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An advanced student, who has acquired the idea in question, will find nothing very objectionable in thus expressing himself, for he knows what is to be described, and mentally assigns a due scientific meaning to the general term of common parlance. But to the beginner, there appears something as vague in the word "inclination," as in the term "direction," when applied to a straight line. It even appears more vague, for the genus of a straight line is given—it is the line of direction: but the angle, is it then inclination itself? The student is apprised, that his attention is to be confined to points, lines, surfaces, and solids, things of which he has definite conceptions: but here at the very outset is a subject introduced, which appears to be distinct from all, and to be a quality of figure rather than an existence. It afterwards turns out that the only practically useful explanation relative to an angle requires merely, that it should measure this quality of position.
Considerations of this nature have induced some distinguishedly successful elementary writers, to deviate from the usual custom in seeking for such a definition of an angle as should appear to be a natural description of it, to be free from metaphysical objections, and to permit of immediate use in the investigation of the properties of angles, or failing that, through medium of such simple considerations as may appear almost axiomatically deducible from the definition.
Of this class is Bossat's statement, that the angle is the opening between two lines, with an explanation impressing the definiteness of the conception, and the mode of comparison naturally resulting from it. This was followed by Professor Young in England. But the nature of the idea thus attempted to be expressed by the word opening, did not seem to be yet satisfactorily developed, and Legendre, accordingly ventured to substitute "quantity" for " opening." The American edition of Brewster's translation calls an angle, "the quantity by which two intersecting lines are separated from each other;" and Francosur, I presume after Legendre, adopts a similar definition in his admirable course. It is, however, easily seen, that very little is gained by this step, on the score of clearness or precision, as the kind of quantity is not specified.
Leslie attempted quite another path, suggested by the cerelations of angles and arcs ; viz., that angular magnitude is generated by the revolution of a line round a fixed point: but we are not told what angular magnitude is. All these definitions then fail in strictly fulfilling their object. Each has been in turn severely criticised by following reviewers, anxious to establish the validity of the most infallible of all—their own. But all agree in this; that they are descriptions of different characteristics of the same idea. If from any one we can obtain a definite conception of what is intended, we immediately perceive that all are sufficiently correct to recall it to our minds. All agree in understanding angular quantity to be "something," or if the expression be too bold, "that" which lies between two intersecting straight lines. All of them agree further in considering, that for purposes of comparison as to magnitude, angles must be estimated crosswayB, or by the width between the lines, and not with any reference whatever to the longitudinal extension in the direction of the sides.
Now, if we analyse the various definitions of an angle in this manner, it is, I think, impossible to come to any other conclusion than that an angle is the plane surface between two lines; of a peculiar nature, partly bounded and partly unlimited, whose value could consequently be only estimated by reference to the bounded direction, that is, the width between the sides. And the neatest and shortest mode of expressing this will apparently best solve our difficulty, as it connects axomatically an explicit definition with the working one.
The first place in which I believe this idea was embodied, was Bertrand's celebrated solution of the difficulty in the theory of parallels. The principle of that demonstration is as follows: Any angle, however small, can by repeated reduplication be made to exceed any given angle however great, but the band of unlimited space between two parallel lines, though repeated ever so often, will never fill up that given angle. Hence an inter-parallel space is less than any assignable angle in value, and therefore a line which cuts one of two parallels, must also cut the other, otherwise the angle which it makes with the one it does cut, would be wholly contained within the inter-parellel space, and be less than it. The stress of this demonstration evidently rests on the comparison of surfaces, and it is surprising that its extreme elegance did not lead Geometers earlier to seek the solution of the problem in that direction. The truth is, that the new "unlimited spaces" were treated as interlopers in the science of figure, and the demonstration rejected, a; "wearing only the semblance of geometrical accuracy."
In Col. Peyronnet Thomson's* Geometry without Axioms, these unlimited spaces are for the first time distinctly enunciated. The fourth edition of that work contains the following paragraphs :—
"The latest innovation has been the assertion, that an angle (or *' the thing spoken of by Geometers under that item, whether they knew "it or not) is a plane surface." Pref. page x.
"The plane surface (of unlimited extent in some directions, but limit"ed in others) passed over by the radius vectus in travelling from one "of the divergent straight lines to the other, is called the angle between "them." "Hence," adds the Colonel, "angles are compared together "by their extension sideways only, without reference to the greater or "smaller length of the straight lines between which they lie."
After making this decided step however, Colonel Thomson stops; the definition is registered in his Book of Nomenclature, but he establishes the properties of angles by the old criterion of supposition. Not only indeed does the definition remain a dead letter, but the gallant radical reformer in Geometry as in Politics,
"Astonished at the sound himself had made,"
virtually doubts its correctness, when at page 14, reviewing the proof of M. Bertrand, he says, " All references to the equality of magnitude "of infinite areas are intrinsically paralogisms."
