Elementa

Euclid

Elementa

Euclid · Elem 7 · perseus-eng2 · More by Euclid

Choose a text id="mobileReaderAuthor" class="mobile-reader-author" href="/authors/euclid" aria-label="More works by Euclid"> â€”

â‹¯

Primary version Secondary version

Original / primary: Perseus Eng2

7.def.1.p1952

An unit is that by virtue of which each of the things that exist is called one.

7.def.2.p1953

A number is a multitude composed of units.

7.def.3.p1954

A number is a part of a number, the less of the greater, when it measures the greater;

7.def.4.p1955

but parts when it does not measure it.

7.def.5.p1956

The greater number is a multiple of the less when it is measured by the less.

7.def.6.p1957

An even number is that which is divisible into two equal parts.

7.def.7.p1958

An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

7.def.8.p1959

An even-times even number is that which is measured by an even number according to an even number.

7.def.9.p1960

An even-times odd number is that which is measured by an even number according to an odd number.

7.def.10.p1961

An odd-times odd number is that which is measured by an odd number according to an odd number.

7.def.11.p1962

A prime number is that which is measured by an unit alone.

7.def.12.p1963

Numbers prime to one another are those which are measured by an unit alone as a common measure.

7.def.13.p1964

A composite number is that which is measured by some number.

7.def.14.p1965

Numbers composite to one another are those which are measured by some number as a common measure.

7.def.15.p1966

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.

7.def.16.p1967

And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

7.def.17.p1968

And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.

7.def.18.p1969

A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

7.def.19.p1970

And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

7.def.20.p1971

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.

7.def.21.p1972

Similar plane and solid numbers are those which have their sides proportional.

7.def.22.p1973

A perfect number is that which is equal to its own parts.

7.prop.1.p1974

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 is left, the original numbers will be prime to one another.

7.prop.1.p1975

For, the less of two unequal numbers AB, CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left; I say that AB, CD are prime to one another, that is, that an unit alone measures AB, CD.

7.prop.1.p1976

For, if AB, CD are not prime to one another, some number will measure them.

7.prop.1.p1977

Let a number measure them, and let it be E; let CD, measuring BF, leave FA less than itself, let AF, measuring DG, leave GC less than itself, and let GC, measuring FH, leave an unit HA.

7.prop.1.p1978

Since, then, E measures CD, and CD measures BF, therefore E also measures BF.

7.prop.1.p1979

But it also measures the whole BA; therefore it will also measure the remainder AF.

7.prop.1.p1980

But AF measures DG; therefore E also measures DG.

7.prop.1.p1981

But it also measures the whole DC therefore it will also measure the remainder CG.

7.prop.1.p1982

But CG measures FH; therefore E also measures FH.

7.prop.1.p1983

But it also measures the whole FA; therefore it will also measure the remainder, the unit AH, though it is a number: which is impossible.

7.prop.1.p1984

Therefore no number will measure the numbers AB, CD; therefore AB, CD are prime to one another. [VII. Def. 12] Q. E. D.

7.prop.2.p1985

Given two numbers not prime to one another, to find their greatest common measure.

7.prop.2.p1986

Let AB, CD be the two given numbers not prime to one another.

7.prop.2.p1987

Thus it is required to find the greatest common measure of AB, CD.

7.prop.2.p1988

If now CD measures AB—and it also measures itself—CD is a common measure of CD, AB.

7.prop.2.p1989

And it is manifest that it is also the greatest; for no greater number than CD will measure CD.

7.prop.2.p1990

But, if CD does not measure AB, then, the less of the numbers AB, CD being continually subtracted from the greater, some number will be left which will measure the one before it.

7.prop.2.p1991

For an unit will not be left; otherwise AB, CD will be prime to one another [VII. 1], which is contrary to the hypothesis.

7.prop.2.p1992

Therefore some number will be left which will measure the one before it.

7.prop.2.p1993

Now let CD, measuring BE, leave EA less than itself, let EA, measuring DF, leave FC less than itself, and let CF measure AE.

7.prop.2.p1994

Since then, CF measures AE, and AE measures DF, therefore CF will also measure DF.

7.prop.2.p1995

But it also measures itself; therefore it will also measure the whole CD.

7.prop.2.p1996

But CD measures BE; therefore CF also measures BE.

7.prop.2.p1997

But it also measures EA; therefore it will also measure the whole BA.

7.prop.2.p1998

But it also measures CD; therefore CF measures AB, CD.

7.prop.2.p1999

Therefore CF is a common measure of AB, CD.

7.prop.2.p2000

I say next that it is also the greatest.

7.prop.2.p2001

For, if CF is not the greatest common measure of AB, CD, some number which is greater than CF will measure the numbers AB, CD.

7.prop.2.p2002

Let such a number measure them, and let it be G.

7.prop.2.p2003

Now, since G measures CD, while CD measures BE, G also measures BE.

7.prop.2.p2004

But it also measures the whole BA; therefore it will also measure the remainder AE.

7.prop.2.p2005

But AE measures DF; therefore G will also measure DF.

7.prop.2.p2006

But it also measures the whole DC; therefore it will also measure the remainder CF, that is, the greater will measure the less: which is impossible.

7.prop.2.p2007

Therefore no number which is greater than CF will measure the numbers AB, CD; therefore CF is the greatest common measure of AB, CD.

7.prop.2.p2008

Porism. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.

7.prop.2.trailer

Q. E. D.

7.prop.3.p2010

Given three numbers not prime to one another, to find their greatest common measure.

7.prop.3.p2011

Let A, B, C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A, B, C.

7.prop.3.p2012

For let the greatest common measure, D, of the two numbers A, B be taken; [VII. 2] then D either measures, or does not measure, C.

7.prop.3.p2013

First, let it measure it.

7.prop.3.p2014

But it measures A, B also; therefore D measures A, B, C; therefore D is a common measure of A, B, C.

