to a third: the firft parallelogram is to the fecond, as the firft Book VI. of the first AC to the second CH, and D E H G F 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 fame with the ratios of the fides. and to this Demonstration agrees the Enuntiation which is at present in the text, viz. equiangular parallelograms have to one another the ratio which is compounded of the ratios of the fides. for the vulgar reading " which is com"pounded of their fides" is abfurd. But in this Edition we have kept the Demonftration which is in the Greek text, tho' not fo fhort as Candalla's; because the way of finding the ratio which Book VI. is compounded of the ratios of the fides; that is, of finding the ratio of the parallelograms, is fhewn in that, but not in Candalla's Demonstration; whereby beginners may learn, in like cafes, how to find the ratio which is compounded of two or more given ratios. From what has been faid it may be observed, that in any magitudes whatever of the fame kind A, B, C, D, &c. the ratio compounded of the ratios of the firft to the fecond, of the fecond to the third, and fo on to the laft, is only a name or expreffion by which the ratio which the firft A has to the last Dis fignified, and by which at the fame time the ratios of all the magnitudes A to B, 'B to C, C to D from the first to the last, to one another, whether they be the fame, or be not the fame, are indicated; as in magnitudes which are continual proportionals A, B, C, D, &c. the Duplicate ratio of the first to the second is only a name, or expreffion by which the ratio of the first A to the third C is fignified, and by which, at the fame time, is fhewn that there are two ratios of the magnitudes from the first to the last, viz. of the first A to the second B, and of the second B to the third or laft C, which are the fame with one another; and the Triplicate ratio of the first to the second is a name or expreffion by which the ratio of the first A to the fourth D is fignified, and by which, at the fame time, is fhewn that there are three ratios of the magnitudes from the first to the last, viz. of the first A to the second B, and of B to the third C, and of C to the fourth or laft D, which are all the fame with one another; and fo in the cafe of any other Multiplicate ratios. And that this is the right explication of the meaning of these ratios is plain from the Definitions of Duplicate and Triplicate ratio in which Euclid makes ufe of the word ayer, is faid to be, or is called; which word, he no doubt made use of alfo in the Definition of Compound ratio which Theon, or some other, has expunged from the Elements; for the very fame word is still retained in the wrong Definition of Compound ratio which is now the 5. of the 6. Book. but in the citation of these Definitions it is fometimes retained, as in the Demonftration of Prop. 19. B. 6. "the first is faid to have, xew λysta, "to the third the Duplicate ratio," &c. which is wrong tranflated by Commandine and others" has" instead of " is faid to have;" and fometimes it is left out, as in the Demonftration of Prop. 33. of the 11. Book, in which we find "the firft has, xe, to “the third the Triplicate ratio;" but without doubt xe, " has," in this place fignifies the fame as xew λyer, is faid to have. fo likewise in Prop. 23. B. 6. we find this citation " but the ratio of Book VI. "K to M is compounded, synes, of the ratio of K to L, and the " ratio of L to M," which is a fhorter way of expreffing the fame thing, which, according to the Definition, ought to have been expreffed by συγκείσθαι λέγεται, is faid to be compounded. From these Remarks, together with the Propofitions subjoined to the 5. Book, all that is found concerning Compound ratio either in the antient or modern Geometers may be understood and explained. PROP. XXIV. B. VI. It seems that fome unskilful Editor has made up this Demonstration as we now have it, out of two others; one of which may be made from the 2. Prop. and the other from the 4. of this Book. for after he has from the 2. of this Book, and Composition and Permutation, demonstrated that the fides about the angle common to the two parallelograms are proportionals, he might have immediately concluded that the fides about the other equal angles were proportionals, viz. from Prop. 34. B. 1. and Prop. 7. B. 5. this he does not, but proceeds to fhew that the triangles and parallelograms are equiangular, and in a tedious way, by help of Prop. 4. of this Book, and the 22. of B. 5. deduces the fame conclufion, from which it is plain that this ill composed Demonstration is not Euclid's. thefe fuperfluous things are now left out, and a more fimple Demonftration is given from the 4. Prop. of this Book, the fame which is in the Tranflation from the Arabic, by help of the 2. Prop. and Compofition; but in this the Author neglects Permutation, and does not shew the parallelograms to be equiangular, as is proper to do for the fake of beginners. PROP. XXV. B. VI. It is very evident that the Demonftration