70 Mathematics, [Chap. VI. 



new. It will be proper briefly to mention some of 

 the more important of these. 



In 1717 Dr. Brooke Taylor invented a new 

 branch of analysis, which he called the Method of 

 hicrements, in which a calculus is founded on the 

 properties of the successive values of variable quan- 

 tities, and their diiferences or increments. This 

 method is nearly allied to Newton's doctrine of 

 Fluxions, and arises out of it ; insomuch, tliat 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 fmding any term of a proposed series, and also 

 in fmding the sums of a given series. In 1763 an 

 ingenious and instructive treatise on this new me- 

 thod was published by Mr. Emerson, w^ho threw 

 further light upon it. The Differential Method x)f 

 Mr. Stirling, which he applied to the summation 

 and interpolation of series, is also 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 denominated 

 Goniometr}}.; by means of which he was enabled 

 to ascertain the measure of angles, without the 

 use of either scales or tables, and with great ex- 

 actness: but this method, though very ingenious, 

 has not been found to be applicable to any pur- 

 poses of practical utility. 



In 1746 the rev. Dr. Stewart, of Scotland, pub- 

 lished new and elegant Theorems, of great value to' 

 the mathematician, by which he extended the ap^ 



