



ANOLE (RECTILINEAR). 



IK TII.IXEAR). 



an vnir angle u that which w lew than * right angle ; aa atliut angle 

 U that which lie* between one and two right angles. Oomplrmmlal 

 angle* are two which together make a right angle ; infftemntat angle* 

 are two which together make two right angle*. Two line* which meet 

 and make a pair of angle*, one lee* than two right angle*, are called 

 alttmt : the other greater than two right angle*, are called rr-ntraxi. 

 or n tulirimf. None but salient angle* are mentioned by Ku. h.l. Fur 

 Uent and re-entering (which are borrowed from fortification), dirert 

 and rttnjlrrttit have *ometime* been lined. 



The angle* which two line* make with the *au>e part of a third, on 

 opposite aide* of it, are called alirrxnlt. Two line* which cro* one 

 another make two pair uf irrf iVu7/y opposite angle*. The angle* made 

 by adjacent aide* of a figure are called ittlmal ; thoee mode by any 

 de* with adjacent aide* produced are t.rtrr.i.il. When the angular 

 (mint i* the centre or on the circumference of a circle, the amjte u laid 

 to be a/ the centre, or at the circumference. Beginner* often confound 

 the angle with the angular point. 



The angle of cotin;ir*re or of contact it an old notion of the opening 

 mad* by two curve*, or a curve and a line, which touch. In modern 

 mathematics, when a curve u *uppond to be composed of infinitely 

 mall rectilinear elements, the infinitely small acute angle made by one 

 element with the production of the next doe* duty for thin old angle of 



A tphrrical angle i* made by two circle* (usually great circle*) of a 

 sphere. When the circle* meet at the pole of the equator, and one of 

 them is the meridian, the angle in an liurnry or hour-aaylt ; and when 

 neither U the meridian, the angle is frequently called horary. The 

 am-jte of petition of a star is that made by the circled drawn to it from 

 the pole* of the equator and ecliptic. The angle of eleratioa U the 

 angle made by a line drawn from the eye to any object with the hori- 

 zontal line which in in the name vertical plane OB the first line ; but 

 when the object is below the horizon, the term i.< angle of depreitiun. 

 When lines are drawn from two points to a third, those two points, and 

 also the line joining them, are said to milileail the angle which is made 

 at the third point. The angle which two objects subtend at the eye is 

 their angle of etomiatinu. The angle of the vertical is a name given to 

 the angle which a line drawn to the spectator's zenith makes with bin 

 radius of the earth produced : it is taken as nothing when the earth is 

 supposed to be a perfect sphere. Angular terms, such as right ascension, 

 longitude, Ac., with which the word angle is not usually coupled, are 

 not considered here. The parallarlic angle is simply the PARALLAX. 



When one line falls upon another, the angle of incidence is the acute 

 angle which the incident line makes with the perpendicular to the 

 other. When the incident line is thrown off again on the some side an 

 that from which it came, the acute angle made with the perpendicular 

 if called the angle of reJUxiott ; when on the opposite side, the angle of 

 refraction. These terms are nearly confined to optics. 



A dihedral angle is the opening made by two planes. It is measured 

 by a rectilinear angle, namely, that mode by two lines drawn in the 

 two planes perpendicular to their common intersection. But the recti- 

 linear measure is not the same thing as the dihedral angle, though the 

 two are often confounded. We might just as well say that the pressure 

 of the air is the same thing as the number of inches in the barometrical 

 column of mercury. 



A tolid angle is said to exist when three or more straight lines, not 

 in the same plane, meet at a point. It is a complex idea, and the best 

 notion of it as a magnitude is derived from considering it as measured 

 by the area of the spherical triangle which subtends it. 



For the most important properties of angles see TRIANGLE; PARALLEL; 

 POLYOOX; TRIGONOMETRY. 