which Euclid had given of this Propofition, has been vitiated by fome unfkilful hand. for after this Editor had demonstrated that " as the rectilineal figure "ABC is to the rectilineal KGH, fo is the parallelogram BE to the "parallelogram EF," nothing more fhould have been added but this," and the rectilineal figure ABC is equal to the parallelogram "BE, therefore the rectilineal KGH is equal to the parallelogram «EF," viz. from Prop. 14. B. 5. but betwixt these two sentences he has inserted this, "wherefore, by Permutation, as the rectilineal Book VI. «figure ABC to the parallelogram BE, fo is the rectilineal KGH "to the parallelogram EF;" by which, it is plain, he thought it was not fo evident to conclude that the fecond of four proportionals is equal to the fourth from the equality of the first and third, which is a thing demonstrated in the 14. Prop. of B. 5. as to conclude that the third is equal to the fourth, from the equality of the first and fecond, which is no where demonftrated in the Elements as we now have them. but tho' this Propofition, viz. the third of four proportionals is equal to the fourth, if the first be equal to the second, had been given in the Elements by Euclid, as very probably it was, yet he would not have made use of it in this place, because, as was faid, the conclufion could have been immediately deduced without this fuperfluous ftep by Permutation. this we have fhewn at the greater length, both because it affords a certain proof of the vitiation of the Text of Euclid, for the very fame blunder is found twice in the Greek Text of Prop. 23. B. 11. and twice in Prop. 2. B. 12. and in the 5. 11. 12. and 18. of that Book; in which places of B. 12. except the last of them, it is rightly left out in the Oxford Edition of Commandine's Tranflation. and also that Geometers may beware of making ufe of Permutation in the like cafes, for the Moderns not unfrequently commit this mistake, and among others Commandine himself in his Commentary on Prop. 5. B. 3. p. 6. b. of Pappus Alexandrinus, and in other places. the vulgar notion of proportionals, has, it seems, pre-occupied many so much, that they do not sufficiently understand the true nature of them. Befides, tho' the rectilineal figure ABC, to which another is to be made fimilar, may be of any kind whatever, yet in the Demonstration the Greek Text has "triangle" inftead of " rectilineal "figure," which error is corrected in the above-named Oxford Edition. PROP. XXVII. B. VI. The fecond Cafe of this has danas, otherwife, prefixed to it, as if it was a different Demonstration, which probably has been done by fome unfkilful Librarian. Dr. Gregory has rightly left it out. the fcheme of this fecond Cafe ought to be marked with the fame letters of the Alphabet which are in the scheme of the firft, as is now done, PROP. XXVIII. and XXIX. B. VI. These two Problems, to the firft of which the 27. Prop. is neceffary, are the most general and ufeful of all in the Elements, and are most frequently made ufe of by the antient Geometers in the folution of other Problems; and therefore are very ignorantly left out by Tacquet and Dechales in their Editions of the Elements, who pretend that they are scarce of any ufe. the Cafes of these Problems, wherein it is required to apply a rectangle which shall be equal to a given fquare, to a given ftraight line, either deficient or exceeding by a fquare; as alfo to apply a rectangle which fhall be equal to another given, to a given ftraight line, deficient or exceeding by a fquare, are very often made ufe of by Geometers. and on this account, it is thought proper, for the fake of beginners, to give their constructions, as follows. 1. To apply a rectangle which fhall be equal to a given square, to a given straight line, deficient by a square. but the given square must not be greater than that upon the half of the given line. Let AB be the given ftraight line, and let the fquare upon the given straight line C be that to which the rectangle to be applied must be equal, and this square by the determination, is not greater than that upon half of the straight line AB. Bisect AB in D, and if the fquare upon AD be equal to the fquare upon C, the thing required is done. but if it be not equal to it, AD must be greater than C, according to the determination. draw DE at right angles to AB, L H K F and make it equal to C; pro duce ED to F, fo that EF be A equal to AD or DB, and from the center E, at the distance C EF defcribe a circle meeting E Book VI. AB in G, and upon GB describe the fquare GBKH, and complete the rectangle AGHL; also join EG. and because AB is bifected in D, the rectangle AG, GB together with the fquare of a DG is equal to (the square of DB, that is of EF or EG, that is a. 5. ãs to) the fquares of ED, DG. take away the fquare of DG from each of thefe equals, therefore the remaining rectangle AG, GB is equal to the fquare of ED, that is, of C. but the rectangle AG, GB is the rectangle AH, because GH is equal to GB. therefore the rec |