RIGHT. (Mathematics) ) This term is applied in mathematical language to anything which is imagined to be the moat simple of its kind, to distinguish it from others. Thus a right line is a STRAIGHT line ; a right angle is the most simple and well-known of the angles used by Euclid - a right cone is one in which the axis is at right angles to the tease; and so on.
It !WIT ANGLE. When two lines, at first coincident, are made to separate so that one of them revolves about their common extremity, the revolving line will in time become the euntinnation 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 Itleter, be called a right angle. But in the geometry of Euclid the word angle seems to have been essentially connected with the idea of a pointed corner, and we have no means of finding out that he considered a line and its continuation as making any angle at all. Instead of this angle, made by o a and o a, or the angle of opposite directions, he introdueca its half, and calls it a right angle. Let a o c and 0 o a be equal angles, that is let 0 C bisect the angle A 0 R, and each half is called a right angle. When the angle A o a is mentioned, it is as two right angles. All that is necessary as to the magnitude of a right angle has been given under Aei.te : we propose here to point out the effects of the forced manner in which Euclid avoids the angle a 0 B. • It is sufficiently evident that nothing can lame its right to be con sidered as a magnitude by augmentation : so that the opening of a o and o is, which is double that of a 0 and o C, must really be a magni tude of the same kind as the angle A 0 e. Nuw the consequences; of pr -(erring a o c to A o a, as a fundamental angle of reference, are as follows : 1. The intrauction of the apparently vary arbitrary axiom, that " all right angles are equal," instead of the more simple and natural one that " two straight lines which coincide in any two points coincide Is-yend those points." It is as evident as that " two straight lines can not inclose a apace," or " two straight lines which coincide in two points, coincide between those points,' that the same also takes place &reel these points." A munient's examination will show that this :Laid!) immediately gives as a consequence that the angle A 0 II in any one straight line Is equal to tho angle A'o'B' in any other ; or as Euclid would express it, the double. of all right angles are equal, whence all right angles are equal. And it is one consequence of leaving the Aural route, that Enclid himself has assumed both the more com plicated axiom which he has expressed, and also the more simple one by which he might base avoided it : for he nowhere shows that if o A be monde to coincide with o' A', then o n coincides with o'n'. Some of his editors have supplied the defect by making it a consequence of " all right angles are equal," that " no two lines can have a common segment :" that ia, by limiting the cart draw the hone.
2. The necessity of appearing to prove a particular case of a pro position which is taken as self-evident in all other cases. Thus Euclid never proves that COD and u o a are together equal to C 0 a ; while he has to spend a proposition in proving that AOD and D011 are togethir 3. The necessity of to proves particelarease after the general case has beenproved. Thus to bisect a given angle is the general propo sition, of which to draw a line perpendiculer to a given line from a given point within it. is the particular case. The construction of the latter is precisely that of the former : but the two results have to be obtained in two distinct propositions : it would be right enough to make them cases of one 4. The habituation of the student to neglect the angles greeter than
two right angles, by his never meeting with one as ;peat. Two lines which end at the mune point make two openings, one greater and the ether lase than two right eagles; except in the intermediate case when both are equal to two right angles. Now Euclid does not positively reject the angle greater than two right angles, nor does ho my that of two hoes which meet, the angle which they make shall be always taken to be that which is Ices than two right angles. Had he hail ausli 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 segment which comicial; as eagle less 'a right eagle, the angle at the centre is double of Oust at the circum ference." It la true that his proposition is," In a circle, the angle at the centre is deubla of the angle at the circumference when they hare the same circumference for abase :" 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 on 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 circumference." If no mention of angles greater than two right angles be intended in the provide; proposition, than the one before us is not completely proved, but only when the angle at the circumference is lees 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 cases iu 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 bast proposition of the sixth book, iu which it is shown that in equal circles angles arc to one another as their subtending arcs. Now the criterion of Puorontatsti, as given by Euclid, requires that, 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 hook should have been unable to bear in mind that the establishment of proportion demands that every possible multiple of the quantities asserted to be proportional should be admitted and compared with every other : and thus it is certain that Euclid must have meant to consider angles not only gikster than two right angles, but even greater than four, or any other number. Some commentators have supposed that Euclid meant to omit all lairs of right angles from such multiples, and all seinicireninferenees from the 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 quantities mentioned to be comparable magnitudes : and that Euclid does consider them as such, is all that is contended for.