7.prop.3.p2015

I say that it is also the greatest.

7.prop.3.p2016

For, if D is not the greatest common measure of A, B, C, some number which is greater than D will measure the numbers A, B, C.

7.prop.3.p2017

Let such a number measure them, and let it be E.

7.prop.3.p2018

Since then E measures A, B, C, it will also measure A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]

7.prop.3.p2019

But the greatest common measure of A, B is D; therefore E measures D, the greater the less: which is impossible.

7.prop.3.p2020

Therefore no number which is greater than D will measure the numbers A, B, C; therefore D is the greatest common measure of A, B, C.

7.prop.3.p2021

Next, let D not measure C; I say first that C, D are not prime to one another.

7.prop.3.p2022

For, since A, B, C are not prime to one another, some number will measure them.

7.prop.3.p2023

Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B. [VII. 2, Por.]

7.prop.3.p2024

But it measures C also; therefore some number will measure the numbers D, C; therefore D, C are not prime to one another.

7.prop.3.p2025

Let then their greatest common measure E be taken. [VII. 2]

7.prop.3.p2026

Then, since E measures D, and D measures A, B, therefore E also measures A, B.

7.prop.3.p2027

But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C.

7.prop.3.p2028

I say next that it is also the greatest.

7.prop.3.p2029

For, if E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C.

7.prop.3.p2030

Let such a number measure them, and let it be F.

7.prop.3.p2031

Now, since F measures A, B, C, it also measures A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]

7.prop.3.p2032

But the greatest common measure of A, B is D; therefore F measures D.

7.prop.3.p2033

And it measures C also; therefore F measures D, C; therefore it will also measure the greatest common measure of D, C. [VII. 2, Por.]

7.prop.3.p2034

But the greatest common measure of D, C is E; therefore F measures E, the greater the less: which is impossible.

7.prop.3.p2035

Therefore no number which is greater than E will measure the numbers A, B, C; therefore E is the greatest common measure of A, B, C. Q. E. D.

7.prop.4.p2036

Any number is either a part or parts of any number, the less of the greater.

7.prop.4.p2037

Let A, BC be two numbers, and let BC be the less; I say that BC is either a part, or parts, of A.

7.prop.4.p2038

For A, BC are either prime to one another or not.

7.prop.4.p2039

First, let A, BC be prime to one another.

7.prop.4.p2040

Then, if BC be divided into the units in it, each unit of those in BC will be some part of A; so that BC is parts of A.

7.prop.4.p2041

Next let A, BC not be prime to one another; then BC either measures, or does not measure, A.

7.prop.4.p2042

If now BC measures A, BC is a part of A.

7.prop.4.p2043

But, if not, let the greatest common measure D of A, BC be taken; [VII. 2] and let BC be divided into the numbers equal to D, namely BE, EF, FC.

7.prop.4.p2044

Now, since D measures A, D is a part of A.

7.prop.4.p2045

But D is equal to each of the numbers BE, EF, FC; therefore each of the numbers BE, EF, FC is also a part of A; so that BC is parts of A.

7.prop.4.p2046

Therefore etc. Q. E. D.

7.prop.5.p2047

If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.

7.prop.5.p2048

For let the number A be a part of BC, and another, D, the same part of another EF that A is of BC; I say that the sum of A, D is also the same part of the sum of BC, EF that A is of BC.

7.prop.5.p2049

For since, whatever part A is of BC, D is also the same part of EF, therefore, as many numbers as there are in BC equal to A, so many numbers are there also in EF equal to D.

7.prop.5.p2050

Let BC be divided into the numbers equal to A, namely BG, GC, and EF into the numbers equal to D, namely EH, HF; then the multitude of BG, GC will be equal to the multitude of EH, HF.

7.prop.5.p2051

And, since BG is equal to A, and EH to D, therefore BG, EH are also equal to A, D.

7.prop.5.p2052

For the same reason GC, HF are also equal to A, D.

7.prop.5.p2053

Therefore, as many numbers as there are in BC equal to A, so many are there also in BC, EF equal to A, D.

7.prop.5.p2054

Therefore, whatever multiple BC is of A, the same multiple also is the sum of BC, EF of the sum of A, D.

7.prop.5.p2055

Therefore, whatever part A is of BC, the same part also is the sum of A, D of the sum of BC, EF. Q. E. D.

7.prop.6.p2056

If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.

7.prop.6.p2057

For let the number AB be parts of the number C, and another, DE, the same parts of another, F, that AB is of C; I say that the sum of AB, DE is also the same parts of the sum of C, F that AB is of C.

7.prop.6.p2058

For since, whatever parts AB is of C, DE is also the same parts of F, therefore, as many parts of C as there are in AB, so many parts of F are there also in DE.

7.prop.6.p2059

Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of DH, HE.

7.prop.6.p2060

And since, whatever part AG is of C, the same part is DH of F also, therefore, whatever part AG is of C, the same part also is the sum of AG, DH of the sum of C, F. [VII. 5]

7.prop.6.p2061

For the same reason, whatever part GB is of C, the same part also is the sum of GB, HE of the sum of C, F.

7.prop.6.p2062

Therefore, whatever parts AB is of C, the same parts also is the sum of AB, DE of the sum of C, F. Q. E. D.

7.prop.7.p2063

If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.

7.prop.7.p2064

For let the number AB be that part of the number CD which AE subtracted is of CF subtracted; I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD.

7.prop.7.p2065

For, whatever part AE is of CF, the same part also let EB be of CG.

7.prop.7.p2066

Now since, whatever part AE is of CF, the same part also is EB of CG, therefore, whatever part AE is of CF, the same part also is AB of GF. [VII. 5]

7.prop.7.p2067

But, whatever part AE is of CF, the same part also, by hypothesis, is AB of CD; therefore, whatever part AB is of GF, the same part is it of CD also; therefore GF is equal to CD.

