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nothing more solidly established, and more accurately handled, than the doctrine of proportionals.” And there is some ground to hope, that geometers will think that this could not have been said with as good reason, since Theon's time till the present.
DEF. II. and V. of B. VI.
THE 2d definition does not seem to be Euclid's, but some unskilful editor's : for there is no mention made by Euclid, nor, as far as I know, by any other geometer, of reciprocal figures : it is obscurely expressed, which made it proper to render it more distinct : it would be better to put the following definition in place of it, viz.
Two magnitudes are said to be reciprocally proportional to two others, when one of the first is to one of the other magnitudes, as the remaining one of the last two is to the remaining one of the first.
But the fifth definition, which, since Theon's time, has been kept in the Elements, to the great detriment of learners, is now justly thrown out of them, for the reason given in the notes on the 23d prop. of this book.
To the first of these a corollary is added, which is often used: and the enunciation of the second is made more general.
PROP. III. B. VI.
A second case of this, as useful as the first, is given in prop: A: viz. the case in which the exterior angle of a triangle is bisected by a straight line : the demonstration of it is very like to that of the first case, and upon this account may, probably, have been left out, as also the enunciation, by some unskilful editor : at least, it is certain, that Pappus makes use of this case as an elementary proposition, without a demonstration of it, in prop. 39 of his 7th book of Mathematical Collections.
PROP. VII. B. VI.
To this a case is added which occurs not unfrequently in demonstration.
PROP. VIII. B. VI. It seems plain that some editor has changed the demonstration that Euclid gave of this proposition : for, after he has demonstrated, that the triangles are equiangular to one another, he particularly shows that their sides about the equal angles are proportionals, as if this had not been done in the demonstration of the 4th prop. of this book; this superfluous part is not found in the translation from the Arabic, and is now left out.
PROP. IX. B. VI.
This is demonstrated in a particular case, viz. that in which the third part of a straight line is required to be cut off'; which is not at all like Euclid's manner: besides, the author of the demonstration, from four magnitudes being proportionals, concludes that the third of them is the same multiple of the fourth, which the first is of the second: now, this is no where demonstrated in the 5th book, as we now have it: but the editor assumes it from the confused notion which the vulgar have of proportionals: on this account, it was necessary to give a general and legitimate demonstration of this proposition.
PROP. XVIII. B. VI.
The demonstration of this seems to be vitiated : for the proposition is demonstrated only in the case of quadrilateral figures, without mentioning how it may be extended to figures of five or more sides : besides, from two triangles being equiangular, it is inferred that a side of the one is to the homologous side of the other, as another side of the first is to the side homologous to it of the other, without permutation of the proportionals; which is contrary to Euclid's manner, as is clear from the next proposition; and the same fault occurs again in the conclusion, where the sides about the equal angles are not shown to be proportionals, by reason of again neglecting permutation. On these accounts, a demonstration is given in Euclid's manner, like to that he makes use of in the 20th prop. of this book : and it is extended to fivesided figures, by which it may be seen how to extend it to figures of any number of sides.
PROP. XXIII. B. VI.
Nothing is usually reckoned more difficult in the Elements of geometry by learners, than the doctrine of compound ratio, which Theon has rendered absurd and ungeometrical, by substituting the 5th definition of the 6th book in place of the right definition, which without doubt Eudoxus or Euclid gave, in its proper place, after the definition of triplicate ratio, &c. in the 5th book. Theon's definition is this: a ratio is said to be compounded of ratios οταν αι των λογων πηλιοτητες εφ' εαυτας πολλαπλαolaoseloai nowwol Tiva: which Commandine thus translates; “quando rationem quantitates inter se multiplicatæ aliquam efficiunt rationem ;" that is, when the quantities of the ratios being multiplied by one another make a certain ratio. Dr. Wallis translates the word amacxoentes " rationem exponentes," the exponents of the ratios : and Dr. Gregory renders the last words of the definition by illius facit quantitatem,” makes the quantity of that ratio ; but in whatever sense the quantities,” or “exponents of the ratios," and their “multiplication” be taken, the definition will be ungeometrical and useless : for there can be no multiplication but by a number. Now the quantity or exponent of a ratio (according as Eutochius in his Comment. on prop. 4. book 2. of Arch. de Sph. et Cyl. and the moderns explain that term) is the number which multiplied into the consequent term of a ratio produces the antecedent, or which is the same thing, the number which arises by dividing the antecedent by the consequent; but there are many ratios such, that no number can arise from the division of the antecedent by the consequent: ex. gr. the ratio which the diameter of a square has to the side of it; and the ratio which the circumference of a circle has to its diameter, and such like. Besides, that there is not the least mention made of this definition in the writings of Euclid, Archimedes, Apollonius, or other ancients, though they frequently make use of compound ratio; and in this 23d prop. of the 6th book, where compound ratio is first mentioned, there is not one word which can relate to this definition, though here, if in any place, it was necessary to be brought in; but the right definition is expressly cited in these words: “But the ratio of K to M is compounded
of the ratio of K to L, and of the ratio of L to M.” This definition therefore of Theon is quite useless and absurd : for that Theon brought it into the Elements can scarce be doubted; as it is to be found in his commentary upon Ptolemy's Meyaan Evvrašus, page 62, where he also gives a childish explication of it, as agreeing only to such ratios as can be expressed by numbers ; and from this place the definition and explication have been exactly copied and prefixed to the definitions of the 6th book, as appears from Hervagius's edition : but Zambertus and Commandine, in their Latin translations, subjoin the same to these definitions. Neither Campanus, nor, as it seems, the Arabic manuscripts, from which he made his translation, have this definition. Clavius, in his observations upon it, rightly judges, that the definition of compound ratio might have been made after the same manner in which the definitions of duplicate and triplicate ratio are given; viz. “That as in several magnitudes that are continual proportionals, Euclid named the ratio of the first to the third, the duplicate ratio of the first to the second, and the ratio of the first to the fourth, the triplicate ratio of the first to the second, that is, the ratio compounded of two or three intermediate ratios that are equal to one another, and so on; so, in like manner, if there be several magnitudes of the same kind, following one another, which are not continual proportionals, the first is said to have to the last the ratio compounded of all the intermediate ratios-only for this reason, that these intermediate ratios are interposed betwixt the two extremes, viz. the first and last magnitudes; even as, in the 10th definition of the 5th book, the ratio of the first to the third was called the duplicate ratio, merely upon account of two ratios being interposed betwixt the extremes, that are equal to one another : so that there is no difference betwixt this compounding of ratios, and the duplication or triplication of them which are defined in the 5th book, but that in the duplication, triplication, &c. of ratios, all the interposed ratios are equal to one another ; whereas, in the compounding of ratios, it is not necessary that the intermediate ratios should be equal to one another.” Also Mr. Edmund Scarburgh, in his English translation of the first six books, page 238. 266, expressly affirms, that the 5th definition of the 6th book is supposititious, and that the true definition of compound ratio is contained in the 10th definition of the 5th book, viz. the definition of duplicate ratio, or to be understood from it, to wit, in the same manner as Clavius has explained it in the preceding citation. Yet these, and the rest of the moderns, do notwithstanding 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 s. 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, 2 (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. hook 8. should have a place in the Elements, as the same thing is demonstrated in it concerning 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, 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 sixth 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 a third. The first parallelogram is to the second, as the first straight line to the third. And the