IRRATIONAL QUANTITY. 



IRRATIONAL QUANTITY. 



Urn* ; \ ' A. s n. Ac., those connected with them ; r, r, w, x, r, I, the 

 six apeitntn*; \/t'. s v - Ac., those connected witli them. 



It if. however, to be noticed, that Kuclid !.>( not uw the term unit 

 but supposes a rational line, t.> whieh In- makes reference. Thus when 

 be mentions in one place a rational line and a fourth I. in. .ini.il. he mean* 

 that the fourth binmni.il shall be related to that rational line In the 

 Mine manner a* our following definition will connect it with the modern 

 phrase, the standard unit. 



i, 4, ftr., line* commensurable with the unit. 



(21. \ <i. v'*. * c -p Mi" 8 commensurable in pwer with the unit. 

 These two heads include the rational lines. 



(8). Va. V*. *c-. medial line*, described by Euclid aa lines equal in 

 power to the rectangle of incommensurable rational lines. 



( ). N /A "" the form V+ V*. A binomial line generally. Thin 

 case contains all the six hereafter described and numbered, for which 

 reason the numbering is here left blank. There is a proposition which 

 we should now enunciate by saying that the square root of a bino- 

 mial of the first species (A) is one or other, and may be either, of tb,e 

 aix binomial*, 



(4). V B has t" 8 f onn ( v' a + V*) "\ X where rAx w & square number. 

 It is the first species of line composed of two medtals, or a first 

 bi-medial, and is compounded of two medials, which moke a rational 

 rectangle (eWi~a in tin piavr w/x6n)V 



(5). V c h tk* ' orm (V + V*)^*, where abx Is not a square 

 number. It u the second Bjwcies of line composed of two medials, or 

 a second bi-medial, and is compounded of two medial lines, which 

 make a medial rectangle (W!a in Svo n<auv ttvrlpa). 



(6). V D has the * orm V( + V* ) + V( V*)> where a* 6 is not a 

 square. It is described by Euclid as composed of two xtraight lines, 

 incommensurable in power, whose squares together make a rational 

 space, but whose rectangle is a medial space, and is called by him a 

 greater line (iv6tM lulfav). 



(7). V* has the form V ( V + V&)+ V(V V*). w herea ftis a 

 square. It is described by Euclid as composed of two straight lines 

 incommensurable in powef, whose squares together make a medial 

 pace, but whose rectangle is a rational space ; and it is called by him 

 ' a line in power making a rational and a medial space " (cv0cia pirriy 

 leal fUaor 8i/f <mT) > . The nomenclature is not here quite correct, for the 

 preceding line, called a greater line, is also a line in power equal to a 

 rational and medial space. 



(8). V has tne form V( V + V*) + V ( V" V*), where 06 i* 

 not square. It is described by Euclid as composed of two lines incom- 

 mensurable in power, making both the sum of their squares and their 

 rectangle medial spaces incommensurable with one another; and it is 

 called " a line in power equal to two medial spaces " (4v9t!a Siio yult 



We now come to the description of the six binomial lines them- 

 selves. 



(9). A has the form a-rl + 2v / (J). It is described by Euclid as 

 baring the greater term commensurable with the standard unit, and 

 more in power than the less by the square of a line commensurable 

 with itself in length ; and it is called the Bret binomial line (<60<?a in 

 tin ofo/uiruy wpvni). 



(10). B has the form (a +6 + 2 v^o*) ) V*i V*'|" lrt ' "'<.'' in a square. It 

 is described as differing from the first binomial only in having the 

 leaser term commensurable with the standard unit ; and is the second 

 binomial line. 



(11). c hasthefonn(o-r& + 2V("''> ) \>. where '../ is not a square. 

 It liiffers from the two preceding only in neither term being commen- 

 surable with the standard unit. 



In the first three binomial lines, reduced to the form \?p + V 1 ?, 

 ^(py) u commensurable with \Jp, the greater term: in the last 

 three these two are incommensurable. 



(12). n has the form a + \l>, where a is greater than ^fb, and a 1 6 

 U not a square. Euclid describes it as having the greater term more 

 in power than the lens by the square of a line incommensurable to 

 iUcIf in length, the same greater term being commensurable with the 

 standard unit : it is the fourth binomial line. 



(13). t has the form a + v''', where y& is greater than a, and 6 o* 

 U not a square. It is described as differing from the preceding by 

 having the less term commensurable with the standard unit : It is the 

 fifth binomial line. 



(14). r has the form Va+ V* where a-'- i* not a square. It 

 is described as differing from the two preceding by neither term 

 being commensurable with the standard unit ; and u the sixth binomial 



We now come to the lines derived from the apotomm, ntid 

 wards to the apotonuc themselves. The descriptions might ). 

 shortened by allusion to the corresponding binomial lines, but this 

 would impede the speedy reference to the complete meaning of any 

 one term. 



( ). V has the form \/a~ \/4. An tpolnmf generally. The 

 numbering is left blank, as this clam of lines is afterwards subdivided. 

 A proposition is proved, which we should now cnuiieiatv b\ 

 bat the square root of an apototno' of the first kind is one or other, 

 and may be any of the six apotonuc. 