7.prop.7.p2068

Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD.

7.prop.7.p2069

Now since, whatever part AE is of CF, the same part also is EB of GC, while GC is equal to FD, therefore, whatever part AE is of CF, the same part also is EB of FD.

7.prop.7.p2070

But, whatever part AE is of CF, the same part also is AB of CD; therefore also the remainder EB is the same part of the remainder FD that the whole AB is of the whole CD. Q. E. D.

7.prop.8.p2071

If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.

7.prop.8.p2072

For let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted; I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD.

7.prop.8.p2073

For let GH be made equal to AB.

7.prop.8.p2074

Therefore, whatever parts GH is of CD, the same parts also is AE of CF.

7.prop.8.p2075

Let GH be divided into the parts of CD, namely GK, KH, and AE into the parts of CF, namely AL, LE; thus the multitude of GK, KH will be equal to the multitude of AL, LE.

7.prop.8.p2076

Now since, whatever part GK is of CD, the same part also is AL of CF, while. CD is greater than CF, therefore GK is also greater than AL.

7.prop.8.p2077

Let GM be made equal to AL.

7.prop.8.p2078

Therefore, whatever part GK is of CD, the same part also is GM of CF; therefore also the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD. [VII. 7]

7.prop.8.p2079

Again, since, whatever part KH is of CD, the same part also is EL of CF, while CD is greater than CF, therefore HK is also greater than EL.

7.prop.8.p2080

Let KN be made equal to EL.

7.prop.8.p2081

Therefore, whatever part KH is of CD, the same part also is KN of CF; therefore also the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD. [VII. 7]

7.prop.8.p2082

But the remainder MK was also proved to be the same part of the remainder FD that the whole GK is of the whole CD; therefore also the sum of MK, NH is the same parts of DF that the whole HG is of the whole CD.

7.prop.8.p2083

But the sum of MK, NH is equal to EB, and HG is equal to BA; therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD. Q. E. D.

7.prop.9.p2084

If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

7.prop.9.p2085

For let the number A be a part of the number BC, and another, D, the same part of another, EF, that A is of BC; I say that, alternately also, whatever part or parts A is of D, the same part or parts is BC of EF also.

7.prop.9.p2086

For since, whatever part A is of BC, the same part also is D of EF, therefore, as many numbers as there are in BC equal to A, so many also are there in EF equal to D.

7.prop.9.p2087

Let BC be divided into the numbers equal to A, namely BG, GC, and EF into those equal to D, namely EH, HF; thus the multitude of BG, GC will be equal to the multitude of EH, HF.

7.prop.9.p2088

Now, since the numbers BG, GC are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of BG, GC is equal to the multitude of EH, HF, therefore, whatever part or parts BG is of EH, the same part or the same parts is GC of HF also; so that, in addition, whatever part or parts BG is of EH, the same part also, or the same parts, is the sum BC of the sum EF. [VII. 5, 6]

7.prop.9.p2089

But BG is equal to A, and EH to D; therefore, whatever part or parts A is of D, the same part or the same parts is BC of EF also. Q. E. D.

7.prop.10.p2090

If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.

7.prop.10.p2091

For let the number AB be parts of the number C, and another, DE, the same parts of another, F; I say that, alternately also, whatever parts or part AB is of DE, the same parts or the same part is C of F also.

7.prop.10.p2092

For since, whatever parts AB is of C, the same parts also is DE of F, therefore, as many parts of C as there are in AB, so many parts also of F are there in DE.

7.prop.10.p2093

Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of DH, HE.

7.prop.10.p2094

Now since, whatever part AG is of C, the same part also is DH of F, alternately also, whatever part or parts AG is of DH, the same part or the same parts is C of F also. [VII. 9]

7.prop.10.p2095

For the same reason also, whatever part or parts GB is of HE, the same part or the same parts is C of F also; so that, in addition, whatever parts or part AB is of DE, the same parts also, or the same part, is C of F. [VII. 5, 6] Q. E. D.

7.prop.11.p2096

If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.

7.prop.11.p2097

As the whole AB is to the whole CD, so let AE subtracted be to CF subtracted; I say that the remainder EB is also to the remainder FD as the whole AB to the whole CD.

7.prop.11.p2098

Since, as AB is to CD, so is AE to CF, whatever part or parts AB is of CD, the same part or the same parts is AE of CF also; [VII. Def. 20]

7.prop.11.p2099

Therefore also the remainder EB is the same part or parts of FD that AB is of CD. [VII. 7, 8]

7.prop.11.p2100

Therefore, as EB is to FD, so is AB to CD. [VII. Def. 20] Q. E. D.

7.prop.12.p2101

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.

7.prop.12.p2102

Let A, B, C, D be as many numbers as we please in proportion, so that, as A is to B, so is C to D; I say that, as A is to B, so are A, C to B, D.

7.prop.12.p2103

For since, as A is to B, so is C to D, whatever part or parts A is of B, the same part or parts is C of D also. [VII. Def. 20]

7.prop.12.p2104

Therefore also the sum of A, C is the same part or the same parts of the sum of B, D that A is of B. [VII. 5, 6]

7.prop.12.p2105

Therefore, as A is to B, so are A, C to B, D. [VII. Def. 20]

7.prop.13.p2106

If four numbers be proportional, they will also be proportional alternately.

7.prop.13.p2107

Let the four numbers A, B, C, D be proportional, so that, as A is to B, so is C to D; I say that they will also be proportional alternately, so that, as A is to C, so will B be to D.

7.prop.13.p2108

For since, as A is to B, so is C to D, therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. [VII. Def. 20]

7.prop.13.p2109

Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [VII. 10]

7.prop.13.p2110

Therefore, as A is to C, so is B to D. [VII. Def. 20] Q. E. D.

