. Mathematics, 35^ 



Within the period under consideration several 

 new and valuable branches of mathematics, now 

 in use, have been either wholly discovered, or 

 placed on a footing, in a great measure, if not 

 entirely, new. It will be proper briefly to men- 

 tion some of the more important of these. 



In 1717 Dr. Brooke Taylor invented a new 

 branch of analysis, which he called the Method of 

 Increments, in which a calculus is founded on the 

 properties of the successive values of variable 

 quantities, and their differences or increments. 

 This method is nearly allied to Newton^s doctrine 

 of Fluxions, and arises out of it; insomuch, that 

 many of the rules formed for one serve also, with 

 little variation, for the other. By means of the 

 Method of Increments many curious and useful 

 problems are easily solved, which scarcely admit of 

 a solution in any other way. It is, particularly, 

 of great use in finding any term of a series proposed, 

 and also in finding the sums of a series given. In 

 1763 an ingenious and instructive treatise on this 

 new method was published by Mr. Emerson, who 

 threw further light upon it. The Differential 

 Method of Mr. Stirling, which he applied to the 

 summation and interpolation of series, is of the 

 same nature with the Method of Increments, but 

 not so general and extensive. 



In 1724 M. Lagny, of France, discovered a 

 new mode of measuring angles, which he deno- 

 minated Goniometry. By means of this method 

 he was enabled to ascertain the measure of angles, 

 without the use of either scales or tables, and with 

 great exactness; a method which exceedingly ab- 

 breviated, or rendered w-holly unnecessary, many 

 tedious calculations. 



In 1746 the Rev. Dr. Stewart, of Scotland, 

 published new and elegant Theorems, of great 

 value to the mathematician, by which he extended 



