PROP.II.—Theorem.
PROP.II.—Theorem.
If two magnitudes of the same kind(a,b)be the same multiples of another(c)which two corresponding magnitudes(a′,b′)are of another(c′), then the sum of thetwo first is the same multiple of their submultiple which the sum of theircorresponding magnitudes is of their submultiple.
Dem.—Letmandnbe the multiples whichaandbare ofc.
Then we have
a = mc and a′ = mc ′, b = nc and b′ = nc′.
Therefore
(a + b) = (m +n )c, and (a′ + b′) = (m + n)c′.
Hencea+bis the same multiple ofcthata′+b′is ofc′.
This Proposition is evidently true for any number of multiples.
PROP.III.—Theorem.
PROP.III.—Theorem.
If two magnitudes(a,b)be equimultiples of two others(a′,b′); then anyequimultiples of the first magnitudes(a,b)will be also equimultiples of the secondmagnitudes(a′,b′).
Dem.—Letmdenote the multiples whicha,bare ofa′,b′; then we have
a = ma ′, b = mb ′.
Hence, multiplying each equation byn, we get
na = mna ′, nb = mnb ′.
Hence,na,nbare equimultiples ofa′,b′.
PROP.IV.—Theorem.
PROP.IV.—Theorem.
If four magnitudes be proportional, and if any equimultiples of the first and third betaken, and any other equimultiples of the second and fourth; then the multiple of thefirst:the multiple of the second::the multiple of the third:the multiple of thefourth.
Leta:b::c:d; thenma:nb::mc:nd.
Dem.—We havea:b::c:d(hyp.);
thereforema:nb::mc:nd.
PROP.V.—Theorem.
PROP.V.—Theorem.
If two magnitudes of the same kind(a,b)be the same multiples of another(c)which two corresponding magnitudes(a′,b′)are of another(c′), then thedifference of the two first is the same multiple of their submultiple(c), which thedifference of their corresponding magnitudes is of their submultiple(c′)(comparePropositionii.).
Dem.—Letmandnbe the multiples whichaandbare ofc.
Therefore (a−b) = (m−n)c, and (a′−b′) = (m−n)c′.Hencea−bis the samemultiple ofcthata′−b′is ofc′.
Cor.—Ifa−b=c,a′−b′=c′; for ifa−b=c,m−n= 1.
PROP.VI.—Theorem.
PROP.VI.—Theorem.
If a magnitude(a)be the same multiple of another(b), which a magnitude(a′)taken from the first is of a magnitude(b′)taken from the second, the remainder isthe same multiple of the remainder that the whole is of the whole (comparePropositioni.).
Dem.—Letmdenote the multiples which the magnitudesa,a′are ofb,b′; then we have
Prop.A.—Theorem(Simson).
Prop.A.—Theorem(Simson).
If two ratios be equal, then according as the antecedent of the first ratio is greaterthan, equal to, or less than its consequent, the antecedent of the second ratio isgreater than, equal to, or less than its consequent.
Dem.—Leta:b::c:d;thena- b=c d;and ifabe greater thanb,a b-is greater than unity; thereforec dis greater than unity, andcis greater thand.
In like manner, ifabe equal tob,cis equal tod, and if less, less.
Prop.B.—Theorem(Simson).If two ratios are equal their reciprocals are equal(invertendo).
Prop.B.—Theorem(Simson).If two ratios are equal their reciprocals are equal(invertendo).
Leta:b::c:d,thenb:a::d:c.
Prop.C.—Theorem(Simson).If the first of four magnitudes be the same multiple of the second which thethird is of the fourth, the first is to the second as the third is to the fourth.
Prop.C.—Theorem(Simson).If the first of four magnitudes be the same multiple of the second which thethird is of the fourth, the first is to the second as the third is to the fourth.
Leta=mb,c=md; thena:b::c:d.
Dem.—Sincea=mb, we havea -- b=m.
