17 



Nov. I, 1878.] ■** [Haupt. 



On the Scales of Maps. By L. M. Haupt, Prof, of Civil Engineering 

 Towne Scientific School. 



(Read before the American Philosophical Society, Nov. 1, 1878.) 



The object of this paper is to attempt if possible the removal of the am- 

 biguities existing in regard to the use of ratios as expressing the scales of 

 maps and degrees of slopes. 



Mathematical authorities are by no means agreed concerning the defini- 

 tion of the term ratio. They all maintain that it is an expression for the 

 relation existing between two quantities, but differ in the manner of deter- 

 mining the value of this relation ; some, as Peck, Davies, Robinson and 

 others, divide the second quantity or consequent by the first or antecedent ; 

 some, as Hutton, Alsop, Ray and others, divide the first by the second 

 quantity, and still a third class, as Chauvenet and others, define it as being 

 the quotient obtained by dividing one quantity by another. It may there- 

 fore be either | or £, 2,000,000, or ., lin } ,. 



The same confusion is found to exist in designating the scales of maps 

 and drawings. Some publishers and engineers giving it as so many miles, 

 or other denomination, to the inch ; others, as so many inches to the mile. 

 Again in expressing slopes many authorities use the tang, of the angle 

 made with the horizon, that is the height divided by the base (^-) while 

 others use the co-tang, or — . 



s a 



Now if we consider the manner of obtaining the value of the ratio in a 

 Geometrical Series or progression where no ambiguity exists, we find that 

 as each subsequent term is obtained from its predecessor by multiplying 

 by a constant factor called the ratio, so to obtain this factor or ratio we 

 must necessarily divide any term by the preceding one, and as this is the 

 only way in which its value can be determined, it establishes a rule which 

 should be made to apply to all other cases. 



We should then define a, ratio as being the expression for the value of the 

 relation existing between tiro quantities, and as obtained by dividing the 

 second by the first. 



The query then arises as to which quantity should be considered the first 

 and which the second, and we answer that the given material object to be 

 represented by the map or drawing is the Unit or measure with which the 

 other is to be compared. The map or drawing may be made of any con- 

 venient size, but the object to be represented is already fixed or constant 

 in its dimensions, and hence, as the unit or standard of comparison, should 

 be made the divisor, or denominator of the quantity expressing the ratio ; 

 it is consequently the antecedent or first quantity. To illustrate, let it be 

 required to determine the ratio between a map and its original in nature. 



The tract to be delineated in miniature is the fixed object, invariable in 

 size, which is to be compared with the plot representing it, and which may 

 be made larger or smaller according to circumstances, hence it become the 

 unit of comparison, and is the antecedent or first quantity, and as such the 

 denominator of the fraction expressing the ratio. The formula will then be: 



