268 



THE POPULAE EDUCATOE. 



horizontal line or height of the eye, B P base of picture. Let 

 a? be the point where the line commences, and from which 

 it retires ; and, to simplify the matter, let P s also be the V P. 

 (The pupil will remember that all retiring lines vanishing at the 

 point of sight, are lines going off at a right angle with our 

 position, or with the picture plane. We advise him to turn 

 to page 72, and read the perspective rules and axioms again.) 

 Make the distance from P s to r> equal to P s s p. Draw a line 

 from a? to P s, and on B P make the distance a? b 2 equal to the 

 given line a b ; draw a line from b 2 to D, which will cut off the 

 space a?c; a? c is then the perspective length of o,l. The 

 lengths of the retiring sides of planes are determined by the 

 same rule. Let it be required to draw a series of retiring 

 square slabs (Fig. 68). On the base of the picture B P, beginning 

 at a, set off any required number of divisions to represent the 

 length of the side of each slab ; from these points, a, I, c, etc., 

 draw lines to P s. Find the distance point, D, as in the last 

 case ; draw lines from b, c, d, etc., to D, cutting a P s in g h i. 

 From g, h, i draw lines parallel to the base of the picture, which 

 will complete the squares required ; for as a b of the first square 

 is parallel with our position, and touching the picture plane, 

 its true length is therefore shown, whilst ag is its retiring or 

 perspective length. 



Having now shown, as we promised, how the retiring 

 horizontal distances of objects may be faithfully represented 

 on paper, we will give some examples as subjects for exercises. 

 Fig. 69 is an example of a retiring row of posts, their distances 

 being purposely shown by the geometric method of the last two 

 problems. It is almost needless to direct the attention of the 

 pupil to the diminishing retiring spaces between the posts ; 

 however, he will see, as we have previously endeavoured to 

 make clear to him, that those retiring distances can be satis- 

 factorily proved. Fig. 70 is given as an exercise, including 

 many of the principles we have before explained viz., angular 

 perspective, horizontal retiring lines, inclined lines of the roofs, 

 and horizontal retiring distances, all of which the pupil, we 

 trust, will now be able to arrange for himself, and to find his 

 vanishing points. 



LESSONS IN ARITHMETIC. XVII. 



DECIMALS (continued). 

 24. To reduce a given Circulating Decimal to a Vulgar Fraction. 



Take the decimal -34567. 



Denote the true value of the equivalent fraction by /. Then 

 J = -34567567567 , the period 567 being supposed con- 

 tinued ad infinitum. 



If we multiply /by 100000, and also the decimal by 100000, 

 the results will still be equal. 



Hence 100000 / = 34567'567567567 



The decimal place being moved five places to the right, and the 

 period 567 being still continued ad infinitum on the right of the 

 decimal point as before. 



Similarly, 100 / = 34-567567567 



Now the difference of 100000 /and 100 f i.e., 99900 /must 

 be equal to the difference of the decimals to which they are 

 respectively equal. Now this difference is 34567 34, because 

 the infinite recurrence of the period '567 on the right of the 

 decimal point is the same in each decimal, and therefore vanishes 

 when the subtraction is performed. 



Hence 99900 / = 34567 - 31; 

 and/, the fraction required, = 3<5 T ~ 



Now observe carefully how each part of this fraction has 

 arisen. The numerator is obtained by writing down the figures 

 of the decimal as far as the end of the first period without the 

 decimal point, and then subtracting from the number so obtained 

 the figures which occur before the period, or, as we may call it, 

 the non-recurring part. The denominator 99900 arises from 

 subtracting 100 (i.e., 10 raised to the same power as the number 

 of figures in the non-recurring part) from 100000 {i.e., 10 raised 

 to the same power as there are figures in the non-recurring part 

 and period together). 



This subtraction will necessarily produce a number 99900, 

 Containing, that is to say, as many nines as there are figures in 



the period, and as many ciphers as there are figures in the non- 

 recurring part. 



25. It will be seen from the above detailed explanation of the 

 method by which the equivalent vulgar fraction may be deter- 

 mined, that an analogous method would apply to any circulating 

 decimal whatsoever. 



Hence we get the following 



Rule for reducing a Circulating Decimal to a Vulgar Fraction. 



Subtract the number formed by the figures of the non-recur. 

 ring part from the number formed by the figures taken to the 

 end of the first period, and set down this difference as a nume- 

 rator. Take as many nines as there are figures in the period, 

 and, annexing to them as many ciphers as there are figures in 

 the non-recurring part, set down the number so formed as a 

 denominator. 



26. We have proved the rule in the case of a mixed circulat- 

 ing decimal. The case of a pure circulating decimal is included 

 in it; for in a pure circulating decimal there is no non-recurring 

 part, and therefore nothing to be subtracted, and the denomina- 

 tor will consist wholly of nines, their number being equal to the 

 number of figures in the period. 



Thus 67 = f, -053 = JJ> a -. 



27. For the sake of clearness, however, we will perform the 

 process for a pure circulating decimal. Take "67, for instance. 



Let, as before, / = '676767 . . . . ; 



Then, 100 / = 67'676767 ..... , 

 and therefore subtracting, as in the previous case, 



99 / = 67, 

 Or,/ =11; 



and it is evident, from the way in which they arise, that the 

 number of nines in the denominator is equal to the number of 

 figures in the period. 



28. Of course, if there is an integral part in the original 

 decimal, that will remain unaltered, and the required answer 

 will be a mixed number, which may be reduced to an improper 

 fraction if necessary. 



EXAMPLE. 3'14i5. 



Taking the decimal part separately, '145 = 



Hence 3'l-d.5 = SJ^gJ = 3 u Vo(? expressed as an improper fraction. 

 Or it may be expressed as an improper fraction at once : 



s-uis = ^VfoV 14 = V..MM, 1 - 



The truth of this latter method may be established exactly in 

 the same way as the two cases we have already explained. 



29. The learner is recommended at first, in reducing circulat- 

 ing decimals to vulgar fractions, to perform the operation in the 

 way we have indicated in the examples already given i.e., by 

 multiplying by the requisite powers of 10, subtracting, etc. He 

 will thus better appreciate the truth of the rule which he will 

 afterwards employ. It is evident that the equivalent fractions 

 found by the rule will often not be in their lowest terms. 



EXERCISE 35. 



Eeduce to their equivalent vulgar fractions the following 

 decimals : 



1. -3. 



2. -03. 



3. -032. 



4. -523. 



30. Approximation. 

 places, etc. 



Decimals correct to a given number oj 



We have already remarked, that if we take only a limited 

 number of the figures of a decimal, we approach nearer and nearer 

 to the true result as we continue to take in more figures. 



We give an example, taken from De Morgan's " Arithmetic," 

 which shows this clearly. 



\ = -142857 a circulating decimal 



Now taking successively one, two, three, etc., figures of the 

 decimal, we have 



JL is less than \ by -f s which is less than T V 



f To"oo 

 $ ToSiTo 



etc. 



etc. 



Tcfeo- 

 ToSoo' 

 > 00*000- 

 Too^ooo 



etc. 



