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Book VI. standing retain this 5th def. of the 6th book, and illustrate and
explain it by long commentaries, when they ought rather to have taken it quite away from the Elements.
For, by comparing def. 5, book 6, with prop. 5, book 8. it will clearly appear that this definition has been put into the Elements in place of the right one which has been taken out of them : because, in prop. 5, book 8, it is demonstrated that the plane number of which the sides are C, D has to the plane number of which the sides are E, Z, (see Hergavius' 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 of 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 concerning plane numbers, which is demonstrated in prop. 23d, 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 wnich 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, circumlocutions 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 sxith 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. Book VI. 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 last, 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 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 especially, agree to the demonstration which is now in the Greek. The proposition may be more briefly demonstrated, as Candalla does, thus : let ABCD, CEFG be two equiangular parallelograms, and complete the parallelo, gram 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 A
D H which is compounded of the ratio of the first AC to the second CH, and of the ratio of CH, to the third CF; but B
с the paralleogram AC is to the parallelogram CH, as the straight line BC to CG; and the parallelogram CH is to CF, as the straight line
F 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 com“ pounded 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 Candalla's ; because the way of finding the ratio which is compounded of the ratios of the sides, that is, of finding the ratio of the parallelograms, is shown in that, but not in Candalla's demonstration ; whereby beginners may learn, in like
Book VI. cases, how to find the ratio which is compounded of two or more
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 oniy 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 whịch, 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 se. cond 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 76767&s, 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 expung. ed 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 some times retained, as in the demonstration of prop. 19, book 6, “ the first is said to have, 'exsan AEX ET ell, to the third the “ duplicate 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 11th book, in which we find “the first has, '%*, to the " third the triplicate ratio ;" but without doubt 's**,“ has," in this place signifies the same as 'syetv syetan, is said to have: so likewise in prop. 23, B. 6, we find this citation, “but the “ ratio of K to M is compounded, cuyautas, 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 our reso das 18yoru, is said so be Book VI. compounded.
From these remarks, together with the propositions subjoined to the 5th book, all that is found concerning compound 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 demonstration 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 tnis book : for after he has, from the 2d of this book, and composition and permutation, demonstrated that the sides about the angle common to the two parallelograms are proportionais, he might have immediately concluded that the sides about the other equal angles were proportionals, viz. from prop 34, B. 1, and prop. 7, book 5. This he does not, 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 demonstraiion is given from the 4th prop. of this book, the same which is in the translation from the Arabic, by the help of the 2d prop. and composition ; but in this the author neglects permutation, and does not show the parallelograms to be equiangular, as is proper to do for the sake of beginners.
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
as the “ rectilineal figure ABC is to the rectilineal KGH, so is the " parallelogram BE to the parallelogram EF;" nothing more should have been added but this, " and the rectilineal figure “ ABC is equal to the parallelogram BE: therefore the recti“ lineal KGH is equal to the parallelogram EF,” viz. from prop. 14, book 5. But betwixt these two sentences he has inserted this ; “ wherefore by permutation, as the rectilineal fi
gure ABC to the parallelogram BE, so is the rectilineal KCH
Book VI. “ to the parallelogram EF;" by which, it is plain, he thought
it was not so evident to conclude that the second of four pmportionals 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 proposition, viz. the third of four proportionals is equal to the fourth, if the third 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 certain 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, B. 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 rightiy left out in the Oxford 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 Alexandrinus, and in other places : the vulgar notion of proportionals has, it seems, pre-occupied 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 " recti“ lineal figure,” which error is corrected in the above named Oxford edition.
The second case of this has 'anaws, otherwise, prefixed to it, as if it was a different demonstration, which probably has been done by some unskilful librarian. Dr. Gregory has rightly left it out : the scheme of this second case ought to be marked with the same letters of the alphabet which are in the scheme of the first, as is now done.
PROP. XXVIII and XXIX. B. VI.
These two problems, to the first of which the 27th prop. is necessary, are the most general and useful in all the Elements.