IRRATIONAL QUANTITY. 



IRRATIONAL QUANTITY. 



988 



it either before or after the rays in each have converged to a point ; 

 the humours of the eye not permitting that convergence to take place 

 exactly on the membrane. A similar effect may be produced, to a 

 greater or less degree, in a telescope, in consequence of defects in the 

 object-glass, the irrationality of dispersion [DISPERSION], and diffraction, 

 from all which the image of a luminous point is not a mere point, 

 but has an apparent magnitude. 



Before the invention of telescopes, the apparent magnitudes of 

 celestial bodies were very erroneously estimated : thus, Tycho Brahe' 

 made the diameter of Venus twelve times and Kepler made it seven 

 times as great as it is now known to be. Telescopes do not entirely 

 remove the cause of such error, but, by increasing the diameters under 

 which the bodies are seen, without at the same time magnifying the 

 effect of irradiation, the error in the estimation of their apparent 

 magnitudes is proportionally diminished. 



It is a consequence of irradiation, that objects which are in reality 

 of equal magnitudes appear frequently to differ in size according to 

 their colour or to the quantity of light which falls upon them. Sir 

 William Herschel remarked (' Phil. Trans.,' 1783) that when a bright 

 circle was viewed together with a dark one on a bright ground, the 

 latter always appeared smaller than the other ; and, hi order to correct 

 the erroneous estimate of the magnitudes of the columns about temples 

 when they are seen against a bright ground, it appears that the ancients 

 made the thickness of the columns to increase proportionally to the 

 distance between them. The reason assigned for this practice by 

 Vitruvius (' De Architecture," lib. 3, cap. 2) is, that the columns with 

 wide intervals, being more surrounded by the air than those which are 

 closer, appear on that account to be more slender. It must be observed, 

 however, that the perceptions of magnitude depend partly on those of 

 distance ; and a contrary effect frequently takes place with objects 

 viewed against the sky when conceived to be more remote than they 

 really are. 



IRRATIONAL QUANTITY. The distinction between quantity La 

 general and number, or rather between the ratio of quantity to quantity, 

 and that of number to number, has begun to appear in the article 

 INCOMMENSURABLE, of which the present may be taken as a continuation. 

 It there appears that there are such things as magnitudes which are 

 not in the proportion of any one number to Any other ; though, if we 

 may use numbers as great as we please, we can find a pair which shall 

 be as nearly as we please in the ratio of any two given incommensurable 

 quantities. 



According to the modern use of the term irrational, it simply means 

 not expressible by a finite fraction. The word ratio, or its equivalent 

 \dyos, does not here mean reason, in the common sense of the word, 

 but mathematical proportion. A quantity whose ratio to the unit of 

 quantity cannot be expressed arithmetically, that is, by a whole number 

 or a fraction, is " inexpressible by an arithmetical ratio," or " arith- 

 metically irrational," abbreviated into " irrational." This explanation 

 is very important, since the student might otherwise be led to suppose 

 that irrational meant unreasonable, or absurd. Looking at the manner 

 in which the common meaning of the word irrational is fixed in our 

 minds, it would be well if the mathematical word were connected with 

 its cognate ratio, by being pronounced irrational. Suppose for example 

 that we have a geometrical problem which we solve by the application 

 of arithmetic, taking a certain line to be unity, and applying the funda- 

 mental principles explained in Ri:c TANGLE. Suppose the problem thus 

 reducible to the solution of .;- = 2, or the quantity sought is such a 

 fraction as multiplied by itself will give 2. The arithmetical answer is 

 very simple ; there is no such fraction. But is the problem therefore 

 impossible ? By no means ; for the line required must be the diagonal 

 of a square whose side is the linear unit. What then is the reason for 

 our not being able to produce an arithmetical solution ? Because the 

 ratio of the line sought to the linear unit given is not to be expressed 

 arithmetically, or is in the preceding sense irrational. The student 

 has now arrived at the point where he must be taught (if he have not 

 learnt it before) that common arithmetic is not the science of all ratios 

 or relative magnitudes, but only of the ratios or relative magnitudes of 

 those quantities which are made by putting together quantities which 

 are all equal to one another. The senses alone would never make this 

 distinction, and those who desire nothing more than sensible evidence 

 in their mathematical studies need not attend to it : unfortunately the 

 present bent of such pursuits tends to inexactness, not explicitly 

 avowed, but wearing the appearance of absolute rigour. 



