On the Division of a Right Line. 159 



this last proportion the value of BH found above, we shall have 

 AB(m — l) . gQ . . (^^ _^ i^ . ,„jj. Obtaining from this propor- 



m 

 tion the value of BG, which is expressed by AB and the known 



quantities m, n, I, we shall have BG = ^^~/ , a formula 



mn — I 



which indicates the ratio of BG to AB, according to the particular 

 values given to I, m, n. 



Although in varying the value of n very interesting results may 

 be obtained, I shall confine myself to the examination of the for- 

 mula in those cases where n = 1. as bemg that which is most con- 

 veniently applicable in practice, and in which case 

 BG = AB {m — I) j^^^^ j^ jg required to divide AB into 2. 3. 



m+ I 

 4. 5. orp equal parts, it is evident that to make BG one of those 



parts we should have generally ( ^" ~ ) = -, from which may 

 ^ \ m + I / p 



be obtained successively (m — l)p = m + I, or mp -- Ip = jn + I, 

 and by transposition p« — m = pi + I, or, which is the same 

 thing,7w(p— 1) = Z(p + 1), from whence maybe deduced the pro- 

 portion m : Z :: (p + 1) : (p. 1). On now giving particular values to 

 p, the ratio will be obtained which should exist between ?w and I for 

 the different cases which may be proposed. It must be remem- 

 bered here that n has been made equal to 1, i. e., that the line BD 

 is to be extended by a quantity equal to its own. To illustrate 

 what has been said by an example, let it be proposed to divide the 

 line AB, fig. 2, into four equal parts. The proportion m '.I:: 

 (p + 1) : (p — 1) becomes, by making p = 4, ?n : Z : : 5 ; 3, so 



that five equal parts must be laid down on AC, and from the ex- 

 tremity F of the line DF = 2 DB a right line must be drawn to the 

 extremity of the third part on the line AC, which right line will 

 divide AB at G, GB being equal to a fourth part of the whole. 



When AB is to be divided into an unequal number of parts, and 

 consequently (p + 1) and(p— I) arc both even numbers, the ratio 

 of TO to /may be simplified by dividing {p + 1) and (/j— 1) by 2. 



