Sll 



ANGLE OF CONTINGENCE. 



ANGLE (RECTILINEAR). 



312 



ployed. It is impossible to discuss here the remedies which the 

 physician should resort to, the reasons which should determine his 

 choice, and the different states which should modify the treatment in 

 adaptation to individual cases. But it is very important to state, that 

 angina pectoris is one of those diseases in which the concurrence of the 

 patient with the efforts of the physician is indispensable. Unless the 

 patient resolve and firmly adhere to his resolution strictly to conform 

 to the plan prescribed in diet, in exercise, in every locomotive move- 

 ment, in aleep, temperature, and medicine, but above all in the regula- 

 tion of the mind, the physician can do but very little for him. 



ANGLE OF CONTINGENCE, or CONTACT, the opening made 

 by a curve and itg tangent. [CURVATURE.] 



ANGLE (CURVILINEAR), the rectilinear angle made by the 

 tangents of two curves at the point where they meet, as A B C. 



ANGLE (HORARY), the angle formed with the meridian of any 

 place by a great circle, which j a.-.sf.-i through a star and the pole. 



ANGLE OF INCIDENCE, REFLECTION, REFRACTION, 

 ELONGATION, ELEVATION, THE VERTICAL. See these 

 several terms. 



ANGLE, PLANE, SPHERICAL, SOLID, PARALLACTIC. See 

 these terms. 



ANGLE OF POSITION, usually the curvilinear angle made by 

 two great circles drawn through a star to the poles of the ecliptic and 

 equator. It may be used to signify the angle made by lines drawn 

 from any point under consideration to any two points which are used 

 in determining th<- fn^iti'm of others. 



ANGLE (RECTILINEAR), from the Latin word anyulus, of the 

 same signification. The notion (for it can hardly be called definition) 

 is, the opening made by two straight lines which cut one another. The 

 term inclinatirm is also used synonymously with angle ; thus, the angle 

 or opening of two lines is called their inclination to one another. 



To investigate a more precise definition for this word, we must 

 recollect that any species of relation is entitled to the term ;/></ . 

 and becomes the object of arithmetic or geometry, BO soon as it can be 

 shown that the notion implied in one or other of the words equal, 

 greater, or less, is always derivable from the consideration of two such 

 relations. Take the two angles or openings made at the points A and 

 B by the straight lines A p and A q at A, and by B B and B s at B, and 

 transfer the first figure to the second, DO that the point" A shall fall 

 upon B, and the straight line A q upon B s ; or rather, let as much of 

 A q as is equal to B 8 fall upon B s, and let the remainder of A q 

 form a continuation of B s : also let A P and B p. be made to lie 



upon the same side of B s. We have now no longer any control over 

 the position of A p with respect to A Q, since the first figure is not to 

 undergo any change except that of simple removal into another position. 

 If, after A q hag been placed upon B s, A p then fall upon B B, the two 

 openings or angles at A and B are the same. If A P, in its new position, 

 fall between B and B it, the opening or angle at A is lew than that at 

 B ; and if A P fall further from B s than B R does, the angle at A is 

 greater than that at B. The angle at A is called the angle P A q, and 

 that at B, the angle R B s. Hence the notion of one angle being twice 

 or three times, &c., as great as another may be fixed. For example 



t the angle MAP being made up of the two MAN and NAP, each of 

 which is equal to the angle D B c, is twice D B c ; the angle q A M is 

 three times DEC; BAM is four times D B c ; and so on. Similarly, 

 the angle D B c is one-half of p A a, one-third of q A M, &c. The angle 

 made by two lines does not depend upon the length of these lines ; if 

 a part D E be cut off from B D, the angle is not altered, that is, the 

 angle r. B c is the same as D B c. If B e and B d be respectively equal 



to B E and B D, and if B c e d turn round B, the same quantity of turning 

 which brings B e into the position B E, will bring B d into that of B D. 



