210 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



raw hide taken from a beast just dead, or putting a pustule into the neck, they 

 should either have passed in the dewlap cotton or silk dipped in well digested 

 pus, or have inserted in proper incisions cotton-thread or silk soaked with pus, 

 either on the shoulders or buttocks ; the true way of inoculating in the English 

 manner. Some persons have indeed thought, that to inoculate with the blood 

 of the infected would answer the intention ; but most of the modem practi- 

 tioners chuse to depend on digested matter. 



Several constitutions would not receive infection, though inoculated ever so 

 judiciously. A Ranby, a Hawkins, a Middleton, and other inoculators, would 

 tell us that the incisions had sometimes suppurated much, and pustules had 

 appeared round the edges of the wound without any other particular marks of 

 the disease ; and yet the patient had never had the small-pox afterwards. The 

 Marquis mentioned an instance somewhat of the same kind in his first Me- 

 moir, p. 147. 



LXX. Trigonometry Abridged. By the Rev. Patrick Murdoch, A.M., F.R.S, 



p. 538. 



The cases in trigonometry, that can properly be called different from one an- 

 other, are no more than 4 ; which may be resolved by 3 general rules or theorems, 

 expressed in the sines of arcs only ; using the supplemental triangle as there is 

 occasion. 



Case 1 . When of three given parts two stand opposite to each other, and the 

 third stands opposite to the part required. Then, 



Theorem I . The sines of the sides are proportional to the sines of angles op- 

 posite to them. fit »% 



Cnses 2 and 3. When the three parts are of the same name. And, when 

 two given parts include between them a given part of a different name, the part 

 required standing opposite to this middle part. Then, 



Theorem 2. Let s and s be the sines of two sides of a spherical triangle, d the 

 sine of half the difference of the same sides, a the sine of half the included angle, 

 b the sine of half the base ; and writing unity for the radius, we have s*a^ -f 

 (P -^ b^ = O; in which a or b may be made the unknown quantity, as the case 

 requires. 



Note. 1 . If this, or the preceding, be applied to a plane triangle, the sines of 

 the sides become the sides themselves ; the triangle being conceivetl to lie in the 

 surface of a sphere greater than any that can be assigned. 



Note 2. If the two sides be equal, d vanishing, the operation is shorter : as it 

 also is when one or both sides are quadrants. 



Note 3. By comparing this proposition with that of the Lord Napier, which 



