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Definitions 1. A less magnitude is said to be a part of a greater one, when the less is a measure of the greater, or is contained a certain number of tiines in it.
2. A greater magnitude is said to be a multiple of a less one, when the greater is measured by the less, or contains it a certain number of times.
A. There is a series of multiples; as, the first, second, third, &c., of which, waving the etyinology of the word, the magnitude itself is the first, its double is the second, its triple the third, &c.
B. A magnitude may have one, two, or three dimensions, as the case may be; and the proper unit of measure will be a line, square surface, or cube.
3. Ratio is the numerical relation of antecedent and consequent, or the number of times, or parts of times, which the latter contains the former.
Or, Ratio is the numerical relation of measure and magnitude, or the number of times which the measure, or a part of it, may be applied to the magnitude.
Note.—This value of ratio prevails; and the words of several propositions are here changed to correspond with it.
4. Magnitudes of the same kind only, or having some common property, can have a ratio to one another.
5. The first of four magnitudes has the same ratio to the second which the third has to the fourth, when equimultiples of the first and third, also of the second and fourth, being taken ; if the multiple of the first be greater than that of the second, the multiple of the third is greater than that of the fourth; if equal, equal; and if less, less.
6. Magnitudes which have the same ratio are called proportionals; of which it is usually said, “ the first is to the second as the second is to the third ; or, the first is to the second as the third is to the fourth.
Note.-The is to, as above, is expressed by a colon, thus, (:), the as by two colons, thus (::).
7. The ratio of one couplet (or antecedent and consequent) is less than the ratio of another couplet, when the quotient of the former consequent, divided by its antecedent, is less than the quotient of the latter consequent divided by its antecedent.
8. When three terms or magnitudes are proportionals, the ratio of the first to the third is the duplicate, or square of the ratio of the first to the second.
9 When four terms are continued proportionals; that is, when
the second is the consequent of the first, the third that of the second, and the fourth that of the third ; then the ratio of the first to the fourth is the triplicate or cube of the ratio of the first to the second. Such ratios are called compound.
10. And when any number of magnitades of the same kind are in a certain order, however different the ratios of the couplets may be, the ratio of the first to the last of them is the continual product of all the ratios; namely, the product of all the antecedents for an antecedent, and the product of all the consequents for a consequent.
11. In proportionals, taken, two and two, from different series, or from remote terms of the same series, the odd terins, namely, the first, third, fifth, &c., are the antecedents; and these are said to be homologous; so, likewise, the even terms, viz. the second, fourth, sixth, &c., which are the consequents.
Geometers use the terms permutando, or alternando, invertendo, componendo, dividendo, convertendo, ex æquali distantia, ex æquo, and ex æquali, in proportione perturbata, vel inordinata, to siguify various changes in the order, or magnitude of proportionals, and still preserving the equality of the ratios, in which proportion consists. The sense of these terms is expressed in the following examples:
The use of the marks t; -, X, :,:,::, and =, is generally known.
Example 1. By permutation, or alternately ; when, of four proportionals, as A: B::C: D, comes A :C::B: D. See p. 16 of 6.5.
Ex. 2. By inversion; when, of four proportionals, as A : B:: C: D, comes B : A::D:C. See p. B. of b. 5.
Ex 3. By composition; when, of four proportionals, as A :B.:: C:D, comes A + B:B::C + D:D. See p. 18 of b. 5.
Ex. 4. By division; when, of four proportionals, as A:B::C:D, comes A - B:B::C-D:D. See p. 17 of b.5.
Ex. 5. By conversion; when, of four proportionals, as A:B::C: D, comes A : A – B::C:C – D. See p. E of b. 5.
Ex. 6. From equal distance in order; when, of two ranks of proportionals, as A, B, C, D, and E, F, G, H, taken, two and two, in order, namely, A : B :: E: F;-B:C:: F:G; and C:D:: G: H;-it comes to be inferred, that A:D::E: H. See p. 22 of b. 5.
Ex. 7. From equal distance out of order; when, of two ranks of proportionals, as A, B, C, D, and E, F, G, H, taken, two and two, in à cross order, namely, Á : B::G:H;-B:C:: F:G; and C:D :: E:F; it comes to be inferred, that A:D::E:H. See p. 23 of b. 5.
1. Equimultiples of the same, or of equal magnitudes, are equal to one another.
2. Those magnitudes are equal to one another, of which the same, or equal magnitudes, are equimultiples.
3. A multiple of a greater magnitude is greater than the same multiple of a less.
4. One magnitude is greater than another, of which a multiple is greater than the same multiple of the other.
Propositions. 1 Th. If two or more magnitudes be equimultiples of as many parts, each of each; what multiple soever any one of them is of its part, the same shall the sum of all the magnitudes be of the sum of all the parts.
Let AB, CD be two magnitudes, and E, F, two parts ; so that AB be the second multiple of E, and CD the second multiple of F (a); AB+CD is the A second multiple of E+F.
For since in AB there are two magnitudes AG,
there are two, CH, C
In like manner, it AB, CD were third multiples of E, F, their sum would be third multiples of the sum of E and F; and so of any equimultiples whatever.
