PROP. XXXV. B. III. As the 25th and 33d propofitions are divided into more cafes, fo this 35th is divided into fewer cafes than are neceffary. Nor can it be fuppofed that Euclid omitted them because they are eafy; as he has given the cafe, which by far, is the easiest of them all, viz. that in which both the ftraight lines país through the centre: And in the following propofition he feparately demonftrates the cafe in which the ftraight line paffes through the centre, and that in which it does not pafs through the centre: So that it seems Theon, or fome other, has thought them too long to infert: But cafes that require different demonftrations, fhould not be left out in the elements, as was before taken notice of : These cafes are in the translation from the Arabic, and are now put into the text At the end of this, the words, "in the fame manner it may "be demonftrated, if the centre be in AC," are left out as the addition of fome ignorant editor. W DEFINITIONS of BOOK IV. HEN a point is in a straight, or any other line, this point is by the Greek geometers faid 11, to be upon, or in that line, and when a ftraight line or circle meets a circle any way, the one is faid azred to meet the other: But when a ftraight line or circle meets a circle fo as not to cut it, it is faid Tid, to touch the circle; and these two terms are never promifcuously used by them: Therefore, in the 5th definition of B. 4. the compound spars must be read, inftead of the fimple TT: And in the 1ft, 2d, 3d, and 6th definitions in Commandine's translation, "tangit," must be read inftead of "contingit:" And in the 2d and 3d definitions of Book 3. the fame change must be made: But in the Greek text of propofitions 11th, 12th, 13th, 18th, 19th, Book 3. the com pound verb is to be put for the fimple. PROP. IV. B. IV. In this, as alfo in the 8th and 13th propofitions of this book, it is demonftrated indirectly, that the circle touches a ftraight line; whereas in the 17th, 33d, and 37th propofitions of book 3. the fame thing is directly demonftrated: And this way we have U 4 Book III. Book IV. Book IV. have chofen to use in the propofitions of this book, as it is fhorter. PROP. V. B. IV. The demonstration of this has been spoiled by fome unskilful hand: For he does not demonftrate, as is neceffary, that the two ftraight lines which bifect the fides of the triangle at right angles, muft meet one another; and, without any reason, he divides the propofition into three cafes; whereas, one and the fame conftruction and demonftration ferves for them all, as Campanus has obferved; which useless repetitions are now left out: The Greek text also in the Corollary is manifeftly vitiated, where mention is made of a given angle, though there neither is, nor can be any thing in the propofition relating to a given angle. PROP. XV. and XVI. B. IV. In the corollary of the first of thefe, the words equilateral and equiangular are wanting in the Greek: And in prop. 16. inftead of the circle ABCD, ought to be read the circumference ABCD: Where mention is made of its containing fifteen equal parts. Book V. M DEF. III. B. V. ANY of the modern mathematicians reject this definition : The very learned Dr Barrow has explained it at large at the end of his third lecture of the year 1666, in which also he anfwers the objections made against it as well as the subject would allow Ánd at the end gives his opinion upon the whole, as follows: "I fhall only add, that the author had, perhaps, no o"ther defign in making this definition, than (that he might "more fully explain and embellifh his fubject) to give a gene"ral and fummary idea of ratio to beginners, by premifing "this metaphyfical definition, to the more accurate defini"tions of ratios that are the fame to one another, or one of which is greater, or lefs than the other: I call it a meta" phyfical, for it is not properly a mathematical definition, "fince nothing in mathematics depends on it, or is deduced, "nor, as I judge, can be deduced from it: And the defini❝tion of analogy, which follows, viz. Analogy is the fimi "litude "litude of ratios, is of the fame kind, and can ferve for no purpose in mathematics, but only to give beginners fome "general, tho' grofs and confufed notion of analogy: But the "whole of the doctrine of ratios, and the whole of mathema"tics, depend upon the accurate mathematical definitions which "follow this: To thefe we ought principally to attend, as the "doctrine of ratios is more perfectly explained by them; this "third, and others like it, may be entirely fpared without any "lofs to geometry; as we fee in the 7th book of the elements, "where the proportion of numbers to one another is defined, "and treated of, yet without giving any definition of the ratio "of numbers; tho' fuch a definition was as neceffary and ufe"ful to be given in that book, as in this: But indeed there is "scarce any need of it in either of them: Though I think that "a thing of fo general and abftracted a nature, and thereby the "more difficult to be conceived and explained, cannot be more "commodiously defined than as the author has done: Upon "which account I thought fit to explain it at large, and defend "it against the captious objections of those who attack it." To this citation from Dr Barrow I have nothing to add, except that I fully believe the 3d and 8th definitions are not Euclid's, but added by fome unskilful editor. DE F. XI. B. V. pro It was neceffary to add the word "continual" before " portionals" in this definition; and thus it is cited in the 33d prop. of book 11. After this definition ought to have followed the definition of compound ratio, as this was the proper place for it; duplicate and triplicate ratio being fpecies of compound ratio. But Theon has made it the 5th def. of B. 6. where he gives an abfurd and entirely useless definition of compound ratio: For this reason we have placed another definition of it betwixt the 11th and 12th of this book, which, no doubt, Euclid gave; for he cites it exprefsly in prop. 23. B. 6. and which Clavius, Herigon, and Barrow, have likewife given, but they retain alfo Theon's, which they ought to have left out of the elements. DE F. XIII. B. V. This, and the reft of the definitions following, contain the explication of fome terms which are used in the 5th and following books; which, except a few, are eafily enough understood from the Book V. the propofitions of this book where they are first mentioned: They seem to have been added by Theon, or fome other. However it be, they are explained something more diftinctly for the fake of learners. PROP. IV. B. V. In the construction preceding the demonftration of this, the words Tox, any whatever, are twice wanting in the Greek, as alfo in the Latin tranflations; and are now added, as being wholly neceffary. Ibid. in the demonstration; in the Greek, and in the Latin tranflation of Commandine, and in that of Mr Henry Briggs, which was published at London in 1620, together with the Greek text of the first fix books, which translation in this place is followed by Dr Gregory in his edition of Euclid, there is this fentence following, viz. " and of A and C have been taken e"quimultiples K, L; and of B and D, any equimultiples "whatever (a ruxs) M, N;" which is not true, the words any whatever," ought to be left out: And it is strange that neither Mr Briggs, who did right to leave out these words in one place of prop. 13. of this book, nor Dr Gregory, who changed them into the word " fome" in three places, and left them Out in a fourth of that fame prop. 13. did not alfo leave them out in this place of prop. 4. and in the second of the two places where they occur in prop. 17. of this book, in neither of which they can ftand confiftent with truth: And in none of all these places, even in those which they corrected in their Latin tranflation, have they cancelled the words & rugs in the Greek text, as they ought to have done The fame words & Tux are found in four places of prop. 11. of this book, in the first and laft of which they are neceffa ry, but in the fecond and third, though they are true, they are quite fuperfluous; as they likewife are in the fecond of the two places in which they are found in the 12th prop. and in the like places of prop. 22. 23. of this book; but are wanting in the last place of prop. 23. as alfo in prop. 25. Book 11. COR. IV. PROP. B. V. This corollary has been unskilfully annexed to this propo fition, and has been made instead of the legitimate demonftration, which, without doubt, Theon, or fome other editor, has taken away, not from this, but from its proper place in this this book: The author of it defigned to demonftrate, that if four Book V. magnitudes E, G, F, H be proportionals, they are also propor tionals inversely; that is, G is to E, as H to F; which is true; but the demonftration of it does not in the leaft depend upon this 4th prop. or its demonftration: For, when he says, “be"cause it is demonstrated that if K be greater than M, L is " greater than N," &c. This indeed is fhewn in the demonstration of the 4th prop. but not from this, that E, G, F, H are proportionals; for this laft is the conclufion of the proposition. Wherefore these words," becaufe it is demonftrated," &c. are wholly foreign to his defign: And he fhould have proved, that if K be greater than M, L is greater than N, from this, that E, G, F, H are proportionals, and from the 5th def. of this book, which he has not; but is done in propofition B, which we have given in its proper place, inftead of this corollary; and another corollary is placed after the 4th prop. which is often of ufe; and is neceffary to the demonftration of prop. 18. of this book. PROP. V. B. V. In the conftruction which precedes the demonftration of this propofition, it is required that EB may be the fame multiple of CG, that AE is of CF; that is, that EB be divided into as many equal parts, as there are parts in AE equal to CF: From which it is evident, that this conftruction is not Euclid's; for he does not show the way of dividing ftraight lines, and far lefs other magnitudes, into any number of equal parts, until the 9th propofition of B 6. ; and he never requires. any thing to be done in the conftruction, of which he had not before given the method of doing For this reafon, we have changed the conftruction to one, which, without doubt, is Euclid's, in which nothing is required but to add a magnitude to itself a E certain number of times; and this is to be found in the translation from the Arabic, though the enunciation of the propofition and the demonftration are there very much fpoiled. Jacobus Peletarius, who was the first, as far as I know, who took notice of this error, gives alfo the right conftruction A G C F D in his edition of Euclid, after he had given the other which he blames He fays, he would not leave it out, because it was fine, and might fharpen one's genius to invent others like it; whereas there |