Book VI. the Elements in place of the right one, which has been

taken out of them: Because, in prop. 5, book 8, it is de-
monstrated, that the plane number of which the sides are
C, D, has to the plane number of which the sides are E,
Z, (see Hervagius's or Gregory's edition,) the ratio which,
is compounded of the ratios of their sides; that is, of the
ratios of C to E, and D to Z; and by def. 5, book 6, and.
the explication given of it by all the commentators, the
ratio which is compounded of the ratios of C to E, and D
to Z, is the ratio of the product made by the multiplication
of the antecedents C, D, to the product by the consequents
E, Z, that is, the ratio of the plane number of which the
sides are C, D, to the plane number of which the sides are
E, Z. Wherefore the proposition which is the 5th def. of
book 6, is the very same with the 5th prop. of book 8, and
therefore it ought necessarily to be cancelled in one of
these places; because it is absurd that the same proposition
should stand as a definition in one place of the Elements,
and be demonstrated in another place of them. Now, there
is no doubt that prop. 5, book 8, should have a place in
the Elements, as the same thing is demonstrated in it con-
cerning plane numbers, which is demonstrated in prop. 23,
book 6, of equiangular parallelograms; wherefore def. 5,
book 6, ought not to be in the Elements. And from this
it is evident that this definition is not Euclid's, but Theon's,
or some other unskilful geometer's.

But nobody, as far as I know, has hitherto shown the true use of compound ratio, or for what purpose it has been introduced into geometry; for every proposition in which compound ratio is made use of, may without it be both enunciated and demonstrated. Now the use of compound ratio consists wholly in this, that by means of it, circumlocuțions may be avoided, and thereby propositions may be more briefly either enunciated or demonstrated, or both may be done; for instance, if this 23d proposition 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 che ancient geometers, when they observed this enuncia

tion could be made shorter, by giving a name to the ratio Book Vla which the first straight line has to the last, by which name m or 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 last, the ratio compounded of the ratio of the first to the second, and of the second to the third straigbt 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 thus: If there be two equiangular parallelograms, they have to one another the ratio which is the same with that which is compounded of ratios that are the same with the ratios of the sides ; which is shorter than the preceding enunciation, but has precisely the same meaning. Or yet shorter thus; Equiangular parallelograms have to one another the ratio which is the same with that which is compounded of the ratios of their sides. And these two enunciations, the first espeeially, agree to the demonstration which is now in ihe Greek. The proposition may be more briefly demonstrated, as Candalla does, thus : Let ABCD, CEFG, be two equiangular parallelograms, and complete the parallelogram CDHG: then, because there are three parallelograms, AC, CH, CF, the first AC (by the definition of compound ratio) has to the third CF, the ratio which is compounded of the A ratio of the first AC to the second

SH CH, and of the ratio of CH to the third CF; but the parallelogram AC is to the parallelogram CH, as the straight line BC to CG: And the parallelogram CH : is to CF, as the straight line CD is to CE; therefore the parallelogram AC has to CF the ratio which is compounded of ratios that are the same with the ratios of the sides. And to this demonstration agrees the enunciation which is at present in the text, viz. equiangular parallelograms have to one another the ratio which is compounded of the ratios of the sides; for the vulgar reading, " which is compounded of their sides,” is absurd. But, in this edition, we have kept the demonstration which is in the Greek text, though not so short as Çandalla's ; because the way of finding the ratio which is compounded of the ratio of the sides, that is, of finding the ratio of parallelograms, is shown in that, but not in Candalla's de.

Book VI. monstration; whereby beginners may learn, in like cases,

how to find the ratio which is compounded of two or more given ratios.

. . From what has been said, it may be observed, that in any magnitudes whatever of the same kind, A, B, C, D, &c. the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is only a name or expression by which the ratio which the first A has to the last D is signified, and by which at the same time the ratios of all the magnitudes A to B, B to C, C to D, from the first to the last, to one another, whether they be the same, or be not the same, are indicated; as in magnitudes which are continual proportionals A, B, C, D, &c. the duplicate ratio of the first to the second is only a name, or expression by which the ratio of the first A to the third C is signified, and by which, at the same time, is shown, that there are two ratios of the magnitudes from the first to the last, viz. of the first A to the second B, and of the second B to the third or last C, which are the same with one another; and the triplicate ratio of the first to the second is a name or expression by which the ratio of the first A to the fourth D is signified, and by which, at the same time, is shown, that there are three ratios of the magnitudes from the first to the last, viz. of the first A to the second B, and of B to the third C, and of C to the fourth or last D, which are all the same with one another; and so in the case of any other multiplicate ratios. And that this is the right explication of the meaning of these ratios is plain from the definitions of duplicate and triplicate ratio, in which Euclid makes use of the word AEYETAI, is said to be, or is called; which word he, no doubt, made use of also in the definition of compound ratio, which Theon, or some other, has expunged from the Elements; for the very same word is still retained in the wrong definition of compound ratio, which is now the 5th of the 6th book : But in the citation of these definitions it is sometimes relained, as in the demonstration of prop. 19, book 6, “ the “ first is said to have, čxaiv Asystat, to the third the dupli“ cate ratio,” &c, which is wrong translated by Commandine and others, “ has” instead of " is said to have,” and sometimes it is left out, as in the demonstration of prop. 33, of the Ilth book, in which we find,“ the first has, é rare " to the third the triplicate ratio;" but without doubt mei; o las,” in this place signifies the same as : Xely Asyete), is uid to have; so likewise in Prop. 23, B. 6, we find this

