180 On the Simplification of Mathematical Analyses. 
do not allow them to form one continued right line, then the 
locus of their intersection is the curve of an ellipse or hyperbola, 
according as the sum or difference be given; which locus being 
described, simplifies the problem to one consisting of a less num-~ 
ber of data. 
Rule 2. If the ratio of any two lines proceeding from two 
known points be given with any other data, the construction of 
the locus is, by joining the two given points, and finding two others 
(one within and the other without), such that the ratio of the 
segments intercepted between the two given points may be equal 
to the given ratio; then describing a circle to pass through the 
two last found points, and whose ceutre shall be in the continua- 
tion of the right line passing through the given points :—that 
circle will be the required locus. 
Rule 3. In case of the difference of the squares of two lines 
proceeding from two known points being given, to determine their 
locus, we connect the two points by a right line, in which we find 
a third point, such that the difference of the squares of the inter- 
cepted segments may be equal to the given difference of squares, 
and erecting a line from that point perpendicular to the before- 
mentioned line, it will be the required locus. 
Observation \. If there be given the difference of squares, to — 
which. the squares of the lines under consideration have given 
ratios, the locus is obtained on the same principle by merely al- 
tering the situation of the given points, according to the ratio the 
squares bear to each other, on the principle of similar triangles. 
Oliservation 2. If the lines in question do not proceed from 
two given points, but are tangents to two circles, whose centres 
and radii are known, the problem is solved in nearly the same 
manner as before shown, on the principle of a tangent being 
perpendicular to the radius, passing through the point ‘of contact, 
and from that arguing upon the Pythagorean proposition. 
Rule 4. But if the sum of the squares, along with any other 
data, be given, to determine the locus of lines proceeding 
from two given points,—Bisecting the right line that connects 
the two given points, and finding another line, such that double 
of its squares, together with double the square on half the given 
line, may be equal to the sum of the squares of the two lines 
whose locus is required; then describing a circle with a radius equal 
to the line just found, and whose centre shall be the middle of the 
line, joining the given points,—then will that circle be the locus 
that was required to be described. 
This rule will, as well as the last, admit of two additional 
cases. 1. When there is given the sum of squares having given 
ratios to the squares of the lines intercepted between the two 
given 