7.prop.14.p2111

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.

7.prop.14.p2112

Let there be as many numbers as we please A, B, C, and others equal to them in multitude D, E, F, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F; I say that, ex aequali, as A is to C, so also is D to F.

7.prop.14.p2113

For, since, as A is to B, so is D to E, therefore, alternately, as A is to D, so is B to E. [VII. 13]

7.prop.14.p2114

Again, since, as B is to C, so is E to F, therefore, alternately, as B is to E, so is C to F. [VII. 13]

7.prop.14.p2115

But, as B is to E, so is A to D; therefore also, as A is to D, so is C to F.

7.prop.14.p2116

Therefore, alternately, as A is to C, so is D to F. [id.]

7.prop.15.p2117

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 times that the second measures the fourth.

7.prop.15.p2118

For let the unit A measure any number BC, and let another number D measure any other number EF the same number of times; I say that, alternately also, the unit A measures the number D the same number of times that BC measures EF.

7.prop.15.p2119

For, since the unit A measures the number BC the same number of times that D measures EF, therefore, as many units as there are in BC, so many numbers equal to D are there in EF also.

7.prop.15.p2120

Let BC be divided into the units in it, BG, GH, HC, and EF into the numbers EK, KL, LF equal to D.

7.prop.15.p2121

Thus the multitude of BG, GH, HC will be equal to the multitude of EK, KL, LF.

7.prop.15.p2122

And, since the units BG, GH, HC are equal to one another, and the numbers EK, KL, LF are also equal to one another, while the multitude of the units BG, GH, HC is equal to the multitude of the numbers EK, KL, LF, therefore, as the unit BG is to the number EK, so will the unit GH be to the number KL, and the unit HC to the number LF.

7.prop.15.p2123

Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [VII. 12] therefore, as the unit BG is to the number EK, so is BC to EF.

7.prop.15.p2124

But the unit BG is equal to the unit A, and the number EK to the number D.

7.prop.15.p2125

Therefore, as the unit A is to the number D, so is BC to EF.

7.prop.15.p2126

Therefore the unit A measures the number D the same number of times that BC measures EF. Q. E. D.

7.prop.16.p2127

If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

7.prop.16.p2128

Let A, B be two numbers, and let A by multiplying B make C, and B by multiplying A make D; I say that C is equal to D.

7.prop.16.p2129

For, since A by multiplying B has made C, therefore B measures C according to the units in A.

7.prop.16.p2130

But the unit E also measures the number A according to the units in it; therefore the unit E measures A the same number of times that B measures C.

7.prop.16.p2131

Therefore, alternately, the unit E measures the number B the same number of times that A measures C. [VII. 15]

7.prop.16.p2132

Again, since B by multiplying A has made D, therefore A measures D according to the units in B.

7.prop.16.p2133

But the unit E also measures B according to the units in it; therefore the unit E measures the number B the same number of times that A measures D.

7.prop.16.p2134

But the unit E measured the number B the same number of times that A measures C; therefore A measures each of the numbers C, D the same number of times.

7.prop.16.p2135

Therefore

7.prop.16.p2135

C

7.prop.16.p2135

is equal to

7.prop.16.p2135

D

7.prop.16.p2135

. Q. E. D.

7.prop.16.p2135

1

7.prop.17.p2136

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

7.prop.17.p2137

For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E.

7.prop.17.p2138

For, since A by multiplying B has made D, therefore B measures D according to the units in A.

7.prop.17.p2139

But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D.

7.prop.17.p2140

Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]

7.prop.17.p2141

For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E.

7.prop.17.p2142

Therefore, alternately, as B is to C, so is D to E. [VII. 13] Q. E. D.

7.prop.18.p2143

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

7.prop.18.p2144

For let two numbers A, B by multiplying any number C make D, E; I say that, as A is to B, so is D to E.

7.prop.18.p2145

For, since A by multiplying C has made D, therefore also C by multiplying A has made D. [VII. 16] For the same reason also C by multiplying B has made E.

7.prop.18.p2146

Therefore the number C by multiplying the two numbers A, B has made D, E.

7.prop.18.p2147

Therefore, as A is to B, so is D to E. [VII. 17]

7.prop.19.p2148

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 first and fourth be equal to that produced from the second and third, the four numbers will be proportional.

7.prop.19.p2149

Let A, B, C, D be four numbers in proportion, so that, as A is to B, so is C to D; and let A by multiplying D make E, and let B by multiplying C make F; I say that E is equal to F.

7.prop.19.p2150

For let A by multiplying C make G.

7.prop.19.p2151

Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E.

7.prop.19.p2152

Therefore, as C is to D, so is G to E. [VII. 17]

7.prop.19.p2153

But, as C is to D, so is A to B; therefore also, as A is to B, so is G to E.

7.prop.19.p2154

Again, since A by multiplying C has made G, but, further, B has also by multiplying C made F, the two numbers A, B by multiplying a certain number C have made G, F.

7.prop.19.p2155

Therefore, as A is to B, so is G to F. [VII. 18]

7.prop.19.p2156

But further, as A is to B, so is G to E also; therefore also, as G is to E, so is G to F.

7.prop.19.p2157

Therefore G has to each of the numbers E, F the same ratio; therefore E is equal to F. [cf. V. 9]

7.prop.19.p2158

Again, let E be equal to F; I say that, as A is to B, so is C to D.

7.prop.19.p2159

For, with the same construction, since E is equal to F, therefore, as G is to E, so is G to F. [cf. V. 7]

7.prop.19.p2160

But, as G is to E, so is C to D, [VII. 17] and, as G is to F, so is A to B. [VII. 18]

7.prop.19.p2161

Therefore also, as A is to B, so is C to D. Q. E. D.

7.prop.20.p2162

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.

