7. But when amongst the equimultiples [of four magnitudes] the multiple of the firft [magnitude] fhall exceed that of the second, but the multiple of the third shall not exceed that of the fourth; then the firft magnitude is faid

to

ons of the fifth book. For fince it is demonftrated in the fourth propofition 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 fhewn 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 lefs, lefs. Therefore when four magnitudes are proportional, if any equimultiple of the first be greater than that of the third, any other equimultiple of the fecond will be greater than that of the fourth. If equal, equal; if lefs, lefs. Wherefore, on the contrary, when there are four propofed magnitudes, and any equimultiples of the first and third be taken, as alfo any other equimultiples of the fecond and fourth; and the equimultiple of the first be always greater than that of the third; and that of the fecond at the fame time always greater than that of the fourth; if that of the first be equal to that of the third; that of the fecond always equal to that of the fourth; or if that of the first be less than that of the third, that of the fecond always less than that of the fourth. Then it neceffarily follows that those four propofed magnitudes will be proportional; fo that from hence it moft clearly appears how Euclid obtained this definition, and that it is a very fimple, natural, and eafy fign of proportionality, derived from the before mentioned two propofitions. However otherwise it may at firft appear to thofe who will not be at the pains to confider it. It is true it is not fo fimple and plain as the definition of proportional numbers, or that which might be given of commenfurable magnitudes, Nor does it at all agree with the common notion that the gene nerality of mankind conceive of proportionals. Yet in ufe and practice it is moft plain and easy. Euclid could not have given any other fo elegant and general a definition that would take in incommenfurable magnitudes, as well as numbers and commenfurable 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 Ir. concludes in these words: There is nothing extant in the whole work of the Elements of Euclid more fubtilely invented, more folidly 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 fimilitude of ratios.

clid; and Malcolm, in his Arithmetic. But as I have already faid, Euclid's is the best of them all.

I have fometimes thought that the doctrine of proportionality, in all quantities whatfoever, might be easily derived from the following pofitions, and Euclid's feventh book.

1. That thofe quantities that differ from one another by magnitudes lefs than any affignable magnitudes, may be taken for equals, or reprefent one another.

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

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

4. Whatsoever properties of proportionality are deduced and demonftrated from the definition of proportional numbers, or from Euclid in his feventh book, will hold good of the proportionality of any magnitudes foever which they reprefent or measure.

• Euclid here declares what condition four magnitudes ought to have when the ratio of the first to the fecond is greater than that of the third to the fourth, faying that taking equimultiples of the first and third, and of the fecond and fourth. If it fhall be at any time found (altho' not always) that the multiple of the first is greater than the multiple of the fecond; but the multiple of the third not greater than that of the fourth, but lefs than it, or equal to it; then the ratio of the first magnitude to the fecond is faid 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 first magnitude A, and the third c, and the magnitudes G, H, four times the fecond B, and the fourth D. And becaufe E, the multiple of the firft magni

tude, is greater than G, the multiple of EABGFCDH the fecond; but F the multiple of the third is not greater than H the multiple of the fourth, but lefs; the ratio of the firft magnitude A to the fecond B, is faid to be greater than the ratio of the third c to the fourth D.

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 faid to be the duplicate of the ratio of the first to the second 8.

II. If four magnitudes be [continual] proportionals, the ratio of the firft to the fourth is faid to be triplicate of the ratio of the firft to the fecond, and fo forwards always more by one, as long as the proportion fhall be continued.

12. Magnitudes are homologous, when the antecedents are to the antecedents, as the confequents to the confe quents h

13. Alternate ratio is the affumption of the antecedent to the antecedent, and of the confequent to the confequent i.

14. Inf Both this definition and the last appear to be fuperfluous, and of no manner of ufe; and therefore I rather take them to be remarks of fome learner put into the text, than genuine de finitions of Euclid himself.

8 As fubduplicate and fubtriplicate ratio are not mentioned by Euclid, and as these are very useful, I thought it might not be amifs to define them here. Accordingly if three magnitudes be proportional, the ratio of the first to the second, or that of the fecond to the third, is faid 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 fecond to the third, or that of the third to the fourth, is faid to be fubtriplicate of the ratio of the first to the fourth.

I think this definition is not clearly exprefs'd; nay, I have often thought it to be fcarcely fenfe; and therefore instead thereof I fhould rather fay, When four magnitudes are proportional, as the first is to the fecond, fo is the third to the fourth; the firft magnitude and the third, or the antecedents, as alfo the Lecond and the fourth, or the confequents of the two equal ra tios, are called homologous or co-rational terms. But this I fubmit to the learned,

As let the magnitude a be to the magnitude в, as the magnitude c is to the magnitude D. Then alternately, or by permutation [fee prop. 16.] it will be as a to C, fo is в to D.

Alternate proportion cannot take place, unlefs the four propofed magnitudes be all of the fame 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, fo is the line в to the number D, because there is no ratio

A

between

14. Inverfe ratio is the affumption of the confequents, as the antecedent to the antecedent taken as the confequent *.

15. Compofition of ratio is an affumption of the antecedent, together with the consequent as one to the fame confequent1.

16. Divifion of ratio is the affumption of the excess, whereby the antecedent exceeds the confequent, to the fame confequent m.

17 Converfe ratio, is the affumption of the antecedent to the excess whereby the antecedent exceeds the confequent ".

18. Ratio of equality is, when there are feveral 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 firft rank as the firft magnitude is to the laft, fo in the fecond rank fhall the first magnitude be to the laft. OR ELSE it is the affumption 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 inverfe ratio, compofition, divifion, and converfion, the two firft magnitudes may be of one kind, and the two last of another, as is evident by the demonftrations of this book.

Let the magnitude A be to the magnitude в, as the magnitude c is to the magnitude D. Then inversely [by cor. 4. 5.] as в is to A, fo is D to c.

the magnitude B, as the magThen by compofition [18. 5.] and D together to D.

As let the magnitude A be to nitude c is to the magnitude D. as A and B together is to B, fo is c m As let the magnitude A be to the magnitude в, as the magnitude c is to the magnitude D. Then by divifion [17. 5.] as the difference between A and B is to в, fo 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 converfely [by cor. 19. 5.] as a is to the difference between A and B, fo 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 D is to E, and в be to c, as E is to F. Then by equality [by 22. 5.] will it be as A is to c, fo is n to F.

P 2

19. Ordi.

19. Ordinate proportion is, when it shall be as an antecedent to a confequent, fo is an antecedent to a confe quent; and as a confequent is to fome other magnitude, fo is a confequent to fome other magnitude P.

20. Inordinate or perturbate proportion is, when there are three magnitudes, and the fame number of others, it fhall be in the first magnitudes as an antecedent is to a confequent; fo in the fecond magnitudes is an antecedent to a confequent: But as in the firft magnitudes a confequent is to fome other magnitude, fo in the fecond magnitudes is some other magnitude to the antecedent 9.

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 confequents B be to fome other magnitude c, as the other confequent E is to fome other magnitude F. Then is this or dinate proportionality; and it fhall be true [by 22. 5.] that a îs to c, as D is to F. I take this definition to be almost uselefs; it being in a manner contained in the last.

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 nagnitudes, the confequent B is to fome other magnitude c; fo in the fecond magnitudes is fome other magnitude D to the antecedent magnitude E. This fort 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, fo is D to F.

PROPOSITION I.

THEOREM.

If there be how many foever magnitudes equimultiples of as many other magnitudes, each of each; the fame multiple one magnitude is of cne, all shall be of all.

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

For because A B is the fame multiple of E, as e D is of ☛; as many magnitudes as there are in A B equal to E,

fo

many