PROP. XVIII. B. V.

The demonstration of this is none of Euclid's, nor is it legitimate; for it depends upon this hypothesis, that to any three magnitudes, two of which, at least, are of the same kind, there may be a fourth proportional: which, if not proved, the demonstration now in the text is of no force but this is assumed without any proof; nor can it as far as I am able to discern, be demonstrated by the propositions preceding this so far is it from deserving to be reckoned an axiom, as Clavius, after other commentators, would have it, at the end of the definitions of the 5th book. Euclid does not demonstrate it, nor does he show how to find the fourth proportional, before the 12th prop. of the sixth book: and he never assumes any thing in the demonstration of a proposition, which he had not before demonstrated: at least, he assumes nothing the existence of which is not evidently possible; for a certain conclusion can never be deduced by the means of an uncertain proposition: upon this account, we have given a legitimate demonstration of this proposition instead of that in the Greek and other editions, which very probably Theon, at least some other, has put in the place of Euclid's, because he thought it too prolix: and as the 17th prop. of which this 18th is the converse, is demonstrated by help of the first and second propositions of this book; so, in the demonstration now given of the 18th, the 5th prop. and both cases of the 6th are necessary, and these two propositions are the converse of the 1st and 2d. Now the 5th and 6th do not enter into the demonstration of any proposition in this book as we now have it: nor can they be of use in any proposition of the Elements, except in this 18th, and this is a manifest proof, that Euclid made use of them in his demonstration of it, and that the demonstration now given, which is exactly the converse of that of the 17th, as it ought to be, differs nothing from that of Eudoxus or Euclid: for the 5th and 6th have undoubtedly been put into the 5th book for the sake of some propositions in it, as all the other propositions about equimultiples have been. Hieronymus Saccherius, in his book named Euclides ab omni nævo vindicatus, printed at Milan, anno 1733, in 4to, acknowledges this blemish in the demonstration of the 18th, and that he may remove it, and render the demonstration we now have of it legitimate, he endeavours to demonstrate the following proposition, which is in page 115 of his book, viz.

"Let A, B, C, D be four magnitudes, of which the two first are of the one kind, and also the two others either of the same kind with the two first, or of some other, the same kind with one another. I say the ratio of the third C to the fourth D, is either equal to, or greater, or less than the ratio of the first A to the second B."

And after two propositions premised as lemmas, he proceeds thus:

"Either among all the possible equimultiples of the first A, and of the third C, and at the same time, among all the possible equimultiples of the second B, and of the fourth D, there can be found some one multiple EF of the first A, and one IK of the second B, that are

equal to one another; and also, in the same case, some one multiple GH of the third C equal to LM the multiple of the fourth D, or such equality is no where to be found. If the first case happen [i. e. if such equality is to be found] it is manifest from what is be

E

A

-F

-K

-H

-M

be not to be found upon both sides, it

will be found either upon one side, as upon the side of A [and B ;] or it will be found upon neither side; if the first happen; therefore (from Euclid's definition of greater and lesser ratio foregoing) A has to B a greater or less ratio than C to D; according as GH the multiple of the third C is less, or greater than LM the multiple of the fourth D: but if the second case happen: therefore upon the one side, as upon the side of A the first and B the second, it may happen that the multiple EF, [viz. of the first] may be less than IK the multiple of the second, while, on the contrary, upon the other side, [viz. of C and D] the multiple GH [of the third C] is greater than the other multiple LM [of the fourth D] and then (from the same definition of Euclid) the 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 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 demonstrations which some give of the 1st 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 shows that the 5th book has been vitiated by edi

tors 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 a corollary; and another 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 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 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 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 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 to be 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 subtle invention, 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.

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 reason 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 Pappas 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 pro

* See page 336.

portionals, 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 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 πολλαπλασιασθεσαι ποιωσι τινα: 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 anλixontes "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