 REMARKS on the 



the fquare root, which is the meafure of the femiduration *. 

 The fame thing is praclifed in lunar eclrpfes f* Thefe opera- 

 tions are all founded on a very diilincl conception of what hap- 

 pens in the cafe of an eclipfe, and on the knowledge of this 

 theorem, that, in a right-angled triangle, the fquare on the 

 hypothenufe is equal to the fquares on the other two fides. It 

 is curious to find the theorem of Pythagoras in India, w r here, 

 for aught we know, it may have been difcovered, and from 

 whence that plrMofopher may have derived fome of the folid, as 

 well as the vifionary fpeculations, with wliich he delighted to 

 inftruc"t. or amufe his difciples. 



49. We have mentioned the ufe that is made of the femi- 

 diameters of the fun and moon in thefe calculations, and the 

 method of afcertaining them, is deferving of attention. For 

 the fun's apparent diameter, they take four-ninths of his 

 diurnal motion, and for the moon's diameter, one twenty- 

 fifth of her diurnal motion. In an eclipfe, they fuppofe the 

 feclion of the fhadow of the earth, at the diftance of the moon, 

 to have a diameter five times that of the moon ; and in all 

 this, there is confiderable accuracy, as well as great fimpli- 

 city. The apparent diameters of the fun and moon, increafe 

 and diminifn with their angular velocities ', and though there 

 be a miftake in fuppofing, that they do fo exactly in the fame 

 proportion, it is one which, without telefcopes and microme- 

 ters, cannot eafily be obferved. The fection of the earth's fha- 

 dow, likewife, if the fun's apparent diameter be given, in- 

 creafes as the moon's increafes, or as her diftance from the 

 earth diminifhes, and nearly enough in the fame ratio to juftify 

 the rule which is here laid down. 



50. The hiftorian of the Academy of Sciences, in giving an 

 account of M. le Gentil's Memoir, has juftly obferved, that 



the 



* Mem. Acad, des Scien. 1772, II. P. 259. 

 f Ibid. 241, 



