as useful as the proposition, and the words, “any whatever," Book V. are supplied near the end of Prop. 23, which are wanting in the Greek text, and the translations from it. In a paper written by Philippus Naudæus, and published after his death, in the history of the Royal Academy of Sciences of Berlin, anno 1745, page 50, the 23d Prop. of the 5th Book is censured as being obscurely enunciated, and, because of this, prolixly demonstrated: The enunciation there given is not Euclid's, but Tacquet's, as he acknowledges, which, though not so well expressed, is, upon the matter, the same with that which is now in the Elements. Nor is there any thing obscure in it, though the author of the paper has set down the proportionals in a disadvantageous order, by which it appears to be obscure: But, no doubt, Euclid enunciated this 23d, as well as the 22d, so as to extend it to any number of magnitudes, which, taken two and two, are proportionals, and not of six only; and to this general case, the enunciation which Naudæus gives, cannot be well applied. The demonstration which is given of this 23d, in that paper, is quite wrong; because, if the proportional magnitudes be plane or solid figures, no rectangle (which he improperly calls a product) can be conceived to be made by any two of them: And if it should be said, that in this case straight lines are to be taken which are proportional to the figures, the demonstration would this way become much longer than Euclid's: But, even though his demonstration had been right, who does not see that it could not be made use of in the 5th Book ? PROP. F, G, H, K. B. V. These propositions are annexed to the 5th Book, because they are frequently made use of both by ancient and modern geometers: And, in many cases, compound ratios cannot be brought into demonstration without making use of them. Whoever desires to see the doctrine of ratios delivered in this 5th Book solidly defended, and the arguments brought against it, by And. Tacquet, Alph. Borellus, and others, fully refuted, may read Dr Barrow's Mathematical Lectures, viz. the 7th and 8th of the year 1666. The fifth book being thus corrected, I most readily agree to what the learned Dr Barrow says *, “ That there is nothing “ in the whole body of the Elements of a more subtile inven“ tion, nothing more solidly established, and more accurately Page 336. Book V. “ 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. Book 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. DEF. II. 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 reasons given in the notes on the 23d Prop. of this Book. PROP. I. and II. B. VI. 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 demonstrations. Book VI. 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. fluous part is not found in the translation from the Arabic, and is now left out. This super 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. Book VI. 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 πολικότητες έφ' εαυτός πολλαπλασιασθείσαι ποιασί 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 nixótaTIS, “ 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 bas 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 scarcely be doubted; as it is to be found in his commentary upon Ptolemy's Meycéan Łórtažus, 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 66 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 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 interme“diate 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 66 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 duplica6 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 |