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and if a and b be the numbers of times any
A+B C+D common measure of the first and second adding 1 to each side,
B D is contained in them respectively, and which is the theorem cited by the Latin c and d the 'numbers of times any com
word componendo; again_subtracting mon measure of the second and fourth is
which is contained in them, The converse
B D od
that cited by the word dividendo : di
'C, is likewise true, that is, if
B D 6 d viding 1 by each side, and mul
А four magnitudes must be proportionals:
A B for it may easily be shown that fractions tiplying into each,
which are do not represent the same part of the whole unit
, or, which is the same thing, the theorems cited by the words inverare not equal to one another, except tendo and alternando respectively. And they be reducible to the same lowest with equal readiness the rest may be
derived from the same equation. As terms, as * and these lowest terms they are all however sufficiently conwill express, at once, the ratio of the sidered in their several places, the refirst to the second, and of the third to mainder of this scholium will be confined the fourth (1. Cor. 3.),
to the explanation of certain rules comThe same conclusion may be stated in
monly practised in the treatment of proother words, as follows:
portions, which suppose the terms A, B,
C, D to be numbers, and are at once deIf four magnitudes be proportionals,
А с. and if A, B, O, D, represent those mag- rivable from the equation nitudes numerically, that is, if A and B
Bū represent the numbers of times the unit Rule. 1. If four magnitudes, which are of their kind is contained in the two first, numerically represented by A, B, C, D, and if C and D represent the numbers be proportionals, the product of the exof times the unit of their kind is con- tremes will be equal to the product of tained in the two last, the quotient or the means, i. e. if A:B::C:D, AXD A
=BxC: and conversely if AXD=BxC fraction shall be equal ta : and con- the magnitudes represented by A, B, B Ō
C, D will be proportionals. versely.
is the founda. For, since B=multiplying each side țion of the theory of proportion as it by B*D, AXD=BxC: and hence, conis treated in Arithmetic or Algebra, versely, dividing each side by BxD,
A с (see Arithmetic, art. 127 and 128.) and i.e. A, B, C, D are proportionals. leads with great facility to all the theo- BD rems of the foregoing section, Thus,
Rule 2. The same things being sup
posed, any common factor M, which is * It is shown in the treatise on Arithmetic (art, found in both the antecedents, or in prime factors: from this it follows, that if a Xd=bxe antecedent and consequent of the same 61.) that the same number can have but one set of both the consequents, or in both the (as is the case when Arith, art. 127.), and ratio, may be expunged, if a be prime to b, the other factor d on the first side
Κx must contain b, and therefore must be of the form
dividing both I Xb, where l is some whole number multiplying the
B D whole number b; and hence it is evident that c is
K L likewise of the form 1 X a, where l is the same whole sides by M, number multiplying the whole number a. Therefore
KXMLXM' multiplying e
a equimultiples of a and b, that is is reducible to
A С d
7 both sides by M, by dividing and d by their greatest common factor
: (see Prop. 5. Scholium); and hence likewise if
Κx M o K C. and if a be not prime to b, nor e prime to do
L XM D' b
Κx M K the fractions and and must be reducible to the same
LXMI ' lowest terms by dividing a and b by their greatest Or the Rule may be thus stated : common factor, and u and d by their greatest com mon factor,
Expunge all common factors, except
This equation =D
they be common only to the two ex- Thus if K:B::C:D tremes, or to the two means. For, in
and B:B'::C' :D all other cases, if, as before, KXM and
and B': L ::C":D". Lx M be the terms having a common by compounding the proportions, KB factor, and if P and Q be the terms not
XB: B x B'XL:: CX CX C" : Dx D' having a common factor, by multiplying xD", and hence by Rule 2, K:L:: extremes and means, K x P x M =
Cx C'xC":DX D'xD".
Section 3.—The General Theory of Of these three cases, the one which
Proportion. occurs most frequently is the last, viz. The foregoing theorems have been esthat in which the common factor is tablished upon the supposition that the found in the terms of the same ratio.
magnitudes spoken of are commensuRule 3. If there be two or more pro- rable. This, however, is not always the portions, the products of the corre- case with magnitudes : there are some sponding antecedents and consequents (examples will appear in a future page) shall constitute a proportion.
