July 19, 1888] 



NATURE 



281 



to the Landerkunde. Although fully recognizing the difficulty 

 of having lectures in all the above-named subjects especially 

 appropriated to the needs of geography, the Council suggest 

 that privat-docents might supply the new want. But if this 

 is found to be impossible, they advise that the students who 

 wish to take either geography or anthropology as their specialty 

 should be left to select in the above-named group of sciences 

 those subjects which would best suit them. Students might 

 thus take any one of the three chief directions opened to the 

 geographer — namely, that of the geologist-geographer, the 

 biologist-geographer, or the anthropologist-geographer. 



THE MULTIPLICATION AND DIVISION OF 

 CONCRETE QUANTITIES* 



T HAVE recently been laying stress on the fact that the funda- 

 mental equations of mechanics and physics express relations 

 among quantities, and are independent of the mode of measure- 

 ment of such quantities ; much as one may say that two 

 lengths are equal without inquiring whether they are going to be 

 measured in feet or metres ; and indeed, even though one may 

 be measured in feet and the other in metres. Such a case is, of 

 course, very simple, but in following out the idea, and applying 

 it to other equations, we are led to the consideration of products 

 and quotients of concrete quantities, and it is evident that there 

 should be some general method of interpreting such products 

 and quotients in a reasonable and simple manner. To indicate 

 such a method is the object of the present paper. 



For example, I want to justify the following definition, and its 

 consequences : Average velocity is proportional to the distance 

 travelled and inversely proportional to the time taken, and is 

 measured by the distance divided by the time, or, in symbols, 

 v — s -r- /. As a consequence of. this, the distance travelled is 

 equal to the average velocity multiplied by the time, or s — vt. 

 The following examples will serve to illustrate what I mean : — 



(i.) If a man walks 16 miles in 4 hours, his average speed is 



16 miles 1 mile -, . ..■ u 1 I mile 



— = 4 x — . = 4 nules an hour, the symbol — 



4 hours I hour I hour 



denoting a speed of a mile an hour, in accordance with the 



definition. 



Similarly, , or shortly, — - , denotes a velocity of 



I second sec. 



a foot per second. The convenience of this notation is that it 

 enables us to represent velocities algebraically, and to change 

 from one mcde of measurement to another without destroying 

 the equation. 



16 miles _ 4 miles _ 4 x 1760 x 3 feet _ ft. 



4 hours 1 hour 60 x 60 seconds. sec. 



Thus !2J 



= 5 "9 feet per second, 

 (ii.) The distance travelled in 40 minutes by a person walking 



at the rate of 4A miles an hour = — ^ x 40 minutes = 



4^ miles 



I hour 



2 = 3 miles. 



Such concrete equations are used by a considerable number 

 of people, I believe, but I have not seen any attempt at a 

 general method of interpreting the concrete products and 

 quotients involved. 



Now, I think I cannot do better by way of clearing the 

 ground before us than quote what Prof. Chrystal says in his 

 " Algebra" about multiplication and division. He begins by 

 saying that multiplication originally signified mere abbreviation 

 of addition ; and then (on p. 12) he says : — 



" Even in arithmetic the operation of multiplication is 

 extended to cases which cannot by any stretch of language be 

 brought under the original definition, and it becomes important 

 to inquire what is common to the different operations thus com- 

 prehended under one symbol. The answer to this question, 

 which has at different times greatly perplexed inquirers into the 

 first principles of algebra, is simply that what is common is the 

 formal laws of operation [the associative, commutative, and dis- 

 tributive laws]. These alone define the fundamental operations 

 of addition, multiplication, and division, and anything further 



1 Paper read at the General Meeting of the Association for the Improve- 

 ment of Geometrical Teaching, on January 14, 1888, by A. Lodge, Cooper's 

 Hill, Staines. 



that appears in any particular case is merely a matter of some 

 interpretation, arithmetical or other, that is given to a symbolical 

 result, demonstrably in accordance with the laws of symbolical 

 operation." 



" Division, for the purposes of algebra, is best defined as the 

 inverse operation to multiplication." 



I will begin by considering instances, and then go on to the 

 general case. 



A product of a number and a concrete quantity presents sio 

 difficulty. All that is necessary is to define that the orchrf of 

 stating the product shall not alter its meaning — that is, tKat the 

 commutative law shall hold — that, 



e.g., 2xi foot = 1 foot x 2 = 2 feet. 



The distributive law is satisfied ; thus, 



2 feet -f 3 feet — (2 + 3) feet 

 = 5 feet. 



In interpreting the meaning of the product of two concrete 

 quantities, we have to be careful that in the interpretation 

 nothing shall violate the laws of numerical multiplication ; i.e. 

 if any numerical factors occur, they must be able to be multiplied 

 in the ordinary way, and placed before the final concrete pro- 

 duct, which must, of course, represent something which varies 

 directly with both quantities. 



Thus 4 feet x 2 yards must be equal to 8 x 1 foot x 1 yard. 



Now a rectangle, whose sides are 4 feet and 2 yards, is eight 

 times the rectangle whose sides are I foot and I yard, so that, 

 if we define the product of two lengths as representing a rect- 

 angle whose sides are these lengths respectively, we are not 

 violating any multiplication law as regards the numerical multi- 

 pliers ; and we can compare one such rectangle with any other 

 whose sides are of different lengths, by ordinary multiplication 

 and division among such numbers as arise, and by interpretation 

 of the concrete products in accordance with the definition. 



Thus, 4 feet x 2 yards = 8x1 foot x 1 yard, 

 = 24 x 1 foot x 1 foot, 

 = 24 square feet, 

 = 24 x 12 inches x 12 inches, 

 — 3456 square inches, 

 &c. 



Here we have applied the commutative law so as to bring 

 the numerical factors together for multiplication, and have in- 

 terpreted the lemaining concrete products in accordance with 

 the definition. 



The general result is that ab = a0 . a'b', if a — aa', and b = 0b', 

 i.e. a rectangle whose sides are a, b is a0 times a rectangle with 

 sides a', b', if a = aa', and b = 0b'. 



From this example I think we can see that a concrete product 

 may properly be used to represent any quantity that varies 

 directly as the several concrete factors, and that, being so repre- 

 sented, it may, by use of the ordinary rules of multiplication, 

 be compared with any other concrete product of the same kind ; 

 that is to say, that, generally, ab = a.0 . a'b', if a = aa', and 

 b — 0b', where a, are numerical factors, and a, a' are different 

 amounts of one kind of quantity, and b, b' of another kind. 



Similarly, a concrete quotient may be used to represent a 

 quantity which varies directly as the concrete numerator and 

 inversely as the concrete denominator, and may, by the ordinary 

 rules of multiplication and division, be compared with any other 

 quantity of the same kind. 



Indeed, I may go further and assert that a concrete product 

 or quotient (the latter including the former) must, if it is to 

 have any meaning at all, represent a quantity varying directly as 

 the concrete factors in the numerator and inversely as those in 

 the denominator, and that the general use of such representation 

 is for comparison of the complex quantity with a standard 

 of the same kind. Or, generally, we may say it should be 

 used, whenever we wish, in our work, to give as full and explicit 

 a representation to the complex quantity as possible. 



The operation of multiplying [and dividing] concretes may be 

 separated into two parts : the formation of the products, and the 

 simplification of them ; and this latter process may be again 

 considered in two parts : the simplification of the numerical 

 factors, i.e. ordinary multiplication and division, and the simpli- 

 fication of the concrete factors, i.e. cancelling where possible, 

 and, finally, interpretation. 



