833 



ELIMINATION. 



ELIMINATION. 



834 



and Persephone, Iris with her veil inflated, Victory winged, the 

 Fates, and the head of one of the Horses of Night. Those toward 



[One of the Horses of Night.] 



the northern end, Nos. 99 to 106, are from the western pediment, the 

 allegory of which represented the Contest of Pallas Athene and 



Poseidon for the guardianship of Attica. With the exception of the 

 first figure, the IHssus, or River God, which is one of the finest Greek 

 statues remaining, they are mere fragments, and were broken in an 

 effort to remove them from the pediment after the siege of 1687. The 

 greater part were recovered by Lord Elgin by excavating below a 

 house which had been built out of the ruins beneath the pediment, 

 and which he had purchased. They consist of the torso of Cecrops, a 

 fragment of the face and the chest of Athene, the upper part of the 

 torso of Poseidon, a fragment of the Ericthonian Serpent, the torso of 

 Wingless Victory, and the lower part, or rather lap, of Leto. 



Between, and separating the sculptures of the two pediments (No. 

 119), stands an imperfect statue of a youth of the size of life, of 

 exquisite workmanship, supposed to have formed part of a group of 

 Diedalus and Icarus ; all that is certainly known of it is, that it came 

 in fragments, and Irom the Acropolis. 



Casts from the temple of Theseus, and others from the bas-reliefs 

 of the choragic monument of Lysicrates, are let into the eastern wall 

 of the Elgin Room, the first above ; the latter at the northern end, 

 below the Panathenaic frieze. 



Of the remainder of the Elgin collection, it may be sufficient to 

 name a few marbles of highest character, such as the colossal statue of 

 Dionysus, from the choragic monument of Thrasyllus, No. Ill, and 

 the Caryatide, from the Temple of Pandrosus, No. 128, engraved under 

 CARYATIDES, col. 639. Of architectural fragments, the pieces of frieze 



[Ilissus.] 



from the treasury of Atreus, at Myceme, Nos. 177, 180, are the most 

 ancient ; the capital and portion of a shaft, No. 112, give a notion of 

 the magnitude of the columns of the Parthenon ; and the capital 

 and shaft from the temple of Erectheus, No. 126, present the most 

 beautiful example of the Ionic order now known. The fragment 

 of the frieze of the temple of Erectheus is also one of the most valuable 

 parts of the collection. Among the inscriptions, the Sigean, written 

 in the most ancient Greek characters, and in the Boustrophedon 

 manner, No. 107, claims the first place, followed by numerous others 

 relating to the temples and buildings of Athens, some containing 

 decrees or treaties, and a few inventories of the treasurers of the 

 Parthenon : sepulchral inscriptions, urns, and stelao abound in the 

 collection ; and among these the epitaph on the warriors who fell at 

 PotioUea. (Thucyd. i. 63.) Votive bas-reliefs and offerings are also 

 preserved in it in large number. The last article we shall name is a 

 vessel of beautiful form, an urn of bronze ; it was found in a tumulus 

 situated on the road which leads from the Piraeus to the Salaminian 

 ferry and Eleusis, enclosed within the marble vase in which it now 

 stands, fad in it was a deposit of burnt bones, a lachrymatory of 

 alabaster, and a sprig of myrtle in gold. It is supposed, from 

 the last article, to have contained the ashes of some amatory poet. 



The possession of the Elgin collection has established a national 

 school of sculpture in our country, founded on the noblest models 

 which human art has ever produced. A tribute of gratitude is due to 

 the memory of the nobleman to whose exertions the nation is indebted 

 for it. If Lord Elgin had not removed them, the greater part would 

 long since have been totally destroyed. In the last siege of Athens 

 the Parthenon suffered additional damage. 



ELIMINATION. This word is from diminare, to drive out of 

 doors, and it is used in mathematics to signify the formation of an 

 equation or equations which do not contain a certain quantity, by 

 means of given equations which do contain that quantity. 



The simplest case of elimination, and therefore the best adapted 

 for the explanation of the term, is the following : if A be equal to B 

 and B 1*3 equal to c, than A is equal to c. In the result, B is eliminated. 

 Any operation of algebra may produce elimination. We give four 

 instances in which such a result is obtained by addition, subtraction, 

 multiplication, and division. 



1. ar + j/ = 12, x y=8; add,2:r=20: >/ is eliminated. 



2. x + z=a,y + z=6; subtract, x y a b ; i is eliminated. 

 ARTS A*P ci. Drv. vot,. nr. 



*Ji, , 



3. xy=a,^ = l; multiply, ^x"=al : y is eliminated. 



4. .ry = , yz=S; divide, - = r: y is eliminated. 



The process of elimination, in the case of more complicated 

 equations, becomes difficult and frequently impracticable. So much 

 is this subject connected with the theory of equations, that a treatise 

 on the first would be the largest portion of one on the second. The 

 writings on this subject are scattered, but most works on algebra 

 contain all that is absolutely necessary. 



Elimination is an essential part of very many mathematical pro- 

 cesses : and in the present article, we can only attempt to give a few 

 general notions, such as may help a student to connect different 

 portions of his algebraical reading. 



If the solution of equations were perfect, so also would be elimination. 

 Having 10 independent equations for instance, each containing x, y, 

 and z, if we could choose any three, and from these three find x, y, and 

 z, we might substitute these values in the remaining seven equations, 

 and thus form seven equations independent of x, y, and z. The 

 general rule is that from m equations can be formed m n equations 

 with n quantities eliminated. But not only must the equations be inde- 

 pendent of each other, but no two or more must be capable of what 

 we call simultaneous elimination. Suppose, for instance, x and y never 

 enter into a set of equations except in functions of , say, x" + if. If 

 then we make x- + y l =p and substitute, we have a set of equations 

 not containing either x or y, but containing y. If by means of one of 

 these we eliminate p from the rest, the process which does this, 

 applied to the original equations, would allow of our eliminating both 

 .r and y by one equation only. 



As to equations which are not purely algebraical, or which contain 

 more than powers or roots, or combinations of them, we cannot pretend 

 to say that there is any organised method of elimination existing, 

 except that of solution. For example, we can eliminate x between 

 the equations x = log (x + y), and sin (x + y) = t + x, because it so 

 happens that we can find y from the first, as in y = e* x; and this 

 value of y may be substituted in the second. But if the first equation 

 had been x + 2y=log (x + y), we could not have found either quantity 

 in finite terms of the other, from either equation. In such a case, we 

 must have recourse to infinite series; with these instruments, elimi- 