citation, but the ratio of K to M is compounded, ouyneltar, Boon VI: 66 of the ratio of K to L, and the ratio of L to M,” which is a shorter way of expressing the same thing, which, according to the definition, ought to have been expressed by QUY XERChat heyetli, is said to be compounded. :. From these remarks, together with the propositions, subjoined to the 5th book, all that is found concerning com.. pound ratio, either in the ancient or modern geometers, may be understood and explained.

PROP. XXIV. B. VI.

It seems that some unskilful editor has made up this de. monstration as we now have it, out of two others; one of which may be made from the 2d prop. and the other from the 4th of this book. For after he has, from the 2d of this book, and coniposition and permutation, demonstrated, that the sides about the angle common to the two parallelograms, are proportionals, he might have immediately concluded, that the sides about the other equal angles were proportionals, viz. from Prop. 3-4, B. I. and Prop. 7, B. 5. This he does nut, but proceeds to show, that the triangles and parallelograms are equiangular: and in a tedious way, by help of Prop. 4. of this book, and the 22d of book 5, deduces the same conclusion: From which it is plain, that this ill-composed demonstration is not Euclid's: These superfluous things are now left out, and a more simple demoustration is given from the 4th prop. of this book, the saine which is in the translation from the Arabic, by help of the 2d prop. and composition ; but in this the author neglects permutation, and does not show the parallelograins to be equiangular, as is proper to do for the sake of beginners.

:: PROP. XXV. B. VI.

. It is very evident, that the demonstration which Euclid had given of this proposition has been vitiated by some unskilful hand: For, after this editor had demonstrated, that 6 as the rectilineal figure ABC is to the rectilineal figure ' “KGH, so is the parallelogram BE to the parallelogram EF;" nothing more should have been added but this," and the

Book VI. " rectilineal figure ABC is equal to the parallelogram BE:

is therefore the rectilineal KGH is equal to the parallelo-
“ gram EF,” viz. from prop. 14, book 5. But betwixt these
two sentences he has inserted this, " wherefore, by permu-
“ tation, as the rectilineal figure ABC to the parallelogram
“ BE, so is the rectilineal KGH to the parallelogram ĚF;"
by which it is plain, he thought it was not so evident to
conclude, that the second of four proportions is equal to the
fourth from the equality of the first and third, which is a
thing demonstrated in the 14th prop. of B. 5, as to conclude
that the third is equal to the fourth, from the equality of
the first and second, which is no where demonstrated in
the Elements as we now have them: But though this pro-
position, viz. the third of four proportionals, is equal to the
fourth, if the first be equal to the second, had been given
in the Elements by Euclid, as very probably it was, yet he
would not have made use of it in this place, because, as was
said, the conclusion could have been immediately deduced
without this superfluous step by permutation : This we have
shown at the greater length, both because it affords a cer-
tain proof of the vitiation of the text of Euclid; for the
very same blunder is found twice in the Greek text of prop.
23, book 11, and twice in prop. 2, book 12, and in the 5,
11, 12, and 18th of that book'; in which places of book
12, except the last of them, it is rightly left out in the Ox-
ford edition of Commandine's translation: And also that
geometers may beware of making use of permutation in the
like cases: for the moderns not unfrequently commit this
mistake, and among others Commandine himself in his
commentary on prop. 5, book 3, p. 6.b. of Pappus Alexan-
drinus, and in other places: The vulgar notion of propor-
tionals has, it seems, preoccupied many so much, that they
do not sufficiently understand the true nature of them.

Besides, though the rectilineal figure ABC, to which another is to be made similar, may be of any kind whatever': yet in the demonstration the Greek text has “ triangle" instead of “ rectilineal figure,” which error is corrected in the above-named Oxford edition. *6.ARENA 18:13

PROP. XXVII. B. VI. The second case of this has draws, otherwise, prefised to it, as if it was a different demonstration, which probably has been done by some unskilful librarian. - Dr. Gregory