-108 G 1‘) 0 )1 E T lt Y PART II.——-.'.LYTICAL Cl7.O.l1ZTRY. This will be here treated as a method. The science is Geoinetry ; and it would be possible, analytically, or by the. method of coordinates, to develop the truths of geometry in a systematic course. hit it is p1‘oposed not iii any way to attempt this, but simply to explain t-he method, giving such examples, interesting (it may be) i11 themselves, as are suitable for showing how the method is employed in the demonstration and solution of theorems or problems. Geometry is one-, two-, or three-dimensional, or, what is the same thing, it is li11eal, plane, or solid, according as the. space dealt with is the line, the plane, or onlinary (three- dimensional) space. X0 more general view of the subject need here be taken :—but in a certain sense one—dimensional geometry does not exist, inasnmch as the geometrical con- structions for points in a linecan only be performed by travel- ling out of the line into other parts of a plane which contains it, and conformablyto the usual practice . -alytical Geometry will be treated under the two divisions, Plane and Solid. It is proposed to consider Cartesian coordinates almost exclusively; for the propel‘ development of the science homogeneous coordinates (three and four in plane and solid geometry respectively) are required ; and it is more- over 11ccessary to have the correlative li11e- and plane- coordinates ; and in solid geometry to have the six coordinates of the line. The most comprehensive linglish works are those of Dr Salmon, The Com'cs(.3tl1 edition, 1869), Hi3/her Plane C'm'z'es (2d edition, 1873), and (}comefr3/ of T /tree 1)z'nIens2'0ns (3d edition, 1874) ; we have also on plane geometry Clebsch’s l'0rlesm2_r/I-H. zib/37' 1-'e0mch'z'e, posthumous, edited by Dr F. Lindemann, Leipsie, 1875, not yet complete. I. PLANE A.'.u.vT1cAL Gnonrgrar 1-25). 1. It is assumed that the points, lines, and figures con- sidered exist in one an(l the same plane, which plane, therefore, need not be in any way referred to. The position of a point is determined by means of its (Cartesian) eo- ordinates; 2'.e., as explained N ............... -.P un(ler the article CURVE, we take the two lines :c'O.r and 3/'03/, called the axes of :r and 3/ respectively, intersecting in a point 0 called the origin, and de- termine the position of any ‘other point 1’ by means of yr its coordinates .r=O)l (or ,_ N1’), and 3/ = M1’ (or ox). " '3' ‘- The two axes are usually (as in fig. 1) at right angles to each other, and the lines PM, PN are t.hen at right angles to the axes of .2: and 3/ respectively. Assuming a scale at pleasure, the coordinates .33, 3/ of a point have numerical values. It is necessary to attend to the signs: .1.‘ has opposite signs according as the point is o11 one side or the other of the axis of 3/, and similarly 3/ has opposite signs according as the point is 011 the one side or the other of the axis of 7;. Using the letters N, E, S, V as in a map, and con- sidering the plane as divided into four quadrants by the axes, the signs are usually taken to bc— 317’ 0 91‘ 3/ for quadt. + + .' F. + — S E — + l' ' - — S V A point is said to have the coordinates (a, L), and is referred to as the point (:1, Z»), when its coordinates are
- L'=a, 3/=1); the coordinates .17, 3/ of a variable point, or
of a point. which is for the time being regarded as variable, are said to be current coordinates. 2. It is sometimes convenient to use oblique coordi- nates ; the only difference is that the axes are not at right. angles to each other; the lines l’.I, PX are drawn parallel to the axes ofy and .2: respectively, and the llgllru ().l l’.' is thus a parallelogram. But in all that follows the Cartesian coordinates are taken to be rectangular; polar coordinates and other systems will be brietly 1-eferred to in the sequel. 3. If the coordinates (.r, 3/) of a point are not given, but only a relation between them f(.c, 3/)=0, then we have a curve. For, if we consider .1: as a real quantity varying continuously from — co to + do , then,for any given value of at, 3/ has a value or values. If these are all imaginary, there is not any real point; b11t if one or more of them be real, we have a real poi11t or points, which (as the assumed value of .r varies continuously) varies or vary continuously therewith; and the locus of all these real points is a curve. The equation completely defines the curve; to trace the curve directly from the equation, nothing else being known, we obtain as above a series of poi11t.s sufiiciently near to each other, and draw the c11rve through them. For instance, let this be done in a simple case. Suppose 3/ = 2.:; — 1 ; it quite easy to obtain and lay down a series of points as near to each other as we please, and the application of a ruler would show that these were in a line; that the curve is a line depends upon something more than the equation itself, viz.., the theorem that every equation of the form 3/=a.:'+7I represents a line ; supposing this known, it will be at onee understood how the process of tracing the curve may be abbreviated ; we have .r:= 0, 3/: — 1, and .1‘:-1., 3/ = 0; the curve is thus the line passing through these two points. But in the foregoing example the notion of a line is taken to be a known one, and such notion of a line does in fart precede the consideration of any equation of a curve what- ever, since the notion of the coordinates themselves ]'c.~t.~'. upon that of a line. 111 other cases it may very well be that the equation is the definition of the curve; the 1|()llll.' laid down, although (as finite in number) they do not actually determine the curve, determine it to any degree of accuracy ; and the equation thus enables us to construct the curve. A curve may be determined in another way; viz., the coordinates .1‘, 3/ may be given each of them as a function of the same variable parameter 6; .r, 3/ =_7'(0), <;b(6) re- spectively. Here, giving to 6 any number of values in succession, these equations determine the values of .r_. 3/,‘ that is, the positions of a series of points on the curve. The ordinary form 3/=gb(.2:), where 3/ is given explicitly as a function" of .r, is a particular case of each of the other two forms: we have f(a:, 3/),= 3/— <;b(.r),=0 ; and J‘: 6: 3/ =(ib(6)‘ 4. As remarked under CURVE, it is a useful exer- cise to trace a considerable number of curves, first taking equations which are purely numerical, and then equations which contain literal constants (representing numbers); the equations most easily dealt with are those wherein one coordinate is given as an explieit function of the other, say 3/=gb(.r) as above. A few examples are here given, with such explanations as seem
proper.