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Book V. ratio of the first A to the second B, is less than the ratio of the
" third C to the fourth D; or on the contrary.
“ Therefore the Axiom, [i.e. the Proposition before set down,] “ remains demonstrated" &c.
Not in the least; but it remains still undemonstrated: for what he fays may happen, may in innumerable cases never happen, and therefore his demonstration does not hold. for example, if A be the side and B the diameter of a square ; and C the side and D the diameter of another square; there can in no case be any multiple of A equal to any of B; nor any one of C equal to one of D, as is well known; and yet it can never happen that when any multiple of A is greater or less than a multiple of B, the multiple of C can, upon the contrary, be less or greater than the multiple of D, viz. taking equimultiples of A and C, and equimultiples of B and D. for A, B, C, D are proportionals, and so if the multiple of A be greater &c. than that of B, so must that of C be greater &c. than that of D. by 5. Def. B. 5.
The same objection holds good against the Demonstration which some give of the 1. Prop. of the 6. Book, which we have made against this of the 18. Proposition, because it depends upon the tane insufficient foundation with the other.
PROP. XIX. B. V. A Corollary is added to this, which is as frequently used as the Proposition itself. the Corollary which is subjoined to it in the Greek, plainly shews that the 5. Book has been vitiated by Editors who were not Geometers, for the conversion of ratios does not depend upon this 19. and the Demonstration which several of the Commentators on Euclid give of Conversion, is not legitimate, as Clavius has rightly observed, who has given a good Demonstration of it which we have put in Proposition E; but he makes it a Corollary from the 19. and begins it with the words, “ Hence it ea“sily follows,” tho' it does not at all follow from it.
PROP. XX, XXI, XXII, XXIII, XXIV. B. V. The Demonstrations of the 20 and 21. Propositions are shorter than those Euclid gives of easier Propositions, either in the preceeding, or following Books. wherefore it was proper to make them more explicit. and the 2 2. and 23. Propositions are, as they ought to be, extended to any number of magnitudes. and in like manner
may the 24. be, as is taken notice of in a Corollary; and another Book v. Corollary is added, as useful as the Proposition. and the words “ any whatever” are supplied near the end of Prop. 23. which are wanting in the Greek text, and the translations from it.
In a paper writ by Philippus Naudaeus, and published, after his death, in the History of the Royal Academy of Sciences of Berlin, Ann. 174 5. page 50. the 23. Prop. of the 5. Book is censured as being obscurely enuntiated, and, because of this, prolixly demonstrated. the Enuntiation there given is not Euclid's but Tacquet's, as he acknowledges, which, tho' not fo well expressed, is, upon the matter, the same with that which is now in the Elements. Nor is there any thing obscure in it, tho’ 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 enuntiated this 23. as well as the 22. so as to extend it to any number of magnitudes, which taken two and two, are proportionals, and not of fix only; and to this general case the Enuntiation which Naudaeus gives, cannot be well applied.
The Demonstration which is given of this 2 3. in that paper, is quite wrong; because if the proportional magnitudcs be plane or solid figures, there can no rectangle (which he improperly calls a Product) 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 tho' his Demonstration had been right, who does not see that it could not be made use of in the 5. Book?
PROP. F, G, H, K. B. V. These Propositions are annexed to the 5. Book, because they are frequently made use of by both antient and modern Geometers, and in many cases Compound ratios cannot be brought into Demonstrations, without making use of them.
Whoever desires to see the doctrine of Ratios delivered in this g. Book folidly 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 7. and 8. of the year 1666.
The 5. Book being thus corrected, I most readily agree to what the learned Dr, Barrow says *, “That there is nothing in the whole • Page 336.
Book V. body of the Elements, of a more subtile invention, nothing more W“ folidly 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.
skilful 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.
DE F. 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 firft.
But the 5. 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 23. 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 Enuntiation 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, tiz. 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 Enuntiation, by some unskilful Editor. as least it is certain, that Pappus makes use of this case, as an Elementary Propofition, without a Demonstration of it, in Prop. 39. of his 7. Book of Mathem. Collections,
Book VI. PRO P. VII. B. VI. To this a case is added which occurs not unfrequently in Demonstrations.
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 shews that their sides about the equal angles are proportionals, as if this had not been done in the Demonstration of the 4. Prop. of this Book. this superfluous part is not found in the Translation from the Arabic, and is now left out.
PROP. IX. B. VỊ. 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 fecond; now this is no where demonstrated in the 5. Book, as we now have it. but the Editor asumes it from the confused notion which the vulgar have of proportionals. on this account cessary 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 fides. 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 shewn 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 20. Prop.
Book VI. 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 5. Definition of the 6. 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 5. Book. Theon's Definition is this; a Ratio is said to be compounded of ratios ötay αι των λόγων πηλικότητες εφ' εαυτας πολλαπλασιασθεσαι ποιωσι τινα, which Commandine thus translates, “ quando rationum quantitates “ inter fe multiplicatae 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 annixótntis, “num 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 “quan“tities” 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 Eutocius in his Comment, on Prop. 4. Book 2. of Arch. de Sph. et Cyl. and the moderns explain ihat 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 arisus 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 bias to the side of it; and the ratio which the circumference of a circle has to it diameter, and such like. Besides, there is not the least mention made of this Definition the writings of Euclid, Archimedes, Apollonius, or other antients, tho’ they frequently make use of Compound ratio. and in this 23. Prop. of the 6. Book, where Compound ratio is first menticned, there is not one word which can relate to this Definition, tho' here, if in any place, it was neceffary to be brought in; but the right Definition is exprefly 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