RECURRING SEHI1S. 



portional to three concrete quantities, the first of which is a concrete 

 t this be the meaning of multiplication, then six yards and 

 three yards can be multiplied together; for as one yard is to three 

 yard. 10 11 six yards to eighteen yards, and eighteen yards is the 

 product But this product is a line, not an area. 



The pertinacity with which some writers still persist in calling the 

 product of two lines the area of a rectangle (not only as a practical 

 rule of mensuration, in which it is a desirable mode of expression but 

 m matters of reasoning) is the result of a long-continued habit formed 

 i the first instance by the study vt the Greek writers. For though 

 then do not confound the product with the area ; yet, on account of 

 the deficiencies of their arithmetical system, they used the area 

 " v th P roduot . ><1 gave the names of spaces to the results of 

 numbers. Thiu the product of two numbers was called plane that of 

 three equal numbers solid, that of two equal numbers a square that 

 of three equal numbers a cube, and the difference of two square 

 numbers a gnomon. To these we may add the titles of polygonal 

 pyramidal, 4c. numbers [NUVBERS, APPELLATIONS OF], and others 

 which it is needless to mention. All arithmetical propositions were 

 made to take the form of geometrical ones ; thus to multiply two 

 lumbers was to form the rectangle of two given lines; to divide one 

 number by another was, given the area of a rectangle and one of iU 

 sides, to find the other side. We have seen it stated that the word 

 "Va3oM (parabola) was sometimes used for quotient, and, it was said 

 Diophautus. We cannot find it there, though it may be used by the 

 scholiast, whom we have not examined. Adrianus Romanus in his 

 Jloga pro Archimede,' says that the Greeks use TrapapoM for 

 ision as well as pmciUt. But most certainly the explanation of the 

 meaning of a parabola, as applied to the well-known curve, comes from 

 some such signification. The term parabola means a thing laid near 

 to or by the side of another; for comparison, for instance, as in the 

 >mmon word parable, or for any other purpose. Now in the conic 

 ctiou in question the square on the ordinate being converted into a 

 ictangle one of whose sides is the abscissa, the remaining side (being 

 that which must be laid by the first side before the figure ran be drawn 

 the rap<&o*4) is always of the same length. If modern writers had 

 applied the term parabola to this remaining side, they would probably 

 nave called the curve an isoparabolic section ; but the Greeks, who 

 the curve in which a certain defect is always in the same pro- 

 'rtion to the whole by the simple name of defect (ellipse), and one 

 having .the same sort of excess by the simple name of excess (hyperbola) 



! the isoparabolic curve simply a parabola. 



f the geometrical system which pervaded the Greek arithmetic we 



lave permanently retained only the words square and cube ; rectangle 



as frequently used for product, but rarely at present. These 



rdsare the causes of much confusion to students who begin to 



apply anthmetic to geometry. Thus in algebra the square of a sum is 



equal t:> the sum of the squares of the two numbers, together with 



twice their product. In geometry the square or the sum of two lines 



. the sum of the squares on the lines, together with twice 



iir rectangle. Those who are not made to see clearly the distinction 



. propositions confound them together. A sufficient distinc- 



aon might be made by a little variation in phraseology : speak of the 



juare on a line, and the square of a number. Thus 49 is the square 



: erect two perpendiculars each equal to A B at the two extremi- 



>f A B, and joining their other extremities completes the square on 



t is already customary to speak of the rectangle whose con- 



tiguous sides are A B and A c, as the rectangle under A B and A c 



The second book of Euclid is devoted to the properties of the 



iangle, as they arise from subdivision into other rectangles Some 



persons advocate what is called the arithmetical proof of thes 



propositions, namely, the substitution of the analogous properties of 



tSSTS. "* f rectan S lJar Pace s . This question must be 



that'" me manner M *! IBt of I'KOPOHTION, and the renu:: ks 



! j *PP ?' " a 'l pairs of lines were commensurable no I 

 objection could be token against the rigour of the substitution ; 'but 

 MB a theory of incommensurable^ and a modification of- the 

 utionof multiplication to suit them, be formally introduced, the 

 method of Luchd sound, and the substitute for it unsound ; though 

 proper enough for the adoption of those who, as explained in the article 

 ited, only wish to become mathematicians to a certain number of 

 decimal places. 