(16}. V has the form ( \/a V6) 4/x where abx is a square- 

 Euclid describes it a* the difference of two medial lines which are 



commensurable in power, and whose rectangle is a rational space. 

 He calls it the first kind of apotom< of a medial line (M'""> 



(18). Vw has the form ( V vWV*, where abx 1s not a square. 

 It is described as differing from the former only in the medial line* 

 containing a medial space, and i* the second apotome' of a medial line. 



(17). V h ne form V (o + V*) - V(o - V *) where o - 6 is 

 not a square. Knelid describes it a* the difference of two straight 

 lines incommensurable in power, the sum of whose squares is ra- 

 tional, and their rectangle medial : and he calls it a lessor line (iuStai 



(18). V T has the form V(v'<+ V*) V(V V*) where 0-6 is 

 a square. It is described as the preceding, except that the sum 

 squares is medial, and the rectangle rational : and Euclid call* it " a 

 line which with a rational space mokes a medial space " (tiitiia iirrk 

 fi-nrov ftirrov fb tt\ov TOIOWO) meaning that a certain rational space 

 added to the square on it makes a whole space uliirli is medial. 

 There i not here the defect of nomenclature mentioned in ( 7 1. : 

 preceding line here con only be called " a line which with a medial 

 space makes a rational space." 



(19). V 1" the form V( V+ V*) V( V"- V*) where 

 not a square. It is described by Euclid as the difference, of tw. 

 incommensurable in power, having the sum of their squares an 

 rectangle both medial : and it is called " a line which with a i 

 space makes a medial space" (tiiStat utrii niaov niaov T& 8Aor *t>iou<ra). 



The six apotomic now follow, all in the form \/u \/6; '" tli 

 three \/(a 6) is commensurable with \'a, in the second three, i ' 

 meiiHiirable. And V ! called the whole, but ^4 is called tin- 

 fitted or adapted line. 



tr has the form a + i 2V(oi). The whole is commensurable 

 with the standard unit, and exceeds the adapted line in power by the 

 square of a line commensurable with itself. Euclid calls this a first 

 apotome'. 



(21). v has the form (o+i 2V(o*) ) V* where abx is a square, 

 Described as the preceding, except that only the adapted line is corn- 

 mensural ilc with the standard unit; and is the second apot 



(22). w has the form (0+6 2 V(a*)) V*> where aloe is not a square. 

 Here neither the whole nor the adapted line is commensuralil. 

 the standard unit ; this is the third apotorae'. 



(23). x has the form a v* where a*b is not a square. Knelid 

 describes it by saying that the whole is commensurable with the 

 standard unit, and exceeds the adapted line in power by the square of 

 a line incommensurable with itself; and calls it the fourth apot. 

 (24). T has the form V* o whore 60* is not a square. Dei- 

 OH the lost, excepting that only the adapted line is commensurable with 

 the standard unit : it is the fifth apotomd. 



(25). z has the form ^b *Ja where 60 is not a square. It differs 

 from the two preceding by neither term being commensurable \\ith 

 the standard unit : and is the sixth apotomr. 



Besides obtaining this classification, Knelid ]irove, firstly, that every 

 one of these species is distinct from every other, and that every line 

 which is commensurable with a line of any one specie* is itself a line 

 of the game species. He shows also how to find lines of every specie* 

 in which he directly applies the theory of numbers obtained in the 

 seventh, eighth, and ninth books. He also demonstrates tl 

 straight line can belong to one species in two different ways : proving, 

 for example, an equivalent to the following, that \a + \/li, if the 

 be incommensurable, cannot be equal to V* + vV, where x differs 

 from o, and y from li : which lie expresses thus : " a binomial line is 

 divided into it names (or terms) in one point only." He then proves 

 that the lines which we have denoted by -\/A, v' B . *c., are derived 

 from A, B, Ac., in the manner which justifies our notation. l-\, r in- 

 M.-meM, "if a space be contained by a rational and a fourtli l.in..nii.il 

 line, the line equal in power to the space is the irrational line ca 

 greater line." Now, r representing a rational line, a fourth binomial 

 derived from it has the form a+ *Jb where a is commensurable with c, 

 and greater than \ '>. and \'(a* 4) is not commensurable with r, 

 His proposition then amounts to this, that V (co-rcy*) has the form 



where 2* is a rational space (or the number 2z commensurable with c"), 

 and x* y is an irrational space, or that number is incommensurable 

 with r'. This involves the algebraical proposition, that the square 

 root of *o + cV*i> 



ind In showing the Identity of the forms, Euclid arrives at the 

 manner df deriving one from the other. He also shows, in two pro- 

 ...ition*. that the form \/ (a + V') gives either a binomial lino, or (4), 

 ii). or (7) of the preceding enumeration, and that V( v'+ V*) gives 

 itlier i :" i or (8). In three more he shows that V (o v4) gives cither 

 an \potomf or (17) of the enumeration, that V( V* ) gives 

 (18) or (18), and that V ( V- V*) givef either <!> " ili'i. He 

 'urther shows the equivalent of the following algebraical j im- 

 position : 



1 _ Va-ry't 



V" V5 o 4 



