821 MULTIPLE, SUBMULTIPLE, MULTIPLICATION. 



MULTIPLE POINTS. 



tion requires that every question should contain a number of times 

 which another number, abstract or concrete, is to be repeated ; and 

 this number of times or repetitions cannot be a number of anything 

 else. Thus, to talk of multiplying 10 feet by 7 feet is a contradiction 

 in terms; if it mean that 10 feet is to be multiplied by 7, or that 7 

 repetitions of 10 feet are to be made, 10 feet is multiplied seven times) 

 not seven-feet times. But if it be meant that 10 feet is to be repeated 

 as often as 7 feet contains one foot, the question has three data, and 

 belongs to a class which will be considered in PROPORTION : it is in 

 fact a question of multiplication in which the number of repetitions is 

 not given, but is to be extracted from the result of a question in 

 division. On this subject see also RECTANGLE. 



It being now distinctly understood that a number of times or 

 repetitions is an essential element of every question of multiplication, 

 the extension is obvious by which a fraction of a time, or a fraction of 

 a repetition, is allowed to enter. Thus 12 + 12 + 12 + 6 is 12 repeated 

 three times and half a time, or 12 multiplied by 34 is 42. Similarly 

 24 + 24 + 24 + 1}, or 8J, is 2J taken 34 times. Up to this point there is 

 no violation of etymology ; the multiplicand (multiplicandwm, number 

 to be multiplied) is taken manifold times. "But [NUMBER] by the same 

 sort of extension of language by which 1, and even 0, are called 

 numbers, the mere exhibition of a multiplicand is called multiplying it 

 by one : thus 7 ia 7 taken once, or 7 multiplied by 1, though, etymologi- 

 cally, multiplication does not take place. Again, when the half of a 

 number is taken, or when it is taken half a time, it is said to be multi- 

 plied by 4 ; and so on for any other fraction. The advantage of such 

 extension in practice more than counterbalances its obvious defect, 

 namely, that the beginner must, without great care, be confused by the 

 application of a word in a sense diametrically opposed to its literal 

 meaning. 



The abbreviated process of multiplication rests upon the following 

 principles. (1.) If the parts of a number be multiplied, and the results 

 added together, the whole is multiplied ; thus 18, composed of 13 and 

 5, is taken 7 times by taking 13 and 5 each 7 times, and adding the 

 results. (2.) Multiplication by the parts of any number, and addition 

 of the results, is equivalent to multiplication by the whole : thus 13 

 taken 7 times and 8 times gives two products, the sum of which is 13 

 taken 7 + 8 or 15 times. (3.) Successive multiplication by two numbers 

 is equivalent to one multiplication by the product of these two num- 

 bers : thus 7 taken 3 times, and the result taken 4 times, is 7 taken as 

 many times as there are units in 4 times 3, or 12 times. (4.) If one 

 number be multiplied by another, the result is the same if the multi- 

 plicand and multiplier be changed : thus 7 times 8 is the same thing 

 as 8 tunes 7. (5.) In the decimal system, the annexing of one cipher 

 multiplies by 10, of two ciphers by 100, &c. 



The application of these principles requires that, in the decimal 

 system of notation, the products of all simple digits up to 9 times 9 

 should be remembered : this is usually done by learning what is called 

 the multiplication table, and this table, which is only absolutely neces- 

 sary up to 9 tunes 9, is usually committed to memory up to 12 times 

 12. This being supposed to be done, we shall now show the process of 

 multiplying 1234 by 5073. By (2.) we must take 1234, 5000 times, 

 70 times, and 3 times, and add the results. To take 1234, 3 times, we 

 subdivide it into 1000, 200, 30, and 4, each of which taken 3 times, 

 and the results added together, gives 



3000 



600 



90 



12 



3702 



from which process the rule for multiplying by a single figure may 

 easily be derived. The next step is to take 1234, 70 times, that is, 

 first 7 times, and the result 10 times. The full process is 



7000 



1400 



210 



28 



86380 



Similarly 1234 taken 6000 times, gives 6,170,000. Now put the three 

 results together, and add them ; which gives the first column follow- 

 ing .*- 



3702 1234 



86880 50*73 



6170000 702" 



6260082 8638 



6170 

 6260082 



The (second column shown the usual manner of performing the 

 operation, which we suppose the reader to know. We have given the 

 preceding detail that he may do what many have never done, namely, 

 compare the common process with the deduction of the result from 

 first principles. 



There are several abbreviations of multiplication which are very 

 valuable, but which are not commonly taught. 



