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In the right angled Triangle ABC, the base

BC and the sum of the perpendicular and
fides, B.A +AC+AD, being given, to find
the Triangle.


Such parts of
this triangle are
to be found as are
necessary for de-
fcribing it: The
perpendicularAD C

will be sufficient
for this purpose, and let it be called x: Let
AB + AC + AD= a, BC=b, therefore
BA+AC=a-x: Let BA-AC be denoted

by y, then BA=+*, and AC=-=-y

But (47. I. El.) BC?=BA? +AC?, which
being expressed-algebraically, becomes 62 =
aty ** +

a- -2ax+x+y?

wise, from a known property of right ang-

led triangles, BCXAD=BAXAC; that


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is bx= (******=)

a-2axtx?—? This last equation being multiplied by 2, and added to the former, gives 62 +2bx =a—2ax+x2, which being resolved according to the rules of Part I. Chap. 5. gives x=a+b-2ab +262.

To construct this: atb is the sum of the perimeter and perpendicular, and is given; 2ab +2b2=vat6 x 2b is a mean proportional between a +b and 2b, and may be found ; therefore, from the sum of the perimeter and perpendicular, subtract the mean proportional between the said fum and double the base, and the remainder will be the perpendicular required.

From the base and perpendicular, the right angled triangle is easily constructed.

In numbers, let BA +AC+AD=18.8 =a; BC= 19 =b; then AD = a +6 10

bzab+262 = 28.8—V 576=4.8=x, and BA+AC=14. By either of the first

equations, ya=262+2ax—a?—*2=4 and

y = BA— AC = 2; therefore BA = 8, and AC=6.




The geometrical expression of the roots of final equations arising from problems, may be found without resolving them, by the intersection of geometrical lines. Thus, the roots of a quadratic are found by the intersections of the circle and straight line, those of a cubic and biquadratic, by the intersection of two conic sections, &c.

The solution of problems may be effected also by the intersections of the loci of two intermediate equations without deducing a final equation : But these two last methods can only be understood by the doctrine of the loci of equations,



Of the Definition of Lines by Equations.


INES which can be mathematically

treated of, must be produced according to an uniform rule, which determines the position of every point of them. This rule consțitutes the definition of

any line from which all its other properties are to be derived.

A straight line has been considered as so simple, as to be incapable of definition. The curve lines here treated of, are supposed to be in a plane, and are defined either from the section of a solid by a plane, or more universally by some continued motion in a plane, according to particular rules. Any of the properties which are shewn to belong peculiarly to such a line, may be assumed also as the definition of it, from which all the others, and even what,



upon other occasions may have been considered as the primary definition, may be demonstrated. Hence lines may be defined in various methods, of which the most convenient is to be determined by the purpose in view. The simplicity of a definition, and the ease with which the other properties can be derived from it, generally give a prefe




I. When curve lines are defined by equations, they are supposed to be produced by the extremity of one straight line, as PM moving in a given angle along another straight line AB given in position, which is called the base.

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