6 AN ACCOUNT OF THE EARLY MATHEMATICAL AND 



drawn a chord BF to cut OC in E, and on the same chord 

 there is taken BD equal to the radius of the semicircle; — it is 

 required to determine the rectangle DE.EF, a maximum." 

 Ans. BD = 1 ; DE = -1974; EF = -4731. 



Gentleman's Diary, Question b^^. By Mt\ John Dalton. 

 " From a given circle to cut off an arc, such, that the 

 rectangle under its versed sine, and the cosine of its double 

 may be a maximum." 



Ladies' Diary. Question 875. By Mr, Alexander Rowe, 

 " It is required to find the diameter of a circular parachute, 

 by means of which a man of 200 pounds weight may descend 

 from a balloon at a great height, with the uniform velocity of 

 only 10 feet in a second of time; the parachute being sup- 

 posed to be made of such materials and thickness that a circle 

 of it 50 feet diameter weighs only 150 pounds." 



Answered by Mr, John Dalton. " If a falling body move 

 with a uniform velocity, it must necessarily meet with a 

 resistance, in the medium it is moving in, equal to its weight. 

 Now it has been proved (Emerson's Mechanics, prop. 1 08, 

 cor, S) that the resistance to a cylinder, moving in a fluid in 

 the direction of its axis, is equal to the weight of a cylinder 

 of that fluid of the same base, whose length is equal to the 

 height a body falls in vacuo to acquire its velocity. Put now 

 g = 32 J feet, v = the velocity ; then g^ ', v^ :: ^g : ^g = the 

 altitude fallen to acquire the given velocity. Call this altitude 

 a, and put also p = '7854 ; b = •0751b. = the weight of a 

 cubic foot of air ; m = 200lbs. = the man's weight ; also 

 X = diameter of the parachute. Then 50^ : x^ :: 150 : ^^ x^ = 

 weight of the same ; which added to 200 or m, must be equal 

 to the resistance. That k — x'^ + m z= a b px^; and hence 



X = IQQj' na^'LG ~ ®^^ ^®®* nearly, the diameter sought." 



On the appearance of the Ladies' Diary for 1 789, Mr. 

 Dalton must have felt himself amply rewarded for all his 



