and that of 16 to 512, any submultiple of the latter antecedent is contained as often in its consequent, as the equi-submultiple of the first antecedent is contained in its consequent. Therefore in both the cases the four quantities compared are proportional. But if the given ratios be that of 8 to 13, and of 16 to 27, although the submultiples, 2, 4, 8 of the latter antecedent are contained as often in the consequent, as the equi-submultiples 1, 2, 4 of the first antecedent are contained in its consequent, yet if the equi-submultiples of the antecedents 1 and be taken, the former shall be oftener contained in 27, than the latter in 13; therefore the ratios of 16 to 27, and of 8 to 13 are not equal, but the first antecedent has a less ratio to its consequent, than the latter has to its consequent.

DEF. 7. B. V.

This definition is substituted for the following: When of the equi-multiples of four quantities, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth then the first is said to have to the second a greater ratio than the third has to the fourth; and on the contrary the third is said to have to the fourth a less ratio than the first has to the second.

:

DEF. 12. B. V.

The definition of compound ratio is generally given among the definitions of the sixth book; but Ř. Simson, having proved that it had been corrupted, restored it to its original form and place, and thus explained the purpose for which it had been introduced into geometry: The use of compound ratio consists wholly in this, that by means of it circumlocutions may be avoided, and thereby propositions may be more briefly either enunciated or demonstrated, or both may be done; for instance, if the 23d prop. of the 6th book were to

be enunciated without mentioning compound ratio, it might be done as follows: If two parallelograms be equiangular, and if as a side of the first to a side of the second, so any assumed straight line be made to a second straight line; and as the other side of the first to the other side of the second, so the second straight line be made to a third; the first parallelogram is to the second, as the first straight line to the third: and the demonstration would be exactly the same as we now have it. But the ancient geometers, when they observed this enunciation could be made shorter by giving a name to the ratio which the first straight line has to the last, by which name the intermediate ratios might likewise be signified, of the first to the second, and of the second to the third, and so on if there were more of them, they called this ratio of the first to the blast, the ratio compounded of the ratios of the first to the second, and of the second to the third straight line; that is, in the present example, of the ratios which are the same with the ratios of the sides: and by this they expressed the proposition more briefly.

The three remarkable species of compound ratios have been called duplicate ratio, triplicate ratio, and sesquialteral or sesquiplicate ratio. The former two have been defined by Euclid, the latter deserves some explanation, as though not used in geometry, it is frequently mentioned in astronomy.

If there be three magnitudes proportional, and also four others continually proportional, and the first be to the last in the one series as the first to the last in the other, the ratio which the first has to the second in the series of three proportionals is said to be sesquialteral or sesquiplicate of the ratio which the first has to the second in the series of four proportionals.

For in the series of four proportionals find a mean between the first and last; the first is to this mean, as the first in the series of three proportionals to the second (1), but the former ratio is composed of the (1) Prop 39. ratios of the first to the second in the series of four 2 5. proportionals, and of the second to the mean, and the former of these ratios is evidently duplicate of the other, and hence the compound ratio is called sesquialteral of the simple.

The ratio of 2 to 16 is sesquialteral of the ratio of 1 to 4: find a mean proportional 8 between 1 and 64; 1 is to 8 as 2 to 16, but the ratio of 1 to 8 is compounded of the ratios of 1 to 4 and of 4 to 8, the former of which is duplicate of the latter.

DEF. 14. B. V.

These different changes of proportional quantities will perhaps be better understood by the assistance of the following table:

Euclid makes no mention of submultiples, but it is evident that all the affections of multiples, which he places among the axioms, must also belong to submultiples.

PROP. 1. B. V.

Although this proposition and some others might be assumed as axioms, I preferred demonstrating them to encreasing the number of axioms.

COR. 1. PROP. 7. B. V.

The word contained is ambiguous, for a magnitude which is contained in another, may either be a submultiple of it or not, and from this ambiguity arises the difficulty of many propositions in this book.

PROP. 32. B. V.

This proposition is Euclid's definition of proportionals.

DEF. 4. B. 6.

This and the two following definitions are added, as necessary for understanding prop. 27, 28 and 29.

PROP. 1. B. VI.

Euclid's demonstration is different, being founded on his definition of proportional magnitudes.

PROP. 3. B. VI.

If the side BA be produced, the external angle CAF Fig. 24. bisected and the bisecting line meet the base, it can be shewn in the same manner that the segments of the base are proportional to the other sides, and reciprocally. The proposition appears to me to have formerly been so expressed as to include both cases, for now it begins thus: "If the angle of a triangle be bisected, and the bisecting line should cut the base;" now the line bisect-" ing the internal angle must cut the base, and therefore Euclid would have expressed the proposition absolutely, if he had not had in view the bisection of the external angle.

Fig. 25.

PROP. 6. B. VI.

If this proposition, and also in the preceding, the construction, which in Euclid is definite, is expressed indefinitely lest any one should imagine there was but one side of the triangle, on which the construction could be made. Euclid says, construct at the side DF, &c. as if it could not be constructed at the other side.

PROP. 11. B. VI.

If the given be a ratio of less inequality LO to LR, the series can be continued until a magnitude be found greater than any assigned.

For let LO, LR, LQ and LI be continually proportional, since LO is to LR as LR to LQ, by conversion, LO is to OR, as LR to RQ; but LR is greater (1)Prop.23, than LO, therefore RQ is greater than OR (1): in the same manner it can be shewn that IQ is greater than RQ; since therefore quantities continually encreasing are added to the first, a magnitude can be attained greater than any assigned.

B. 5.

Fig. 26.

If the given be a ratio of greater inequality, AB to CB, the series can be continued till a magnitude be found less than any assigned.

Let the assigned quantity be OL, as CB is to AB so let OL be to LR, and continue the series till IL be found greater than AB, continue the ratio of AB to CB through as many terms, and let the last be FB, FB is less than OL.

For since there are two series of magnitudes proportional and equal in number, ex æquo AB is to FB as (1) Const. IL to OL, but AB is less than IL (1), therefore FB is (2) Prop.23. less than ÓL (2), and in the same manner a magnitude can be found less than any other given one.

B. 5.

Tacquet has given from Gregorius a S. Vincentio, the following method of finding a series of lines in any given ratio of greater inequality, and of exhibiting the