The edition in question of" Geometry without Axioms," was reviewed in the 13th No. of the Journal of Education, in an article which betrays the sparkling pen of Professor De Morgan. The part relative to angles is noticed thus: "His is the first work, which we know, in which this "idea (that of a plane surface) is fairly brought before the beginner. "We suspect he is quite right, and that in the extension of the term "equal to unlimited figures which coincide in all their parts, will be "found the ultimate resting point of the theory of parallels. Had our "author stuck close to his definition, the demonstration of Euclid's axiom "given by M. Bertrand, ought to have been sufficient." After noticing the neglect of Colonel Thompson to make any use of his definition, as well as his attack on unlimited spaces, the reviewer proceeds: '* We "wonder therefore that the definition should have been inserted, for it "is in the definition only, and the difficulty which a beginner must find
• The well known Editor of the Westminster Review, and author of the Corn Late Catechism.
"in settling his ideas of greater, less, and equal on that definition, that '' the whole objection to M. Bertrand's demonstration turns."
I have been minute in these quotations, not only because they contain all that to my knowledge has been developed on a very interesting subject, but also in the hope that they may draw further attention. Led independently to similar conclusions, by attempting to trace the natural affinities, if we may so term it, of geometrical truths, with the intention of forming a definite arrangement of them, I was induced to trace their consequences in establishing the various relations of angular space. The results of the inquiry may be thrown into a connected chain of propositions, as subjoined.
Definition 1.—The plane surface between two straight lines, bounded in the one direction, unlimited in the other,—is called an angular space.
Definition 2.—When an angular space is bounded on one side by the intersection of the containing lines, it is called an angle.
Definition 3.—The point of intersection is called the vertex, and the containing lines are called the sides of the angle.
Axiom.—From the definition, it will follow that two angular spaces A B C D, and E F G H, must be compared thus : If placing the line F E on B A, we find that, F falling on the point P, G will fall on some point Q in C D, then according as the line G H falls within, upon or without the line C D—is the angular space A B C D, greater than, equal to, or less the angular space E F G H. (Fig. 1.)
Definition 4, 5, 6.—Euclid's definitions of right, acute, and obtuse angles.
Every angular space is equivalent to its angle.
This follows from the axiom, since the sides of the angular space, and of the angle are identical, and may therefore be considered to coincide.
Proposition II. (Tig. 1.)
All right angles are equal to one another.
Let the right angles A B C, E F G, be made respectively by A B with B C and E F with F G; they are equal. Produce C B to D and F G to H, and apply the figures one to the other, so as to make F coincide with B and G H with C D. If then F E do not coincide with B A, let it fall asBK. Then GFE = CBK ^ CBA ^ (its = ) ABD^KBD ^rEFH.
But GFE = EFHby definition, hence the supposition, that F E does not coincide with B A involves absurd consequences, .\ F E does coincide with B A, and G F E with C B A, G F E is therefore equal to C B A.
Cor. 1.—A right angle is therefore a constant in angular magnitude.
Cor. 2.—The space on one side of a straight line, considered as an angle at some given point in the line, is two right angles.
Scholium. (Fig. Z.J
Consider a line O A fixed, and another line O R, having a point O in common with O A, but being itself in a state of rotation round O. When in the position O R, it will have generated an acute angle A 0 B: as it proceeds, it will coincide with the perpendicular O B, and will have described a right angle. In the position 0 R2, the angle generated is an obtuse one. The generating line then coincides in its progress with O C, the continuation of O A. It will in such position have described two right angles (A O B and BOC). Supposing the rotation to continue, O R will fall below O C, as 0 R3, having described the whole of the coloured angular space, which is greater than two right angles. Such angle is called a reverse angle. During the progress of the line, the reverse angle continues to increase, equals 3 right angles, exceeds that amount, and at length equals 4 right angles, when O R has completed an entire circuit. By conceiving the line to move on, still revolving, and with the aid of a contrivance like the spiral twisted palm-leaf fans, used by the Natives, the beginner may obtain the idea of angles greater than four right angles, and generally of (2 n n + A) which some find it difficult to understand in their later trigonometrical studies, and perfect acquaintance with which is so indispensable to the comprehension of periodic functions.
Proposition III. (Fig. 4.J
The vertically opposite angles made by two intersecting straight lines, are equal to one another.
The vertically opposite angles A E D and C E B made by the intersectors A B and C D at E are equal. For C E D being a straight line, the angular space on the side of it towards A is two right angles. For