ALli 



ALG 



conceived to be drawn, as the diameter 

 or some other principal line about the 

 curve ; and upon any indefinite points of 

 this line other lines are erected perpendi- 

 cularly, which are called ordinates, whilst 

 the parts of the first line cut off by them 

 are called abscisses. Then, calling 1 any 

 absciss x, and its corresponding- ordinate 

 y, by means of the known nature, or rela- 

 tions, of the other lines in the curve, an 

 equation is derived, involving 1 x and y, 

 with other given quantities in it. Hence, 

 as x and y are common to every point in 

 the primary line, that equation,so derived 

 will belong to every position or value of 

 the absciss and ordinate, and so is proper- 

 ly considered as expressing the nature of 

 the curve in all points of it; and is com- 

 monly called the equation of the curve. 



In this way it is found, that any curve 

 line has a peculiar form of equation be- 

 longing 1 to it, and which is different from 

 that of every other curve, either asto the 

 number of the terms, the powers of the 

 unknown letters x and y, or the signs or 

 co-efficients of the terms of the equation. 

 Thus, if the curve line HK, (fig. 2.) be a 

 circle, of which HI is part of the diame- 

 ter, and ]K a perpendicular ordinate; 

 then put Hl=x, 1K=?/, and p= the 

 diameter of the circle, the equation of the 

 circle will be p x x2=yz. But if HK 

 be an ellipse, an hyperbola, or parabola, 

 the equation of the curve will be differ- 

 ent, and for all the four curves will be 

 respectively as follows: viz. 



For the circle . 

 For the ellipse. 



. p x #2=^ 

 . p x x2= 



For the hyperbola /> x-f- *2 =^8, 

 For the parabola . . p x . . =y% 



where t is the transverse axis, and p its 

 parameter. And in like manner for other 

 curves. 



This way of expressing the nature of 

 curve lines, by algebraic equations, has 

 given occasion to the greatest improve- 

 ment and extension of the geometry of 

 curve lines ; for thus all the properties 

 of algebraic equations, and their roots, 

 are transferred and added to the curve 

 lines, whose abscisses and ordinates have 

 similar properties. Indeed the benefit of 

 this sort of application is mutual and re- 

 ciprocal, the known properties of equa- 

 tions being transferred to the curves they 

 represent; and, on the contrary, the 



known properties of curves transferred 

 to their representative equations. 



Besides the use and application of the 

 higher geometry, namely of curve lines, 

 to detecting the nature and roots of equa- 

 tions, and to the finding the values of 

 those roots by the geometrical construc- 

 tion of curve lines, even common geome- 

 try made be made subservient to the pur- 

 poses of algebra. Thus, to take a very 

 plain and simple instance, if it were re- 

 quired to square the binomial a + b 

 (fig. 3.) by forming a square, as in the 

 figure, whose side is equal to u-\*b, or 

 the two lines or parts added together de- 

 noted by the letters o and b : and then 

 drawing two lines parallel to the sides, 

 from the points where the two parts join, 

 it will be immediately evident that the 

 whole square of the compound quantity 

 ft-f-6 2 is equal to the squares of both the 

 parts, together with two rectangles under 

 the two parts, or 2 an d 62 and 2 a b, 

 that is, the square of a-\-b is equal to 

 2 +* 2 +2 a ft, as derived from a geo- 

 metrical figure or construction. And in 

 this very manner it was, that the Arabi- 

 ans, and the early European writers on 

 algebra, derived and demonstrated the 

 common rule for resolving compound 

 quadratic equations. And thus also, in 

 a similar way, it was, that Tartalea and 

 Cardan derived and demonstrated all the 

 rules for the resolution of cubic equa- 

 tions, using cubes and parallelopipedons 

 instead of squares and rectangles. Many 

 other instances might be given of the use 

 and application of geometry in algebra. 



ALGOL, the name of a fixed star of 

 the third magnitude in the constellation 

 Perseus, otherwise called Medusa's Head. 

 This star has been subject to singular va- 

 riations, appearing at different times of 

 different magnitudes, from the fourth to 

 the second, which is its usual appear- 

 ance. These variations have been noticed 

 with great accuracy, and the period of 

 their return is determined to be 2 d 20 k 

 48' 56". The cause of this variation, Mr. 

 Goodricke, who has attended closely to 

 the subject, conjectures, may be either 

 owing to the interposition of a large body 

 revolving round Algol, or to some motion 

 of its own, in consequence of which, part 

 of its body, covered with spots or some 

 such like matter, is periodically turned 

 towards the earth. 



ALGORITHM, an Arabic term, not 

 unfrequently used to denote the practical 

 rules of algebra, and sometimes for the 

 practice of common arithmetic ; in which 

 last sense it coincides with logistica nwne- 



