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As the 25. and 33. Propofitions are divided into more cafes, fo this 35. is divided into fewer cafes than are neceffary. Nor can it be supposed that Euclid omitted them because they are easy; as he has given the cafe which by far is the eafieft of them all, viz. that in which both the straight lines pass thro' the center. and in the following Propofition he feparately demonftrates the cafe in which the ftraight line paffes thro' the center, and that in which it does not pass thro' the center. so that it seems Theon, or some other, has thought them too long to infert. but cases 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 center be in AC" are left out as the addition of fome ignorant Editor.



HEN a point is in a ftraight, or any other line, this point is by the Greek Geometers faid as, to be upon, or in that line. and when a ftraight line or circle meets a circle any way, the one is faid to meet the other. but when a straight line or circle meets a circle fo as not to cut it, it is faid pásu, to touch the circle; and these two terms are never promifcuously used by them. therefore in the 5. Definition of B. 4. the compound ἐράπηται muft be read, inftead of the fimple ἅπληται. and in the 1, 2, 3. and 6. Definitions in Commandine's translation « tangit” must be read instead of " contingit." and in the 2. and 3. Definitions of Book 3. the fame change must be made. but in the Greek text of Propofitions 11, 12, 13, 18, 19. B. 3. the compound verb is to be put for the fimple.


In this, as alfo in the 8. and 13. Propofitions of this Book, it is demonftrated indirectly that the circle touches a straight line;

Book III.

Book IV.

Book IV. whereas in the 17. 33. and 37. Propofitions of Book 3. the same thing is directly demonstrated. and this way we have chosen to use in the Propofitions of this Book, as it is fhorter.


The Demonftration of this has been spoiled by fome unskilful hand. for he does not demonftrate, as is neceffary, that the two ftraight lines which bisect the fides of the triangle at right angles, must meet one another; and, without any reason, he divides the Propofition into three cafes, whereas one and the same construction and demonstration serves for them all, as Campanus has obferved; which useless repetitions are now left out. the Greek text also in the Corollary is manifestly vitiated, where mention is made of a given angle, tho' there neither is, nor can be any thing in the Propofition relating to a given angle.

PROP. XV. and XVI. B. IV.

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In the Corollary of the first of these, the words equilateral and equiangular are wanting in the Greek. and in Prop. 16. instead of the circle ABCD ought to be read the circumference ABCD, where mention is made of its containing fifteen equal parts.

Book 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 answers the objections made against it as well as the subject would allow. and at the end gives his opinion upon the whole, as follows.

"I shall only add, that the Author had, perhaps, no other De"fign in making this Definition, than (that he might more fully "explain and embellish his subject) to give a general and fummary "idea of ratio to beginners, by premising this Metaphysical Defi"nition, to the more accurate Definitions of ratios that are the "fame to one another, or one of which is greater, or less than the ❝other. I call it a Metaphysical, for it is not properly a Mathema"tical Definition, fince nothing in Mathematics depends on it, or ❝is deduced, nor, as I judge, can be deduced from it. and the "Definition of Analogy, which follows, viz. Analogy is the fimi

litude of ratios, is of the fame kind, and can ferve for no purpose Book V. "in Mathematics, but only to give beginners fome general tho' "grofs and confused notion of Analogy. but the whole of the doc"trine of Ratios, and the whole of Mathematics depend upon the "accurate Mathematical Definitions which follow this. to these we 66 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 7. 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 useful to be given in that Book, as in this. but " indeed there is scarce any need of it in either of them. tho' I "think that a thing of fo general and abstracted a nature, and there"by 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 3. and 8. Definitions are not Euclid's, but added by fome unfkilful Editor.

DEF. XI. B. V.

It was neceffary to add the word "continual" before " pro"portionals" in this Definition; and thus it is cited in the 33. 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 species of Compound ratio. But Theon has made it the 5. 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 11. and 12. 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.


This and the rest of the Definitions following, contain the explication of fome terms which are used in the 5. and following


Book V. Books; which, except a few, are easily enough understood from the Propofitions of this Book where they are first mentioned. they feem to have been added by Theon or fome other. However it be, they are explained fomething more diftinctly for the fake of learners.


In the construction, preceding the demonftration of this, the words a Tuxe, 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 demonftration; 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 tranflation 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 equimultiples K, L; "and of B and D, any equimultiples whatever (rus) 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 also 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 stand confiftent with truth. and in none of all these places, even in those which they corrected in their Latin translation, have they cancelled the words 'Tuys in the Greek text, as they ought to have done.

The fame words 'ruxe are found in four places of Prop. 11. of this Book, in the first and last of which, they are necessary, but in the second and third, tho' they are true, they are quite fuperfluous; as they likewise are in the second of the two places in which they are found in the 12. 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. B. 11.


This Corollary has been unskilfully annexed to this Propofition, and has been made instead of the legitimate demonstration which

without doubt Theon, or fome other Editor has taken away, not Book V. from this, but from its proper place in this Book. the Author of it defigned to demonftrate that if four magnitudes E, G, F, H be proportionals, they are also proportionals 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 4. Prop. or its demonstration. for when he fays "because it is demonstrated that if K be greater "than M, L is greater than N," &c. this is indeed fhewn in the demonftration of the 4. Prop. but not from this that E, G, F, H are proportionals, for this last is the conclufion of the Proposition. Wherefore these words "because 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 5. Def. of this Book, which he has not; but is done in Propofition B, which we have given, in its proper place, instead of this Corollary. and another Corollary is placed after the 4. Prop. which is often of use, and is neceffary to the Demonftration of Prop. 18. of this Book.

PROP. V. B. V.

In the conftruction which precedes the demonstration 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 fhew the way of dividing straight lines, and far lefs other magnitudes, into any number of equal parts, until the 9. Propofition of B. 6. and he never requires any thing to be done in the construction, of which he had not before given the method of doing. for this reason we have changed the conftruction to A one which without doubt is Euclid's, in which nothing is required but to add a magnitude to itself a certain E



number of times. and this is to be found in the tran-
flation from the Arabic, tho' the enunciation of the
Propofition and the demonstration are there very much
fpoiled. Jacobus Peletarius who was the firft, as far B
as I know, who took notice of this error, gives alfo
the right construction in his edition of Euclid, after he had given.
the other which he blames. he fays he would not leave it out, be-
cause it was fine, and might sharpen one's genius to invent others

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