EQUATION. 



the intersection of the right line with a 

 line of the third order, and so on. But 

 if, instead of the right line, some other 

 line of a higher order be used, then tbe 

 second line, whose intersections with the 

 former are to determine the roots of the 

 equation, may be taken as many dimen- 

 sions lower as the former is taken high- 

 er. And, in general, an equation of any 

 height will be constructed by the inter- 

 section of two lines, whose dimensions 

 multiplied together produce the dimen- 

 sion of the given equation. Thus, the in- 

 tersections of a circle with the conic sec- 

 tions, or of these with each other, will 

 construct the biquadratic equations, or 

 those of the fourth power, because 2x2 

 = 4; and the intersections of the circle, 

 or conic sections, with a line of the third 

 order, will construct the equations of the 

 fifth and sixth power, and so on. For 

 example : 



To construct a simple equation. This is 

 done by resolving the given simple equa- 

 tion into a proportion, or finding a third 

 or fourth proportional, &c. Thus, 1. If 

 the equation be a x = b c; then a:b::c:x 



= , the fourth proportional to a,6,c. 2. 



If a ar=6; then a:b::b:x = ' a third 



a 



proportional to a and b. 3. If a x = b* 

 c* ; then, since b 1 c j = b -}- c X b o, it 

 Will be q:6 



a fourth^ proportional to a, b-\-c, and b c. 

 4. If ax = b* + c 1 ; then construct the 

 right-angled triangle ABC (Plate V. Mis- 

 eel. fig. 5.) whose base is b, and perpen- 

 dicular is c, so shall the square of the hy- 

 pothemise be A 4 +c,' which call h- ; then 



the equation is ax=h* t and = , a third 



a 

 proportional to a and h, 



To construct a quadratic equation. l.If it 

 be a simple quadratic, it may be reduced 

 to this form, x 1 = a b ; and hence a r x :: x 

 -. b, or x = v/ a b, a mean proportional 

 between a and b. Therefore upon a 

 straight line take A B = a, and B C = b; 

 then upon the diameter A C describe a 

 semicircle, and raise the perpendicular 

 B D to meet it in D ; so shall B D be = 

 x, the mean proportional sought between 

 A B and B C, or between a and b. 2. If 

 the quadratic be affected, let it first be x 1 

 -j- 2 ax = b 1 ; then form the right-angled 

 triangle, whose base A B is a, and perpen- 

 dicular B C is b ; and with the centre A 



and radius A C describe the semicircle D 

 C E; so shall D B and B E be the two roots 

 of the given quadratic equation x l + 2 ax 

 = 6 l . 3. If the quadratic be x* 2 a x 

 =b 1 , then the construction willbe the very 

 same as of the preceding one x 1 + 2 a x 

 = b\ 4. Bnit if the form be 2 a x X* 

 = b 1 , form a right-angled triangle (fig. 

 1.) whose hypothenuse F G is a, and per- 

 pendicular G H is b ; then with the radius 

 F G and centre F describe a semicircle I 

 G K ; so shall I H and H K be the two 

 roots of the given equation-2 ax x 1 = 

 6 1 , or x* 2 a x = 6 1 . 



To construct cubic and biquadratic equa- 

 tions. These are constructed by the inter- 

 sections of two conic sections; for the 

 equation will rise to four dimensions, by 

 which are determined the ordinates from 

 the four points in which these conic sec- 

 tions may cut one another ; and the 

 conic sections may be assumed in such 

 a manner as to make this equation co- 

 incide with any proposed biquadratic ; 

 so that the ordinates from these four 

 intersections will be equal to the roots 

 of the proposed biquadratic. When one 

 of the intersections of the conic section 

 falls upon the axis, then one of the or- 

 dinates vanishes, and the equation by 

 which these ordinates are determined 

 will then be of three dimensions only, 

 or a cubic to which any proposed cubic 

 equation may be accommodated ; so that 

 the three remaining ordinates will be 

 the roots of that proposed cubic. The 

 conic sections for this purpose should 

 be such as are most easily described; 

 the circle may be one, and the para- 

 bola is usually assumed for the other. 

 See Simpson's and ^laclaurin's Algebra. 



EQ.UATIONS, nature of. Any equation 

 involving the powers of one unknown 

 quantity may be reduced to the form z n 

 p z"-i -f- q z-2, &c. = 0, here the whole 

 expression is equal to nothing, and the 

 terms are arranged according to the di- 

 mensions of the unknown quantity, the 

 coefficient of the highest dimension is 

 unity, understood, and the coefficients 

 />, 7, r, and are effected with the proper 

 signs. An equation, where the index is 

 of the highest power of the unknown 

 quantity is n, is said to be of n dimen- 

 sions, and in speaking simply of an equa- 

 tion of n dimensions, we understand one 

 reduced to the above form. Any quan- 

 tity z- jfr z"- 1 + q z-2, &c. + P zQ. 

 may be supposed to arise from the multi- 

 plication of z a X z b X - c t &c- 



