

LESSONS IN ENGLISH. 





We thus see that tin- dUbnnM between the ^"HnfO and the 



true value of tin 1 > .' nnuih diminishes. IntheoMeof 



mating decimal this difference becomes sero when we hare 



tiik.'ii all tin- ti'-ruivs iti. In the case of a circulating decimal, 



r actually becomes zero, but we can make it M small as 



we please by taking a sufficient number of decimal places. 



31. When a result ia required correct only to a certain num- 

 ber of decimal places, it ia better, aa wo have already explained 

 (Art. 14), to find one figure more of the result than i* actually 

 required, so as to ascertain whether this figure in greater or less 

 than 5. If it ia greater, we increase the figure in the last place 

 which is required in the result by 1. 



The following ia an example of a decimal continually approxi- 

 mated to in thin way, by taking successive figures, and increas- 

 ing, where necessary, the last figure by unity 

 Let 4-80169 be the decimal. The aucoessive approximations would be 



5, 4-9, 4-89, 4-892, 4 8917, 4*89169. 



Hero 5 ia nearer to the true value than 4 would bo. 

 4-9 4-8 



4-892 4-891 



4-8917 4-8916 



82. Operations in which circulating decimals occur are better 

 conducted by reducing the circulating decimals to their equiva- 

 lent vulgar fractions, if absolute accuracy is required. If an 

 approximate result is desired true to a certain number of decimal 

 places, then, in additions and subtractions, it will be sufficient 

 to take in two or three figures of the period beyond the number 

 of places required, and then add or subtract. For instance, in 



adding '4567 to '812468J correctly to 9 decimal places, we 

 should write the decimals as follows : 



45675675675 



31246894689 



769225703 



In all cases, however, where circulating decimals are involved 

 as multipliers or divisors, it will bo best to reduce them to their 

 equivalent vulgar fractions before performing the multiplications 

 or divisions, and then afterwards to reduce the resulting frac- 

 tions to decimals. 



EXERCISE 36. 



1. Write down the decimals containing respectively one, two, 

 three, four, five, and six decimal places which are the nearest 

 approximation to the decimals -67819473, '203781947. 



2. Find the value correctly to seven decimal places of tho 

 following expressions : 



1. 2-0127 + 89-3897 + '003704. 



2. 15-379 + 2-13159 + 18 + 70'2178 

 -* 5-34567. 



& 27-459 - 3-876439. 



4. 7-28705 - -378 + W34567. 



5. 85'6 *- 7-5. 



. 2} + 5-123 - 2-345. 

 o. 



2-39 + 3-28. 



EXERCISE 37. 

 1. Eeduce tho following decimals to vulgar fractions : 



3. 

 6. 

 18. 



123. 

 297. 



6. -72. 



7. -09. 



8. -045. 



9. -142857. 

 10. -076923. 



11. 



12. 

 13. 

 14. 

 15. 



16. 



8567923. 

 138. 

 53. 

 '5925. 



16. -583. 



17. '0227. 



18. -4745. 



19. -5925. 



20. -008497133. 



3. '27, '3, and -045. 



4. 4-32*., 6-4263, and '6. 



2. Change the following sets of decimals to similar and con- 

 terminous periods : 



L 6-814, 3-26, and -083*. 



2. 46-162, 5.26, 73'423, '486, and 12'5. 



3. Add together the following sets of decimals : 



1. 24-132 + 2'23 85-2*4 + 67'6. 



2. 328-126 + 81*23 + 5'624 + 61'6. 



3. 31-62 + 7-824 + 8'392* + -027. 



4. 462-31 + 60-82 + 71-164 + '35. 



5. 60-25 -t- -34 + 6-435 + -45 + 45'24. 



6. 9-814 + 1-5 + 87-26 + 0'83 * 124-09. 



7. 3-6 + 78-3476 + 735-3 + 375 + '27 + 187'4. 



8. 5391-357 + 72-38 + 187-21 + 4'2965 + 217-8496 -f 42-176 '523 



58-30048. 



9. *i62 + 154-09 + 2-93 + 97'28 + 3-769230 99-083 + 1'5 + -814. 



