101 



RIGHT. 



RIGIDITY OF ROPES. 



102 



of nature ; which theory, though recommended by the deservedly high 

 authority of Locke, has now been abandoned by nearly all political 

 speculators. 



IM'iHT. (Mathematics) Thin term U applied in mathematical 

 language to anything which U imagined to be the most simple of its 

 kind, to distinguish it from others. Thus a right line U a STU.U'.HT 

 line ; a right angle is the most simple and well-known of the angled 

 used by Kuclid ; a right cone is one in which the axis is at right angles 

 to the base ; and so on. 



RKJHT AHOLB. When two lines, at first coincident, are made to 

 separate so that one of them re volves about their common extremity, 

 the revolving line will in time become the continuation of the other. 

 This angle or opening, made by a line and its continuation, would, we 

 might suppose, be one of the principal angles considered in geometry, 

 and should, according to the previously defined meaning of RIGHT, be 

 called a right angle. But in the geometry of Kuclid the word angle 

 seems to hare been essentially connected with the idea of a pointed 

 . and we have no means of finding out that he considered a line 

 and its continuation as mlring any angle at all. Instead of this angle, 



C D 



A' O" ff 



made by o A and OB, or the angle of opposite directions, he introduces 

 it* half, and calls it a right angle. Let A O c and o o B be equal angles, 

 that is, let o c bisect the angle A O B, and each half is called a right 

 angle. When the angle A o B is mentioned, it is as two right angles. 

 All that is necessary as to the magrrfoi'tr of a right angle has been 

 given under ASM.* : we propose here to point out the effects of the 

 forced manner in which Euclid avoids the angle A o B. 



mifficiently evident that nothing can lose its right to be con- 

 sidered as a magnitude by augmentation : so that the opening of .\ ,, 

 and u B, which is double that of AO and oc, must really be a magni- 

 aame kind a* the angle A o c. Now the consequences of 



ing AOC to A OB, as a fundamental angle of reference, are as 



: 



1. The introduction of the apparently very arbitrary axiom, that 

 " all right angle* are equal," instead of the more simple and natural 

 one that " two straight lines which coincide in any two points coincide 

 beyond those points." U is a* evident as that " two straight lines can- 

 lose a space," or " two straight lines which coincide in two 

 point*, coincide hetrm those points, that the same also takes place 

 Uymtd those points." A moment's examination will show that this 

 axiom immediately gives as a consequence that the angle A o B in any 

 on* straight line is equal to the angle A'U'B' in any other ; or & Km -lid 

 : .x press it, the doubles of all right angle* are equal, whence all 

 right angles are equal. And it is on* consequence of leaving the 

 lid himself has assumed both the more com- 

 plicated axiom which be has expressed, and also the more sin. 



; h he might have av.. : he nowhere shows that if o A 



be made to coincide with o'.\', then o B coincides with oV. Home of 

 bis editors have supplied the defect by making it a consequence of 

 " all right angles are equal," that " no two lines can have a common 

 egui .-, by making the cart draw the horse. 



i lie iisnasilj of appearing to prove a particular case of a pro- 

 position whn-li Li taken as self-evident in all other esses. Thus Km lid 



|. rove* that CUD and DO Bare together equal to COB; while he 

 has to spend a proposition in proving that AOD and DOB are together 



i fie necessity of appearing to proves particular case after the general 



case ban been proved. Thus to bisect a given angle is the general propo. 



sition, of which t-. draw a line perpendicular to a given line from a given 



- the particular case. The construction of the latter 



is precisely that of the former : but the two result* have to be 



I in two distinct proposition* : it would be right enough to 



make them cases of on* proposition. 



he uAl.itiiati..ti of the student to neglect the angle* greater than 

 two right angles, by his never meeting with one at yreal . Two lines 

 end at the same point make two openings, one greater and the 

 other lew than two right angles ; except in the intermediate case when 

 both are equal to two right angles. Now Kuclid doe* not p< - 

 reject the angle grater than t r j.-i.t angles, nor doe* he say that of 



.-y make shall be always taken 



to be that which in less than two right angles. Had he had such 

 intention, one of his propositions would have been positively false, to 

 wit. that in any segment of a circle.'the angle at the centre is double 

 of the angle of the circumference. Had such been his intention, he 

 would have said, " in every segnv ., an.///- leu than 



'a riykt anylf, the angle at the centre is double of that at 



It is true that his proposition is, " In a circle, the angle at 

 the centre is double of the angle at the circumference wAot tkey hare 



the same circumference far a lose : " and some may think that the words 

 in italics exclude (as in one sense they certainly do) the segment which 

 has an angle greater than a right angle ; since this angle, and its 

 central angle, that, namely, which is less than two right angles, do not 

 stand on the same circumference as a base. Let this be so, then we 

 throw the difficulty ou another proposition, the 27th. It is there 

 shown that " in equal circles, the angles which stand upon equal 

 circumferences are equal whether they stand at the centre or at the 

 circumfereiue." U no mention of angles greater than two right angles 

 be intended in the previous proposition, then the one before us is not 

 completely proved, but only when the angle at the circumference is 

 less than a right angle. At the same time there seems to be, in some 

 of the subsequent propositions, proof of a desire to avoid the angle 

 greater than two right angles, and to subdivide proofs into particular 

 case* in order to avoid the difficulty. 



