£5() GEOMETRICAL PK OPOS ITlOttS* 



tranfcribing it mod probablv, may tend to puzzle your read- 

 ers, and amohg the reft perhaps Mr. Bofwell hirhfelf, who 

 muft be the mofi interefied : You have faid, " as the radius is 

 to IW, fo is the diameter to IB," (Plate VIII. Fig. 10); 

 but I take it for granted you meant, " as I W is to radius, 

 jo is the diameter to I B," becaufe this analogy is agreeable to 

 Farther elucida- the relult of your calculad'on. And as you have net men- 

 tion of the note. (ioned how (h ' e doclrine of f]m ji ar trian gi es a p p ij es in the con- 



ftruclion of the figure, I hope you and your numerous readers 



will not deem me prefumptuous, if I introduce the diagram 



again with fome additions, particularly as it will give me an 



opportunity of proving more intelligibly the error of the fecond 



proportion already quoted. Any triangle formed in a femi^ 



circle, of which the' diameter is one fide, with its oppofite 



angle at the circumference, is a right angled triangle (Eucl. III. 



Prop, xxxi.) : This will be the cafe in Mr. Bofwell's figure 



if a right line be drawn from B to F, the right angle being at 



B; and as the angle at I is common to the (mail right angled 



triangle I O W, and to the large one I B F, the two triangles 



are fimilar, and therefore the hypothenufe I W of the fmall 



one is to its bafe I O, as the hypothenufe I F (or diameter) is 



to the bafe I B of the large one. Q. E. D. 



How far Mr. With refpeel to the fecond propofition, it is very well 



B.'s fecond pro- known , Euc!i ttt. p rop . xx .) that the angle formed at the 



potation is accu- ' ' „ ° 



-ate. centre of a circle is double of the angle at the circumference, 



if they have the fame common chord as a bafe; now in the 

 triangle I O W (Plate XI. Fig. 4), the fides I O being 

 given = .5, O W = .2.5, and therefore I W == .5590!, ive 

 have, by a fimple cafe in plane trigonometry, the angle at 

 I =26° 33' 5".i, the double of which is 53° 6' lW.86 for 

 the angle B O F at the centre ; but B G, one half of the line 

 B E in quefiion, is the fine of this angle, and in natural num- 

 bers is, by the tables, equal to .7997163, when unity is ra- 

 tlins ; but, in our figure, the diameter reprefents unity, there- 

 fore the double fine, or chord line, B E is = .7997 163, which, 

 according to Mr. Bofwell's affertion, ought to be = .7854, or 

 one-fourth of the circumference 3.1416; hence the error is 

 .0143163, or very nearly ^—th of the whole. 

 Apology fo thefc As I have not the pleafure of knowing Mr. Bofwell, not- 

 remarks. withftanding I have admired feveral fpecimens of his ingenu- 



ity, and as I profel's to be a promoter of all arithmetical ap* 



proximations 



