205 



ALGEBRAIC GEOMETRY. 



ALIEN. 



206 



The following system is sometimes convenient : 

 x = a + Ar, y = 4 + Br, z = c + Cv 



where -/(A 5 + B 3 + C 2 ) is the distance between (6c) and (.cyz). 



7. The plane whose equation is A(JC a) + Ac. =0 may be called 

 the plane (ABCaic), and the straight line just mentioned may be called 

 the straight line (ABC4r). Throughout this article capitals are 

 generally proportional to cosines of angles, and small letters are co- 

 ordinates of points. The order of co-ordinates is xtjz, and all letters 

 connected with co-ordinates ran in consecutive triplets, as ABC, PQR, 

 Ac. But when triplets of pairs are made, as AB, BC, CA, then AB 

 inrticularly belongs to the co-ordinate to which C is attached, BC to 

 that of A, and C'A to that of B. 



8. The angles made by the straight line (ABC, Ac.) with the axes 

 have for their cosines A v'(A'- + B J + C 2 ), Ac. ; and the angle made 

 by the two straight lines (ABC, Ac.), (A'B'C', Ac.), has for its cosine 



AA' + BB' x CC' 



V (A? * S- + C 2 ) V (A 11 + B' a + C' 2 ) 



When the lines are perpendicular, AA' + BB' + CC' = ; and when 

 they are parallel, A, B, C are in the same proportion as A', B', C'. 



9. The angles made by the plane (ABC, Ac.) with the planes of yz, 



1 .ry, are A -f- \/(\* + B 2 + C 2 ), Ac. ; the cosine of the angle made 

 by the planes (ABC, Ac.) and (A'B'C', Ac.) with one another is as in 

 the last, and also the conditions of perpendicularity and parallelism. 



10. The plane (ABCaic) is at right angles to the line (ABCy*?r), 

 whatever a, b, c, p, q, r may be. And the plane (ABC, Ac.) is at right 

 angles to the line (A'B'C', Ac.) whenever A, B, C, are proportional to 

 A', B', C". But the plane (ABC, Ac.) is parallel to the line (A'B'C', Ac.) 

 when AA' + BB' + CC'= 0. 



11. The straight line (ABC4c) lies entirely in the plane 

 when 



P(o j>) + Q (b >j) + R (c 

 AP + BQ + CR = 0. 



r) = 



12. The intersection of the two planes (PQRyxyr) and (P'Q'R'yA/r') 

 is the straight line (ABCaJc) in which A, B, C. are proportional to 

 QR' RQ', RP' PR', PQ' QP' and 



P'(a 



l>) + Q (6 q) 



+ C (c r) = 

 C'(c r-) =0 



13. The intersection of the plane (PQRyjjr) and the straight line 

 (ABC4c) is at the point whose co-ordinates are a + Ar, 6 + Br, c + Cr, 

 where 



AP + BQ + CR 



14. Two straight lines (ABCa4c) and (A'B'C' a' 4V) do not, gene- 

 rally speaking, intersect at all ; their shortest distance is(BC' CB') 

 (a a') + Ac., divided by -/{(BC CB') 5 + Ac.}, and they meet 

 when the numerator of this fraction is nothing. The plane (PQR/x/r) 

 is parallel to both straight lines when P,Q,R, are proportional to 

 BC' CB', CA' AC 1 , AB' BA'. 



15. The equation of a plane which passes through the straight line 

 (ABC'a4c) and is perpendicular to the plane (PQRyjjr) is 



(BR CQ) (x a) + (CP AR) (y 4) 

 + (AQ BP) (z c)= 0. 



16. The perpendicular distance from the point (4c) to the plane 

 (PQRy>?r) is P( p) + Q(6 rj) + R(c r) divided by ^(P 2 + Q 2 + 

 R 2 ), independently of sign. And the perpendicular distance between 

 the parallel planes (PQHyjfyr) and (PQR/jVyV) is P(p /) + Ac., 

 divided by V(P 1 + Ac.) 



17. The perpendicular from the poiuti (ftnu) upon the straight hue 

 (ABCa4r) meets it in a point of which the co-ordinates are a + Ar, Ac., 

 where 



_ A(l a) + B(, M + C (H c) . 



A 2 "+ B 2 + C- 



aud the square of the length of that perpendicular is 



i 6) + C (n c)}t. 



+ C- 



J_)i + Ac - 



This is sufficient for a specimen of the method, and even for a 

 -nmmary of the most important propositions respecting the straight 

 line a'nd plane. On the good effect of symmetry it is hardly necessary 

 to make much remark ; not only are formula; more easily remembered, 

 since the whole can be formed from recollection of a part and more 

 ranily used, since an unsymmetrical result is an indication of error 

 but the actual expression can be shortened in type, of which the above 

 is a sufficient proof. It would have been impossible safely to write 

 down as many result* in twice the space, if the ordinary plan of nota- 

 tion had been adopted. The best mode of treating the ordinary forms 



for the straight line, such as y == ax + a, z =bx + j3, is to reduce them to 

 the preceding, thus : 



x _y a _ z (3 

 ~T~ ~~a~ ~T~ 



ALGORITHM, a corruption from the Arabic, the root being a word 

 which means calculation, or at least refers to calculation or reckoning. 

