PROPOSITION 19.

Given three numbers, to investigate when it is possible to find a fourth proportional to them.

Let A, B, C be the given three numbers, and let it be required to investigate when it is possible to find a fourth proportional to them.

A B C D E

Now either they are not in continued proportion, and the extremes of them are prime to one another ; or they are in continued proportion, and the extremes of them are not prime to one another ; or they are not in continued proportion, nor are the extremes of them prime to one another ; or they are in continued proportion, and the extremes of them are prime to one another.

If then A, B, C are in continued proportion, and the extremes of them A, C are prime to one another,
it has been proved that it is impossible to find a fourth proportional number to them. [IX. 17]

†Next, let A, B, C not be in continued proportion, the extremes being again prime to one another ;

I say that in this case also it is impossible to find a fourth proportional to them.

For, if possible, let D have been found, so that, as A is to B, so is C to D, and let it be contrived that, as B is to C, so is D to E.

Now, since, as A is to B, so is C to D, and, as B is to C, so is D to E,
[VII. 14]therefore, ex aequali, as A is to C, so is C to E.
But A, C are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. [VII. 20]
Therefore A measures C as antecedent antecedent.
But it also measures itself ;
therefore A measures A, C which are prime to one another : which is impossible.
Therefore it is not possible to find a fourth proportional to A, B, C.†

Next, let A, B, C be again in continued proportion, but let A, C not be prime to one another.

I say that it is possible to find a fourth proportional to them.

For let B by multiplying C make D ;

therefore A either measures D or does not measure it.

First, let it measure it according to E ;

therefore A by multiplying E has made D.
But, further, B has also by multiplying C made D ;
therefore the product of A, E is equal to the product of B, C ; therefore, proportionally, as A is to B, so is C to E ; [VII. 19] therefore E has been found a fourth proportional to A, B, C.

Next, let A not measure D ;

I say that it is impossible to find a fourth proportional number to A, B, C.

For, if possible, let E have been found ;

therefore the product of A, E is equal to the product of B, C. [VII. 19]
But the product of B, C is D ;
therefore the product of A, E is also equal to D.
Therefore A by multiplying E has made D ; therefore A measures D according to E, so that A measures D.
But it also does not measure it :
which is absurd.
Therefore it is not possible to find a fourth proportional number to A, B, C when A does not measure D.

Next, let A, B, C not be in continued proportion, nor the extremes prime to one another.

And let B by multiplying C make D.

Similarly then it can be proved that, if A measures D, it is possible to find a fourth proportional to them, but, if it does not measure it, impossible.
Q. E. D.

