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“ definition of Euclid) the ratio of the first A to the 66 second B is less than the ratio of the third C to the 6 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 still remains undemonstrated : for what he says 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 than a multiple of B, the multiple of C can be less than the multiple of D, nor when the multiple of A is less than that of B, the multiple of C can be greater than that 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 5th Def. B. 5.
The same objection holds good against the demonstration which some give of the first Prop, of the 6th Book, which we have made against this of the 18th Prop. because it depends upon the same 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 5th book has been vitiated by editors who were not geometers : for the conversion of ratios does not depend upon this 19th, 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 19th, and begins it with the words, “ Hence it easily follows," though it does not at all follow from it.
PROP. XX. XXI. XXII. XXIII. XXIV. B. V.
The demonstrations of the 20th and 21st Propositions are shorter than those Euclid gives of easier propositions, either in the preceding or following books: wherefore it was proper to make them more explicit, and the 22d and 23d Propositions are, as they ought to be, extended to any number of magnitudes: and in like manner may the 24th be, as is taken notice of in the corollary; and another corollary is added, as useful as the proposition, and the words “any whatever” are applied 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, there can be no rectangle (which he improperly calls a product) 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.
because they are frequently made use of by both 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 5th 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 invention, nothing more solidly esta“blished, and more accurately handled than the doc“ trine 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. ll. 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 5th 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
* Page 336.
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,
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. VI. 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 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.
PROP. XVIII. B. VI.
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 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 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 in 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 όταν αι των λόγων πηλικότητες εφ' εαυτας πολλαπλασιασθείσαι πoιώσι τινα : which Commandine thus translates: “Quando rationum quantitates “ inter se multiplicatæ aliquam efficient rationem;" that is, when the quantities of the ratios being multiplied by one another make a certain ratio. Dr. Wallis translates the word ArnauxÓTITES “ rationem exponentes,” the exponents of the ratios: and Dr. Gregory renders the last words of the definition by “illius facit quantita“ tem,” 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 as the moderns explain that term) is the number