C *74 ] 
XXVI. A Specimen of a new Method of comparing Curvi li- 
neal Areas \ by which many fuch Areas may be co??ipared 
as have not yet appeared to be comparable by any other 
Method, By John Landen, F, R. S. 
Read June 9 ,\TTHEN a body in motion is continually adted upon 
1768. yy by a variable force, the fpace it has pafled over at the 
end of any given time, it is well known, will be expreffed by the area 
of the curve, whofe ordinate expreffes the velocity of the body, whilft 
the time it has been in motion is exprelfed by the correfponding abfcifla. 
Therefore the facilitating the computation of curvilineal areas will mani- 
fefUy contribute to the improvement of the dodtrine of motion. Which 
dodtrine being a branch of philofophy of no lmall importance, fuch im- 
provement will not, i am perfuaded, be looked upon as a trifling fpecu- 
lation, by the Royal Society : to whom, therefore, I do myfelf the 
honour of communicating this fpecimen of a new and ready method of 
computing fuch areas, by means of a given area; prefuming that what 
is here written will not be deemed undeierving of a place in the Philofo- 
phical Tranfadtions. 
I. 
Geometricians have found, that, if A be put to denote the whole area 
of the curve, whofe abfcifla is at. and ordinate 
I — ■ X 
)P I 
X X 
2 r 
the 
f 