Dependency Graph

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"V.def.6" [label="V.def.6", fillcolor="#224422", tooltip="V.def.6: Let magnitudes which have the same ratio be called proportional.", URL="../../../book_v/definitions/index.html#v-def-6", target="_top"]; "V.def.7" [label="V.def.7", fillcolor="#224422", tooltip="V.def.7: When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then…", URL="../../../book_v/definitions/index.html#v-def-7", target="_top"]; "V.def.11" [label="V.def.11", fillcolor="#224422", tooltip="V.def.11: The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.", URL="../../../book_v/definitions/index.html#v-def-11", target="_top"]; "V.def.12" [label="V.def.12", fillcolor="#224422", tooltip="V.def.12: Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.", URL="../../../book_v/definitions/index.html#v-def-12", target="_top"]; "V.def.13" [label="V.def.13", fillcolor="#224422", tooltip="V.def.13: Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.", URL="../../../book_v/definitions/index.html#v-def-13", target="_top"]; "V.def.17" [label="V.def.17", fillcolor="#224422", tooltip="V.def.17: A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to…", URL="../../../book_v/definitions/index.html#v-def-17", target="_top"]; "V.1" [label="V.1", fillcolor="#222244", tooltip="V.1: If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one…", URL="../../../book_v/propositions/01/index.html", target="_top"]; "V.7" [label="V.7", fillcolor="#222244", tooltip="V.7: Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.", URL="../../../book_v/propositions/07/index.html", target="_top"]; "V.8" [label="V.8", fillcolor="#222244", tooltip="V.8: Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.", URL="../../../book_v/propositions/08/index.html", target="_top"]; "V.9" [label="V.9", fillcolor="#222244", tooltip="V.9: Magnitudes which have the same ratio to the same are equal to one another ; and magnitudes to which the same has the same ratio are equal.", URL="../../../book_v/propositions/09/index.html", target="_top"]; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../../book_vii/definitions/index.html#vii-def-2", target="_top"]; "VII.def.3" [label="VII.def.3", fillcolor="#224422", tooltip="VII.def.3: A number is a part of a number, the less of the greater, when it measures the greater.", URL="../../../book_vii/definitions/index.html#vii-def-3", target="_top"]; "VII.def.4" [label="VII.def.4", fillcolor="#224422", tooltip="VII.def.4: but parts when it does not measure it.", URL="../../../book_vii/definitions/index.html#vii-def-4", target="_top"]; "VII.def.5" [label="VII.def.5", fillcolor="#224422", tooltip="VII.def.5: The greater number is a multiple of the less when it is measured by the less.", URL="../../../book_vii/definitions/index.html#vii-def-5", target="_top"]; "VII.def.11" [label="VII.def.11", fillcolor="#224422", tooltip="VII.def.11: A prime number is that which is measured by an unit alone.", URL="../../../book_vii/definitions/index.html#vii-def-11", target="_top"]; "VII.def.12" [label="VII.def.12", fillcolor="#224422", tooltip="VII.def.12: Numbers prime to one another are those which are measured by an unit alone as a common measure.", URL="../../../book_vii/definitions/index.html#vii-def-12", target="_top"]; "VII.def.15" [label="VII.def.15", fillcolor="#224422", tooltip="VII.def.15: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.", URL="../../../book_vii/definitions/index.html#vii-def-15", target="_top"]; "VII.def.20" [label="VII.def.20", fillcolor="#224422", tooltip="VII.def.20: Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.", URL="../../../book_vii/definitions/index.html#vii-def-20", target="_top"]; "VII.1" [label="VII.1", fillcolor="#222244", tooltip="VII.1: Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit…", URL="../../../book_vii/propositions/01/index.html", target="_top"]; "VII.2" [label="VII.2", fillcolor="#222244", tooltip="VII.2: Given two numbers not prime to one another, to find their greatest common measure.", URL="../../../book_vii/propositions/02/index.html", target="_top"]; "VII.4" [label="VII.4", fillcolor="#222244", tooltip="VII.4: Any number is either a part or parts of any number, the less of the greater.", URL="../../../book_vii/propositions/04/index.html", target="_top"]; 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"VII.12" [label="VII.12", fillcolor="#222244", tooltip="VII.12: If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.", URL="../../../book_vii/propositions/12/index.html", target="_top"]; "VII.13" [label="VII.13", fillcolor="#222244", tooltip="VII.13: If four numbers be proportional, they will also be proportional alternately.", URL="../../../book_vii/propositions/13/index.html", target="_top"]; "VII.14" [label="VII.14", fillcolor="#222244", tooltip="VII.14: