VOL. XXXIV.J PHILOSOPHICAL TRANSACTIONS. 145 



of the blood-vessels, were found in each ventricle of the heart. These Dr. V. 

 considers as the principal cause of the patient's asthma and his sudden death. 



The liver and spleen appeared to be free from disease; in the ileum there 

 were several livid spots; and the colon, together with the rectum, was so con- 

 tracted in its whole course on the right side, where it rests upon the liver, that 

 it was scarcely as thick as one's finger, and its cavity was nearly destroyed. 



The kidnies and ureters exhibited nothing preternatural ; but three stones 

 were found in the bladder. They were as large as kidney-beans, not loose but 

 inclosed within a strong membrane, and adhered to the fore part of the bladder, 

 near the sphincter. How the membrane, in which these stones were enveloped, 

 was formed, Dr. V. thinks it difficult to explain; but the situation of the afore- 

 said stones, in the part of the bladder just mentioned, readily accounts, as it 

 appears to him, for the frequent strangury with which the patient was affected, 

 and also, in consequence of the perpetual irritation the stones would occasion, 

 for the preternatural constriction of the colon and rectum. 



Concerning Equations with impossible Roots. By Mr. Colin Mac Laurin, Pro- 

 fessor of Mathematics at Edinburgh. F. R. S. N° 394, p. 104. 



The following is a very easy and simple way of demonstrating Sir Isaac New- 

 ton's rule, by which it may be often discovered when an equation has impossible 

 ^oots. This method requires nothing but the common algebra, and is founded 

 on some obvious properties of quantities demonstrated in the following lemmata, 

 without having recourse to the consideration of any curve whatsoever, which 

 does not seem so proper a method in a matter purely algebraical. 



Lemma 1. — The sum of the squares of two real quantities, is always greater 

 than twice their product. Thus a^ + b"^ is greater than 2ab ; because the ex- 

 cess a^ -f Z)* — lab is equal to a — b , and therefore is positive ; since the 

 square of any real quantity, negative or positive, is always positive. 



Lemma 2. — The sum of the squares of three real quantities, is always greater 

 than the sum of the products, that can be made by multiplying any two of them. 

 Thus a^ + ^^ + c^ 's always greater than ab -\- ac -\- be; for the excess a" + b^ 

 , „ , , 2a'^ + 26^ + 2c* - iab - 2ac' - 2bc 

 -j- c — ab — ac — be = = 



a^ — 2a6 + 6* + a- — 2ac + c" + b'^ — Ibc + c" a — b -\- a — c + b 



_ , that IS, 



2 2 ' 



half the sum of the squares of the differences of the quantities a, b, c: but since 

 these squares are positive, it follows, that the excess of a'- -\- b'^ + c"^ above 

 ab -\- ac + be is positive, and that the sum of the squares of three quantities 



VOL. VII. U 



