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In the right angled Triangle ABC, the bafe BC and the fum of the perpendicular and fides, BA+AC+AD, being given, to find the Triangle.

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for this purpose, and let it be called x: Let 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.) BC2=BA2+AC3, which being expreffed algebraically, becomes b2 = a-x-y a—2ax+x2+y2 Like






wife, from a known property of right angled triangles, BCXAD BAXAC; that



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This last equation being multiplied by 2, and added to the former, gives b2+2bx =a2—2ax+x2, which being resolved according to the rules of Part I. Chap. 5. gives x=a+b−√2ab+2b2.

To conftruct this: a+b is the fum of the perimeter and perpendicular, and is given; √2ab+2b2=√a+bx 26 is a mean proportional between a+b and 2b, and may be found; therefore, from the fum of the perimeter and perpendicular, fubtract the mean proportional between the faid fum and double the base, and the remainder will be the perpendicular required.

From the bafe and perpendicular, the right angled triangle is easily constructed. In numbers, let BA+AC+AD=18.8 a; BC 10b; then AD = a+b— √2ab+2b2=28.8—√576=4.8=x, and BA+AC 14. By either of the first equations, y2=2b2+2ax—a2—x2—4 and y= BA-AC 2; therefore BA = = 8, and AC=6.


The geometrical expreffion of the roots of final equations arifing from problems, may be found without refolving them, by the intersection of geometrical lines. Thus, the roots of a quadratic are found by the interfections of the circle and ftraight line, those of a cubic and biquadratic, by the interfection of two conic fections, &c.

The folution of problems may be effected alfo by the interfections 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 pofition of every point of them.

This rule conftitutes the definition of any line from which all its other properties are to be derived.

A ftraight line has been, confidered as fo fimple, 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 univerfally by fome continued motion in a plane, according to particular rules. Any of the properties which are fhewn to belong peculiarly to fuch a line, may be affumed also as the definition of it, from which all the others, and even what,


have been con

upon other occafions may fidered as the primary definition, may be demonftrated. Hence lines may be defined in various methods, of which the most convenient is to be determined by the purpose in view. The fimplicity 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 fuppofed to be produced by the extremity of one straight line, as PM moving in a given angle along another ftraight line AB given in pofition, which is called the base.

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