If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.", URL="../../../book_vii/propositions/14/index.html", target="_top"]; "VII.15" [label="VII.15", fillcolor="#222244", tooltip="VII.15: If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of…", URL="../../../book_vii/propositions/15/index.html", target="_top"]; "VII.16" [label="VII.16", fillcolor="#222244", tooltip="VII.16: If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.", URL="../../../book_vii/propositions/16/index.html", target="_top"]; "VII.17" [label="VII.17", fillcolor="#222244", tooltip="VII.17: If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.", URL="../../../book_vii/propositions/17/index.html", target="_top"]; "VII.18" [label="VII.18", fillcolor="#222244", tooltip="VII.18: If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.", URL="../../../book_vii/propositions/18/index.html", target="_top"]; "VII.19" [label="VII.19", fillcolor="#222244", tooltip="VII.19: If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the…", URL="../../../book_vii/propositions/19/index.html", target="_top"]; "VII.20" [label="VII.20", fillcolor="#222244", tooltip="VII.20: The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.", URL="../../../book_vii/propositions/20/index.html", target="_top"]; "VII.21" [label="VII.21", fillcolor="#222244", tooltip="VII.21: Numbers prime to one another are the least of those which have the same ratio with them.", URL="../../../book_vii/propositions/21/index.html", target="_top"]; "IX.17" [label="IX.17", fillcolor="#222244", tooltip="IX.17: If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the…", URL="../17/index.html", target="_top"]; "IX.19" [label="IX.19", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="IX.19: Given three numbers, to investigate when it is possible to find a fourth proportional to them.", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.13" -> "I.def.3"; "I.def.14" -> "I.def.13"; "V.9" -> "I.def.14"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.4" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "VII.def.11" -> "V.def.1"; "VII.def.12" -> "V.def.1"; "VII.1" -> "V.def.1"; "VII.2" -> "V.def.1"; "VII.15" -> "V.def.1"; "VII.20" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "VII.5" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.8" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "VII.14" -> "V.def.5"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "VII.20" -> "V.def.5"; "VII.21" -> "V.def.5"; "V.7" -> "V.def.6"; "VII.13" -> "V.def.6"; "VII.19" -> "V.def.6"; "IX.17" -> "V.def.6"; "IX.19" -> "V.def.6"; "V.8" -> "V.def.7"; "V.def.12" -> "V.def.11"; "V.def.13" -> "V.def.11"; "VII.12" -> "V.def.11"; "VII.9" -> "V.def.12"; "VII.10" -> "V.def.12"; "VII.13" -> "V.def.12"; "VII.15" -> "V.def.12"; "V.7" -> "V.def.13"; "VII.14" -> "V.def.17"; "V.8" -> "V.1"; "VII.19" -> "V.7"; "V.9" -> "V.8"; "VII.19" -> "V.9"; "VII.def.2" -> "VII.def.1"; "VII.def.11" -> "VII.def.1"; "VII.def.12" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.1" -> "VII.def.1"; "VII.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "VII.1" -> "VII.def.2"; "VII.2" -> "VII.def.2"; "VII.4" -> "VII.def.2"; "VII.5" -> "VII.def.2"; "VII.6" -> "VII.def.2"; "VII.9" -> "VII.def.2"; "VII.10" -> "VII.def.2"; "VII.12" -> "VII.def.2"; "VII.13" -> "VII.def.2"; "VII.14" -> "VII.def.2"; "VII.15" -> "VII.def.2"; "VII.16" -> "VII.def.2"; "VII.17" -> "VII.def.2"; "VII.18" -> "VII.def.2"; "VII.19" -> "VII.def.2"; "VII.20" -> "VII.def.2"; "VII.21" -> "VII.def.2"; "IX.17" -> "VII.def.2"; "IX.19" -> "VII.def.2"; "VII.def.20" -> "VII.def.3"; "VII.4" -> "VII.def.3"; "VII.5" -> "VII.def.3"; "VII.9" -> "VII.def.3"; "VII.10" -> "VII.def.3"; "VII.def.20" -> "VII.def.4"; "VII.4" -> "VII.def.4"; "VII.6" -> "VII.def.4"; "VII.9" -> "VII.def.4"; "VII.10" -> "VII.def.4"; "VII.def.20" -> "VII.def.5"; "VII.5" -> "VII.def.5"; "IX.17" -> "VII.def.11"; "VII.1" -> "VII.def.12"; "VII.2" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "IX.17" -> "VII.def.12"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "VII.20" -> "VII.def.20"; "VII.2" -> "VII.1"; "VII.4" -> "VII.2"; "VII.20" -> "VII.4"; "VII.6" -> "VII.5"; "VII.9" -> "VII.5"; "VII.10" -> "VII.5"; "VII.12" -> "VII.5"; "VII.9" -> "VII.6"; "VII.10" -> "VII.6"; "VII.12" -> "VII.6"; "VII.10" -> "VII.9"; "VII.13" -> "VII.10"; "VII.15" -> "VII.12"; "VII.20" -> "VII.12"; "VII.14" -> "VII.13"; "VII.17" -> "VII.13"; "VII.20" -> "VII.13"; "IX.17" -> "VII.13"; "IX.19" -> "VII.14"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.19" -> "VII.18"; "IX.19" -> "VII.19"; "VII.21" -> "VII.20"; "IX.17" -> "VII.20"; "IX.19" -> "VII.20"; "IX.17" -> "VII.21"; "IX.19" -> "VII.21"; "IX.19" -> "IX.17"; }

See Commentary on Proposition IX.19.