378 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739. 



in equations to cosines, observe the following rule: if n be an even number, 

 there will be as many affirmative roots as negative. But if n be an odd num- 

 ber, but such that — ^ be an even number, the number of affirmative roots 



will be ^^^^^, and the number of negative — ^. 



But if ^ be an odd number, the number of affirmative roots will be "\-, 



and the number of negative 



n — 1 



2 

 END OF THE FORTIETH VOLUME OF THE ORIGINAL. 



A Catalogue of the Fifty Plants from Chelsea Garden, presented to the Royal 

 Society by the Company of Apothecaries, for tfw Year 1737, pursuant to the 

 Direction of Sir Hans Sloane, P. R. S. By Isaac Rand, F. R. S. N" 432, 

 p. 1. Vol.XLL 



This is the l6th annual presentation of this kind, making the number of 

 800 different plants. ^ 



Of the Measure and Motion of Effluent Water. By James Jurin, M. D. F. R. S. 

 ^c. N° 452, p. 5. From the Latin. 



Essay I. Of Water issuing from a Vessel kept always full, through a 

 round Hole ; and of its Resistance arising from a Defect of Lubricity. — ^The 

 ancients had no other measure of effluent water, than that uncertain and falla- 

 cious one, which, having no regard to the velocity, depended wholly on the 

 perpendicular section of the stream. The first who opened a way to the truth, 

 was Castelli, an Italian, and the friend of Galileo. He, having discovered that 

 the quantity of water flowing through a given section of a stream, is not 

 given, as the ancients thought, but that it is proportional to the celerity with 

 which the water is carried through it ; by this noble discovery he laid the foun- 

 dation of a new and most useful hydraulic science. This discovery therefore 

 engaged the philosophers to study this doctrine so carefully, that after Castelli's 

 time there was hardly any eminent mathematician, who did not endeavour to 

 add something to it, either by experiments, or by reasonings and argument* 

 k priori. 



But most of them, notwithstanding their great abilities, had no success in 

 it, because of the exceeding difficulty of the work. For those who studied 

 only the theory, laid down such theorems as were found to be false, when 



