IRREDUCIBLE CASE. 



IRREDUCIBLE CASE. 



The preceding enumeration points to one of the most remarkable 

 pages in the history of geometry. The question immediately arises, 

 had Euclid any substitute for algebra ? If not, how did he contrive to 

 pick out, from among an infinite number of orders of incommensurable 

 lines, the whole, and no more than the whole, of those which were 

 necessary to a complete discussion of all lines represented by V ( V<* 

 + V6), without one omission or one redundancy ? He had the power 

 of selection, for he himself has shown how to construct an infinite 

 number of other species, and an algebraist could easily point out many 

 more ways of adding to the subject, which could not have been beyond 

 Euclid. If it be said that a particular class of geometrical questions, 

 involving the preceding formula and that one only, pointed out the 

 various cases, it may be answered that no such completeness appears in 

 the thirteenth book, in which Euclid applies his theory of incommen- 

 surables. It is there proved that each of the segments of a line 

 divided in extreme and mean ratio is an apotome" that the side of an 

 equilateral pentagon inscribed in a circle is, relatively to the radius, 

 the irrational line called a lesser line, as is also the side of an icosahe- 

 dron inscribed in a sphere and that the side of a dodecahedron is an 

 apotome. The apotome then and the lesser line are the only ones 

 applied. 



It seems probable that the distinction of commensurable and 

 incommensurable, and even a notion of different species of incom- 

 i rabies, was familiar to the geometer before Euclid wrote : aud 

 this may be inferred, almost with certainty, from what is said by 

 Proclug. Had it been otherwise, we must suppose that the defini- 

 tions of the fifth book would have been accompanied by some little 

 account of their necessity, and also that the absolute determination of 

 two incommensurable magnitudes would not have been postponed till 

 the last proposition of the tenth book. But it is impossible to draw 

 any very positive conclusion on this subject. Owing to the loss of 

 Euclid's book on Fallacies [GEOMETBY], we are probably left without 

 those notions which he intended to be preliminary to the elements. 



The most conspicuous propositions of elementary geometry which 

 are applied in the tenth book are the 27th, 28th, aud 29th of the sixth 

 book, of which it may be useful to give the algebraical signification. 

 The first of these (the 27th) amounts to showing that Ixjp has its 

 greatest value when .r = l,and contains a limitation necessary to the 

 conditions of the two which follow. The 28th proposition is a solu- 

 tion of the equation ax x*=l>, upon a condition derived from the 

 preceding proposition, namely, that \a- shall exceed b. It might 

 appear more correct to say that the solution of this equation is one 

 particular case of the proposition, namely, where the given parallel- 

 ogram is a square ; but nevertheless the assertion applies equally to all 

 cases. Euclid however did not detect the two solutions of the 

 question ; though if the diagonal of a parallelogram in his construction 

 be produced to meet the production of a line which it does not cut, 

 the second solution may be readily obtained. This is a strong pre- 

 sumption against his having anything like algebra ; since it is almost 

 impossible to imagine that the propositions of the tenth book, deduced 

 from any algebra, however imperfect, could have been put together 

 with< ml the discovery of the second root. The remaining proposition 

 (the 29th) is equivalent to a solution of ax-\-x* = b : but the case of 

 if 1 ajc=b ia wanting, which is another argument against Euclid 

 having known any algebraical reasoning. 



IRREDUCIBLE CASE (that is, of cubic equation*), the common 

 name of a particular class of cubic equations, to which Cardan did not 

 succeed in applying his celebrated rule. Bombelli however showed 

 that the reason of this was the reality of all the three roots. The 

 following is the sketch both of the method and the difficulty. 

 [BuMBELi.i ; CARDAS ; TARTAOLIA, in BIOG. Div. THEORY OF EQUA- 

 TIONS ; NEGATIVE AND IMPOSSIBLE QUANTITIES.] 



Unity has three cube roots, 1, 4(1 y^3), and J(l + V-3), 

 of which, the product of the second and third is possible and equal to 

 unity. Calling these 1, r, and r 1 , it is next shown that o a has three 

 cube roots, namely, a, ra, and r'a. Now, let there be a cubic equation 

 (A, B, and c being real quantities) 



and, by the method explained in INVOLUTION and EVOLUTION, find 

 another equation which has each root greater than a root of the 

 preceding by JA. We have then 



= ... (1) 



P=B-JA S 

 Let x be v + w : then x'=v* + w* + Zmcx, and (1) becomes 



v*+w* + (3vw + T)x + <) = . . (2). 

