On the Simplification of Mathematical Analyses. 181 
given points and the point of their intersection with the required 
locus. 2d. When the two lines do not proceed from the two fixed 
points, but are tangents to two circles whose radii and centre 
are known. Both these cases will admit of circular loci, on the 
same principle as the similar cases adduced to Rule 3. 
If you have the ratio of two right lines with other data, varying 
according to the circumstances of the case, there are a great 
number of loci that may be applied, according to the different 
forms the problem may assume. One of the most. generally useful 
of that species of loci, is given in the second rule; but the fol- 
lowing are some that apply to problems involving other con- 
ditions or data. 
Rule 5. If it be required to determine the locus of one ex- 
tremity of a line revolving round a point as a centre, and meeting 
a right line given in position when the ratio of the segments in- 
tersected by that line is a given one,——Draw any line from the 
fixed point to the line given by position, and continue it till the 
parts obtain the given ratig; then drawing through its extremity 
a line parallel to that first given, it will be the locus required, as 
will be evident by drawing any other line through the given point 
to meet the two parallels. 
Rule 6. If it be required to determine the locus of one ex- 
tremity of a line, which being parallel to a line given in position 
is always interposed between two other lines given in position, and 
continued, so that the ratio of the segments may be equal to the 
ratio of two known lines,—Draw any line parallel to the line 
given in position, and continue it so that the ratio of the seg- 
ments may be equal to the ratio of the known lines; join the ex- 
tremity of the line so continued, and the point of intersection of 
the two lines given by position ; then will the line so drawn con- 
stitute the required locus. 
Much more might undoubtedly have been added to the above 
rules on so important a subject as the geometrical loci; but as 
most of the higher and therefore more interesting geometry 
concerns the conic sections, &c. which are so much more diffi- 
cult in their enumeration as hardly ever to admit of being de- 
scribed by mere words without diagrams, I have therefore here 
emitted them. 
M 3 XXXV. Notes 