which have no common measure, and A C
A' C' which are therefore said to be incomFor if
and if also BD'
B' =D mensurable. multiplying together the quantities upon lations of such magnitudes will be briefly
In the present section, the similar rethe left hand, and also those upon the considered ; a new definition will be laid right of the two equations,
down, comprehending that already given AXA CXC'
(def. ) of the proportion of commenBxB=DxD
surable magnitudes, at the same time i.e. if A :B::C:D
that it does not require that the magniand A' : B'::C': D'
tudes which satisfy it shall be commenAXA' : BB'::CxC': DxD'. surable ; and to this new definition the Again, if there be a third proportion
theorems of the preceding section will A":B" :: C":D", the terms of this be shown to apply equally as to the
former. being multiplied into those of the preceding, AX AX A" : B x B'x B'' :: Cx
In the first place, then, it is evident C'xC" : Dx D'xD"; and so on, if that, incommensurable magnitudes havthere be any number of proportions.
ing no common part, their ratio can When this Rule is applied, the result: Numbers may nevertheless be obtained
never be exactly expressed by numbers. ing proportion is said to be compounded of the others, and hence the rule is which shall serve to compare two such called, “ compounding the proportions." magnitudes A and B to any required deWe may observe that the compound gree of accuracy, proportion commonly admits of reduc- ber of equal parts, a million for ex
Let' B be divided into any large numtion by Rule 2., on account of the same term or terms occurring in more than ample: then À will contain a certain
number of these parts with an excess one of the component proportions.
which is less than one of them, less, If, for example, A:B :: 0 :D and B' :A::D :C
that is, than a millionth part of B; so
that if we take no account of this excess and A": C':: B' :D",
in our estimate of their relative magni. the proportion which is compounded of tude, we shall commit an error of less these is Ax B'x A": BXAXC':: C* than one-millionth. And it is plain that, DxB': Dx C'xD": which is reducible in this manner, by dividing B into a still by Rule 2. to A" : B::C:D".
greater number of equal parts, the error The terms are, however, seldom (or of our estimate may be made as small never) so intermixed as in this example. as we please. The end which is usually proposed in
It is found, for example, (by methods the compounding of proportions is to which will be noticed hereafter) that, obtain the ratio of one magnitude K to if the diameter of a circle be divided into another L by means of a number of 7 equal parts, the circumference will intermediate magnitudes; in order to contain not quite 22 of those parts; if, which, K is made the antecedent of the again, the diameter be divided into 113 leading ratio of the first proportion, and equal parts, the circumference will conL the consequent of the like ratio in the tain not quite 355 of those parts : if into last.
10,000,000, the circumference will con
tain not quite 31,415,927; and so on. cond which the third has to the fourth, Therefore the ratio of the circumference when any like parts whatsoever of the to the diameter is expressed by the second and fourth are contained in the ratio 22:7 nearly; more nearly by 355: first and third the same number of times 133; still more nearly by 31,415,927 : exactly, or the same number of times 10,000,000; and so on: nor is there with corresponding remainders less than any limit to the accuracy of this ap- the parts. proach, although there should (as is For example: let ABCD, EFGH be really the case) be no two numbers by two rectangles having the same altitude, which it can be expressed exactly, and let A B, EF be their bases. Let
This consideration brings us directly the base E F be divided into any number to the only case in which, consistently of equal parts E f, &c., and the rectwith the view already taken of the sub- angle E F G H into as many equal rectject of equal ratios, two magnitudes may angles, E f g H, &c. by lines drawn be said to be similarly related (or in the sume ratio) to two others of the same kind respectively, with which they are incommensurable. The ratios of the former to the latter, each to each, must admit of being approximately represented by the same numbers, to how great an through the points of division parallel extent soever the degree of approxima- to EH. Then if A b, &c. be taken tion may be carried : in other words, any equal to E f, and if straight lines be like parts whatsoever of the two latter drawn through the points b, &c. parallel magnitudes, however minute they may to AD; the base A B, and the rectangle be taken, must be contained in the two ABCD, will contain, the one a certain former, each in each, the same number number of parts equal to E f, and the of times, with corresponding* remain- other the same number of rectangles ders less than the parts. It has been already observed that equal to E f g H, either exactly, or with
corresponding remainders less than Ef this obtains with regard to the propor- and Efg H. And this will always be the tionals of def. . When four magni- case, whatsoever be the number of parts tudes are proportionals by that defini, into which E F is divided. Therefore, tion, which supposes the first two, and according to def. 7., the two rectangles second two to be commensurable, there and their two bases are proportionals. are, indeed, some like parts of the second and fourth which are contained in tudes contain any part of the second a
Def. 8. If the first of four magnithe first and third the same number greater number of times, with or withof times without remainders; viz. the out a remainder, than the third congreatest common measures of the first tains the like part of the fourth, the first two and second two, and any like parts is said to have to the second a greater of the greatest common measures : it is ratio than the third has to the fourth : easy to perceive, however (and the same
also, in this case, the third is said to has been demonstrated at large in Prop. have to the fourth a less ratio than the ), that any other like parts of the first has to the second. second and fourth will be contained in
As from Prop.  with regard to the first and third the same number of
commensurable proportionals, so from times, with corresponding less remain- the terms of our new general definitions ders. The following, therefore, is to be
7. and 8. with regard to the proporconsidered as the general test of two tionals described in def. 7. it is at once magnitudes A and C, having the same
evident that of four magnitudes, A, B, ratio to two others B and D, of the C, D, the first A cannot be said to same kind with the former two re
have to the second B the same ratio spectively.