 RECTIFICATION. [DISTILLATION! 



5CTIFICATION means the finding of a straight line equal in 



ratiut Ma* "ued * CU " e * RO] ' ^ * . anal 8 OUB * the term quad" 



considered to be known when a square eqtri to tt to ^WHtedT.0 



exhibited a " ar " " a " traight n e e( l" al * J * * 



Of the celebrated problem of the rectification of the circle we have 



enough raider QUADRATURE OF THE CIRCLE, in which article it 



SI ^'^IS l hC l tl0n J FT* th .r-*"*"": o that the latte 



RECUSANTS. 



<l iN/)mtnm, in which the coefficient* <!, a,, *c. can each be Mpiumd 

 y means of certain preceding coefficients and constants in one uniform 

 manner; and it is usual to consider only such series as will admit of a 



" >>r ne ' n Which coefflciento enter ) : 



1 + x + 4if + 



+ 43* + 



+ . . . . 



followsthe U nearl,wa.= 3^, + a_,(4 = 3.I + 1,13 = 8.4 + 1, 



, th t tk^ ,' " What " common 'y <*"l recurring 

 scries, though the following 



in which a, = a;., + a'_j.is equally recurring, according to the de- 

 ,^' , ;" recurrence alluded " i not that of ternn. but of 



S, h ), 'n g **?**'' a " d !t would be desirable that the 



os which are ,,snaly called recurring should bt ,.,,rring 



called' recurVin '" " ^^ recurrence ( of '^) ^ould b 



Ky.'ry linearly recurring series is the development of an algebraic 

 function with a ntional and integral numerator and denominatoV and 

 ever}- such function can be developed into a linearly recurring series. 

 have"' 8CTIea mentioned > wh '<='' = 80.-I +*.-,, we 



a. x* = 3d, 3? + a,, x 

 a, x = 3a, x* + , j 

 o. X* = 3o, 3* + a, x>, &c. 



8 ~ - 



s; 



_ 



+ (a, - 3o ) x 



- :;. c 



We have here the value of any series in which this law of reci 



; .-eva.18 for all terms after the second; and it cannot prevail 

 before, since two terms must exist before a third can be expS 

 In the case we chose, =!,,= 1, whence the function of which 

 the series was the development is (1 - 2.r) *. (1 _ 3 + "') 

 'neraUy, a linear recurring series having the law of recurrence 



" = Pi .-, + P, a. 

 is the development of the function 



+ Pt <*-< 



A,, + A, .r + A, a;' + ---- + A ,_, tf 

 1 ~ Pi x - P 2 & - ----- p, x' 



where A, = B..A, = , _ Pi Oat Aj = flj _ ^ ^ _ ft ^ 



from which the inverse theorem may easily be derived, namely, that 

 A +A l a; + A.,a? + .. . . + A( _, ^.f-i 



B + BI X + 3, & + . . . . ~B,.i X '-' + B * 



+ &c., in which tl 



0, 



S a._ 2 



and the terms up to a,_i are determined by 



B O a () , A, = B O a, + BI a , 

 A,-, = B a f _ + B a 



= B O o a + BI o, + B a 



NO. [DISTILLING. 

 KK'TOR. UKOTOKY. []; 

 ^ RKCURRIXU SERIES. By a recurring series is meant one of the 



a + a. x + a. 



+ o,, x> + a, ^ + 



.. 7,. uuo w iuuuu to tne solution < 



ruig s ^ D f . dlSeTeace> >: Some use may thus be made of .. 



^^p^^^^^i.tel i ^i^^ 



The most simple mode of finding the law of the terms of a 

 ring series is by the solution of the equation of d^renl Xh 

 expresses the relation of the coefficients. This may be "erified ht 



.mnot be traced higher than the 16th century. By the 1 E!" 

 c. 2, all persons having no lawful exc.se are to resort to their 



