page note 200 * it is necessary to insist that the (so-called) “unreal” aspects of the algebra are not applicable to the plane geometry. They are, of course, applicable to the four-dimensional geometry which fully corresponds to the algebra—and includes the plane geometry (“section” fashion: see the second part of Note 1317, XXII, No. 251, p. 394).

page note 200 † Vectors of this type are the main theme of this article; in what follows, the arrowhead vector sign is to be “understood”, throughout.

page note 200 ‡ Opinions vary as to the quite fundamental importance of this proposition; but the fact, as stated, is common ground. The present writer has made plain his own position, in relevant communications; vide, in particular, XXII, No. 250, pp. 225-233.

page note 200 § Some of what immediately follows was set out more fully in Xi, No. 165, pp. 330-1 (July, 1923). A minimum of repetition is unavoidable. (i have in my copy of that number a letter from E. H. N. on the subject—which was the beginning of a personal contact i have greatly valued.)

page note 200 || The trigonometric functions “cos” and “sin” are, thus, a specially important particular case of Rectangular Cartesian coordinates.

page note 201 * in “equation” form, (x-x ₁)/l = (y-y ₁/m (= p); also, y=μ x +β, or x = v y + α—where μ(= m/l) is the gradient, v(=l/μ) the co-gradient, of the line, and (α, 0), (0, β) its axial points. (The “Change-of-origin” transformation of coordinates is, of course, used; vide § 6.2.1, infra.)

page note 201 † in “equation” form,

also, x/α + y/β = 1, with notation as above.

page note 201 ‡ Comparing with the footnotes to § 3.4, supra—

—in relation to which the vanishing of one, or of two, of the coefficients a, 6, c is properly interpretable.

page note 201 § Questions of perpendicularity, or of distance—or of the use of trigonometric methods—will obviously be simpler when the frame of reference is rectangular, But it is a profound mistake to introduce Analytical Geometry, generally, in terms of the Rectangular Frame. Most of the elementary theory can—and should—be expressed in terms of the general Cartesian Frame (which should not be stigmatized by the term “oblique”!). This point becomes of first-rate importance in the treatment of 3-dimensional geometry (See next footnote.)

page note 202 * This is not quite obvious, in the case of 2-dimensional geometry—because this case (of n =2) is too special and too simple, and the methods used in it not, therefore, sufficiently “systematic” What this means becomes clear when we compare with the 3-dimensional case—where the methods used depend much less specially on the particular value of n (=3). Some parallel treatment—however elementary—of the cases of n=2 and n=3 is of great value, for sound grasp of the theory. in the latter case, the “Perpendicular Form” of the equation to the Plane is, of course, quite fundamental—for the general Cartesian frame. (it is to be specially noted that the mistaken emphasis on the Rectangular Frame (see previous footnote) has led to over-emphasis upon direction-ccmnes—where these are, however important, properly subordinate to the so-called “direction-rata’os”, which are better thought of as direction-coordinates. Very commonly the direction-cosines, in the case of a Rectangular Frame, are really “direction-ratios”: e.g. § 3.4.1, supra. Here, of course, they are not.)

page note 202 † The choice is arbitrary—but is “followed through” As between positive and negative, there is (caeteris paribus) general priority of the former. it is the correlation of x and p

page note 202 ‡ This is a good method for the case of the rectangular frame; but not nearly so good for the general frame—whereas the method of § 4.1 is not much less simple for the general frame: the “Perpendicular Form” equation being then

in terms of “direction-cosines” for the perpendicular. For § 4.1.2 we have then the relations

(it is to be noted that the symbols β, β are here used with a different—standard—meaning from that of the footnotes to § 3.4 and the first footnote to § 4. Practically all attempts at systematic notation, in terms of alphabetical symbols, almost inevitably break down; but we should, nevertheless, do the best we can.)

page note 203 * Phrases like “the line a.x +b.y+c =0”, or “the line L”, are typical of a sketchy unprecision which is unworthy of the most exact of all branches of learning. This kind of unprecision has had very serious consequences in the Physical Sciences; and these consequences are becoming of critical importance, in the utter inadequacy of many forms of loose expression to the extreme difficulty and subtlety of modern physical theory.

page note 203 † The notation (a, b, c) for lines is, of course, analogous to the standard notation (x, y) for points. Complete analogy can be achieved in one or other of two ways: (1) by using equations of some such form as a.x +b.y ±1 =0 and calling the line “(a, b)”—then a unique specification; this is. sometimes useful; (2) by using homogeneous forms, and corresponding specification of points (in the plane) as “(x, y, z)”; this is related to matters of major theoretical importance upon which it is not possible to elaborate here.

page note 203 ‡ Though we may, of course, make use of these equivalents as freely as we please. (Vide, e.g., the numerical detail in § 5.3.2, infra.)

page note 203 § Vide footnotes to § 3.4, for the notation.

page note 203 || “To”, perhaps, rather than on—which is commonly used for the points which constitute the line.

page note 203 ¶ These are, of course, consistent with one another. Taken together, they cover the special cases of the vanishing of one, or of two, of the three coefficients. it should be observed, as characteristic of the “analytical” theory, that the apparent unsymmetry of the three modes of discrimination is resolved when Cartesian Frame is realised as “limiting case” of Triangle of Reference. (The cases of § 5.3 infra— as shown in the figure—may be taken as examples.)

page note 205 * Drawn for me—as on previous occasions where skill was required—by Mr. J. M. Allen, M.A., B.Sc, of Melbourne Technical College.

page note 206 * From the point of view of the general method used, note the importance of writing h + ξ as contrasted with (the algebraically equivalent) ξ +h, etc., in § 6.2.1; and, similarly, φ + ω, etc., in § 6.2.2; and so on. (The method is, of course, used in good textbooks, but is not sufficiently “followed through” in the detail which is so peculiarly important to this kind of discussion.)

page note 206 † See the preceding footnote.

page note 207 * it is worth noting that, here and in § 7—and generally in “elementary” theory (as distinguished from “analysis”)—there is no special virtue in using circular measure, except convenience of printing. Always, in such theory, the “argument” of the trigonometric function is the angle itself (rather than its “measure”); the “function” (cos or sin) being a ratio and, therefore, a “number”

page note 207 † i.e. two angles of which the sum = 0.

page note 208 * This is the only satisfactory way of dealing with that standard theory, and that should be recognised once for all. it is set out carefully in Ch. V of my Theory of Elementary Trigonometry: now out of print; there is a copy in the Math. Ass. Library. (At the time when that book was written (1910), i had not clearly realised the better alternative—of “resolutes”—to orthogonal projection.)

page note 208 † Vide the urgent impatience in Note 1595; but, more especially, the Presidential Address, “The Food of the Gods” (Gazette, XiX, No. 232, February 1935, pp. 5-17).