Book VI. ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is only a name or expression by which the ratio which the first A has to the last D is signified, and by which at the same 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 same, or be not the same, 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 expression by which the ratio of the first A to the third C is signified, and by which, at the same time, is shown, 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 last C, which are the same with one another; and the triplicate ratio of the first to the second is a name or expression by which the ratio of the first A to the fourth D is signified, and by which, at the same time, is shown, 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 last D, which are all the same with one another; and so in the case 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 use of the word xyera, is said to be, or is called: which word he, no doubt, made use of also in the definition of compound ratio, but which Theon, or some other, has expunged from the Elements for the very same word is still retained in the wrong definition of compound ratio, which is now the 5th of the 6th Book: But in the citation of these definitions it is sometimes retained, as in the demonstration of Prop. 19, Book 6," the first is said to have, yra to the third the "duplicate ratio," &c. which is wrong translated by Commandine and others "has," instead of "is said to have:" and sometimes it is left out, as in the demonstration of Prop. 33, of the 11th Book, in which we find "the first has, xu to the "third the triplicate ratio;" but without doubt xe, “has," in this place, signifies the same as xeyra, is said to have: So likewise in Prop. 23, B. 6, we find this citation, "but the "ratio of K to M is compounded, ryzura, of the ratio of K "to L, and the ratio of L to M," which is a shorter way of expressing the same thing, which, according to the definition, ought to have been expressed by συγκεῖσθαι λεγέται, is said to be compounded.

From these remarks, together with the propositions subjoined to the fifth book, all that is found concerning compound

ratio, either in the ancient or modern geometers, may be un- Book VI. derstood and explained.

PROP. XXIV. B. VI.

It seems that some unskilful editor has made up this demonstration as we now have it, out of two others; one of which may be made from the 2d Prop. and the other from the 4th of this book: For, after he has, from the 2d of this book, and composition and permutation, demonstrated, that the sides about the angle common to the two parallelograms are proportionals, he might have immediately concluded, that the sides 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 show, that the triangles and parallelograms are equiangular: and in a tedious way, by help of Prop. 4, of this book, and the 22d of Book 5, deduces the same conclusion: From which it is plain, that this ill-composed demonstration is not Euclid's: These superfluous things are now left out, and a more simple demonstration is given from the 4th Prop. of this book, the same which is in the translation from the Arabic, by help of the 2d Prop. and composition; but in this the author neglects permutation, and does not show the parallelograms to be equiangular, as is proper to do for the sake of beginners.

PROP. XXV. B. VI.

It is very evident, that the demonstration which Euclid had given of this proposition has been vitiated by some unskilful hand: For, after this editor had demonstrated, that "as "the rectilineal figure ABC is to the rectilineal KGH, so is "the parallelogram BE to the parallelogram EF;" nothing more should 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, Book 5. But betwixt these two sentences he has inserted this; "wherefore, by permutation, as the recti"lineal figure ABC to the parallelogram BE, so is the recti"lineal KGH to the parallelogram EF;" by which, it is plain, he thought it was not so evident to conclude, that the second of four proportionals is equal to the fourth from the equality of the first and third, which is a thing demonstrated in the 14th Prop. of B. 5, as to conclude that the third is equal to the fourth, from the equality of the first and second,

Book VI. which is no where demonstrated in the Elements as we now have them: But though this proposition, 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 said, the conclusion could have been immediately deduced without this superfluous step, by permutation: This we have shown at greater length, both because it affords a certain proof of the vitiation of the text of Euclid; for the very same blunder is found twice in the Greek text of Prop. 23, Book 11, and twice in Prop. 2, Book 12, and the 5th, 11th, 12th, and 18th of that Book; in which places of Book 12, except the last of them, it is rightly left out in the Oxford edition of Commandine's translation: And also that geometers may beware of making use of permutation in the like cases; for the moderns not unfrequently commit this mistake, and among others Commandine himself in his commentary on Prop. 5, Book 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.

Besides, though the rectilineal figure ABC, to which another is to be made similar, may be of any kind whatever; yet in the demonstration the Greek text has "triangle" instead of "rectilineal figure," which error is corrected in the abovenamed Oxford edition.