The student who begins to extract the square root of numbers is 

 allowed to place the symbol of that process over numbers which do not 

 admit of its performance, as V 2, v 3, &c. These symbols are reasoned 

 on as if they represented fractions, and arithmetical deductions are 

 drawn ; but when it is required to reduce them to practice, then the 

 powibility of determining their arithmetical values is denied, and it is 

 implied that they have an existence which can only be approximately 

 represented. Thus, since 1-4142 multiplied by itself gives 2 very 

 nearly, it is said that 1'4142 is very nearly the square root of 2. This 

 method, which is indispensably necessary in practice, should not be 

 allowed in perfectly strict reasoning. It cannot be just to say that '2 

 ha* no square root, but that since fractions very near to 2 have square 

 therefore these square roots are very near to the non-existent 

 square root of 2. It is only in a properly extended arithmetic, which 

 by express agreement admits of extended symbol) of ratio, that it can 



be lawful to speak of the square root of 2. [RATIO.] Waiving this 

 point for the present, we proceed to further considerations, confining 

 ourselves to those irrational quantities which arise from taking the 

 square roots of numbers, but premising that similar remarks might be 

 made on cube, fourth, &c., roots. If we take the series of numbers 

 1, 2, 3, &c., and extract the square root of each, we thereby obtain (1.) 

 the original series 1, 2, 3, &c., by means of V 1, V 4, V 9, &c. ; (2.) a 

 series of multiples of V 2, namely, V 2, V 8, */ 18, &c., which are 

 V 2, 2 V 2, 3 V 2, &c. ; (3.) a similar series of multiples of V 3 ; and 

 so on ad infinitum. The primitive numbers are either prime numbers 

 or products of different prime numbers. Thus we have a series of 

 multiples of V (7 x 5), but not of V (7 x 7 x 5), since this last is 7 >/ 5, 

 and, with its multiples, is included in those of V 5. Any two quantities 

 in the same series are commensurables ; thus 7 V 10 and 12 V 10 are 

 in the proportion of 7 to 10, and have V 10 for a common measure : 

 but any two which are in different series are iucommensurables ; thus 

 V 10 and V 11 have no common measure whatsoever. And the sum 

 or difference of any two incommensurable quantities is incommensurable 

 with either ; thus we can form infinite sets of binomials, such as 

 V2+-/3, V10+VH, V19 V5, &c., no one of which shall be 

 commensurable with any other. 



The square root of any arithmetical fraction is commensurable with 

 that of the product of its numerator and denominator : thus V (J) is 

 $ V 15- And the reciprocal of any square root is commensurable with 

 that square root : thus l-i-V7 is $ V 7. Also the fraction made 

 by any two of the binomials just described is commensurable with the 

 product of some similar pair : thus 



If we take the square root of one of the preceding binomials, a 

 V ( V 3 + V 5) we have a new quantity, not commensurable with any 

 of those just mentioned, except only in certain cases pointed out by 

 the following theorem. Let a and 6 be two numbers, of which a, is 

 the greater : 



If a and a b be both square numbers, let a=p 1 , o l = j 5 , and we 

 have 



Though Euclid was not acquainted with any direct algebraical pro- 

 cess, yet he carried the distinction of incommensurable quantities to 

 the length of a complete subdivision of all the possible oases which can 

 be contained in the formula V ( Va + V )- We are induced to give 

 an account of his tenth book, because Tnere does not, to our knowledge, 

 exist any such thing in a form accessible to the student. Indeed, we 

 do not know where to find a description of its details in any form what- 

 soever. In old geometrical writings references to the classification of 

 this book are not unfrequently met with. If we take any given line to 

 represent the unit of length, and if a, b, c, &c., represent lines commen- 

 surable with this unit, arithmetically expressed, it is well known that 

 the most common geometry shows how to find the lines expressed by 

 \/, V > Ac. All such lines Euclid terms rational, all others irrational 

 (^IJTO? and 4*0705) ; and any area which being formed into a square has 

 a rational side, he calls a rational area ; that is, in fact, any area which 

 is commensurable (aiVjuerpos) with the square unit is rational. The 

 term for the square on a line is its power (Xvmfus), and from this comes 

 the algebraical use of the word power. Thus, when he says that two 

 lines are only commensurable in power, he means that the squares on 

 them are commensurable, but not the lines themselves. A mean, or 

 medial line (/itVoj), is the mean proportional between two iucommen- 

 Burable rational lines, and is such as can be represented in algebra by 

 V, where a is commensurable with the unit ; and a medial area is 

 the mean proportional between two rational areas, and its number of 

 square units may be represented by \/ <* 



A line which is made by putting together (ffdnBeau) two incommen- 

 surable rational lines is called a line of two names (4K Svo ofofjuirutf), or 

 a binomial line ; while one which is made by taking away (cupaipfffis 1 ) 

 the lesser of two incommensurable rational lines from the greater is 

 called an apotomd (iiroro^), literally, off-cut. The binomial therefore 

 has one of the forms a+ V b, and V a *Jb, while the apotome' has 

 one of the forms \/a \Jb,a\/b, \/b a. Six distinct species of 

 each sort of line are found, and in connection with each set of six is 

 another similar set, which a modern mathematician would describe as 

 composed of the square roots of the first set. But Euclid describes 

 the square roots as we should call them, previously to the lines them- 

 selves ; and in order to render this article more available to those who 

 look through the tenth book, we shall do the same. The whole 

 amounts to this : that, taking a given line as the unit and standard, 

 Euclid separates the lines represented by \J(\Ja + V*)> where a and 

 b are commensurable with the standard unit, into twenty-five distinct 

 classes, no one of which contains any lines commensurable with those 

 of any other class. The following enumeration contains the order in 

 which they make their appearance : a, b, &c., representing lines com- 

 mensurable with the standard unit; A, B, c, D, E, F, the six binomial 



