ALGEBRA. 



ALGEBRAIC GEOMETRY. 



for all integer values of m: and thin not merely from equation () but 

 from comparison of what co 1 + sin 1 . .' 1 must mean in the 

 definition* jirior to that of the ex|x>nent, with the complete exponential 



meaning of, . Heooe we must have , the base of the logarithm*, 

 connected with the angular unit by the equation 

 v'-l 

 - cosl+ainl. </ \ 



and any tM and angular unit which satisfy thU condition will ilo. 

 Tin- niott Dimple way f lining thi< in to take t = 271828 . . . tut usual, 

 and the angular unit >udi that there shall be w or :>-141.V.i .... units 

 in two right angles : but if any one should prefer ;/ (271828 . . . ) for 

 a baae, and x 3-14159 . . . for the number of uniU in two right angle*, 

 he might get into trouble, l.ut m.t into error. 



28. Another difficulty of the sort which arisen when the result seems 

 above the means employed in the demonstration is, that we have a 

 complete ayatem of trigonometry ready for demonstration by mere 

 algebraical mechanism, without casting a further thought nn the mean- 

 ing of the ymbols COB and sin 8, or more on those of tan 9. cot 9, 

 ate 0, cosec t, than to make them, by definition, severally mean, sin 

 -}- tat t, cos t -i- sin I, 1 4- cos t, and 1 -=- sin 0. All we have d..ne 

 with cos and sin 9 is to take them into our system as expressing the 

 numerical values of the projections of a unit inclined at the angle upon 

 the axes of length ami direction. \Ve have not even directly used 

 in* + cos' = 1. But it should be remembered, that in proving 

 generally A (B + C) rrAB+ AC, we have used the property of rimilar 

 fgxrti ; an assumption which is quite sufficient to be the basis of the 

 demonstration of Euclid I. 47, un which sin' + cos' 0=1 depends. 

 And those who attentively read Euclid see that he does, in the sixth 

 book, prove L 47 over again, without any use of it, in showing that all 

 similar figures described on the three sides of a right-angled triangle 

 have the two smaller together equal to the greater. [HvruTiii 



In the above definition of sin and cos it is clear that COM ( 0) = 

 co and sin ( 9) = - sin 0, whence the first of the following. Wins,' 

 universally true, gives the second : 



I S - 1 _ 

 = cos + uin . \/ 1 



* ^l 



t = cost sin 9 . / 1 



Whence we have ', 



(cos + sin / 1) (cos sin 0. J 1) = 

 Now ", by definition, lias Ox A, or x 1, or 0, for its logometer, and 

 U therefore (1, 0) or 1. And the first side is, by common application 

 of rule*, cos* + sin* 0. If any one, instead of merely applying rules 

 to the equation 



(cos0 + sin0. / 1) (cos sin*. / 1) = 1, 

 should proceed to demonstrate the rules upon this instance, he would, 

 in a circuitous way, be led to a perfect geometrical demonstration of 

 sin* -f cos 1 = 1. 



29. No equation of this system, which hitherto admits of expression, 

 presents any difficulty as to the meaning of its sides, or any combination 

 of symbols for which the meaning is to be found by interpretation. 

 Perhaps one of the most remarkable results of the ancient system of 

 algebra is the equation 



log(-l) 

 * = Sl 



Some algebraical writers have stated that 1 has neither square 

 root nor logarithm, and without further warning, have afterwards 

 made the non-existent logarithm, divided by the non-existent square 

 root, give the ratio of the circumference of a circle to its diameter. 

 Others have given fair warning that, in using what they called imagi- 

 nary quantities, they were appealing, more or leas, to experience ; seeing 

 that operations no conducted always M to truth, when, by the ultimate 

 disappearance of */ l.the result could be tntsrarttod. They were 

 content to use such an equation as T </ 1 = log ( 1) as an instru- 

 ment of which the power was known, though its mechanism was con- 

 cealed. In the complete system it is visibly obvious that * / 1 ia 

 one of the logometen of 1, or of (1, w). 



30. In ordinary working, there is no objection to dropping the dis- 

 tinction between the logometer and the logarithm, there being no 

 difference between the two in "iterations. 



For fuller account of the whole of this system, we refer to De Mor- 

 gan, ' Trigonometry and Double Algebra.' 



That this system will finally be introduced into elementary instruction 

 we entertain n doubt whatever. But how soon will this take place ? 