PROP. XXVII. B. VI.

THE second case of this has anλws, otherwise, prefixed to it, as if it was a different demonstration, which probably has been done by some unskilful librarian. Dr Gregory has rightly left it out: The scheme of this second case ought to be marked with the same letters of the alphabet which are in the scheme of the first, as is now done.

PROP. XXVIII. and XXIX. B. VI.

THESE two problems, to the first of which the 27th Prop. is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the ancient geometers in the solution 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 use: The cases of these problems, wherein it is required to apply a

rectangle which shall be equal to a given square; to a given Book VI. straight line, either deficient or exceeding by a square; as also to apply a rectangle which shall be equal to another given, to a given straight line, deficient or exceeding by a square; are very often made use of by geometers: And, on this account, it is thought proper, for the sake of beginners, to give their constructions as follow:

1. To apply a rectangle which shall 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 straight line, and let the square 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 the straight line AB.

Bisect AB in D, and if the square upon AD be equal to the square 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, and make it equal to C; produce ED to F, so that EF may be equal to AD or DB, and from the centre E, at the distance EF, describe a circle meeting AB in G, and

a

L

HK

F

A

D

G B

C

E

upon GB describe the square GBKH, and complete the rectangle AGHL; also join EG: And because AB is bisected in D, the rectangle AG, GB, together with the square of DG is equal to (the square of DB, that is, of EF or EG, that is a 5. 2. to) the squares of ED, DG: Take away the square of DG from each of these equals; therefore the remaining rectangle AG, GB is equal to the square of ED, that is, of C: But the rectangle AG, GB is the rectangle AH, because GH is equal to GB; therefore the rectangle AH is equal to the given square upon the straight line C. Wherefore the rectangle AH, equal to the given square upon C, has been applied to the given straight line AB, deficient by the square GK. Which was to be done. 2. To apply a rectangle which shall be equal to a given square, to a given straight line, exceeding by a square.

Let AB be the given straight line, and let the square upon the given straight line C be that to which the rectangle to be applied must be equal.

X

Book VI.

a 6. 2.

a 3. 3.

a

L

E

KH

FAD B G

Bisect AB in D, and draw BE at right angles to it, so: that BE may be equal to C; and having joined DE, from the centre D at the distance DE describe a circle meeting AB produced in G; upon BG describe the square BGHK, and complete the rectangle AGHL. And because AB is bisected in D, and produced to G, the rectangle AG, GB together with the square of A DB is equal to (the square of DG, or DE, that is, to) the squares of EB, BD. From each of these equals take the square of DB; therefore the remaining rectangle AG, GB is. equal to the square of BE, that is, to the square upon C. But the rectangle AG, GB is the rectangle AH, because GH is equal to GB. Therefore the rectangle AH is equal to the square upon C. Wherefore the rectangle AH, equal to the given square upon C, has been applied to the given straight line AB, exceeding by the square GK. Which was to be done.

C

3. To apply a rectangle to a given straight line which shall be equal to a given rectangle, and be deficient by a square; but the given rectangle must not be greater than the square upon the half of the given straight line.

Let AB be the given straight line, and let the given rectangle be that which is contained by the straight lines C, D, which is not greater than the square upon the half of AB; it is required to apply to AB a rectangle equal to the rectangle C, D, deficient by a square.

Draw AE, BF at right angles to AB, upon the same side of it, and make AE equal to C, and BF to D: Join EF and bisect it in G; and from the centre G, at the distance GE, describe a circle meeting AE again in H: Join HF, and draw GK parallel to it, and GL parallel to AE, meeting AB in L.

Because the angle EHF in a semicircle is equal to the right angle EAB, AB and HF are parallels, and AH and BF are. parallels, wherefore AH is equal to BF, and the rectangle EA, AH equal to the rectangle EA, BF, that is, to the rectangle C, D: And because EG, GF are equal to one another, and AE, LG, BF parallels: therefore AL and LB are equal, also EK is equal to KHa, and the rectangle C, D, from the determination, is not greater than the square of AL, the half of AB; wherefore the rectangle EA, AH is not greater