OF QUANTITY AND NyMBER. 



23. Theorem, A quantity divided by a 

 mnltiple fraction is equal to the same quan- 

 tity multiplied by the denominator, and di- 

 vided by the numerator. 



b h / 1 \ , 1 



Or a: — zzac:l-, for a: — r: a:l b. — ]—a:b: — — 

 c c \ c / c 



a: l.c[l\), ZZac -.1. 



Scholium. A beginner may perhaps render these de- 

 monstrations more intelligible by substituting any numbers 

 at pleasure for the characters. For example, the demon- 

 stration of the first theorem may be written thus. Twelve 

 fourths, y, are equal to 12 divided by 4 ; for, by the defi- 

 nition of a multiple fraction, 'Jnia.i, and multiplying 

 these equals by 4, 4.^:^4.12.^ ; but by the definition of a 

 simple fraction 4.i3:i, therefore iAi-^^xt, whence 4.u.= 

 12, and by the definition of division, 12:4z:l^. But, in 

 fact, the proposition is too evident to admit much demon- 

 strative confirmation. 



24. Theorem. A positive number or 

 quantity being multiplied by a positive one, 

 the product is positive. 



For the repeated addition of a positive quantity must 

 make the result actually greater. What is true of numbers, 

 may practically be affirmed of quantities in general (lo). 



25. Theorem. A negative number or 

 quantity being multiplied by a positive one, 

 the product is negative. 



For since adding a negative quantity is equivalent to sub- 

 tracting a positive one, the more of such quantities that are 

 added, the greater will the whole diminution be, and the 

 sum of the whole, or the product, must be negative. 



26. Theorem. A negative number or 

 quantity being multiplied by a negative one, 

 the product is positive. 



Oi —a.—h=:al: For a — l-=.—aT]{ib): that is, when 

 the positive quantity a is multiplied by the negative h, the 

 product indicates that a must be subtracted as often as there 

 are units inb: but when a is negative, its subtraction is 

 equivalent to the addition of an equal positive number ; 

 therefore in this case an equal positive number must be add- 

 ed as often as there are units in I. 



27. Definition. If the quotients of two 

 pairs of numbers are equal, the numbers are 

 proportional, and the first is to the second, 

 it-s the third to the fourth ; and any quantities 



expressed by such numbers, are also pro- 

 portional. 



If a : lyZlc : d, a is to b is c to d, or a, : b : : c ■ d. 



28. Theorem. Of four proportionals, the 

 product of the extremes is equal to that of 

 the means. 



Since a:b~c : d, a : b. bd::Zc : d. bd. (l7), or adzzcb. 



29. Theorem. If the product of the ex- 

 tremes of four numbers is equal to that of the 

 means, the numbers are proportional. 



If adZZjcb, ad : bd^Zcb : bd (is), and a : bzZc : d ; also ad i 

 cdZZcb : cd, and a : c~t : d. 



30. Theorem. Four proportionals are 

 proportional alternately. 



If a : b::c:d, adZZbc (as), therefore a:c::b:d (29). 



31. Theorem. Four proportionals are 

 proportional by inversion. 



1( a : b : : c : d, ad'ZZhc, ad : ac':^bc : ac, and d : c^Zfc : a. 



32. Theorem. Four proportionals are 

 proportional by composition. 



If a: b:: c: d, a+b : b: : c + d : d ; fo; since ad^bc, 

 ad+bd::ibc-{-bd (15), or {a+b). d::z(c+d). b, therefore 

 a+b:b::c-^-d:d (29). 



33. Theorem. Four proportionals are 

 proportional by division. 



If ab::c:d, a — b:b::c — d:d; for since ad^bc, 

 ad—bd:=.bc—bd (16), (a—b). d=:(c— d). i, and a—b:b:: 

 c — d : d (29). 



34. -Theorem. If any number of quan- 

 tities are proportional, the sum of the ante- 

 cedents is in the same ratio to the sum of the 

 consequents. 



11 a . b : : c : d, a: b : : a+c : i+d ; for since adZZbc, 

 ab+adz:ab-\-bc, a.[b+d):^b.(a+c), and a: b : : a+c:b+d 



(29). 



35. Theorem. If any number of ante- 

 cedents and any number of consequents be 

 added together, the ratio of the sums will be 

 less than the greatest of the single ratios, 

 when those ratios are unequal. 



a c a4-c a 



b d b+d b 



{ori[-=-, e>c. 



,«-f e a + c , ^ , a a-\-c 



and— -->— — (34) ; consequently —> ——-.The same 

 b+a b+d b t'-J-ci 



demonstration may be extended to any number of ratios. 



