315 



ANGLE (RECTILINEAR). 



ANHYDRIDES. 



346 



an angle, to express the same in theoretical units, the following table is 

 given : 



EXAMPLE. It is required to express in theoretical units the angle 

 89 52' 34". Take out the row corresponding to each figure from the 

 column having the same denomination, taking seven places only for a 

 unit's figure, and the whole eight places for the tens, increasing the 

 last figure when necessary, as before : add and make seven decimal 

 place-;. 



1-3902C34 

 1570796 

 00145444 

 0005818 

 0-0001454 

 0000194 



For 80' 



50' 



2' 



30" 



4" 



1-5686340 

 and the answer is 1-5686340. 



Given any angle and a radius, required the circular arc subtended 

 by that angle ; proceed as above, and then multiply by the radius. 

 Thus , to a radius of 100 feet, the arc which subtends an angle of 

 89 52' 34" is 



1 5686340 x 100 or 156 86340 feet. 



In the attempt to effect a universal change of weights and measures, 

 which followed the French Revolution, the circle was divided into 400 

 degrees, each degree into 100 minutes, each minute into 100 seconds, 

 and so on. This innovation obtained only a partial introduction, and is 

 iinw almost entirely abandoned. When used, it is customary in this 

 country to distinguish the French degrees by the name of GRADES, and 

 to denote one grade by 1, or !" The convenience of this method, 

 from its close affinity with the decimal system, is certainly great : for 

 example, grades and decimals of grades, such as 12 r - 1329 are converted 

 into grades, minutes, and seconds, by mere separation of the figures : 

 thus, 12'- 13' 29'. 



It is not necessary to give complete tables of reduction from the new 

 French to the ancient system, as they would so seldom be useful ; the 

 following is all that is necessary : 



l"is 0'9 or 54' or 3240" 



1' '009 (K-54 32"'4 

 1" -00009 0'-0054 0"'324 



The third method of measuring angles, in which they are said to be 

 measured in that, is confined to astronomy, and is derived from the 

 complete apparent revolution of the heavens which takes place in 

 24 hours. That is, if a line revolve round a point at the rate of a 

 whole revolution in 24 hours, or a right angle in 6 hours, the times of 

 moving through different angles are made the measures of their com- 

 parative magnitudes. Thus 4 b 32 60" is the angle moved through in 

 4 hours, 32 minutes, and 60 seconds. The following tables are useful 

 in turning angles measured in degrees, tc., of space into the correspond- 

 ing measures in time, and the converse : 



In these tables, where there are two headings, either the upper or 

 under of both must be used. The following are examples : 



To turn 18 h 11 35"'3 into degrees, &c., of space. From the first 

 table: 



is 150 



120 



10 h 



8" 

 10- 



l" 1 

 30' 



5 



0-3 



0' 

 

 30 

 15 

 7 

 1 



0" 













 30 

 15 



4-5 



18 h 11" 35"3 is 272 53' 49"'5 

 To turn 97 54' 23" into hours, &P. From the second table, 



90 



r 



50' 

 4' 



20'' 

 3" 



97 54' 23" is 



6 h O m O f 



28 



3 20 



16 



1-333 

 0-200 



6 h 31 m 371-533 



In astronomy 30 is sometimes called a sign, in allusion to the arc 

 of the ecliptic, through" which one of the signs of the zodiac extends : 

 Thus 2- 3 5 4' 12" means 63 4' 12". 



The anyk is distinguished from all other magnitudes whatsoever in 

 a very remarkable way. It is the only magnitude which is a function 

 of a number, and of which a number is a function. Let any one single 

 magnitude, not" angle, say a length, be presented : without reference to 

 other magnitude, it is impossible to assign a number with which that 

 length is necessarily connected, so that by merely giving the length, 

 the number is deducible. But an angle, when given, determines ratios 

 of lilies, and so determines number. To one angle there is but one 

 ratio of arc to radius, one sine, one cosine. &c., and all these are num- 

 bers. Consequently, a relation may exist between tmc angle and numbers ; 

 but the idea of a relation between one length and numbers is absurd. 



ANGLE, TRISECTION OF. [TRISKCTION.] 



ANHYDRIDES, or ANHYDROUS ACIDS. A class of chemical 

 compounds of great theoretical interest. They are represented both 

 amongst organic and inorganic bodies, but it is especially the organic 

 anhydrides discovered by Gerhardt, which, in conjunction with 

 Williamson's compound ethers, have exercised so profound an influence 

 upon the development of theoretical chemistry. The anhydrides bear 

 the same relation to the hydrated acids as the ethers bear to the 

 alcohols, and they may be regarded as formed from the hydrated acids, 

 by the substitution in the latter, of a second equivalent of a negative 

 radical for the remaining single equivalent of hydrogen, a transforma- 

 tion which will be more readily understood by reference to the 

 following comparison : 



Vinlc alcohol. 



ir ?O, 



Acetic acid. 



Vinic ether. 



(C.H 4 > \ 

 (C 4 II 5 )) 



Acetic anhydride. 

 (C.H,0 2 )1 

 (C.H.O,)) ' 



The organic anhydrides are best obtained by acting upon the potash 

 salt of an organic acid by the chloride of a negative radical. Thus 

 acetic anhydride is produced by distilling a mixture of solid and dry 

 acetate of potash with chloride of othyl. The following equation 

 represents the nature of the reaction : 



. (C t H 3 0,) 

 Cl 

 Acetate of potash. Cloridc of othyl. 



K \ 



Cl ] 



(0,11,0,) 



Anhydrous acetic Chloride 

 acid. of potassium. 



The anhydrides are neutral bodies, generally liquid, although some 

 are solid. In contact with water they are gradually re-converted into 

 hydrated acids. Acted upon by ammonia they give either neutral 

 amides or ammoniacal salts of amidated acids. The anhydrides may 

 either contain two equivalents of the same negative radical, as acetic 

 anhydride, or they may be formed by the juxtaposition of two different 

 negative radicals as, for instance, aceto-benzoic anhydride, which 

 contains both othyl (C.H.,0.,) and benzoyl (C J4 H,0,). 



The organic anhydrides may be divided into two classes, namely, 

 1st, The anhydrides of monobasic acids; and 2nd, the anhydrides of 

 bi basic acids. A third class would be formed by the anhydrides of the 

 tribasic acids, but no organic compound of this class has yet been 

 formed. The following is a list of the principal anhydrides hitherto 

 obtained : 



I. 



Acetic anhydride . 



Valeric 



\HIIYDBIDEX OF MONOBASIC ACIDS. 



C,H,0, 



' ' ' C 4 H 3 2 



C.oH.Oj 



