OF QUANTI.Xr AND NUMBER. 



poses a unit, and the number of such frac- 

 tions contained in a unit, is denoted by the 

 denominator, or number below the line. 



Thusi+i+i=l. 



9. Definition. A number composed of 

 such simple fractions by continual addition, 

 may properly be termed a multiple fraction ; 

 the number of simple fractions composing 

 it is denoted by the upper figure or nu- 

 merator. 



In this sense |, -3, i, are multiple fractions, and j— 1, 

 5=Hi=l+i,ori.. 



10. Definition. Sucli quantities as are 

 expressible by the relations denoted by whole 

 numbers, or fractions, are called commen- 

 surable quantities. 



Scholium. All quantities may, in practice, be con- 

 sidered as commensurable, since all quantities are expres- 

 sible by numbers, either accurately, or with an error less 

 than any assignable quantity. 



11. Definitjon, Multiplication is the 

 adding together so many numbers equal to 

 the multiplicand as there are units in the 

 multiplier, into one sum, called the product. 



Scholium. Multiplication is expressed by an oblique 

 cross, by a point, or by simple apposition ; a x ''—a. t—a('. 



12. Definition. Division is the sub- 

 traction of a number from another as often 

 its it is contained in it; or the finding of that 

 quotient, which, when multiplied by a given 

 divisor, produces a given dividend. 



Scholium. Division is denoted by placing the divi- 

 dend before the sign -r- or :, and the divisor after it ; as 

 a—b^:a : I. 



13. Axiom. When no difl^erence can be 

 shown or imagined between two quantities, 

 they are equal. 



14. Axiom. Quantities equal to the same 

 quantity, are equal ^o each other. 



If azzl and c~?', then azzc. 



15. Axiom. If to equal quantities equal 

 quantities be added, the wholes will be equal. 



If azzb, then a+c'Zll+c; if a—lzzc, then adding b, 

 a—l+c; if a-fi— c=d, then adding c, a+i:=c+<2. 



16. Axiom. If from equal quantities 

 equal quantities be subtracted, the re- 

 mainders will be equal. 



If azzb, a—cizb — c, if a-f-ir:i-f-c, a~c. 



17. Axiom. If equal quantities be mul- 

 tiplied by equal numbers, the products will be 

 equal. 



If aZZ.b, aa^-Zb ; if a:^b : 3, 3a:z.b ; and if aZZb, 

 caZZcb. 



18. Axiom. If equal quantities be divided 

 by equal numbers, the quotients will be 

 equal. 



If 50^10/', azzai ; and if ca~cb, aZZb. 

 Scholium. Articles 16, 17, 18, might have been dedu- 

 ced from art. 15, but they are all easily admitted as axioms. 



19. Theorem. A multiple fraction is 

 equal to the quotient of the numerator di- 

 vided by the denominator. 



a a 1 , , " , 1 



Or, — r:a:t,for ■— = -T-.a (9); and (•.— -=:i.— -a 

 b b b b b 



(17); but ;•.-—=: 1 (8) ; and i.-:-.a:=i.a=a, therefore 



b.—r=.a (14), s^nAa-.b—— (12). 



Scholium. Hence — is a common symbol for a : b. 



b 



20. Theorem. A quantity multiplied by 

 a simple fraction, is equal to the same quan- 

 tity divided by its denominator. 



1 la a 



Or a. — "ZZa : b; for a. — =1 — (9), and -—ZZa :b(lii), 

 b b b V 



therefore a.-r-= — (l*)- 

 b b 



21. Theorem. A quantity divided by a 

 simple fraction, is equal to the same quantity 

 multiplied by its denominator. 



-= ab, for if a : -^=c,a=c.—{l2')=Z——c b 



Or a ; 



b ' ~ b^ ■ b 



1 



(20), and multiplying by b, ab — c — a : —. 



22. Theorem. Aquantity multiplied by a 

 multiple fraction is equal to the same quan- 

 tity multiplied by the numerator, and then di- 

 vided by the denominator. 



Or a.— — ah:c; for a.—-=.aji. —— at. — =:ai;f (aa). 

 c ' c c t 