7.prop.20.p2163

For let CD, EF be the least numbers of those which have the same ratio with A, B; I say that CD measures A the same number of times that EF measures B.

7.prop.20.p2164

Now CD is not parts of A.

7.prop.20.p2165

For, if possible, let it be so; therefore EF is also the same parts of B that CD is of A. [VII. 13 and Def. 20]

7.prop.20.p2166

Therefore, as many parts of A as there are in CD, so many parts of B are there also in EF.

7.prop.20.p2167

Let CD be divided into the parts of A, namely CG, GD, and EF into the parts of B, namely EH, HF; thus the multitude of CG, GD will be equal to the multitude of EH, HF.

7.prop.20.p2168

Now, since the numbers CG, GD are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of CG, GD is equal to the multitude of EH, HF, therefore, as CG is to EH, so is GD to HF.

7.prop.20.p2169

Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. [VII. 12]

7.prop.20.p2170

Therefore, as CG is to EH, so is CD to EF.

7.prop.20.p2171

Therefore CG, EH are in the same ratio with CD, EF, being less than they: which is impossible, for by hypothesis CD, EF are the least numbers of those which have the same ratio with them.

7.prop.20.p2172

Therefore CD is not parts of A; therefore it is a part of it. [VII. 4]

7.prop.20.p2173

And EF is the same part of B that CD is of A; [VII. 13 and Def. 20] therefore CD measures A the same number of times that EF measures B. Q. E. D.

7.prop.21.p2174

Numbers prime to one another are the least of those which have the same ratio with them.

7.prop.21.p2175

Let A, B be numbers prime to one another; I say that A, B are the least of those which have the same ratio with them.

7.prop.21.p2176

For, if not, there will be some numbers less than A, B which are in the same ratio with A, B.

7.prop.21.p2177

Let them be C, D.

7.prop.21.p2178

Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [VII. 20] therefore C measures A the same number of times that D measures B.

7.prop.21.p2179

Now, as many times as C measures A, so many units let there be in E.

7.prop.21.p2180

Therefore D also measures B according to the units in E.

7.prop.21.p2181

And, since C measures A according to the units in E, therefore E also measures A according to the units in C. [VII. 16]

7.prop.21.p2182

For the same reason E also measures B according to the units in D. [VII. 16]

7.prop.21.p2183

Therefore E measures A, B which are prime to one another: which is impossible. [VII. Def. 12]

7.prop.21.p2184

Therefore there will be no numbers less than A, B which are in the same ratio with A, B.

7.prop.21.p2185

Therefore A, B are the least of those which have the same ratio with them. Q. E. D.

7.prop.22.p2186

The least numbers of those which have the same ratio with them are prime to one another.

7.prop.22.p2187

Let A, B be the least numbers of those which have the same ratio with them; I say that A, B are prime to one another.

7.prop.22.p2188

For, if they are not prime to one another, some number will measure them.

7.prop.22.p2189

Let some number measure them, and let it be C.

7.prop.22.p2190

And, as many times as C measures A, so many units let there be in D, and, as many times as C measures B, so many units let there be in E

7.prop.22.p2191

Since C measures A according to the units in D, therefore C by multiplying D has made A. [VII. Def. 15]

7.prop.22.p2192

For the same reason also C by multiplying E has made B.

7.prop.22.p2193

Thus the number C by multiplying the two numbers D, E has made A, B; therefore, as D is to E, so is A to B; [VII. 17] therefore D, E are in the same ratio with A, B, being less than they: which is impossible.

7.prop.22.p2194

Therefore no number will measure the numbers A, B.

7.prop.22.p2195

Therefore A, B are prime to one another. Q. E. D.

7.prop.23.p2196

If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.

7.prop.23.p2197

Let A, B be two numbers prime to one another, and let any number C measure A; I say that C, B are also prime to one another.

7.prop.23.p2198

For, if C, B are not prime to one another, some number will measure C, B.

7.prop.23.p2199

Let a number measure them, and let it be D.

7.prop.23.p2200

Since D measures C, and C measures A, therefore D also measures A.

7.prop.23.p2201

But it also measures B; therefore D measures A, B which are prime to one another: which is impossible. [VII. Def. 12]

7.prop.23.p2202

Therefore no number will measure the numbers C, B.

7.prop.23.p2203

Therefore C, B are prime to one another. Q. E. D.

7.prop.24.p2204

If two numbers be prime to any number, their product also will be prime to the same.

7.prop.24.p2205

For let the two numbers A, B be prime to any number C, and let A by multiplying B make D; I say that C, D are prime to one another.

7.prop.24.p2206

For, if C, D are not prime to one another, some number will measure C, D.

7.prop.24.p2207

Let a number measure them, and let it be E.

7.prop.24.p2208

Now, since C, A are prime to one another, and a certain number E measures C, therefore A, E are prime to one another. [VII. 23]

7.prop.24.p2209

As many times, then, as E measures D, so many units let there be in F; therefore F also measures D according to the units in E. [VII. 16]

7.prop.24.p2210

Therefore E by multiplying F has made D. [VII. Def. 15]

7.prop.24.p2211

But, further, A by multiplying B has also made D; therefore the product of E, F is equal to the product of A, B.

7.prop.24.p2212

But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [VII. 19] therefore, as E is to A, so is B to F.

7.prop.24.p2213

But A, E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [VII. 21] and 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, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures B.

7.prop.24.p2214

But it also measures C; therefore E measures B, C which are prime to one another: which is impossible. [VII. Def. 12]

7.prop.24.p2215

Therefore no number will measure the numbers C, D.

7.prop.24.p2216

Therefore

7.prop.24.p2216

C

7.prop.24.p2216

,

7.prop.24.p2216

D

7.prop.24.p2216

are prime to one another. Q. E. D.

7.prop.24.p2216

1

7.prop.25.p2217

If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

7.prop.25.p2218

Let A, B be two numbers prime to one another, and let A by multiplying itself make C: I say that B, C are prime to one another.

