MATHEMATICS. 23 



strictly speaking, the representations of magnitudes ; 

 they denote ratios, or abstract numbers. 



Unknown Quantities. 



A quantity that is unknown is only to be found 

 from the relations it bears to quantities that are 

 known. By reasoning on these relations we come 

 at last to one so simple, that the thing sought is 

 thereby determined. 



Quantities Less than Nothing. 



The phrase of " quantities less than nothing" has 

 been censured, yet there are correct ideas, which 

 correct language can hardly be made to express. 



There is a common paradox, that nothing divided 

 by nothing, may be equal to various finite quantities. 



Continued Quantity. 



There are proportions of continued quantity which 

 cannot be expressed by numbers, such as that be-* 

 tween the diagonal and side of a square, and many 

 others. 



Asymptotes. 



Right lines which approach nearer and nearer to 

 some curve, but which, if it extended for ever, would 

 never meet. 



Logarithms. / 



From Xoyoc and api^/zoc the index of the ratios 

 of numbers to one another. 



If, for example, the third term of the progression 

 were to be multiplied by the 7th, the product must 

 be the 10th; and if the 12th were to be divided by 

 the 4th, the quotient must be the 8th ; so that the 

 multiplication and division of such terms is reduced 

 to the addition and subtraction of the numbers that 

 indicated their places in the progression. 