which the third C has to the fourth Def. 7. The first of four magnitudes D, according to def. 7, and at the same is said to have the same ratio to the se- time a greater or a less ratio than C
has to D, according to def. 8.: much * By the word “corresponding" here used, it is less can A be said to have to B at the merely intended to point out the fact of there being
same time both a greater and a less ratio two remainders, i.e. a remainder in the comparison of the two first magnitudes, and a remainder corre- than C has to D.* sponding to it in the comparison of the two last. And
* See note at Prop. 19.] subsequent passages on the same subject.
in the same sense the word is to be understood in
We proceed to the properties of this B, C will contain, also, by the supposimore general description of propor- tion, more than m' and less than m' + 1 tionals, which will be found the same of the like parts of D, that is, the same with those already demonstrated in the number m' of these with a remainder, preceding Section of commensurable and this, whatever be the value of n. proportionals. They will be considered Therefore, (def. 7.) A, B, C, D are proaccordingly in the same order, and will portionals. have the same numbers affixed to them. Therefore, &c. It will be observed, also, that they are Cor. 1. The same had been expressed stated in the same words, with the excep by saying, “if A, B, C, D be proportion of Prop. 9., which is little more than another form of expressing def. 7., and tionals, and if A be greater than B, its corollaries, which again express the same thing in different terms, the 2d. C shall be likewise greater than
n of them being, in fact, Euclid's celebrated definition of proportionals. if equal equal, and if less less, whatso
ever values may be given to m and n: PROP. 9.
and, conversely, if this be the case with
four magnitudes, they shall be proporIf four magnitudes A, B, C, D be tionals.” proportionals, the first and third shall
Cor. 2. (Euc. v. def. 5.) Or, again, contain respectively ths of the second
“if A, B, C, D he proportionals, and if n A be greater than m B, n C shall like
wise be greater than m D, if equal equal, and fourth, or both more than ths, or and if less less, whatsoever values may
be given to m and n: and, conversely, if both less than ths, whatsoever values
this be the case with four magnitudes,
they shall be proportionals." may be given to m and n; and conversely, For n A, m B, in C, m D are equimulif this be the case with four magnitudes, they shall be proportionals.
tiples of A,
B, C, D; and of the for, by def. 7., if A contain exactly equimultiples of two magnitudes, one
. ths of B, C must contain exactly will be greater than, or equal to, or less
than the other, according as the correths of D: or, again, if A contain more
sponding magnitude is greater than, or
equal to, or less than the corresponding than m of the nth parts into which B magnitude of the other; and conis divided, as m' (suppose), or m' with a versely (ax. 1, 2, 3, 4).* remainder, where m' is greater than m, C must also contain m' parts, or m' with
* “The first of four magnitudes is said to have the a remainder, that is, more than m of the
same ratio to the second, which the third has to the ntla parts into which D is divided : and fourth, when any equimultiples whatsoever of the
first and third being taken, and any equimultiples in like manner, if A contain less than
whatsoever of the second and fourth ; if the multiple
of the first be less than that of the second, the multiths of B, C will likewise contain less ple of the third is also less than that of the fourth ; n
or, if the multiple of the first be equal to that of the
second, the multiple of the third is also equal to than "ths of D.
that of the fourth ; or, if the multiple of the irst be
greater than that of the second, the multiple of the Next, let this be the case with four
third is also greater than that of the fourth."
Euc. y. def. 5. magnitudes, A, B, C, D, whatsoever This detinition of proportionals has been sometimes numbers be substituted form and n:
found fault with as too abstrnse and recondite for be
ginners ;-which would not perhaps have been the A, B, C, D shall be proportionals. case, had its connexion with the more obvious but For, if A and C contain exactly m of contined view of def. 17 ] been always pointed out.