Also, if there were three, four, or more magnitudes, equimultiples of as many parts; the sum of all the magnitudes would be the same multiple of the sum of all the parts that each magnitude would be of Wherefore, if two, or more, &c.
Q. E. D. Recite (a) def. 1, 2, A of b 5; (6) ax. 2 of b 1.
2 Th. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fo th; then the sum of the first and fifth, and of the third and sixth are equimultiples of the second and fourth respectively.
Let AB, C, DE, F, BH, EL be six magnitudes, A in order: and since multiples are a numerical series of magnitudes (a); let AB=1C, and DE=1F; also, let BH=2C, and EL=2F: therefore the sum of AB, BH=3C, and the sum of DE, EL--3F, the same multiple of C and F. Wherefore, if the first, &c.
Q. E. D. Recite (a) def. 1, 2, A of b 5.
Scholium. It advances no general principle to use letters instead of numbers in the case of equimultiples. There is no variety in the case: it is only necessary to understand that the multiples are the same in every set of magnitudes 80 compared ; and there is no better way to express the same multiples than by the same numbers undisguised with symbols.
3 Th. If the first be the same multiple of the second which the third is of the fourth; and if of the first and third equimultiples be taken, these shall be equimultiples, one of the second the other of the fourth.
Let A, B, C, D be four magnitudes, in order; F such that A is the second multiple of B, and C the second multiple of D; and if EF be taken the second multiple of A, and GH the second multiple of C: then will EF be the fourth mul- kt. tiple of B, and GH the fourth multiple of D (a).
In like manner, if A be the third multiple of B, and C the third multiple of D; and if EF be taken the second multiple of A, and GH the second multiple of C; then EF is the sixth multiple of B, and GH the sixth multiple of D.
And, in general, because B measures A as often as D measures C; and A measures any multiple of itself as often as C measures the same multiple of itself: therefore B measures any multiple of A, as often as D measures the same multiple of C (a).
Wherefore, if the first be the same multiple, &c. Q. E. D.
4 Th. If the first of four magnitudes have the same ratio to the second which the third has to the fourth ; then any equimultiples of the antecedents shall have the same ratio as any equimultiples of the consequents.
Let A, B, C, D, be four magnitudes; such that A is to B as C is to D; of which A and C are the antecedents, and B and D the consequents (a). Take E, F equimultiples of A, C, and G, H equimultiples of B, D, It is inferred that È is to F as G is to H (6).
Take K, L equimultiples of E, F, and M, N equimultiples of G, H. Then, because E, F are equimultiples of A, C, and K, L of E, F, the same K, L are equimultiples of A, C (c). ' Likewise, because G, H are equimultiples of B, D, and M, N of G, H, the same M, N are equimultiples of B, D (c).
And since A is to B as C is to D, by hypothesis, K is to L as M is to N. Therefore, if K be greater than M, L' is greater than N; if equal, equal; and if less, less (). And K, L are equimultiples of E, F, and M, N of G, H: therefore E is to F as G is to H. Wherefore, if the first of four, &c.
Q. E. D. Cor. And, if A is to B as C is to D; that is, if 1A is to 1B as 1C is to 1D, then 2A is to B as 2C is to D; or A is to 2B as C is to 2D; and 80 of any equimultiples whatever.
Recite (a) def. 3, 5; (6) def. 5, 5; (c) p. 3, 5;
5 Th. If one magnitude be the same multiple of an. other, that a part of the one is of a part of the other, the remainder shall be the same multiple of the remainder that the whole is of the whole.
Let AB be the same multiple of CD that the part AE is of the part CF; the other part EB is the same multiple of the part FD that AB is of ĈD.
Make AG the same multiple of FD that AE is of CF; then AE is to CF as EG is to CD, or AB to CD: hence AB Ç and EG are equimultiples of CD, and are equal to each other (a). Take AE from both; then AG and EB are left equal (B). But AG is to FD as AE is to CF, or as AB is to CD: therefore EB is to FD as AB is to CD; or, in other words. EB is the same multiple of FD that AB is of CD.
Q. E. D. Recite (a) p. 1, and ax. 1, 5; (6) ax. 3, 1.
6 Th. If two magnitudes be equimultiples of other two; and if equimultiples of the latter be taken from the former; the parts left, if any, will be equals, or equimultiples of the other two.
Let the two magnitudes AB, CD, be equimultiples of IK the two, E, F; and the parts AG, CH equimultiples of the same E, F; the remainders GB, HD, are equal to E, F, or they are equimultiples of them.
1. If GB equal E, HD will equal F: make CK=F. Then, since AG, CH are equimultiples of E, F; and GB=E, and CK=F; therefore AB, KH are equimultiples of E, F; but AB and CD are equimultiples of E, F; therefore KH equals CD (a). Take CH from both; the remainders KC, HD are equal: but KC=F.
Therefore HD is equal to F
Q. E. D. Recite (a) ax. 1, 5; (6) p. 2, 5.