in Orthographic Projection. 385 







Minor diameter of ellipse in aob z= 88.19 



in 6oc or coa= 47.14 



Major diameter, . . . =: 110.55 



The minor diameter of an ellipse, representing a circle lying 

 in the plane, bounded by two of the axes, is always parallel 

 to the representation of the other one. 



The method of using these data is obvious : Thus, lay off 

 three lines to represent o a, ob^ oc, at the proper angles by 

 means of a protractor ; or, if there be none at hand, by the 

 dimensions a 0, be, c a ; then proceed exactly as in the iso- 

 metrical projection, making all the lengths, breadths, or 

 heights severally parallel to <? «, o b, or o c, as will best suit 

 the subject, and observing to make the measurements with 

 the proper scales.* 



The annexed figures, Plate VIII. represent the same object 

 in each of the arrangements described. 



* Formulae for computing the numbers. 



A, B, C, being the angles which the axes make with the orthographic 

 plane, there is 



(1) cos 2A + cos 2B + cos ' C = 2 



(2) cos a 6 = tan A tan B 



(3) rtc 2 = ( V 2 — sin A + sin C) (V 2 + sin A — sin C ) 



cos 2 A 

 opfOq, or being the lines in which the planes aoh,bo c,coa intersect the 

 orthogi-aphic plane, and P, Q, R, being the angles which these lines make 

 with the axes represented by o a, o 6, o c, there is 



(4) poo = qoa = rob = 90^ 



. ^ sin A . ^ sin B . ^ sin C 



o n being the representation of any other line inclined to o a, lying in the plan e 

 aob,v its real length, N the angle which it makes with the orthographi c 

 plane, and n the angle which it makes with op, there is 



(6) tan poll ^^ tan n sin C 



(7) sin N = sin n cos C 



/ox cos N 



(8) on = , T- 



cos A 



D being the diameter of a circle in the object represented there is 



sinC 

 Minor diameter of ellipse in plane a o 6 = D ^^^ ^ 



boc — D tan A 



sin B 



coa = D 



cos A 



Major diameter of ellipse in any plane = D —, 