When we cast our eyes on two angles, the sides containing which 

 are nearly equal in both, we judge of their comparative magnitude by 

 the spaces which are included between the lines. But this ia is not a 

 notion capable of being rendered rigorous, because one boundary of 

 the space is indefinite. Nevertheless we may correct this method of 

 judging, and produce a precise idea of an angle, if we admit the pro- 

 priety of comparing with one another spaces which are absolutely 

 infinite in extent. The longer the lines are, the more nearly is the 

 preceding notion absolutely correct, because the space at and near the 



mouth of the angle, which for want of a definite boundary is doubtful 

 as to whether it is or is not to be considered a part of the angular 

 opening, becomes less and less with respect to that about which there 

 is no doubt. If then we suppose the lines which contain the angle 

 to be produced without end, the infinite spaces so imagined will be 

 correctly in the same proportion to one another as the angles. The 

 objection to introducing this into geometry is the real or supposed 

 want of rigour in the comparison of unbounded spaces. [INFINITE.] 

 It must be remarked, however, that the disputed theory of parallels 

 follows immediately and rigorously from the preceding (see ' Library 

 of Useful Knowledge,' Study of Mathematics, pp. 77, 78 ; and Lacroix, 

 ' Siemens de Geome'trie,' p. 23, note), and it is therefore in the choice 

 of every person to decide for himself whether he will add the words 

 in italics to the first of the two following axioms, and prove the second, 

 or omit the words in italics, and astume the second. 



1. Two spaces, whether of finite or infinite extent, are equal when the 

 one can be placed upon the other, so that the two shall coincide in all 

 their parts. 



2. Through a given point, not more than one parallel can be drawn 

 to a given straight line. 



In order to bound the preceding spaces, and compare angles by 

 means of spaces or lines, it is necessary to draw arcs of circles having 

 equal radii through the two points. 



Let p q and B s be arcs of circles having the equal radii A Q, B s. 



Then the angles p A Q and R B s are in the same proportion as the 

 spaces (called sectors) P A q, and BBS, and also as the lengths of the 

 arcs p q and R s. This proposition, which is Euclid, vi. 33, is not so 

 far from first principles as its position would appear to indicate. For 

 the fifth book, on proportion, is entirely independent of, and might be 

 considered as antecedent to, the first four books : if this were supposed, 

 the preceding proposition might be easily made to follow book i. 23, 

 or even i. 8. We might even place it immediately after the doctrine 

 of proportion, by a proof founded on simple superposition, provided 

 we assume (what is tacitly assumed in various parts of the first book 

 of the elements, i. 4, for example) that an angle may be conceived 

 equal to another angle before we know how to construct equal angles. 



If a line setting out from A B be conceived 

 to revolve round the point A, it will in every 

 position form two openings or angles with its 

 original position A B. For example, in the 

 position A c, A B and A c will form the smaller 

 angle BAG, and the larger angle made up of 

 the angles c A F, v A K, and K A B. Only the 

 former of these is usually considered in 

 geometry, but the latter is frequently used 

 in analysis. When half a revolution has been 

 made, and A B has come to A F, at first sight 

 we might say there was no angle formed ; 

 but on looking at the preceding position A E, we see that the opening 

 of B A and A F is greater than that of B A 'and A E. The half of thin 

 opening B A F, that is, B A D, is called a right angle. A whole revolu- 

 tion makes A B pass through four right angles, and, in analysis, if we 

 wish to point out that the line A c is supposed to have made a com- 

 plete revolution, and to have come into the position B A C f or the second 

 time, the angle made with A B is said to be 



4 right angles + B A c 



An angle is the opening of two lines ; rectilinear, of two straight 

 lines ; curvilinear, of two curves ; mixtilinear, of a straight line and a 

 curve. But, in truth, angle always means rectilinear angle ; and when 

 a curve enters, its tangent is the straight line which is used in deter- 

 mining the angle. 



A riyht angle is half the opening of a straight line and its continuation; 



