366 



THE rOPULAE EDUCATOE. 



ascertain its embouchure. After encountering many and great 

 dangers, they reached, the sea by the central or principal branch 

 of the Niger, which is the river called Nun, and which disem- 

 bogues itself into the Atlantic Ocean, between the Bight of 

 33cnin and the Bight of Biafra. The source of this river, as 

 determined by Laing, is at the foot of Mount Loma, in the 

 Kong Mountains. From this point to Timbuctoo its course was 

 known ; but the brothers Lander made it known from Boussa 

 to the ocean, and so solved a part of the geographical problem 

 which had so long existed without a satisfactory solution. 



LESSONS IK ARITHMETIC. XXI. 



CONCRETE OK COMMERCIAL ARITHMETIC. 



1. WE have hitherto been concerned with what are called 

 abstract numbers that is to say, numbers abstracted from their 

 connection with any special thing, object, or magnitude ; and 

 we have established all the principles connected with them 

 which are necessary to be known by the student of elementary 

 arithmetic. Wo now proceed to apply these principles to co- 

 crcte numbers that is to say, to numbers which indicate some 

 actual magnitude, object, or thing as, for instance, time, money, 

 length, etc. 



Theoretically, we are already in possession of principles which 

 enable us to perform any calculation with reference to any con- 

 crete number. Take length, for instance. Suppose that we fix 

 upon a certain length, and call it a mile. By means of this mile 

 we could measure any other length whatever. Foe by fractions 

 or decimals we could express any part or parts of a mile whatso- 

 ever ; we could add, subtract, multiply, or divide any number of 

 miles or parts of a mile, etc. etc. But it is manifest that, 

 although this could be done, great inconvenience would arise 

 from the cumbrous nature of the operations. In treating, for 

 instance, of fractional parts of a mile, it would bo often very 

 difficult to realise the length indicated. What idea would most 

 people have of -^ of a mile ? But if they were told that this 

 length is very nearly indeed equal to a foot, they would form a 

 very clear conception of the length. Hence, in measuring all 

 magnitudes, the method of subdivision has been employed. 

 Certain magnitudes have been fixed upon and named, and then 

 these again divided and subdivided, and names given to the 

 divisions, as convenience best suggested. 



Quantities expressed in this way by means of different sub- 

 divisions arc called compound quantities. Thus, a sum of money, 

 expressed in pounds, shillings, and pence, is a compound quan- 

 tity. The names of the various subdivisions are generally called 

 denominations. 



2. Accurate Standard or Unit. 



On proceeding to measure any magnitude or quantity, it is 

 evident that it is of the utmost importance to come to an exact 

 definition of some one fixed magnitude of the same kind, with 

 which we may compare all such magnitudes. Such a fixed 

 magnitude is called a standard. When this has been done, 

 then the standard can bo subdivided, or multiples of it can be 

 taken, as we please, and names given to the subdivisions or 

 multiples. 



The subdivisions which are employed in England in the 

 coinage and weights and measures are, as might be expected, 

 not founded upon one carefully prepared and philosophical 

 system, but have gradually grown up during long centuries, 

 having often been suggested by special convenience or local 

 usage. The subject, has of late received much attention, and 

 the possibility and advantage of establishing a uniform decimal 

 system of coinage, weights, and measures, have been discussed 

 with considerable warmth. 



On July 29th, 1864, an Act of Parliament was passed to 

 render permissive the use of a decimal system of weights and 

 measures called the "Metric System." Contracts r.nd transac- 

 tions, therefore, based on this system are now legal. We shall, 

 however, return to this subject hereafter. 



We proceed now to treat of the subdivisions of various con- 

 crete quantities which are now generally in use. 



MEASURES OP TIME. 



3. The time of the revolution of the earth in its orbit can be 

 shown by the calculations of astronomical science to be an 

 unvarying quantity, or, at any rate, to be subject to no appreci- 



able variation for an immense number of centuries. Now, it is 

 found that this time is 3G5-24224 (i.e., about 365-25, or 365) 

 mean solar days, a solar day being the interval which elapses 

 between noon and noon that is, between the times when the 

 sun is successively highest in the heavens.* 



The year is made to consist of 365 days i.e., about | of a day 

 less than the time of the revolution of the earth in its orbit. 

 To every fourth year (Bissextile or leap year, as it is called) one 

 day is added, and thus at the end of every four years the earth 

 is again very nearly in the same part of its orbit as it was at the 

 beginning of them. We say very nearly, because the earth 

 actually revolves round the sun in 365'24224 days, which is less 

 than 365i days by -00776 of a day. This error in excess amounts 

 to a day in about 128 years i.e., to very nearly 3 days in 4 

 centuries. Hence, to make our reckoning still more accurate, 

 we omit 3 days in 4 centuries ; and this is done by making the 

 year which completes every century not a leap year, except such 

 centuries as are divisible by 4. Thus A.D. 1700, 1800, and 1900 

 are not leap years, but A.D. 2000 i.e., the year completing the 

 twentieth century is a leap year. 



The establishment of the leap year is due to Julius Ceesar ; 

 that of the omission of the leap year three times in 400 years to 

 Pope Gregory XIII., who, in the year A.D. 1582, when the error 

 amounted to ten days, caused the ten days which followed 

 October 4th to be omitted in the reckoning. October 5th con- 

 sequently was called October 15th. 



This latter system, the New Style, as it is called, was not 

 adopted in England until A.D. 1752, when the difference between 

 this and the old mode of reckoning amounted to about eleven 

 days. Tho difference between the Old and Now Style amounts 

 at present to about twelve days. Thus any fixed day 

 Christmas Day and Lady Day, for instance Old Style, would 

 occur twelve days later than our present Christmas and Lady 

 Day. Russia is now the only country in Europe which retains 

 the Old Style. 



Having, then, thus established a fixed invariable standard 

 whereby to measure time, we are enabled to make any further 

 subdivisions for convenience. 



DIVISIONS OF TIME. 



= 1 minute, written thus, 1m., or 1'. 



= 1 hour 1 hr. 



= 1 day 1 d. 



= 1 week ,, 1 wk. 



= 1 common month 1 mo. 



} = i yt 



60 seconds 



60 minutes 



2t hours 



7 days 



4 weeks 



12 calendar months, or 

 305 days 



Any number of seconds arc written either thus 35", 23'', or 

 35 sec., 23 sec. 



It is better, however, in indicating time, to use the abbrevia- 

 tions sec. and min. for seconds and minutes, inasmuch as the 

 same names and the marks ' and " are used for certain divisions 

 of the circle (Art. 18). 



The Calendar months into which the year is divided do 

 not each contain the same number of days. The number in 

 each month, however, may be remembered by the following 

 lines: 



Thirty days lias September, 

 April, June, and November ; 

 February twenty-eight alone- 

 All the rest have thirty-one ; 

 But leap year comes one year in four, 

 And February then has one day more. 



MEASURES OF LENGTH. 



4. Having determined, as above explained, an exact measure 

 of time, we are enabled, curious as it may appear, to deduce 

 from it a fixed and invariable measure of length. We might, of 

 course, take any object a piece of metal, say and, giving to 

 its length a particular name, thus obtain a means of measuring 

 all other magnitudes. But this object, whatever it might be, 

 and however carefully preserved, would be liable to be lost, to 

 alteration from decay, variation of temperature, etc. It is 

 therefore very desirable to have some invariable and independent 



* A solar day is not actually of unvarying duration, but is at some 

 times in the year rather longer, and at others rather shorter, than its 

 average length. It is this average length of the solar day which is 

 called the mean solar day, and is divided into 24 hours. 