For, we have seen that a general theory of propor. the n th parts into which B and D are
tion, which shall embrace indifferently all magnidivided, the four A, B, C, D are com- tudes, whether commensurable or otherwise, admits mensurable proportionals, according to
of no test essentially different from that which is
here adopted. The greatest geometers in dwelling def. 7; and, therefore, also (by Prop. upon this part of the Elements have ever found  of the last Section, as has been al- cause to admire the profoundness and sagacity of
Witness the energetic testimony of ready observed) proportionals accord
Barrow, “ That there is nothing in the whole body ing to def. 7. Again, if A contain more of the Elements of a more subtile invention, nothing than m, as m' and a remainder, that is
more solidly established, or more accurately handled,
than the doctrine of proportionals.” Euclid has, more than m' and less than m' +1 parts indeed, left little in this respect, as in others, to be
Cor. 3. (Eue. v. def. 7.) In like man- Let D be the difference of A and B : ner, it may be shown, that, if there be then, whether B and D be or be not both four magnitudes A, B, C, D, and if a of them greater than C, multiples m B, multiple of A, as mA, can be found m D may be taken of them which are both which is greater than a multiple of B greater than C. And, because C is less as n B, while the corresponding multiple than m B, let multiples of C be taken, as of C, viz. m C, is not greater than the 2C, 3 C, &c. until a multiple be found, corresponding multiple of D, viz. n D,
as p+1.C, which is the first greater than A has to B a greater ratio than C has
m B. Then, because m B is not less than to D, For, A will be greater than
the preceding multiple p C, and because
m Dis greater than C, the two m B, m D of B, but C will not be greater than together are greater than p C and C togeof D; that is, the nth part of B ther, that is, than p+1.C. But, because ;
A is equal to B and D together, it is eviwill be contained in A m times or more dent that m A is equal to m B and m D with a remainder, but the nth part of together. Therefore m A is greater than D will not be contained in C so much as p+1. C, and mB less than p+1.C. m times with a remainder. (See def. 8.)
Therefore the p+1th part of A is conPROP. 10. (Euc. v. 7.)
tained in C, not so much as m times, Equal magnitudes have the same ratio and the p+Ith part of B is contained in to the same magnitude : anil the same
C, m times with a remainder. There. has the same ratio to equal magnitudes. fore (def. 8.) C has a less ratio to A than
For, any the same part of the same it has to B, or a greater ratio to B than magnitude will be contained the same it has to A. (See also 9. Cor. 3.) number of times in equal magnitudes
Therefore, &c. with corresponding less remainders. Cor. 1. (Euc. v. 9.) Magnitudes which And, again, any like parts of equal have the same ratio to the same magnimagnitudes, being equal to one another tude are equal to one another : as like(ax. 2), will be contained the same num
wise those to which the same magnitude ber of times with the same remainder in has the same ratio. the same magnitude.
Cor. 2. A ratio which is compounded Therefore, &c.
of two ratios, one of which is the reciCor. If a ratio which is compounded procal of the other, is a ratio of equality: of two ratios be a ratio of equality, one for, A having to Ca ratio which is com. of these must be the reciprocal of the pounded of the ratios of A to B and other. (See  Cor.) ;
of B to C, if the latter be the same
with the ratio of B to A, A must be PROP. 11. (Euc. V. 8.)
equal to C. of two unequal magnitudes, the Cor.3. (Euc. v. 10.) If one of two maggreater has a greater ratio to the same nitudes have a greater ratio to the same magnitude : and the same magnitude has magnitude than the other has, the first a greater ratio to the lesser of the two. must be greater than the other: and if
The first is evident: for, of any the the same magnitude have a greater ratio same magnitude a part may be found to one of two magnitudes than it has to less than the difference of two unequal the other, the first must be less than the magnitudes, which part must evidently other. be contained a greater number of times
PROP. 12. (Euc. y. 11.) in the greater than in the lesser of the two. In the second place, therefore, let A
Magnitudes A, B and C, D, which and B be any two magnitudes of which have the same ratio with the same magA is the greater, and let C be any third nitudes P, Q, have the same ratio with magnitude: C shall have to the greater one another. A a less ratio than the same C has to For any part of B will be contained the other B.
in A exactly or with a rema nder, as
often as a like part of Q is contained in done by succeeding writers, but to follow his steps P exactly or with a remainder, because as closely as they are able: the principles, the A:B:: P:Q, that is, as often as a theorems, the demonstrations they are in search of are all to be found in this one masterpiece, and for like part of D, is contained in C exactly, the most part under the simplest form.
or with a remainder, because C:D:; it will, of course, be easily understood how large a
P:Q. portion of the present section is borrowed from the 5th Book of the Elements,