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Let D N be a right line drawn on a plane, and a 
point given A at any diftance from the line, from 
which having raifed, above the plane, the right line 
AM in any given angle, and drawing from the fame 
point A the line AD, meeting the line DN on the 
plane in D, make another plane M N R to move pa- 
rallel to itfelf, and to that line A D in a given angle 
to the firft plane; and then, fuppofe the interfedtions 
of it M, N, zzz, zz, &c» with the lines A M and D N, 
to be continually joined with right lines M N, zzz zz, 
&c. and there will be generated an incurvated fur- 
face by them, and bounded by the lines AM, MN, 
N D, A D. Plate X. Fig. i. 
The fame furface will be defcribed, if a line MN 
be fuppofcd to move in fuch manner, along the lines 
A M and D N, that the parts A M and DN be con- 
tinually in a given ratio. For let a plane be drawn 
thro’ the raifed line A M, perpendicular to the given 
plane A D N, and their common interfedtion be the 
line A R ; then having joined the points A and D, 
from the point N, while the line M N is moving, let 
there be continually drawn a line N R parallel to 
D A, meeting the line of interfedtion A R of the 
planes in R, and the proportion of D N to AR will 
always be given. But as the ratio of D N to A M is 
given, the ratio of A M to AR will alfo be given; 
and the line RM being drawn, the angle ARM will 
be given ; and therefore the plane of the triangle 
M R N will move parallel to itfelf, and to the line 
A D, making the given angle M R A with the plane 
A D N, and the line M N will generate the fame 
furface as before. 
It 