1. Five times is half of ten times : to multiply by 5 annex a cipher 

 and divide by 2 : thus 76783 x 5 is most easily done as follows : 



2 ) 767830 

 383915 



2. Nine times is one less than ten times, so that 76783 x 9 can bo 

 found as follows : 



767830 

 76783 



691047 



This may be best done by subtracting every figure of the multipli- 

 cand from the preceding, carrying and borrowing where necessary, in 

 the usual way, on the supposition that the first figure is to be sub- 

 tracted from ten. Thus the process of multiplying 27293 by 9 is as 

 follows : 



27293 



245637 



3 from 10, 7, carry 1 ; 1 and 9 is 10, 10 from 13, 3, carry 1 ; 1 and 2 

 is 3, 3 from 9, 6 ; 7 from 12, 5, carry 1 ; 1 and 2 is 3, 3 from 7, 4 ; 

 from 2, 2. 



3. Eleven times is one more than ten times ; so that the addition 

 corresponding to the preceding subtraction must be made. Thus to 

 multiply 62781 by 11, proceed as follows : 



62781 

 690591 



Let 1 remain ; 1 and 8 is 9 ; 8 and 7 is 15, carry 1 ; 1 and 7 is 8 and 

 2 are 10, carry 1 ; 1 and 2 is 3 and 6 are 9 ; 6 and is 6. 



4. To multiply by any number from 12 to 19 inclusive, multiply by 

 the last figure, and to the carrying figure add the figure of the multi- 

 plicand which is just done with. Thus 



2734 



17 



46478 



7 times 4 is 28, carry 2, adding 4, or carry 6 ; 7 x 3 is 21, and 6 is 27, 

 carry (2 + 3 or) 5 ; 7 x 7 is 49 and 5 is 64, carry (5 + 7 or) 12 ; 7 x 2 is 

 14 and 12 is 26, carry (2 + 2 or) 4. 



5. To multiply by 25, annex two ciphers and divide by 4 : to 

 multiply by 125 annex three ciphers and divide by 8. 



6. In multiplying by a number of two figures, ending with 7 or 8, as 

 68, it may be advisable to take the multiplicand 70 times, and subtract 

 it twice, in preference to taking it 60 times, and adding it 8 times. 



The following rules are taken from the ' Risala Hisab.' (Taylor's 

 ' Liliwati,' Introduction, p. 17.) The first at least can easily be done 

 without paper. 



1. To multiply two numbers together, each of which is between 11 

 and 19 ; to the whole of one number add the units of the other; ten 

 times this, together with the product of the units' places, is the product 

 required. Thus, 17 times 14 is 21 times 10 and 28, or 238. 



2. To multiply two numbers together, each of which has only two 

 places : to the whole of one factor, multiplied by the tens of the other, 

 add the tens of that factor multiplied by the units of the other ; ten 

 times the result, together with the product of the units, is the product 

 required. Thus 76 x 38 is done as follows: 76x3 is 228, which in- 

 creased by 7 x 8, or 56, is 284, and 2840 increased by 48 is 2888, the 

 answer required. 



The multiplication of sums of money is facilitated by a process 

 known by the name of PRACTICE. 



The multiplication of fractions offers no difficulty when the extension 

 of the word multiplication, already described, is understood and 

 admitted. For instance, when we have to multiply | by > r , or to take 

 J 4-elevenths of a time, we see that being | [FRACTIONS], one-eleventh 

 of this is ^, and 4-elevenths is 3 ' 3 : whence the rule commonly given, 

 namely, multiply the numerators together for a numerator, and the 

 denominators fora denominator. In the multiplication of one decimal 

 fraction by another, as 1'23 by '018, the multiplication of the nume- 

 rators gives 123 x 18, or 2214, and that of the denominators 100 x 1000, 

 or 100,000. But a decimal fraction which has 100,000 for its denomi- 

 nator, has as many places as there are in both of the others together, 

 whose denominators are 100 and 1000. From this consideration the 

 common rule immediately follows. 



For a mechanical contrivance for expediting multiplication, see 

 NAPIER'S RODS. 



MULTIPLE POINTS. When two or more branches of a curve 

 pass through the same point, it is called a multiple point ; and this 

 whether the branches touch or cut one another. When two or more 

 branches intersect, it is obvious that as many distinct tangents may bo 

 drawn at the multiple point as there are branches which there intersect, 

 that is, for one value of the abscissa the differential co-efficient of the 

 ordinate may have more values than one. In most cases the points at 

 which this happens may be ascertained by inspection of the equation 

 of the curve. Thus in 