 Determine v and w so that 



by which means (2), and therefore (1), is satisfied. This gives 



t^ (or 

 * (or 



-4 4 + V ( 

 -4 4 - 



V 



from which t and v> can be found. But as each of the two, if and w 9 , 

 bu three cube root* ; and as no reason yet appears for choosing one 



rather than another, it should seem as if the possible combinations by 

 which v + w might be made would be nine in number. But on looking 

 back we find the condition 3vw= P ; so that the product of v and w 

 must be a possible quantity. And since nothing but p 3 appears in v 

 and 10, the same values of v and w would appear whatever cube root of 

 p 1 might have appeared in the original equation : consequently the six 

 values of v + w which are now to be abandoned belong to the roots of 

 3?+vrx+ Q=0,andx 3 + p/ x+ Q = 0. If then;we signify by v and to 

 the real cube roots of z 3 and u 3 , the others are rv and r'v, rw and r'w ; 

 and the only combinations which satisfy the condition Svw + P=0 are 



v + w, rv + r'w, r'v + rw, 



which are the three roots of the equation (1) : to the exclusion of v + 

 rw, rv + w, r'v + r'w, the roots of a 3 + pr a: + Q = ; and v + r'w, r'v + w, 

 rv + rw, the roots of z 3 + rr'x + <j = 0. So far all is right, and the 

 algebraical solution is complete, and may be represented thus : let p 

 stand for any cube root of unity ; then, the three solutions of (1) are 

 contained in 



where V signifies the real cube'root. 



This is perfectly intelligible when jQ' + sV 3 is a positive quantity: 

 for if we call the real cube roots above mentioned K and L, we find for 

 the three roots of the equation, first, the possible root, K + L ; next, the 

 pair of impossible roots contained in the formula 



If we apply this to the equation a 3 9x 28 = 0, where P = 9, and 

 <J= 28, wejihall find K = 3, L = l, and the roots are 4, 2+ V 3> 

 and 2 V 3- But if it should happen that 'Q' + ^p 3 is negative 

 (which requires that p should be negative and ^p 3 numerically greater 

 than IQ-), we return to the original form of the solution, and find that 

 the roots of the equation are contained in the formula 



where -| j- 3 means any cube root, there being a tacit condition that 



the product of the two cube roots must be possible, v stands for 

 44, and w for the possible (though perhaps irrational) quantity 

 V( i<J' iV p3 )- Now it is shown in books of algebra that every cube 

 root of v + wV 1 i of the same form, say F + uV 1, and that the 

 corresponding cube root of v-wV^l is F ay^l. If, then, we 

 assume 



Iv + wV^lJ =f + a^^T, 



tv-wV^lj =F-G N / :r i> 

 we find by multiplication 



f P 



\ V' + w" / = p + a; 



and by addition of their cubes, and division by 2, 



V = F 3 3FG 8 , 



between which the elimination of o gives 



from which it would seem that we might find F, and then o. But on 

 examining this last equation we find it to be precisely that kind of 

 cubic equation about which the difficulty arose; for the p of this 

 equation is negative, being } %/ (v a + w 2 ), and the q Is J v; and 

 AP 3 , being s 'j (v 2 + W), is numerically greater than J Q 1 , or ri ',v 2 . 

 Whence this case is called irreducible ; for though, as will be shown 

 immediately, there are three possible values of the expression (3), yet 

 every direct algebraical attempt to find them leads to the same diffi- 

 culty in another form. 



If F and o could be determined, one value of (3) is 2 F|; and taking 

 the other cube roots, selecting only those pairs whose products are 

 possible, we find 



El> -i (1- VED (g-oy^T) 



for the other admissible values. These may be reduced to 



which are both possible. Consequently, the irreducible case of a cubic 

 equation is that in which the three roots are all possible. 



Let us apply the preceding to z s 2 Ix + 20 = 0. Here P= 21, 

 4=20, j4* + iV p3 = 243= 81 x 3. Hence the roots are contained in 



