To describe an Hyperbola. 



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in the point 2, and proceed in a similar manner to find the points 3, 

 4j 5, 6, &c. which will be points in the hyperbola. 



To find intermediate points, in BF assume any point «, and with 

 the distance BF find another series of equidistant points h, c, &c. 

 and proceed as at first to find the corresponding points d, e, &;c. in 

 the hyperbola. By assuming other points in BF as many points 

 may be found in the curve as are required. 



Demonstration . 

 . Let it be required to prove that the point 4 is in the hyperbola. 

 Prpduce DB and draw AP parallel to BE ; then by similar triangles 

 PG : PA : : GH : H4, or by substituting equals 

 BH : CF: : BF : H4; hence, (Euc. prop. 16, b. 6,) the paral- 

 lelogram BH4 completed would be equal to the parallelogram BFCL, 

 which is the property of the hyperbola. In the same manner, it may 

 be demonstrated that any other of either series of points is in the 

 curve. In this construction, the continual approximation of the curve 

 to its asymptote, without the possibility of ever arriving at it, is very 

 obvious ; the distance of any point of the asymptote from the curve, 

 measured on the parallel from that point, being always equal to BF^, 

 divided by the distance of the point from the centre of the hyperbola. 

 In the first series, if BF or FC be the unit, the distance of the points 

 in succession will be 1, ^, ^, ^, i, &;c. 