But are we not in fact to assume, without particular inspection, 

 from the general tone of the first six books, that the angle equal to or 

 greater than two right angles was never really meant, and that all 

 propositions are to be taken with such limitations as the above restric- 

 tion would render necessary ? Let those who think so, look at the 

 last proposition of the sixth book, in which it is shown that in equal 

 circles angles are to one another as their subtending arcs. Now the 

 eriterion of PROPORTION, as given by Euclid, requires tliat, in this 

 proposition, any multiple, however great, of the angles may be taken. 

 Now a multiple of an angle may not only be greater than two right 

 angles, but greater than a thousand right angles; and every such 

 multiple must not only be really included in the demonstration, but 

 considered as a magnitude, and compared with other magnitudes of 

 the same kind. It is impossible that the writer of the fifth book 

 should have been unable to bear in mind that the establishment of 

 proportion demands that every possible multiple of the quantities 

 asserted to be pro|>rtional should be admitted and compared with 

 every other : and thus it is certain that Kin-lid must have meant to 

 consider angles not only greater than two right angles, but even greater 

 than four, or any other numlx-r. Some commentators have supposed 

 that Kuclid meant to omit all pairs of right angles from such multiples, 

 and all semiein -mm. reiiee.t from tlie multiples of the arcs; but this 

 would only be a use of the axiom, that if equals be taken from un- 

 equals, the remainders are unequal, which admits the greater of the 

 i|ii iniitiei mentioned to l.e coiu|>ai-:tblu magnitudes: and that Euclid 

 does consider them as such, is all that is contended for. 



III'iHT. 1'KTITMN "I'. [ l'i rm...\ OF KIIJHT.] 



HKillTS. 1JILL UK. [Hiu. UK RumTS.] 



RIGIDITY ()K liol'KS. In estimating the powers of machines, it 

 is frequently necessary to take into consideration the effects arising 

 n.-iu the rigidity or stillness of the ropes which pass over the pulleys 

 or the axles of the wheels ; and, in order to understand how this con- 

 dition affects the relation between the moving power and the resistance, 

 let it be observed that when a stiff rope is bent over the upper ; 

 a wheel or pulley in a vertical plane, for example, the weights or 

 powers applied at its extremities may not be sufficient to draw the 

 descending portions into the positions of two vertical lines. Now, if 

 one of the parts of the rope should take such a direct!" >n that a vert i.-.il 

 line drawn through the weight attached to that part, cuts the hori- 

 zontal diameter of the wheel or pulley at a point between the outre 

 and one extremity of the diameter, and if, at the same time, the other 

 part should take such a direction that a vertical line drawn through ' 

 the attached weight cuts the horizontal diameter at a point beyond 

 the extremity of the latter, the distances of tin se vertical lines from 

 the extremities of the diameter being represented by s and ..-' respec- 

 tively, the corresponding weights by w and w', and the radius of the 

 .-. le.l l.y B, the conditions of equilibrium, instead of being w = w', 

 will be 



w(n jr) = w'(u +x\ 



Hut, if w be the weight which l.y descending raises u 

 value of je is generally so small that it may be disregard 

 have, in the case of e'juilil>rium, 



W'(R + J), or (W-W')R^ '..-', 



the other, the 

 so that we 



WB 



or again, w-w'= ; 



that is, in order to put the system in a state of equilibrium, the excess 



JT 

 of w above w' should be equal to -- 



The formula given by Coulomb to express the force necessary for 



\\ ' i :> 

 overcoming the rigidity of a rope, or the equivalent of 1_, is 



r being the semi iliameU-i of the rope, a the force arising from tin 1 

 warping or twisting of the rope, anil h that which depends on the 

 t. nsioii arising from the weight w ; the values of in, a, and l> may be 

 determined by experiments made with cords of different diameters ; 

 and tliUB ./' may be found. M.<'ouloiub .-i^-.-rt-iin. .1 that, for slender 

 ,.. 1. and that for stiff cordage the .du.- of M varied from 1 -r, 

 to 2 ; also, from some experiments made with ropes consisting oi '','> 