7.prop.25.p2219

For let D be made equal to A.

7.prop.25.p2220

Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.

7.prop.25.p2221

Therefore each of the two numbers D, A is prime to B; therefore the product of D, A will also be prime to B. [VII. 24]

7.prop.25.p2222

But the number which is the product of D, A is C.

7.prop.25.p2223

Therefore

7.prop.25.p2223

C

7.prop.25.p2223

,

7.prop.25.p2223

B

7.prop.25.p2223

are prime to one another. Q. E. D.

7.prop.25.p2223

1

7.prop.26.p2224

If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

7.prop.26.p2225

For let the two numbers A, B be prime to the two numbers C, D; both to each, and let A by multiplying B make E, and let C by multiplying D make F; I say that E, F are prime to one another.

7.prop.26.p2226

For, since each of the numbers A, B is prime to C, therefore the product of A, B will also be prime to C. [VII. 24]

7.prop.26.p2227

But the product of A, B is E; therefore E, C are prime to one another.

7.prop.26.p2228

For the same reason E, D are also prime to one another.

7.prop.26.p2229

Therefore each of the numbers C, D is prime to E.

7.prop.26.p2230

Therefore the product of C, D will also be prime to E. [VII. 24]

7.prop.26.p2231

But the product of C, D is F.

7.prop.26.p2232

Therefore E, F are prime to one another. Q. E. D.

7.prop.27.p2233

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].

7.prop.27.p2234

Let A, B be two numbers prime to one another, let A by multiplying itself make C, and by multiplying C make D, and let B by multiplying itself make E, and by multiplying E make F; I say that both C, E and D, F are prime to one another.

7.prop.27.p2235

For, since A, B are prime to one another, and A by multiplying itself has made C, therefore C, B are prime to one another. [VII. 25]

7.prop.27.p2236

Since then C, B are prime to one another, and B by multiplying itself has made E, therefore C, E are prime to one another. [id.]

7.prop.27.p2237

Again, since A, B are prime to one another, and B by multiplying itself has made E, therefore A, E are prime to one another. [id.]

7.prop.27.p2238

Since then the two numbers A, C are prime to the two numbers B, E, both to each, therefore also the product of A, C is prime to the product of B, E. [VII. 26]

7.prop.27.p2239

And the product of A, C is D, and the product of B, E is F.

7.prop.27.p2240

Therefore D, F are prime to one another. Q. E. D.

7.prop.28.p2241

If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.

7.prop.28.p2242

For let two numbers AB, BC prime to one another be added; I say that the sum AC is also prime to each of the numbers AB, BC.

7.prop.28.p2243

For, if CA, AB are not prime to one another, some number will measure CA, AB.

7.prop.28.p2244

Let a number measure them, and let it be D.

7.prop.28.p2245

Since then D measures CA, AB, therefore it will also measure the remainder BC.

7.prop.28.p2246

But it also measures BA; therefore D measures AB, BC which are prime to one another: which is impossible. [VII. Def. 12]

7.prop.28.p2247

Therefore no number will measure the numbers CA, AB; therefore CA, AB are prime to one another.

7.prop.28.p2248

For the same reason AC, CB are also prime to one another.

7.prop.28.p2249

Therefore CA is prime to each of the numbers AB, BC.

7.prop.28.p2250

Again, let CA, AB be prime to one another; I say that AB, BC are also prime to one another.

7.prop.28.p2251

For, if AB, BC are not prime to one another, some number will measure AB, BC.

7.prop.28.p2252

Let a number measure them, and let it be D.

7.prop.28.p2253

Now, since D measures each of the numbers AB, BC, it will also measure the whole CA.

7.prop.28.p2254

But it also measures AB; therefore D measures CA, AB which are prime to one another: which is impossible. [VII. Def. 12]

7.prop.28.p2255

Therefore no number will measure the numbers AB, BC.

7.prop.28.p2256

Therefore AB, BC are prime to one another. Q. E. D.

7.prop.29.p2257

Any prime number is prime to any number which it does not measure.

7.prop.29.p2258

Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.

7.prop.29.p2259

For, if B, A are not prime to one another, some number will measure them.

7.prop.29.p2260

Let C measure them.

7.prop.29.p2261

Since C measures B, and A does not measure B, therefore C is not the same with A.

7.prop.29.p2262

Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible.

7.prop.29.p2263

Therefore no number will measure B, A.

7.prop.29.p2264

Therefore A, B are prime to one another. Q. E. D.

7.prop.30.p2265

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

7.prop.30.p2266

For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B.

7.prop.30.p2267

For let it not measure A.

7.prop.30.p2268

Now D is prime; therefore A, D are prime to one another. [VII. 29]

7.prop.30.p2269

And, as many times as D measures C, so many units let there be in E.

7.prop.30.p2270

Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15]

7.prop.30.p2271

Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B.

7.prop.30.p2272

Therefore, as D is to A, so is B to E. [VII. 19]

7.prop.30.p2273

But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B.

7.prop.30.p2274

Similarly we can also show that, if D do not measure B, it will measure A.

7.prop.30.p2275

Therefore D measures one of the numbers A, B. Q. E. D.

7.prop.31.p2276

Any composite number is measured by some prime number.

7.prop.31.p2277

Let A be a composite number; I say that A is measured by some prime number.

7.prop.31.p2278

For, since A is composite, some number will measure it.

7.prop.31.p2279

Let a number measure it, and let it be B.

7.prop.31.p2280

Now, if B is prime, what was enjoined will have been done.

7.prop.31.p2281

But if it is composite, some number will measure it.

7.prop.31.p2282

Let a number measure it, and let it be C.

7.prop.31.p2283

Then, since C measures B, and B measures A, therefore C also measures A.

