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4. Magnitudes are said to have a ratio to one another, which being multiplied can exceed each other.
5. Four magnitudes are said to be in the fame ratio, the first to the second, and the third to the fourth: When the equimultiples of the first and third compared with the
equiAs some take the numerical exponent or measure of a ratio to be the quotient of the division of the number expressing the measure of the antecedent term of that ratio by the number expressing the consequent term: As if the magnitude a be the antecedent, and the magnitude B the consequent, and a be three ünites, and B be one unite of the same kind; then will the numerical exponent or measure of the ratio of A to B be 3. And if B be the antecedent, and a the consequent, the numerical exponent of the ratio B to a will be one third.
So there are others who call the logarithms the numerical exponents or measures of ratios, the very meaning of the word implying it, viz. the ratio of numbers. And accords ingly these take the measures of ratios to be either the absciffes of the logarithmic curve, or the sectors of equilateral hyperbolas expressed in numbers. See Mr. Coate's Harmonia menJurarum.
« Magnitudes of different kinds, such as lines and superficies, angles of contact and right lined angles, superficies and solids, lines and solids, cannot have any ratio to one another ; because a line never so often multiplied will never exceed, or ever become a superficies; nor a superficies exceed or become a solid; nor any number of angles of contact become equal to, much less exceed, any the least right lined angle.
Euclid has not expresly told us what the quantity of a ratio is, yet from the fifth definition of the sixth book, one would think he (if he was the author of it) meant by it the quotient of the division of the antecedent term of the ratio by the consequent term. But whether it be taken as this quotient, or whether it be taken as the logarithm of that quotient, there can no error arise from either of these fuppofitions when they are rightly understood. There has indeed been a good deal of difpute amongst mathematicians about this : But it has chiefly arose from both parties not considering either of those fuppofitions might be taken at pleasure, and because some having taken it to be the one, and some the other, they have both reasoned accordingly; and so disagreed in their conclusions, as they certainly must, because of their different premises. I readily own, that the fupposition of the measure of a ratio being a quotient is most fimple and easy. But it is more apt to mislead, and indeed is not quite so accurate as the supposition of its being the logarithm of that quotient, which is a more difficult and remote consideration.
equimultipes of the second and fourth, according to any multiplication whatsoever, are either together deficient or together equal, or together exceeding each other. 4. N. B. Instead of the word equimultiples it would be better to say equal multiples the meaning of this word being easier understood than of that word.
6. Magnitudes which are in, or have the same ratio, are called proportionals. N. B. When four magnitudes are proportionals it is usually expressed by saying, the first is to the second, as the third to the fourth.
Let there be four magnitudes A, B, C, D where the first A is compared to the second B, and the third c to the fourth
Let E be any multiple whatsoever of the first A, and p the same multiple of the third c, as let e be the double, triple, quadruple, &c. of A, and f the double, triple, quadruple, &c. of c. And again let G be any multiple of the fe- A
Eroty cond magnitude B, either the same as
Brey G before E was of A, or F of c, or any o, Cr
Ft ther whatsoever; and let H be the same
H multipleof the fourth magnitude D, as G is of B.
Then whenfoever it is demonstrated that according to any multiplication whatsoever, the equimultiples E, F of the first magnitude A and the third c, compared to the equimultiples G, H of the second B and fourth D, each to each, e to G, and r to H.
I say when these equimultiples are proved to be together, less, or equal, or greater, e than G, and r than . Then these four magnitudes A, B, C, D are said to be in the same proportion A to B, as c to D. And so when in any one particular instance the contrary shall be demonstrated, that the equimultiple either of A exceeds the equimultiple of B, and the other exceeds not, but is either equal or less ; Then the proposed magnitudes A, B, C, D are not in the same proportion, the first to the second as the third to the fourth, because the agreement of the equimultiples in joint defect, equality, or excess ought to hold in any multiplication whatsoever.
Many have disliked this definiton of Euclid, and thought it foreign, difficult, and obscure. But this was for want of a thorough understanding thereof. This definition is plainly and clearly deduced from the fourth and fourteenth propofiti
7. But when amongst the equimultiples (of four magnitudes) the multiple of the first (magnitude] shall exceed that of the second, but the multiple of the third shall not exceed that of the fourth ; then the first magnitude is said
to ons of the fifth book. For since it is demonstrated in the fourth proposition that any equimultiples of the first and third of four proportional magnitdes, are proportional to any equimultiples of the second and fourth, and because it is shewn in the fourteenth propofition, if four magnitudes be proportional, and the first be greater than the third ; the second fall be greater than the fourth ; and if cqual, equal; if less, less.
