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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 sewn in the fourteenth propofition, if four magnitudes be proportional, and the first be greater than the third ; the second shall be greater than the fourth ; and if equal, equal ; if less, less. 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 fecond al. ways less than that of the fourth. Then it necessarily follows that those four proposed magnitudes will be proportional; fo that from hence it moft 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 so simple 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 ufe 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 incommensurable 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 lor. 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 Eu

clid;

10 have a greater ratio to the second, than the third has to the fourth .

8. Proportion is a similitude of ratios. clid; 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 asignable magnitudes, may be taken for iquals, or represent one another.

2. That any four proposed magnitudes may either be accurately expressed in numbers, or else four magnitudes that differ from them by magnitudes lefs than any asignable ones may; and therefore.

3. Any four magnitudes may be taken for proportionols, the first to the second, and the third to the fourth, when the product of the multiplication of the numbers representing or measuring the forf and fourth is equal to the product of the multiplication of the numbers reprefinting or measuring the fecond and third magnitudes; and accordingly.

4. Whatsoever 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 rbey represent or measure.

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 second and fourth. If it Thall be at any time found (altho' not always) that the multiple of the firit 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 manifest by the annex'd example; wherein the magnitudes E, F are taken thrice the first magnitude A, and the third c, and the magnitudes G, A, four times the fe

and the fourth d. And becaufe 2, the multiple of the first magni

III tude, is greater than G, the multiple of EABGFCDH the second; but f the multiple of the third is not greater than

the multiple of the fourth, but less; 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 D.

P

cond B,

9. There

9.

There must be three terms at least to constitute próportion f.

10. If three magnitudes are proportionals, the ratio of the first to the third is said to be the duplicate of the ratia of the first to the second 8.

11. If four magnitudes be [continual] proportionals, the ratio of the first to the fourth is said 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

quents .

13. Alternate ratio is the assumption of the antecedent to the antecedent, and of the consequent to the conse

quenti.

14. Inn

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 de finitions 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 fubduplicate 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.

I think this definition is not clearly express'd; nay, I have often thought it to be scarcely fensé; and therefore instead thereof I hould rather say, When four magnitudes are propose tional, as tbe forft is to the second, so is the third to the fourth; the firfi magnitude and the third, or the antecedents, as also the second and the fourth, or the consequents of the two equal ra, tios, are called homologous or co-rational terms. But this I fube 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 toc, 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 A 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

between

14. Inverse ratio is the assumption of the con eq ients, as the antecedent to the antecedent taken as the conse

quentk

15. Composition of ratio is an assumption of the ante. cedent, together with the consequent as one to the same consequent

16. Division of ratio is the assumption of the excess, whereby the antecedent exceeds the consequent, to the fame consequent m.

17 Converse ratio, is the assumption of the antecedent to the excess whereby the antecedent exceeds the consequent "

18. Ratio of equality is, when there are several magnitudes in one rank or order, and as many others in another rank or order, comparing two to two, being in the fame ratio, it shall be in the first rank as the first mignitude is to the last, so in the second rank shall the first magnitude be to the last. OR ELSE it is the assumption of the extremes by taking away the intermediate terms.

between a line and a number, as is evident from the fifth definition of this fifth book. But in the next following ways of arguing, viz. by inverse ratio, composition, division, and conversion, the two first magnitudes may be of one kind, and the two last of another, as is evident by the demonftrations of this book.

k Let the magnitude a be to the magnitude B, as the maga nitude c is to the magnitude D.

Then inversely (by cor. 4. 5.) as B is to a, so is D to c.

1 As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D. Then by composition (18. 5.] as a and B together is to B, so is c and D together to D.

m As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude d. Then by division (17.5.] as the difference between A and B is to B, so is the difference between c and D to D.

- As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D. Then conversely [by cor. 19. 5.] as a is to the difference between A and B, so is c to the difference between c and D.

• As let there be three magnitudes A, B, C in one rank or order, and three magnitudes D, E, F in another order. And let a be'to B, as p is to e, and B be to c, as e is to F. Then by equality [by 22. 5.) will it be as A is to c, fo is a to F.

P 2

19. Ordi.

19. Ordinate proportion is, when it shall be as an antecedent to a consequent, so is an antecedent to a consequent; and as a consequent is to some other magnitude, ro is a consequent to some other magnitude P.

20. Inordinate or perturbate proportion is, when there are three magnitudes, and the same number of others, it Thall be in the first magnitudes as an antecedent is to a confequent ; so in the second magnitudes is an antecedent to a consequent: But as in the first magnitudes a consequent is to some other magnitude, so in the second magnitudes is some other magnitude to the antecedent %

P As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D : and again let one of the consequents B, be to some other magnitude c, as the other consequent e is to some other magnitude F. Then is this ore dinate proportionality; and it Thall be true [by 22. 5.) that a iš to c, as D is to F. I take this definition to be almost useless ; it being in a manner contained in the last.

9 As let the magnitude a be to the magnitude B, as the magnitude e is to the magnitude F. And again, as in the first inagnitudes, the consequent b is to some other magnitude c; so in the fecond magnitudes is fome other magnitude o to the antecedent magnitude E. This sort of proportion is called inordinate or perturbate, because the fame order is not kept in the proportion of the magnitudes. And it will be [by 23. 5.] as A is to c, so is D to F.

PROPOSITIONI. THEOREM. If there be how many soever magnitudes equimultiples

of as many other magnitudes, each of each ; the same multiple one magnitude is of one, all fall be

of all.

Let any number of magnitudes AP, CD be equimultiples of the fame number of magnitudes E, F, each of each : I say, A B and C D is the same multiple of E and F, as A B is of E.

For because A B is the same multiple of E, as 6 D is of I; as many magnitudes as there are in A B equal to B, lo

many

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