7.prop.31.p2284

And, if C is prime, what was enjoined will have been done.

7.prop.31.p2285

But if it is composite, some number will measure it.

7.prop.31.p2286

Thus, if the investigation be continued in this way, some prime number will be found which will measure the number before it, which will also measure A.

7.prop.31.p2287

For, if it is not found, an infinite series of numbers will measure the number A, each of which is less than the other: which is impossible in numbers.

7.prop.31.p2288

Therefore some prime number will be found which will measure the one before it, which will also measure A.

7.prop.31.p2289

Therefore any composite number is measured by some prime number.

7.prop.31.p2289

1

7.prop.31.p2289

2

7.prop.32.p2290

Any number either is prime or is measured by some prime number.

7.prop.32.p2291

Let A be a number; I say that A either is prime or is measured by some prime number.

7.prop.32.p2292

If now A is prime, that which was enjoined will have been done.

7.prop.32.p2293

But if it is composite, some prime number will measure it. [VII. 31]

7.prop.32.p2294

Therefore any number either is prime or is measured by some prime number. Q. E. D.

7.prop.33.p2295

Given as many numbers as we please, to find the least of those which have the same ratio with them.

7.prop.33.p2296

Let A, B, C be the given numbers, as many as we please; thus it is required to find the least of those which have the same ratio with A, B, C.

7.prop.33.p2297

A, B, C are either prime to one another or not.

7.prop.33.p2298

Now, if A, B, C are prime to one another, they are the least of those which have the same ratio with them. [VII. 21]

7.prop.33.p2299

But, if not, let D the greatest common measure of A, B, C be taken, [VII. 3] and, as many times as D measures the numbers A, B, C respectively, so many units let there be in the numbers E, F, G respectively.

7.prop.33.p2300

Therefore the numbers E, F, G measure the numbers A, B, C respectively according to the units in D. [VII. 16]

7.prop.33.p2301

Therefore E, F, G measure A, B, C the same number of times; therefore E, F, G are in the same ratio with A, B, C. [VII. Def. 20]

7.prop.33.p2302

I say next that they are the least that are in that ratio.

7.prop.33.p2303

For, if E, F, G are not the least of those which have the same ratio with A, B, C, there will be numbers less than E, F, G which are in the same ratio with A, B, C.

7.prop.33.p2304

Let them be H, K, L; therefore H measures A the same number of times that the numbers K, L measure the numbers B, C respectively.

7.prop.33.p2305

Now, as many times as H measures A, so many units let there be in M; therefore the numbers K, L also measure the numbers B, C respectively according to the units in M.

7.prop.33.p2306

And, since H measures A according to the units in M, therefore M also measures A according to the units in H. [VII. 16]

7.prop.33.p2307

For the same reason M also measures the numbers B, C according to the units in the numbers K, L respectively;

7.prop.33.p2308

Therefore M measures A, B, C.

7.prop.33.p2309

Now, since H measures A according to the units in M, therefore H by multiplying M has made A. [VII. Def. 15]

7.prop.33.p2310

For the same reason also E by multiplying D has made A.

7.prop.33.p2311

Therefore the product of E, D is equal to the product of H, M.

7.prop.33.p2312

Therefore, as E is to H, so is M to D. [VII. 19]

7.prop.33.p2313

But E is greater than H; therefore M is also greater than D.

7.prop.33.p2314

And it measures A, B, C: which is impossible, for by hypothesis D is the greatest common measure of A, B, C.

7.prop.33.p2315

Therefore there cannot be any numbers less than E, F, G which are in the same ratio with A, B, C.

7.prop.33.p2316

Therefore

7.prop.33.p2316

E

7.prop.33.p2316

,

7.prop.33.p2316

F

7.prop.33.p2316

,

7.prop.33.p2316

G

7.prop.33.p2316

are the least of those which have the

7.prop.33.p2316

same ratio with

7.prop.33.p2316

A

7.prop.33.p2316

,

7.prop.33.p2316

B

7.prop.33.p2316

,

7.prop.33.p2316

C

7.prop.33.p2316

. Q. E. D.

7.prop.33.p2316

1

7.prop.34.p2317

Given two numbers, to find the least number which they measure.

7.prop.34.p2318

Let A, B be the two given numbers; thus it is required to find the least number which they measure.

7.prop.34.p2319

Now A, B are either prime to one another or not.

7.prop.34.p2320

First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [VII. 16]

7.prop.34.p2321

Therefore A, B measure C

7.prop.34.p2322

I say next that it is also the least number they measure.

7.prop.34.p2323

For, if not, A, B will measure some number which is less than C.

7.prop.34.p2324

Let them measure D.

7.prop.34.p2325

Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [VII. Def. 15] therefore the product of A, E is equal to the product of B, F.

7.prop.34.p2326

Therefore, as A is to B, so is F E. [VII. 19]

7.prop.34.p2327

But A, B are prime, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore B measures E, as consequent consequent.

7.prop.34.p2328

And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [VII. 17]

7.prop.34.p2329

But B measures E; therefore C also measures D, the greater the less: which is impossible.

7.prop.34.p2330

Therefore A, B do not measure any number less than C; therefore C is the least that is measured by A, B.

7.prop.34.p2331

Next, let A, B not be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken; [VII. 33] therefore the product of A, E is equal to the product of B, F. [VII. 19]

7.prop.34.p2332

And let A by multiplying E make C; therefore also B by multiplying F has made C; therefore A, B measure C.

7.prop.34.p2333

I say next that it is also the least number that they measure.

7.prop.34.p2334

For, if not, A, B will measure some number which is less than C.

7.prop.34.p2335

Let them measure D.

7.prop.34.p2336

And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H.

7.prop.34.p2337

Therefore A by multiplying G has made D, and B by multiplying H has made D.

