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a similar reason, the space on that side of A E B towards C is equal to two right angles. These two angular spaces being constantly equal, take away from both the common angular space A E C, therefore the remainders are equal, viz. the angles A E D and C E B.
It may be useful to devote an angular space by two letters, one from each side, if the angle be less than two right angles, or by three if the angle be two right angles or more, to prevent the confusion of direct and reverse angles. Thus in Fig. 4, C B would stand for the angular space corresponding to C E B; A C B for the two right angles between E A and E B: and D A C B for the reverse angular space between E D and E B. The demonstration of HI may then be made shorter, and perhaps clearer, thus: C E D being a straight line, D A C=2 right angles; also because A E B is straight, ACB=2 right angles, Hence D A C=A C B ; take away the common part A C, then A D=C B. that is the angle A E D=C E B.
Proposition IV. (Fig. b.)
If the angle contained by two straight lines is equal to two right angles, those straight lines form but one continued line.
For if A B, A C including an angle equal to two right angles, are not in the same straight line, let AD be the continuation of A B: then D C B is two right angles, hut C B is the same by hypothesis, .'. DCB = CBan absurd result; therefore A C and A B form but one line.
Proposition V. (Fig. 6.J
If any number of straight lines tend towards the same parts, the angle made by the extremes is equal to the sum of the angles made by the successive pairs of lines. •
Let A, B, C and D be straight lines, tending towards the same parts, then the angle A H D is equal to the sum of the angles A E B, B F C, C G D formed by the successive pairs of lines. For the angular space A B C D is equal to the sum of the three angular spaces A B, B C and C D. But A B C D is the angle A H D and the constituent angnlar spaces A B, B C, C D are respectively identical with the angles ABB, BFC.CGD. Hence AHD = AEB + BFC + CGD.
Proposition VI. (Fig. 1.)
The three angles of a triangle are together equal to two right angles.
Let A B C be the triangle, produce AB, AC to E and F and the base BC both ways to D and G. Then since the lines DB,EB,F C, G C all tend towards the same parts, the angular space D E F G = DE+EF+FG. But D E is the angle D B E or its vertically opposite A B C ; E F corresponds to the angle B A C and G F is the angle G C F or A C B. Also D E F G is the angular space contained by tvo portions of the same straight line, it is therefore two right angles. Hence
ABC +BAC+ BC A = 2 right angles.
Cw. 1.—The exterior angle is equal to the two interior and opposite
m the same side, proved by reversing the process of Euclid in the 32.1.
or as well thus (see Fig. 7-) The angular space E G is equal to E F
iadFG:EG = EBG.EF = BACandFG =FCG = ACB
.-. GBE = BAC + ACB.
Ctr. 2.—Euc. I. 16 and 17 are further contained in the last corollary.
Proposition VII. (Fig. 8.J
The interior angle of a polygon of n sides are together equal to (2 B — 4) right angles.
LetABCDEFbethe polygon ; subdivide it into triangles by lines trom one of the points A. Then the angles of the polygon are equal to tie angles of the triangle taken together. Each of the polygon, save >'k two meeting in A, corresponds to one of these triangles, therefore the number of triangles, is n — 2. And the sum of the angles in each is '- right angles, .•. the sum of all the angles is (n — 2) X 2 right angles. That is, (2 n — 4) right angles. Hence the angles of the polygon are equal to (2 n — 4) rigjit angles.
Proposition VIII. (Fig. 9.J
The exterior angles of a polygon, whatever be the number of sides, are together equal to 4 right angles.
The whole angular space FGHKLF is composed of the angular
spaces FG, G H, H K, KL, L F. But the whole space FGHKLF
's the entire angular space on both sides of the line F E, i. e. 4 right
wgles, and each of the constituent angular spaces corresponds to an
exterior angle of the polygon. Hence the exterior angles together amount to 4 right angles.
The above eight Propositions comprise all the properties of intersecting lines which are independent of the consideration of length and size. They shew how possible it is to translate the spirit of the principle of homogeneity from analytical into geometrical inquiries; for our results being altogether free from the comparison of triangles or the length of lines, the interweaving of those subjects in our processes raises a suspicion, that we are not proceeding so simply as we might do, but are embarrassed with matters really foreign to the direct truth. We might extend the same course to parallel lines.
Definition.—Straight lines that never intersect each other, are called parallel lines.