 The school-books hardly yet teach the interpretation of the negative 

 quantity ; so that there is but little hope of the speedy success of the 

 complete system. But truth must conquer at but ; and the respect 

 with which the memory will be preserved of the mathematicians who 

 were neither discouraged by the difficulties nor rendered incredulous 

 by the mysteries of the ancient system, will not protect from ridicule 

 those who shall obstinately refuse to see light because there was once 

 darkness, or shall wilfully continue in the imperfect system from which 

 those who wish the exact sciences to be in all their parts the perfection 

 of reason are most glad to be delivered. With respect to those parts of 

 the doctrine of aeries, and of the integral calculus, which still present 

 difficulties, though of a different character from those here treated, the 



lesson taught by the victory over what was , l ; u , /lfl aMr. 



which many hauls have required many years to gain, is Never refuse 

 to examine, and to continue in the examination of, all consequences of 

 the symbolic laws of algebra : there ia erery reason to hope that the 

 symbols are always right, even though the views of their explanation 

 may require correction. 



ALGEBRAIC. An expression is said to be atgrbralf, as distinguished 

 from tramirr(lrt,tl. when its number of terms is finite, and w |j. 

 term contains only addition, subtraction, multiplication, divisi. 

 extraction of roots, the exponents of which are given. Tl, 

 infinite series, as well as expressions containing 



log. r, a", uin r, cos r, 4c. 



though used in algebra, in the widest sense of the word, are improperly 

 said to be not alytbraie, but tramctndtntal. Similarly, a curve is said 

 to be algebraic when its equation (CURVE) contain*, n.'. t ransceudenUl 

 quantities. 



ALGEBRAIC GEOMETRY. A name given to the application of 

 algebra to the solution of geometrical problem*. For the piinci|..d 

 points of interest connected with it, see ABSCISSA, ORDINAL 

 mi DI SATES, Cl'RVK, t'riiVATriti:, Egi-ATiox, TANGENT. The regular 

 treatises in this, ax in every other cose, cannot be dispensed with by 

 any of our readers who are desirous of acquiring it But in this art id,', 

 M in the last, we are led to write by the jnucity of elementary 

 works which explain a new and useful modification of the n, 

 viewing a port of the subject. 



In geometry of two dimensions, the number of co-ordinates is not so 

 great as to make symmetrical disposition extremely necessary 

 outset. We should not, for instance, gain much by forming the 

 equation of the straight line thus : ay + fcr + c = instead of y = 

 a.r + 4 ; in fact, the writers who have preferred the symmetrical course 

 have rather overloaded themselves with symbols, to an extent which 

 makes the burden thus ini]>osed on the memory greater than that from 

 which symmetry relieves it But it is not so in geometry of three 

 dimensions. There are here three co-odinates to every point ; and, as it 

 happens, symmetry, even when obtained by augmentation of the num- 

 ber of symbols, is found to be, on the whole, on assistance in the 

 remembrance of formulae. Accordingly, the equations of the plane 

 and straight line, and formula: connected with them, are expressed 

 with great convenience in the manner ..f which the following is a sum- 

 mary; the first use of which, as far as we know, is found in the 

 celebrated work of Mains, "Thcorie de la double Refraction,' 4to. 

 Paris, 1810. 



1. In arranging expressions containing three quantities or different 

 sets of three, symmetry requires that no one letter shall 

 at the beginning of one expression without appearing at the end ..f 

 another ; and that in every set of expressions which are formed by 

 combination of pairs, an interchange of any two letters, or of the 

 corresponding pairs taken out of different sets, shall reproduce the 

 same set of expressions in another order, or at most with different 

 signs. Thus, a b, b c, c n, are symmetrical, and so are ay Zir, 

 bz t>i, rr a:. 



1. When there are two equations of the form n.r + by + rz = 0, './ + 

 b'y + e'z = 0, the quantities are not given, but their proportion 

 and r, y, and z are in the proportion of 



be' cb', ca' ac 1 , a.V ba' 

 8. An expression of the form 



(ay bx? + (bz ey)"- + (ex nj)' 

 is identical with 



4. The form of the equation of a plane is A-r + By + Cr + P = ; the 

 common form 2 = A* + By + C is unsymmetrical. The pl.me which 

 passes through the point whose co-ordinates are p, g, r, which call the 

 [Kiint (p q r), is 



Afc 7.) + B(y 7) + C(z r) = 



The plane which passes through the three jwints (;> 9 r), (;>' <,' r'\, 

 (l>" V r"), is 



{(?'-?) (r' r)- (r> r) (9" g)} (* )>) + &c. 



0: 



one term of which is enough; symmetry will point out the rest. 

 Remember the order pq, qr, rp. 



5. When such symmetrical expressions as alf in'.lr' rli',nt' 

 at 1 , are to be constructed, the best way is to write down , li. r. <i, and 

 n',1', r', a', under one another, and to calculate from the left to the 

 end, without returning to the beginning. Thus from 



8138 

 2402 

 we have 



30, 6, 42 



6. The most convenient form of the equations of a straight li nc j a 

 as follows : Let (ale) be a point on the straight line, and let A B C be 

 any three quantities proportional to the cosines of the angles which 

 the straight line makes with the directions of x, y, c. Then the 

 equations of the straight line are 



x a _ y I _ : e 

 ~~ ~~~ ~~ 



