ANCYLOMETRY. 335 



sides are proportional to the sines of the opposite angles. In a right 

 angled spherical triangle, if we omit the right angle, we have five 

 parts left, sides and angles ; one of which being called the middle 

 part, two of the others become adjacent parts, and the other two, the 

 opposite parts ; taking however not the oblique angles and hypothe- 

 nuse themselves, but their complements in their stead. Then, radius 

 into the sine of the middle part, will be equal to the product of the 

 tangents of the adjacent parts, and also equal to the product of the 

 cosines of the opposite parts. These rules, called Napier's Analo- 

 gies^ may be applied to oblique angled spherical triangles ; by divid- 

 ing them each into two right angled triangles, by means of an arc 

 drawn- from one vertex, perpendicularly, to the opposite side. 



3, The invention of Coordinates^ has furnished the means of 

 representing geometrical curves, by the medium of algebraic equa- 

 tions. For this purpose, we imagine two straight lines, XX' and 

 YY', PI. VII., Fig. 15, to be drawn in the plane which contains the 

 given curve ; and these lines are called the axes of coordinates : the 

 origin of coordinates being their point of meeting. Generally, the 

 axis which extends across the figure, from right to left, is called the 

 axis of abscissas ; and the other, the axis of ordinates. If, then, 

 from any given point, w, we draw a vertical line, m n, until it meets 

 the axis of abscissas, this line is called the ordinate of that point ; 

 and the distance, #>i, from the foot of this ordinate, on the axis of 

 abscissas, to the origin of coordinates, is called the abscissa of the 

 same point. Thus, the position of the point, in the plane under con- 

 sideration, is fixed by means of its abscissa and ordinate ; which, 

 being parallel to the axes, are generally perpendicular to each other, 

 and together are called the coordinates of that point. 



Suppose, now, that we imagine a series of points, at different dis- 

 tances from the origin of coordinates, but so situated that the ordinate 

 of each point shall be equal to its abscissa. Then will all these 

 points lie in one and the same straight line, Jl b, Fig. 15, passing 

 through the origin, and making an angle of 45 with each of the axes, 

 when they are rectangular : and the equation of this straight line 

 would be y = x ; calling x the abscissa, and y the ordinate, in gene- 

 ral. By giving any particular value to a?, it determines the corres- 

 ponding value of y, and defines some particular point of this straight 

 line. For the origin itself, we have x = 0, and y = . and in 

 general the abscissa will be for any point situated on the axis of 

 ordinates, and the ordinate will be zero for the axis of abscissas. In 

 like manner as above, the equation y = | #, is that of a line, /?(?, 

 Fig. 15, passing through the origin, and having the ordinate for each 

 point, the double of its abscissa. A line whose equation is y = x-\- 

 10, would be parallel to this last, but would cut the axis of ordinates 

 at a distance from the origin expressed by the number 10. The 

 equation y = \ x + 5, might represent the line d e, Fig. 5 ; the 

 coefficient , determining its oblique direction. 



4. We must pass on to the Conic Sections. If we suppose a 

 cone, (Plate VII., Fig. 16), to be bounded by an infinite number of 

 consecutive straight lines, all passing through its vertex, V, and 

 together composing its convex surface, these lines are called its 



