ANGLE, rectilinear, (Lat. angalus, the elbow,) according to Euclid, " the inclination of two straight lines to one an other, which meet, but are not in the same direction." This definition, if indeed it may he termed such, is so very indis tinet, and even inaccurate, that it has been entirely discarded by modern mathematicians, who have individually given many suggestions for its improvement, but have not agreed so far as to adopt any as a standard definition. We give the following as one of the most eorreet :—" An angle is the ratio of the plane surface bounded by two infinite right lines which meet, to the plane surface on all sides indefinitely extended about the point where they meet." Thus the A B c is the ratio of the plane surface, bounded by the straight lines A n, B c infinitely extended to the unbounded plane of the paper about the point B. Objections, doubtless, may be urged against this, as against all other suggestions; but the subject is unquestionably a difficult one, as it necessarily involves the long-disputed question concerning infinite magnitudes. The following description, though not amounting in preciseness to a definition, affords a very intelligible notion of the idea intended to be conveyed by the term, viz: the opening made by two intersecting right lines.
Comparison of Angles. As every theory respecting the comparison of infinite spaces is attended with considerable difficulty, we shall leave the consideration of the more abstruse points of this subject to works of a different nature, and endeavour to explain, as clearly as possible, the method of comparing angles.
Let A B C, D E r, (see the plate) be two angles formed by the intersection of the straight lines A B e, D E, E F, at the points n E, respectively.
Apply the angle A B C to angle n E F in such a manner, that the points a F., and the lines B c, E F coincide, then the posi tion of E D with respect to B A is determined. Such being the position of the two figures, if E D fall upon B A, the two openings coincide, or, in other words, the angle a B c is equal to the angle D E F. lf, however, E n fall between B c and 13 A, the opening or angle D E F is less than the other A B c; if, on the other hand, a a fall without or beyond B A, the angle n E F is said to be greater than the angle A B c.
Again, supposing the angles to be applied as before, and a a to fall within A B ; let E D remain fixed in that position, butt let E r be turned about E n as an axis, until it fall on the opposite side of it ; then, if E F coincide with B A, it is evident that the angle A B c is equal to twice the angle a a F. In the same manner may be explained the notion of
one angle being three, four, or any number of times greater or less than another.
It may be necessary to observe, that the magnitude of the angle in no wise depends upon the length of the interseeting lines; for, if we suppose a part D d to be cut off from the side n a, upon applying the angle D E F to angle A B c, as above, we shall tied that the line E d will still fall in the same position with respect to A n, as it did before D d was cut off; and will do so, however short E n may become, until the line, and therefore the angle, ceases to exist.
Again, let us suppose a line starting from a certain station A B, to revolve round one of its extremities A as a fixed point or axis, and to arrive at the situation A it will then, with its original position, describe an angle a A a,. Let it now continue its revolution, until it has passed over another space. equal to the preceding, and in so doing has reached the position A B,; it will then be readily understood that the angle n A a, equals twice the angle B A B, and thus we might describe an angle any number of times greater than a A B.
Euclid's notion of an angle has been very much enlarged upon by later mathematicians, as we proceed to illustrate by reference to the lass. diagram. Let us conceive the line A a to continue its revolution to and thence to By; we say then that A forms with its first position the angle B A and thus far Euclid allows; but if the revolution be con tinued until A B arrives in the position A n,, so as to form a straight line with its first position—which event takes place when it has performed half a revolution—Euclid no longer recognizes the opening so ffirmed as an angle. Such, however, it is reckoned to be by the moderns, and that not without reason ; for it will be readily acknowledged that the opening ffirmcd by the lines A B and A is greater than that formed by A ri and A B„ thus showing that such opening is liable to comparison in the same manner as any other angle. The same reasoning will apply to openings formed by a whole revolution or more ; indeed, the moderns do not restrict .the term to any number of revolutions however great.
A Itidur ANGLE is that traced out by A B while perform ing a quarter revolution.