tlN'ETS IN A CIRCLE. 5l 



(ficTcfore the reaangle F I, I G =:: I B*, 8. Ts. VI. ; but I B*= Propt>fkion» 

 (be redangle under I O and the arc I F F by lemma; ^•onfe-^^^^J^^,^.'^*^^^ 

 quentlvas F I (=: 2 I O) : I O : : arc ITF: right hne IG; 

 }IE.VI hence I G ::= i the arc I T F, = the arc TBF, 

 Q. E. D. 



Corollary 1. If I G, a fegment of I F be equal to Ihe arc 

 TB F; draw G B perpendicular to I F; and let it meet (he 

 circle in B; the line I B is the fide of a fquare; which is equal 

 t-o the area I T F C, This is the converfe of the Theorem. 



Cor, 2. U any angle at the centre of the circle as I O B, 



be given in parts of the right angle T O F; and I G be eqaal 



to the arc T B F ; find a right line N; which (hall be a fourth 



proportional to the angles T O F, I O B and the right line I G; 



this line N is equal to the arc I T B; and a mean proportionail 



N 

 betwixt I O, and — ^ is the fide of a fquare which is equal to 



the area I O B I ; join I B, and from the laft mentioned fquare 

 take the trinangle I B O ; the remaining magnitude is equal to 

 the circular fegment I T B. 



Cor, 3. The fquare upon B F is equal to the difference of 

 (he areas of the circle and its circumfcribing fquare. 



Problan. If the circumference of a circle, whofe diameter 

 is unity be denoted by 3.1416; it is required to find a right hne 

 which {hall approximate MQxy nearly to | of this nomber, oc 

 .7854. 



ConfiruHlon. Draw I B, as in Theorem I ft, and make B G 

 perpendicular to I F; then I Gx 1 = I B,^= .8000; confe- 

 quenlly J G= .8000, which is greater than .7854. Let the 

 reader take £^ in I G, fo that I g may be of the required length ; 

 then as 8000 : 7854 : : IG: I^; but 8000 is to 7854 nearly as 

 55^ to 54; therefore divide I G into 55 parts and I^ will be 

 54 of thefe parts. Draw g h perpendicular to I G; join I 6; 

 and the fquare upon I h will be nearer the truth than that upon 

 IB. If a more complete approximation be required, it may 

 be di (covered by the method given in the ninth Problem of 

 Einerfon^s Arithmetic, Q. E. F. 



It is the bufinefs of the pradical geometrician to determine 

 the value of thefe propolitions in practical geometry. Tl)e 

 ingenious Mr. Bofwell confiders the firft theorem to be oC 

 ulilitjj^ ; for which reafon I imagine any improvemcHt in the 



difcover/ 