The methods of measuring an angle, of which we think it necessary 

 to take notice, are three in number. The first is the one universally 

 employed in theoretical investigations, and is as follows : in the last 

 figure but one, the number which expresses what proportion the are 

 r q i of the radius, is the number chosen to represent the angle. It is 

 ahown in geometry that if any number of arcs be drawn with the centre 

 A, subtending the same angle p A q, what port soever any one of them 

 is of its radius, the same part u any other of if* radius ; that is, what- 

 ever circle may be chosen, the preceding measure gives the same num- 

 ber for the same angle. For example, if the arc r q be equal to the 

 radius, the angle r A q is the angle 1 . If r q be two-thirds of the radius, 

 the angle r A q is the angle {. The unit of this measure is tlu-r, I.T,' 

 the angle whose arc is equal in length to its radius. It is customary to 

 ny that an angle or arc (for the terms are frequently confounded) thus 

 measured, is given in port* of the radius ; but this expression doe* not 

 convey much meaning, and we cannot propose any better, unless it 

 might be judged proper to say it is measured in theoretical units, mean- 

 ing thereby, in the units which are always employed in pure theory ; 

 or in arcnal units, derived from use of 'the arc. The theoreliral or 

 armal unit would then be the angle subtended by the arc which U 

 equal to iU radius. 



The semi circumference of a circle contains its radius, 



3 14159, 26536, 89793, 23840 



times, very nearly. This is then the number of theoretical unit* con- 

 tained in two right angles. The right angle is therefore 



1-67079, 63267, 94896, 61923, 



and the following are the angles of one degree, one minute, and one 

 wound, to which we *hall presently come : 



01745, 32925, 19943, 29577 degree 

 00029, 08882, 08665, 72160 minnt.- 

 00000, 48481. 36811, 09536 *ee..i,.l. 



In the second measure, in which angle* are (aid to be measured in 

 iixtce (the word space being here opposed to lime, as we shall nee. and 

 not to leoijth), the whole angle traced out in one revolution, equal to 

 four right angles, is divided into 360 equal part*, each of which i 

 called one dryree, and marked thus, 1. Each degree is divid, 

 60 equal parts, each called one mixHlf (!'), and each minute into 60 

 equal part*, each called one Brand (I 1 ). Formerly, the second was 

 divided into 60 equal parts called third*, and so on ; but it is now usual 

 to use the tenths, huudredths, to., of seconds. The present table 

 therefore stands thus : 



A whole revolution 360* = 21,600' = 1296,000" 

 A right angle 90 = 5,400' = 324,000" 



.Minute-. 



60 



1 



-,,. i.,',-. 

 3600 

 60 



To convert an angle from theoretical unite into degree*, &e., oftpace, 

 observe that the lost-mentioned unit is 



200264 806247096355 in seconds 



3437' -746770784939 in minutes 



57 -295779513082 in degrees 



and multiply the number which expressed the angle in thei.u-ticul unit . 

 by the one among the preceding numW* which ha the same denomi- 

 nation as that to which the angle is to be reduced. As many docimal- 

 niay be taken as shall be considered necessary. The following table 

 however will be found more convenient : 



".729578 

 11459156 

 17188784 

 22918812 

 28647890 

 84377468 

 40107046 

 45836624 

 51506202 



Minute*. 



03437747 

 06875494 

 10313240 

 13750987 

 17188784 

 20626481 

 24064-J-J7 

 27501974 

 30989721 



Second*. 

 "-MU26481 

 041252901 

 061879442 

 082505922 



123758884 

 144385364 



185638325 



KXAHPLE. What number of minutes and decimals of minute* doe* 

 the angle contain which expressed in theoretical unitn is 17900 f 



From the minutes' column take out the rows opposite to 1, 7, 9, and 

 6 ; write them so that the first figure of each shall fall under its corre- 

 sponding figure in 17906, and odd, but take only so many out of each 

 row as will serve to fill up the places under the first row, increasing the 

 last figure of each broken row by 1, when the first neglected figure is 

 5 or upwards : 



17906 



08437747 

 2400423 

 309397 

 2063 



06155630 



Place the decimal poiut three places off the unit's column 

 jii-e for MiHHttt, and $enn for tecundt. This gives 6155''630, since the 

 present calculation U made for minutes. Further to illustrate the 

 placing of the decimal point, let the angle theoretically expressed be 

 096, to be turned into degrees and decimals of degrees, and afterwards 

 to seconds and decimals of seconds : 



0- 096 

 : i 51566202 

 ! 3437747 



055003949 



Bring down the preliminary cyphers, and then cut off three places, 

 which give* 5-500394P. Again for the seconds : 



0- 096 



1 185038325 

 j i 12375388 



Cut off Ki-en places, which gives 19801"-4213. 



Given an arc of a circle and the radius to determine the degrees, 

 minutes, or seconds in the angle at the centre ; divide the arc l.v ;1. 

 radius, and proceed with the quotient a* above. 



For the convene problem, given the degrees, minutes, and seconds in 