7.prop.34.p2338

Therefore the product of A, G is equal to the product of B, H; therefore, as A is to B, so is H to G. [VII. 19]

7.prop.34.p2339

But, as A is to B, so is F to E.

7.prop.34.p2340

Therefore also, as F is to E, so is H to G.

7.prop.34.p2341

But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore E measures G.

7.prop.34.p2342

And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17]

7.prop.34.p2343

But E measures G; therefore C also measures D, the greater the less: which is impossible.

7.prop.34.p2344

Therefore A, B will not measure any number which is less than C.

7.prop.34.p2345

Therefore C is the least that is measured by A, B. Q. E. D.

7.prop.35.p2346

If two numbers measure any number, the least number measured by them will also measure the same.

7.prop.35.p2347

For let the two numbers A, B measure any number CD, and let E be the least that they measure; I say that E also measures CD.

7.prop.35.p2348

For, if E does not measure CD, let E, measuring DF, leave CF less than itself.

7.prop.35.p2349

Now, since A, B measure E, and E measures DF, therefore A, B will also measure DF.

7.prop.35.p2350

But they also measure the whole CD; therefore they will also measure the remainder CF which is less than E: which is impossible.

7.prop.35.p2351

Therefore E cannot fail to measure CD; therefore it measures it. Q. E. D.

7.prop.36.p2352

Given three numbers, to find the least number which they measure.

7.prop.36.p2353

Let A, B, C be the three given numbers; thus it is required to find the least number which they measure.

7.prop.36.p2354

Let D, the least number measured by the two numbers A, B, be taken. [VII. 34]

7.prop.36.p2355

Then C either measures, or does not measure, D.

7.prop.36.p2356

First, let it measure it.

7.prop.36.p2357

But A, B also measure D; therefore A, B, C measure D.

7.prop.36.p2358

I say next that it is also the least that they measure.

7.prop.36.p2359

For, if not, A, B, C will measure some number which is less than D.

7.prop.36.p2360

Let them measure E.

7.prop.36.p2361

Since A, B, C measure E, therefore also A, B measure E.

7.prop.36.p2362

Therefore the least number measured by A, B will also measure E. [VII. 35]

7.prop.36.p2363

But D is the least number measured by A, B; therefore D will measure E, the greater the less: which is impossible.

7.prop.36.p2364

Therefore A, B, C will not measure any number which is less than D; therefore D is the least that A, B, C measure.

7.prop.36.p2365

Again, let C not measure D, and let E, the least number measured by C, D, be taken. [VII. 34]

7.prop.36.p2366

Since A, B measure D, and D measures E, therefore also A, B measure E.

7.prop.36.p2367

But C also measures E; therefore also A, B, C measure E.

7.prop.36.p2368

I say next that it is also the least that they measure.

7.prop.36.p2369

For, if not, A, B, C will measure some number which is less than E.

7.prop.36.p2370

Let them measure F.

7.prop.36.p2371

Since A, B, C measure F, therefore also A, B measure F; therefore the least number measured by A, B will also measure F. [VII. 35]

7.prop.36.p2372

But D is the least number measured by A, B; therefore D measures F.

7.prop.36.p2373

But C also measures F; therefore D, C measure F, so that the least number measured by D, C will also measure F.

7.prop.36.p2374

But E is the least number measured by C, D; therefore E measures F, the greater the less: which is impossible.

7.prop.36.p2375

Therefore A, B, C will not measure any number which is less than E.

7.prop.36.p2376

Therefore E is the least that is measured by A, B, C. Q. E. D.

7.prop.37.p2377

If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.

7.prop.37.p2378

For let the number A be measured by any number B; I say that A has a part called by the same name as B.

7.prop.37.p2379

For, as many times as B measures A, so many units let there be in C.

7.prop.37.p2380

Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, therefore the unit D measures the number C the same number of times as B measures A.

7.prop.37.p2381

Therefore, alternately, the unit D measures the number B the same number of times as C measures A; [VII. 15] therefore, whatever part the unit D is of the number B, the same part is C of A also.

7.prop.37.p2382

But the unit D is a part of the number B called by the same name as it; therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. Q. E. D.

7.prop.38.p2383

If a number have any part whatever, it will be measured by a number called by the same name as the part.

7.prop.38.p2384

For let the number A have any part whatever, B, and let C be a number called by the same name as the part B; I say that C measures A.

7.prop.38.p2385

For, since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore, whatever part the unit D is of the number C, the same part is B of A also; therefore the unit D measures the number C the same number of times that B measures A.

7.prop.38.p2386

Therefore, alternately, the unit D measures the number B the same number of times that C measures A. [VII. 15]

7.prop.38.p2387

Therefore C measures A. Q. E. D.

7.prop.39.p2388

To find the number which is the least that will have given parts.

7.prop.39.p2389

Let A, B, C be the given parts; thus it is required to find the number which is the least that will have the parts A, B, C.

7.prop.39.p2390

Let D, E, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, F, be taken. [VII. 36]

7.prop.39.p2391

Therefore G has parts called by the same name as D, E, F. [VII. 37]

7.prop.39.p2392

But A, B, C are parts called by the same name as D, E, F; therefore G has the parts A, B, C.

7.prop.39.p2393

I say next that it is also the least number that has.

7.prop.39.p2394

For, if not, there will be some number less than G which will have the parts A, B, C.

7.prop.39.p2395

Let it be H.

7.prop.39.p2396

Since H has the parts A, B, C, therefore H will be measured by numbers called by the same name as the parts A, B, C. [VII. 38]

7.prop.39.p2397

But D, E, F are numbers called by the same name as the parts A, B, C; therefore H is measured by D, E, F.

7.prop.39.p2398

And it is less than G : which is impossible.

7.prop.39.p2399

Therefore there will be no number less than G that will have the parts A, B, C. Q. E. D.

Reader

Word analysis

Intertext edge

Note