In like manner,c d=m; thereforea- b=-c d.
Prop.D.—Theorem(Simson).
Prop.D.—Theorem(Simson).
If the first be to the second as to the third is to the fourth, and if the first be amultiple or submultiple of the second, the third is the same multiple or submultiple ofthe fourth.
1. Leta:b::c:d, and letabe a multiple ofb, thencis the same multiple ofd.
Dem.—Leta=mb, thena b-=m;buta- b=-c d; thereforec d=m, andc=md.
2. Leta=b n-, thena b-=1 n-;
PROP.VII.—Theorem.1. Equal magnitudes have equal ratios to the same magnitude.2. The same magnitude has equal ratios to equal magnitudes.
PROP.VII.—Theorem.1. Equal magnitudes have equal ratios to the same magnitude.2. The same magnitude has equal ratios to equal magnitudes.
Letaandbbe equal magnitudes, andcany other magnitude.
Dem.—Sincea=b, dividing each byc, we have
a-= b; c c
thereforea:c::b:c.
Again, sincea=b, dividingcby each, we have
c-= c; a b
thereforec:a::c:b.
Observation.—2 follows at once from 1 by Proposition B.
PROP.VIII.—Theorem.
PROP.VIII.—Theorem.
1. Of two unequal magnitudes, the greater has a greater ratio to any third magnitudethan the less has;2. any third magnitude has a greater ratio to the less of two unequalmagnitudes than it has to the greater.
1. Letabe greater thanb, and letcbe any other magnitude of the same kind, then the ratioa:cis greater than the ratiob:c.
Dem.—Sinceais greater thanb, dividing each byc,
a-is greater than b; c c
therefore the ratioa:cis greater than the ratiob:c.
2. To prove that the ratioc:bis greater than the ratioc:a.
Dem.—Sincebis less thana, the quotient which is the result of dividing any magnitude bybis greater than the quotient which is got by dividing the same magnitude bya;
Hence the ratioc:bis greater than the ratioc:a.
PROP.IX.—Theorem.
PROP.IX.—Theorem.
Magnitudes which have equal ratios to the same magnitude are equal to one another;2. magnitudes to which the same magnitude has equal ratios are equal to oneanother.
1. Ifa:c::b:c, to provea=b.
Hence, multiplying each byc, we geta=b.
2. Ifc:a::c:b, to provea=b.
PROP.X.—Theorem.
PROP.X.—Theorem.
Of two unequal magnitudes, that which has the greater ratio to any third is thegreater of the two; and that to which any third has the greater ratio is the less of thetwo.
1. If the ratioa:cbe greater than the ratiob:c, to proveagreater thanb.
Dem.—Since the ratioa:cis greater than the ratiob:c,
a b -c is greater than c.
Hence, multiplying each byc, we getagreater thanb.
2. If the ratioc:bis greater than the ratioc:a, to provebis less thana.
Dem.—Since the ratioc:bis greater than the ratioc:a,
c c -b is greater than a.
Hence, multiplying each byc, we get
b less than a.
PROP.XI.—Theorem.Ratios that are equal to the same ratio are equal to one another.
PROP.XI.—Theorem.Ratios that are equal to the same ratio are equal to one another.
Leta:b::e:f, andc:d::e:f, to provea:b::c:d.
Dem.—Sincea:b::e:f,
PROP.XII.—Theorem.
PROP.XII.—Theorem.
If any number of ratios be equal to one another, any one of these equal ratios isequal to the ratio of the sum of all the antecedents to the sum of all theconsequents.
Let the ratiosa:b,c:d,e:f, be all equal to one another; it is required to prove that any of these ratios is equal to the ratioa+c+e:b+d+f.
Dem.—By hypotheses,
a-= c = e. b d f
Since these fractions are all equal, let their common value ber; then we have
Cor.—With the same hypotheses, ifl,m,nbe any three multipliers,a:b::la+mc+ne:lb+md+nf.