 When the Indian numerals were introduced from the East, this word 

 came with them, and the new figures were denominated (by Chaucer 

 for example) augrime (or algorithm) figures. The word is tolerably well 

 naturalised among the French mathematicians, as meaning the system 

 of notation : thus there is an algorithm of functions, and an algorithm 

 of the differential calculus, &c. It has also been used by English 

 writers, but our language does not want it; the word notation does 

 just as well. Hard words sometimes lead to misconception ; those who 

 attempt to interpret them find them not only spelt in different ways, 

 but with very different meanings. Daniel Penning (' Young Algebraist's 

 Companion,' 1750) tells us that some writers are so short and intricate, 

 that it is almost impossible to learn the algorithm from them, much 

 less the algorism. In a note he informs us that the first of these 

 hieroglyphics means the first principles, and the second their application 

 to practice. Our old mathematical dictionaries define the word 

 algorithm in probably its real imported sense, namely, as meaning tho 

 great rules of arithmetic. 



ALGUACIL, an officer in Spain, answering to the English bailiff. 

 The name is from the Arabic el-raal, or from the Hebrew verb yazal, 

 which means to catch. The alguacil mayor is a superior officer, whose 

 functions are the same as those of the common alguacil. The duty of 

 an alguacil is at present confined to the apprehension of criminals ; the 

 office of executioner being discharged by the verdugo. 



ALIAS, a term used in legal proceedings to denote a second or 

 further description of a person who has gone by two or more different 

 names. If the same person is known by the name of John Smith as 

 well as the name of John Thomson, he is described in legal language as 

 John Smith, allot diclus (otherwise called) John Thomson. 



ALIBI, a Latin term signifying "in another place," of frequent 

 occurrence in criminal courts. Thus, where a person charged with an 

 offence committed at a certain time and place, shows that he was 

 e/ictcltere at that time, he is said to prove an alibi. If true, this is 

 obviously the best proof of innocence ; but no kind of defence offers so 

 ready an opportunity for false evidence : and the setting lip an alibi is 

 therefore, in practice, always regarded with suspicion. 



ALIEN. An alien is one who is born out of the legiance (allegiance) 

 of the king. (Littleton, 198.) The word is derived from the Latin, 

 alien us ; but the word used by the English or other law writers in Latin 

 is alienegena. The condition of an alien, according to this definition, is 

 not determined by place, but by allegiance [ALLEGIANCE], for a man 

 may be born out of the realm of England, or without the dominions of 

 the king, and yet he may not be an alien. It is essential to alienage 

 that the birth of the individual occurred in a situation .and under 

 circumstances which gave to the sovereign of this country no claim to 

 his allegiance. 



The following instances will serve to illustrate the description of an 

 alien. The native subject of a foreign country continues to be an alien, 

 though the country afterwards becomes a part of the British dominions. 

 Thus, persons born in Scotland before the accession of James I., were 

 aliens in England even after that event ; but those who were born 

 afterwards were adjudged to be natural-born subjects. This question 

 was the subject of solemn discussion in the reign of that prince ; and 

 the reported judgment of the court has guided lawyers in all similar 

 controversies. Persons born in those parts of France which formerly 

 belonged to the crown of England, as Normandy, Guienne, and Gascony, 

 were not considered as aliens so long as they continued so annexed ; 

 and, upon the same principle, persons bom at this day in any of our 

 colonial possessions are considered native subjects. A man, born and 

 settled at Calais whilst it was in the possession of the English, lied to 

 Flanders with his wife, then pregnant ; and there, after the capture of 

 Calais by the French, had a son ; the issue was held to be no alien. 

 If, however, enemies invade the kingdom, and a child is born among 

 them, the child is an alien. 



The children of ambassadors, and other official residents in foreign 

 states, have always been held natives of the country which they repre- 

 sent and in whose service they are. This rule prevailed even at a time 

 when the law was stricter than it now is. It has been since so far 

 extended by various enactments, that all children born abroad, whose 

 fathers, or grandfathers on the father's side, were natural subjects, are 

 now deemed to be themselves natural-born subjects, unless their fathers 

 were liable to the penalties of treason or felony, or were in the service of a 

 prince at war with this country. (25 Ed. III. st. 2; 7Anne,c.5; 4 Geo. II. 

 c. 21 ; 13 Geo. III. c. 21.) The children of a British mother by an alien, 

 though aliens if born out of the king's allegiance, are now enabled to 

 take property by devise, purchase, or succession. (7 A 8 Viet. c. 66.) 



The children of aliens bom in England are, as a general rule, the 

 same as iiatural-boru subjects ; they are entitled to the same rights and 

 owe the same allegiance. 



It follows from the general principles of our law that a man may be 

 subject to a double and conflicting allegiance; for, though he may 

 become a citizen of another state (the United States of America, for 