Proposition IX. (Fig. 10.J
If a straight line meet two others, so as to make the exterior angle equal to the interior and opposite on the same side, these two others shall be parallel.
Let C B E meet A B and D E making A B C = D E B, then D E must be parallel to AB. For the angular space DC = DA-f AC and DC is DEC, and AC is ABC,
.-. DEC = DA + ABC, butDED = ABC.
.•. D A is zero, or D E and A B contain no angle, therefore they never meet, for if they met, they must contain an angle; hence they are parallel.
Cor.—This proposition proves the possible existence of parallels.
Proposition X. (Fig. \0.J
If a straight cuts a pair of parallels, it makes an exterior angle equal to an interior and opposite one on the same side.
For as before DAC = DA + AC, but since D E and A B never meet, they contain no angle, i. e. DA is zero; hence D A C = A C or the angle D E B = A B C.
Cor.—It would be a waste of space to deduce from this, the other usual properties of parallels.
Proposition XI. (Fig. \1.)
A and B being each parallel to C, B is parallel to A. For AC = AB + BC,but A Cand B C are each a zero, .-. A B is also zero, or B parallel to C. ■
Proposition XII. (Fig.
If a straight line cut one of two parallels, it must cut the other.
A C, meeting A B, not meet its parallel G D, parallel to E D, consequently A C, A B being both parallel to E D, are parallel inter se, which is not the case.
The only other property of parallel lines not included in the above is, that two straight lines which are respectively parallel to two others contain an angle equal to the angle of those others. But there is nothing peculiar in its demonstration. These thirteen propositions contain a complete and homogenous geometry of position as contra-distinguished from that of magnitude: I speak of course relatively to lines. It is scarcely necessary to refer the student to the Third Book of Euclid, as far as relates to the consideration of angles in a circle, to shew how much this mode of treatment, and the introduction of reverse angles would simplify the subject, as well as prepare him for analytical inquiries by generalising his ideas on it.
In looking over some of the mathematical articles of the Penny Cyclopedia, written by Professor De Morgan, I have subsequently to the writing of the above, found a confirmation of my views as to the nature of the angle under the heads, " Angle" and " Infinite."
The former proposes to introduce the axiom, that " two spaces whe"ther of finite or infinite extent are equal, when one can be placed upon "the other, so that the two shall coincide in all their parts." After which, it is remarked, that Bertrand's demonstration becomes rigorous. This also considers an interparallel space viewed as an angle to be zero, as I have done, since it is less than any assignable angle.
The latter has the following passage :—
"The comparison of such infinite spaces is therefore possible, con"sistently with perfect clearness in the meaning of the terms employed, "and a simplicity of reasoning which would convince any one who is "capable of the most ordinary thought. Had Euclid been accustomed "to the modes of thinking which involve the idea of infinite magnitude "under any form whatsoever, it may be reasonably suspected that he "would admit the following axiom, Magnitudes which can be made to "coincide in all their parts are equal, as applicable to infiniteas well as to "finite spaces. Not having done so, the adherence to his standard has "to this day excluded the only proof of the theory of parallels, which "does not assume the axiom of Euclid, or an equivalent."
Remarks on the Essay " on the Theory of Angular Geometry." By Capt.
A definition is perfect, when it includes all that has the property intended to be denned, while it excludes all that has it not.
If we would have a true definition of angle, or of any thing else, it is of the utmost importance that we have a clear idea of the thing, and then use such words as plainly to convey the idea. If there be any neglect in either of these, our definition must necessarily be imperfect.
Geometry as commonly defined, treats of figured space. If this definition be correct, (and I find no fault with it), then it is plainly improper to introduce indefiniteness, or boundlessness, or infinity, as part and parcel of the definition of a thing or idea, of which the property signified by these terms, is not necessarily a part. I can conceive of an angle formed by finite lines: unboundedness is therefore not necessary to the idea of angle, and therefore ought not to form a part of the definition.
Since the idea of angle is somehow sooner or later convertible to, and commensurable by that of circular arc, every attempt at defining angle should be made with this in view, otherwise the definer will discover, (or some one will discover it for him), that his definition is not perfect.
As the author of this Essay introduces un limit cdness in the containing lines as part of his definition of angle, I do not see why the plane surface of a hyperbola between its assymptotes may not be angle, as well as the thing intended by him. If it be said, the meaning is the whole plane surface between the lines, I rejoin that the whole plane surface being unlimited, I cannot form an idea of how much it is.