4. Subtract the greater from the less in the following sU of 

 decimals* 



1. 85-02 - 1878431 

 I. 47-3i - 84-7097. 

 8. 3-8504 - -0882*. 



4. 4'128 * 



5. 801 -0-400-75. 



6. 4-7824 - -87. 



7. UW - 1200.9 



5. Multiply together the following decimals 



1. 87-23 x -20. 



2. 'itS x -0. 



3. -245' x 7-3. 



4. 24-0 x 15 7. 



5. 48*23 x 10-13. 

 0. 8574'3 x 87'5. 



7. 9-973 x -8. 



8. 40440-54 x 4MM. 

 *. 7"7t x -W. 



6. Work tho following examples in division of 

 decimals; 



1. 319-28007112 



764-5. 



2. 18-56 + -3. 

 8. -6 -123. 



4. 2-297 + '2*97*. 



8. 750730-518 + ' 8. 24-081 + 191. 

 87'5. j . -36 + -25. 



fl. 54 * -15. -28-4375 + 20-87. 



7. 10-5160533 '. 1L 76*32 -Oisi 

 4-27. 12. 15-379 * 7-28705*. 



LESSONS IN ENGLISH. IX. 



DEBIT ATIUN : PREFIXES 



BEFORE proceeding further with these prefixes, we may now 

 expose a common error. It is generally thought that words 

 have several disconnected significations. Several sipilflnstitsas 

 many words have, but these significations are all allied one with 

 another, and they are allied one with another in such a way 

 that a genealogical connection runs through them alL I mean 

 that the second ensues from the first, and conducts to the third. 

 Tho meanings of words flow from a common source, like the 

 waters of a brook. That common source, or parent-signification, 

 is, in all cases, one that denotes some object of sense, for objects 

 of sense were named before other objects. Our first duty, then, is 

 to ascertain the primary meaning of a word. From that mean- 

 ing the other meanings flow, as by natural derivation. Those 

 secondary or derivative significations, then, can scarcely be 

 termed meanings ; they are not so much meanings as mn/Kfan- 

 tions of the primary import of the root. Certainly they are not 

 independent significations. Thus viewed, words have not two or 

 more senses, but in the several cases the one sense is varied and 

 modified. Even in instances in which opposite meanings are 

 connected with the same word, the filiation may be traced, aa 

 both Jacob and Esau sprang from the same stock. I will take- 

 an example in the word prevent. Prevent means both to guide 

 and to hinder, to lead to, and to debar from. The opposition 

 is sufficiently decided. Tet these two opposed meanings are 

 only modifications of the root-sense of the word. First I will 

 exhibit the diversity, and then explain it 

 Prevent, signifying to guide, aid forward : 

 " Prevent us, O Lord, by thy grace." " Book of Common Prayer." 

 " T- Love celestial, whose pmwMuiU mid 

 Forbids approaching 1 ill." Mattft. 



Prevent, signifying to hinder, obstruct: 



" Where our pretrnfvm ends, danger begins." Cart*. 



" Which, though it be a natural prtttntiv* to some erilm, yet without 

 either stop or moderation, must needs exhaust his spirits." fettf. 

 Wottonianae. 



"Physick IB either curative or preventive; preventive we call that 

 which preventeth sickness in the healthy." Brwc*, " FwJ^or Brron." 



"Prevent us, O Lord, by thy grace," means "aid us forward." 

 "Preventive of sickness," signifies that which causes m'olnae 

 not to 'come. There is the contrariety. Now for the explana- 

 tion. Prevent is made up of two Latin words, namely pne, 

 before, and venio, I come or go. Now. yon may go before a 

 person for two opposite purposes. Toa may go baore him in 

 order to guide, aid, and conduct him onward ; or you may go 

 before him to bar up his way, to hold him back, to prevent hia 

 advance. And as either of these two purposes is prominent in 

 the mind of the speaker, so the word is used by him to signify 

 to guide or to hinder. The proper meaning, then, of prevent is, 

 to come before: hence, 1, to guide, or, as a natural consequence, 

 2, to aid; or again, 1, to obstruct, and, as a natural consequence, 

 2, to stop, etc. And how the moral and spiritual imports come 



