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PROP. VIII. B. VI.
It seems plain, that some editor has changed the demonstration that Euclid gave of this proposition; for, after he bas 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 five sided 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 ratios, &c. in the 5th Book. Theon's definition is this ; a ratio is said to be compounded of ratios, όταν αι των λόγων πηλικότητες εφ' εαυτός πολλαπλασιασθείσαι πoιώσί τινα : Which Commandine thus translates: “ Quando rationum 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 anaixótntes, “ rationum
exponentes,” the exponents of the ratios : And Dr Gregory renders the last words of the definition by “illius facit quan“ titatem,” 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 to 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 Euelid, 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 scarcely be doubted; as it is to be found in his commentary upon Ptolemy's Meydan Eviratus, 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 Book VI. 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 tri
plicate ratio of the first to the second, that is, the ratio com “pounded of two or three intermediate ratios that are equx. « 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 interme“ diate ratios, only for this reason, that these intermediate “ ratios are interposed betwixt the two extremes, viz. the first 6 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 an“ other: So that there is no difference betwixt this compound“ing of ratios, and the duplication or triplication of them " which are defined in the 5th Book, but that in the duplica“ tion, 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 sixth Book is supposititious, and that the
true definition of compound ratio is contained in the 10th Definition of the fifth Book, viz. the definition of ,duplicate ratio, or to be understood from it, viz. 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
Book VI. 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 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 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. 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 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 Book VI. 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 the 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 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
D is compounded of the ratio of the A
H first AC to the second CH, and of the ratio of CH to the third CF; B
G but the parallelogram AC is to the
C parallelogram CH, as the straight line BC to CG; and the parallelo
E F gram 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 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 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