When a problem admits of but one opposition (may it simply refers to time measured future or paet from a given epoch), there is no diffi culty about the interpretation of any result. If this result be positive, it must be such time as was called positive when the operation was put into shape ; and the contrary. But in the application to geometry, an extension of the interpretation of signs enables us to remove some difficulties in the proper expression of angles, which we proceed to describe.
In the rectilinear figures considered by Euclid the sum of the external angles is always equal to four right angles. In these figures there are no re-entering angles. (SALIENT.] The proposition remains true when there are re-entering angles, provided that the angles which then take the place of what were called external angles be counted as negative. In algebraic geometry it is usual to refer all points and lines to two straight lines at right angles to one another. [AnscissA; tIo-onmeAres.] The following conventions must now be employed: I. Let v x and w v be the axes of co-ordinates, meeting at o, the origin. Let o x and o v be positive directions : o v and o w, negative directions.
2. A line drawn from o to any point e is in itself (for the present) neither positive nor negative ; either sign may be given to o P, but the contrary one must be given to o q.
3. The line o e may, keeping its sign, revolve round o; nor, if nega tive, ie it to be counted positive when it comes into momentary coin cidence with o x ; nor, if positive, is It to be counted negative when in coincidence with o v. And generally, a straight line revolving round any point does not take the signs of lines which receive signs on account of the fixed directions in which they are drawn.
4. The angle made by two lines A and n is to have a distinction of sign drawn, according as it hi called the angle made by A with B, or the angle made by B with A (the angles, made by A from B, and by it from A, would be better). If the angle of A with D (say A^e) be posi tive, then the angle made by B with A (B'A) Is negative, and rice void.
5. The positive direction of revolution is that in which a line moves from the positive part of the axis of x to the positive part of the axis of y (as marked by tho arrows).
6. The sign Of any line drawn through e is thus determined. If or be positive, that direction is positive in which the point r must move so as to revolve pcaitirdy; thus, o r being positive, r K is positive and r t, negative. But if o r be negative, the reverse is the case ; but the rules need only be remembered which suppose o r positive.
7. When an angle amounts to more than four right angles, the four right angles may be thrown away ; and generally, four right angles, or any multiple of them, way be added to or subtracted from any angle.
S. In measuring the angles made by two lines passing through P (o r being positive), the positive directions on those lines (found as in 6) must be used : and by Ali), the angular departure of A from a, is understood the amount of positive revolution which will bring n into the position A.
9. Hence it follows that .4.'13 is either equal to A' x-e^x, or differs from it by four right angles, x standing for the axis of x.
10. Ilenee also it follows that iu every closed figure, whether such as those admitted by Euclid or not, some of the angles are negative, if every angle A^B be interpreted as A^x And in every such measurement, the sum of all the angles, with their proper signs, is equal to nothing. But, measured as in (S), the sum of all the angles is nothing, or a multiple of four right angles positive or negative. This ambiguity is wholly indifferent in trigonometrical operations.
To prove the last, let ne consider a four-sided figure, of which the sides are A, B, C, D. The angles of the figure, taken in order, are A'e, s'c, Co, WA, which, measured as in 9, are Is'xs ex, ao'x n'x-A'x, the sum of which is obviously nothing. lint if any one of these angles should differ from the preceding, it can be only by a multiple of four right angles, whence the sum must Lea multiple of four right angles.
We shall now take as an example, the three angles of a triangle, estimating them first by each aide with reference to the next, and then by comparing each of the sides with the axis of x. Tho sides are marked A, B, e; the angles of Euclid, without reference to sign, arc a, 0, 7: and the positive and negative directions arc marked. Four right angles are denoted as usual by 2w. Required A 'is + n'e + eA.
In Ate, the amount of positive revolution by which the positive part of n turns round into that of A is 7+ w. Similarly for Bee, we have a + sr, and for /3+ w. The sum of these, a + +7 being w, is str, multiple of 2w. Now let the angle A•x be O. Then vax will be seen to bo an angle iu this figure greater than two right angles, and will be found to be w + -y, while c'x is greater than three right angles, wel is 2w-se + 0-7. Hence we have A's1=0-(w+ 0-7)=7-w ti"e=w+0-7-(2r-a+ 0-7)=a-or ols..=(2w =11+w.
Only the third angle gives precisely the same in both; in the other two, the second determination gives in each case four right angles less than the first. The sum of the three angles is now 0.
The use of this system lies in enabling us to give in a general form propositions which would otherwise require a large examination of particular cases. This examination is not usually made in elementary works; but instead of it, the true result, derived from the superior knowledge of the writer, is made to the reader the consequence of a particular case. In the consideration of curves, for instance, there arc to be considered, perhaps at the same time, the co-ordinates, the radius vector, the tangent, the perpendicular on it, and the radius of curva ture. The varieties of figure which arise out of these lines are very numerous, and nothing but generalised suppositions, competent to assign definite angles in all cases, can legitimately bring out general propositions. For instances, see TANGENT, Smut; and for further development, see ' Library of Useful Knowledge,' Differential Calcu lus, pp. 341-845.