ALGEBRA. 



=.r- y 



9 12_ 3 

 s ~~4~~ 4 



B . 2 



3-fy/ 3 



*+!/ 2 



j- 3 v/ - 3 



2 



Since the square of every quantity is 

 jiosithe.:' negative quantity has no square- 

 root; the conclusion therefore shews that 

 there are no such numbers as the ques- 

 tion supposes. See Hivovriti. TIIF.O- 

 RKM; EU.I ATIONS, niiture of; SKUIES, 

 Si mis, &c. &c. 



A MI K BRA, nf>/>/ica!ion of to geometry. 

 The first and principal applications of al- 

 gebra were to arithmetical questions and 

 computations, as being the first and most 

 useful science in all the concerns of hu- 

 man life. Afterwards algebra was applied 

 to geometry, and all the other sciences 

 in their turn. The application of algebra 

 to geometry is of two kinds; that which 

 regards the plane or common geometry, 

 and i hat which respects the higher geo- 

 metry, or the nature of curve lines. 



The first of these, or the application of 

 algebra to common geometry, is concern- 

 eel in the algebraical solution of geome- 

 trical problems, and finding out theorems 

 in geometrical figures, by means of alge- 

 braical investigations or demonstrations. 

 Tiiis kind of application has been made 

 from the time ofthc most early writers on 

 algebra, as Diophantus, Cardan, &.c. &c. 

 down to the present times. Some of t he- 

 best precepts and exercises of this kind 

 of application are to be met with in Sir I. 

 Newton's ' I'nivcrsal Arithmetic," and in 

 Thomas Simpson's" Algebra and Select 

 Exercises." lieomctrical problems are 

 commonly resolved more directly and ea- 

 sily by algebra, than by the geometrical 

 analysis, (specially by young beginners ; 

 but then the synthesis, or construction 

 and demonstration, is most elegant as de- 

 duced from the latter method. Now it 

 commonly happens, that the algebraical 

 solution succeeds best in such problems 

 as respect the sieles and other lines in ge- 

 ometrical figures; and, on the contrary, 

 those problems in which angles are con- 

 cerned are best eflected by the geometri- 

 cal analysis. Sir Isaac 'Newton gives 

 .among many otherrcmarks on this 



branch. Having any problem proposed, 

 compare toi;c' l.erthc quantities concern- 

 ed in it; and making no difference be- 

 u\ . en 1 he kiiownaiid in, known quan 

 consider how they depend, or are related 

 to, one another; that we may perceive 

 what quantities, if tin \ an- assumed, will, 

 by proceeding synthetically, give the rest, 

 aiul that in the simplest manner. And in 

 this comparison, the geometrical figure is 

 to be feigned and constructed at random, 

 as if all the parts were actually known or 

 given, and any other linesdrawn, that maj 

 appear to conduce to the easier and sim- 

 pler solution of the problem. Having 

 considered the method of computation, 

 and drawn out the scheme, nam. 

 then to be given to the quantities enter- 

 ing into the computation, that is, to some 

 few of them, both known and unknown, 

 from which the ivsi ma\ most naturally 

 and simply be derived or expressed, bv 

 means of the geometrical properties of 

 figures, till an equation be obtained, by 

 which the value of the unknown quantity 

 may be derived by the ordinary methods 

 of reduction of equations, when only one 

 unknown quantity is in the notation ; or 

 till as many equations are obtained as 

 there are unknown letters in the notation. 

 Forexample suppose it were required 

 to inscribe a square in a given triangle. 

 Let ABC, (Plate Miscellanies, fig. 1.) be 

 the given triangle : and feign DEFGtobe 

 the required square : also draw the per- 

 pendicular BP of the triangle, which will 

 be given, as well as all the sides of it 

 Then, considering that the triangles BAC, 

 BKFare similar, it will be proper to make 

 the notation as follows, vi/. making the 

 base AC=6, the perpendicular HP -_/>, 

 and the side of the square DE or EF=x, 

 Hence then BQ=BP En=/> x, 

 consequently ,by the proportionalityof the 

 parts of those two similar triangles, vi/ 

 BP : AC ::BQ : EF, it is/.: A ::p x : x, 

 then, multiply extremes and means, &c 

 there arises px=bf> b x t or bx-\-px 



=-t> p, andx= ~- ,t.he sideofthcsquan 



sought; that is, a fourth proportional u> 

 the base and perpendicular, and the sum 



of the two, taking this sum for the firs; 

 term, or AC -(-HI 1 : \\\> -.-. AC : EF. 



The other branch of the application <>t 

 algebra to geometry \\ as introduced by 

 .tes, in his (ieometry, which is t| u - 

 IK- w, or higher.geoniet ry, and respects the 

 nature and properties of curve lines. In 

 this branch, the nature of the curve isex- 

 pressed or denoted by an algebraic equa- 

 tion, which is thus derived : A line is 



