PROP.XIII.—Theorem.If two ratios are equal, and if one of them be greater than any third ratio, thenthe other is also greater than that third ratio.
PROP.XIII.—Theorem.If two ratios are equal, and if one of them be greater than any third ratio, thenthe other is also greater than that third ratio.
Ifa:b::c:d, but the ratio ofc:dgreater than the ratio ofe:f; then the ratio ofa:bis greater than the ratio ofe:f.
Dem.—Since the ratio ofc:dis greater than the ratio ofe:f,
c is greater than e-. d f
or the ratio ofa:bis greater than the ratio ofe:f.
PROP.XIV.—Theorem.
PROP.XIV.—Theorem.
If two ratios be equal, then, according as the antecedent of the first ratio is greaterthan, equal to, or less than the antecedent of the second, the consequent of the first isgreater than, equal to, or less than the consequent of the second.
Leta:b::c:d; then ifabe greater thanc,bis greater thand; if equal, equal; if less, less.
and multiplying each byb cwe get
Hence, Proposition [A], ifabe greater thanc,bis greater thand; if equal, equal; and if less, less.
PROP.XV.—Theorem.Magnitudes have the same ratio which all equimultiples of them have.
PROP.XV.—Theorem.Magnitudes have the same ratio which all equimultiples of them have.
Leta,bbe two magnitudes, then the ratioa:bis equal to the ratioma:mb.
Dem.—The ratioa:b=a- b, and the ratio ofma:mb=ma- mb; but since the value of a fraction is not altered by multiplying its numerator and denominator by the same number,
PROP.XVI—Theorem.If four magnitudes of the same kind be proportionals they are alsoproportionals by alternation(alternando).
PROP.XVI—Theorem.If four magnitudes of the same kind be proportionals they are alsoproportionals by alternation(alternando).
Leta:b::c:d, thena:c::b:d.
Dem.—Sincea:b::c:d,
a-= c, b d
and multiplying each byb c, we get
PROP.XVII.—Theorem.
PROP.XVII.—Theorem.
If four magnitudes be proportional, the difference between the first andsecond:the second::the difference between the third and fourth:the fourth(dividendo).
Leta:b::c:d: thena−b:b::c−d:d;
PROP.XVIII.—Theorem.If four magnitudes be proportionals, the sum of the first and second:thesecond::the sum of the third and fourth:the fourth(componendo).
PROP.XVIII.—Theorem.If four magnitudes be proportionals, the sum of the first and second:thesecond::the sum of the third and fourth:the fourth(componendo).
Leta:b::c:d; thena+b:b::c+d:d.
PROP.XIX.—Theorem.
PROP.XIX.—Theorem.
If a whole magnitude be to another whole at a magnitude taken from the first it to amagnitude taken from the second, the first remainder:the second remainder::thefirst whole:the second whole.
Leta:b::c:d,canddbeing less thanaandb;thena−c:b−d::a:b.
Prop.E.—Theorem(Simson).If four magnitudes be proportional, the first:its excess above the second::thethird:its excess above the fourth(convertendo).
Prop.E.—Theorem(Simson).If four magnitudes be proportional, the first:its excess above the second::thethird:its excess above the fourth(convertendo).
Leta:b::c:d; thena:a−b::c:c−d.
PROP.XX.—Theorem.
PROP.XX.—Theorem.
If there be two sets of three magnitudes, which taken two by two in direct orderhave equal ratios, then if the first of either set be greater than the third, thefirst of the other set is greater than the third; if equal, equal; and if less,less.
Leta,b,c;a′,b′,c′be the two sets of magnitudes, and let the ratioa:b=a′:b′, andb:c=b′:c′; then, ifabe greater than, equal to, or less thanc,a′will be greater than, equal to, or less thanc′.
Therefore ifabe greater thanc,a′is greater thanc′; if equal, equal; and if less,less.
PROP.XXI.—Theorem.
PROP.XXI.—Theorem.
If there be two sets of three magnitudes, which taken two by two in transverse orderhave equal ratios; then, if the first of either set be greater than the third, thefirst of the other set is greater than the third; if equal, equal; and if less,less.
Leta,b,c;a′,b′,c′be the two sets of magnitudes, and let the ratioa:b=b′:c′, andb:c=a′:b′. Then, ifabe greater than, equal to, or less thanc,a′will be greater than, equal to, or less thanc′.
Therefore, ifabe greater thanc,a′is greater thanc′; if equal, equal; if less,less.
PROP.XXII.—Theorem.
PROP.XXII.—Theorem.
If there be two sets of magnitudes, which, taken two by two in direct order, haveequal ratios, then the first:the last of the first set::the first:the last of the secondset(“ex aequali,”or “ex aequo”).
Leta,b,c;a′,b′,c′be the two sets of magnitudes, and ifa:b::a′:b′, andb:c::b′:c′, thena:c::a′:c′.
and similarly for any number of magnitudes in each set.
Cor.1.—If the ratiob:cbe equal to the ratioa:b, thena,b,cwill be in continued proportion, and so willa′,b′,c′. Hence [Def.xii.Annotation 3],
a- a2 a′ a′2- c = b2 and c′ = b′2;
Or if four magnitudes be proportional, their squares are proportional.
Cor.2.—If four magnitudes be proportional, their cubes are proportional.
PROP.XXIII.—Theorem.
PROP.XXIII.—Theorem.
If there be two sets of magnitudes, which, taken two by two in transverse order, haveequal ratios; then the first:the last of the first set::the first:the last of the secondset(“ex aequo perturbato”).
Leta,b,c;a′,b′,c′be the two sets of magnitudes, and let the ratioa:b=b′:c′, andb:c=a′:b′; thena:c::a′:c′.
and similarly for any number of magnitudes in each set.
This Proposition and the preceding one may be included in one enunciation, thus: “Ratios compounded of equal ratios are equal.”