l Therefore when four magnitudes are proportional, if any equimultiple of the first be greater than that of the third, any other equimultiple of the second will be greater than that of the fourth. If equal, equal ; if less, less. Wherefore, on the contrary, when there are four proposed magnitudes, and any equimultiples of the first and third be taken, as also any other equimultiples of the second and fourth; and the equimultiple of the first be always greater than that of the third; and that of the second at the same time always greater than that of the fourth; if that of the first be equal to that of the third ; that of the second always equal to that of the fourth; or if that of the first be less than that of the third, that of the second always less than that of the fourth. Then it necessarily follows that those four proposed magnitudes will be proportional ; fo that from hence it most clearly appears how Euclid obtained this definition, and that it is a very simple, natural, and easy Sign of proportionality, derived from the before mentioned two propofitions. However otherwise it may at first appear to those who will not be at the pains to consider it. It is true it is not fo fimple and plain as the definition of proportional numbers, or that which might begiven of commensurable magnitudes, Nor does it at all agree with the common notion that the genenerality of mankind conceive of proportionals. Yet in use and practice it is most plain and easy. Euclid could not have given any other so elegant and general a definition that would take in incommenfurable magnitudes, as well as numbers and commensurable ones; and therefore he did right to give this rather than a worse, See. Dr. Barrow's full and learned defence of this definition of Euclid, in his 21st and 22d Mathematical Lectures ; at the end of the latter whereof the I'r. concludes in these words : There is nothing extant in the whole work of the Elements of Euclid more fubtilely invented, more solidly established, or more accurately handled, than the doctrine of proportionality.
Some have given other definitions of proportional magnitudes, as Borellus, in his Euclides relitutus ; Taquet, in his Ex
10 have a greater ratio to the second, than the third has to the fourth e.
8. Proportion is a similitude of ratios. did; and Malcolm, in his Arithmetic. But as I have already said, Euclid's is the best of them all.
I have sometimes thought that the doctrine of proportionality, in all quantities whatsoever, might be easily derived from the following positions, and Euclid's seventh book.
1. That those quantities that differ from one another by ma nitudes less than any affignable magnitudes, may be taken for iquals, or represent one another.
2. That any four proposed magnitudes may either be accurately expréfed in numbers, or elje four magnitudes that differ from them by magnitudes less than any alignable onės may; and therefore.
3. Any four magnitudes may be taken for proportionols, the sfirit to the second, and the third to the fourth, when the product of the multiplication of the numbers representing or measuring the firA and fourth is equal to the product of the multiplication of the numbers reprefinting or measuring the fecond and third magnitudes; and accordingly. 4. What
foever properties of proportionality are deduced and demonstrated from the definition of proportional numbers, or from Euclid in his seventh book, will hold good of the proportionality of any magnitudes. foever which they represent or measure.
e Euclid here declares what condition four magnitudes ought to have when the ratio of the first to the second is greater than that of the third to the fourth, saying that taking equimultiples of the first and third, and of the fecond and fourth. If it shall be at any time found (altho' not always) that the multiple of the first is greater than the multiple of the second ; but the multiple of the third not greater than that of the fourth, but less than it, or equal to it; then the ratio of the first magnitude to the second is said to be greater than the ratio of the third to the fourth, as is manifeft by the annex'd example; wherein the magnitudes E, F are taken thrice the firit magnitude A, and the third c, and the magnitudes G, #, four times the second B, and the fourth D. And becaufe E, the multiple of the first magni
III tude, is greater than G, the multiple of EABGFCDH the fecond; but r the multiple of the third is not greater than h the multiple of the fourth, but lefs; the ratio of the first magnitude A to the second B, is said to be greater than the ratio of the third c to the fourth p.
9. There must be three terms at least to constitute proportion f.
10. If three magnitudes are proportionals, the ratio of the first to the third is said to be the duplicate of the ratio of the first to the second 8.
11. If four magnitudes be [continual] proportionals, the ratio of the first to the fourth is faid to be triplicate of the ratio of the first to the second, and so forwards always more by one, as long as the proportion shall be continued.
12. Magnitudes are homologous, when the antecedents are to the antecedents, as the confequents to the conse
13. Alternate ratio is the assumption of the antecedent to the antecedent, and of the consequent to the conse
f Both this definition and the last appear to be superfluous, and of no manner of use; and therefore I rather take them to be remarks of some learner put into the text, than genuine definitions of Euclid himself.
8 As subduplicate and subtriplicate ratio are not mentioned by Euclid, and as these are very useful, I thought it might not be amiss to define them here. Accordingly if three magnitudes be proportional, the ratio of the first to the second, or that of the second to the third, is said to be subduplicate of the ratio of the first to the third. And if four magnitudes be continual proportionals, the ratio of the first to the second, or that of the second to the third, or that of the third to the fourth, is said to be subtriplicate of the ratio of the first to the fourth.
n I think this definition is not clearly express'd ; nay, I have often thought it to be scarcely fensé; and therefore instead thereof I should rather say, When four magnitudes are proportional, as tbe forft is to the second, fo is the third to the fourth ; the firf magnitude and the third, or the antecedents, as also the fecond and the fourth, or the consequents of the two equal ra. tios, are called homologous or co-rational terms. But this I fub mit to the learned,
1 As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude d. Then alternately, or by permutation (see prop. 16.] it will be as a to c, so is B to d.
Alternate proportion cannot take place, unless the four proposed magnitudes be all of the same kind. For if a line Å be to à line B, as a number c is to a number D, it would not be right to infer by alteration, that as the line a is to the number c, so is the line 3 to the number D, because there is no ratio