Book XF. quired that B, C, D together be greater than A, from each of thefe taking away B, C, the remaining one D must be greater than the excefs of A above B and C. take therefore any magnitude D which is less than A, B, C together, but greater that the excess of A above B and C. then B, C, D together are greater than A; and because A is greater than either B or C, much more will A and D, together with either of the two B, C be greater than the other. and, by the conftruction, A, B, C are together greater than D.

COR. If befides, it be required that A and B together fhall not be less than C and D together; the excefs of A and B together above C muft not be less than D, that is D must not be greater than that excefs.

PROP. II. PROBLEM.

Four magnitudes A, B, C, D being given of which A and B together are not lefs than C and D together, and fuch that any three of them whatever are greater than the fourth; it is required to find a fifth magnitude E fuch, that any two of the three A, B, E fhall be greater than the third, and alfo that any two of the three C, D, E fhall be greater than the third. Let A be not less than B, and C not lefs than D.

First, Let the excefs of C above D be not lefs than the excess of A above B. it is plain that a magnitude E can be taken which is lefs than the fum of C and D, but greater than the excefs of C above D; let it be taken, then E is greater likewife than the excefs of A above B; wherefore E and B together are greater than A; and A is not lefs than B, therefore A and E together are greater than B. and, by the Hypothefis, A and B together are not less than C and D together, and C and D together are greater than E; therefore likewife A and B are greater than E.

But let the excess of A above B be greater than the excess of C above D. and, becaufe, by the Hypothefis, the three B, C, D are together greater than the fourth A; C and D together are greater than the excefs of A above B. therefore a magnitude may be taken which is less than C and D together, but greater than the excess of A above B. Let this magnitude be E, and becaufe E is greater than the excefs of A above B, B together with E is greater than A. and, as in the preceeding cafe, it may be fhewn that A together with E is greater than B, and that A together with B is greater than E.

therefore

therefore in each of the cafes it has been fhewn that any two of the Book XI. three A, B, E are greater than the third.

And because in each of the cafes E is greater than the excefs of C above D, E together with D is greater than C, and, by the Hypothefis, C is not lefs than D, therefore E together with C is greater than D; and, by the construction, C and D together are greater than E. therefore any two of the three, C, D, E are greater than the third.

PROP. III. THEOREM.

There may be innumerable folid angles all unequal to one another, cach of which is contained by the fame four plane angles, placed in the fame order.

Take three plane angles A, B, C, of which A is not lefs than ei ther of the other two, and fuch, that A and B together are less than two right angles; and by Problem 1. and its Corollary, find a fourth angle D fuch, that any three whatever of the angles A, B, C, D be greater than the remaining angle, and fuch, that A and B together be not less than C and D together. and by Problem 2. find a fifth angle E fuch, that any two of the angles A, B, E be greater than the third, and also that any two of the angles C, D, E

be greater than the third. and because A and B together are lefs than two right angles, the double of A and B together is lefs than four right angles. but A and B together are greater than the angle E, wherefore the double of A and B together is greater than the three angles A, B, E together, which three are confequently lefs than four right angles; and every two of the fame angles A, B, E are greater than the third; therefore, by Prop. 23. 11. a folid angle may be made contained by three plane angles equal to the angles A, B, E, each to each. Let this be the angle F contained by the three plane angles GFH, HFK, GFK which are equal to the angles

Y

Book XI. angles A, B, E, each to each. and because the angles C, D together are not greater than the angles A, B together, therefore the angles C, D, E are not greater than the angles A, B, E. but these laft three are lefs than four right angles, as has been demonstrated, wherefore also the angles C, D, E are together less than four right angles, and every two of them are greater than the third; therefore a folid angle may be made which shall be contained by three plane a. 23. 11. angles equal to the angles C, D, E, each to each. and by Prop.

26. 11. at the point F in the straight line FG a folid angle may be made equal to that which is contained by the three plane angles that are equal to the angles C, D, E. let this be made, and let the angle GFK, which is equal to E, be one of the three; and let KFL, GFL be the other two which are equal to the angles C, D, each to each. thus, there is a folid angle conftituted at the point F contained by the four plane angles GFH, HFK, KFL, GFL which are equal to the angles A, B, C, D, each to each.

N

Again, Find another angle M fuch, that every two of the three angles A, B, M be greater than the third, and alfo every two of the three C, D, M be greater than the third. and, as in the preceeding part, it may be demonftrated that the three A, B, M are lefs than four right angles, as alfo that the three C, D, M are less than four right angles. Make therefore a folid angle at N contained by the three plane angles ONP, PNQ, ONQ, which are equal to A, B, M, each to each. and by Prop. 26. 11. make at the point N in the ftraight line ON a folid angle contained by three plane angles of which one is the angle ONQ equal to M, and the other two are the angles QNR, ONR which are equal

R

P

to

to the angles C, D, each to each, thus at the point N there is a fo- Book XI. lid angle contained by the four plane angles ONP, PNQ, QNR, ONR which are equal to the angles A, B, C, D, each to each. and that the two folid angles at the points F, N, each of which is contained by the above named four plane angles, are not equal to one another, or that they cannot coincide, will be plain by confidering that the angles GFK, ONQ; that is, the angles E, M are unequal by the construction, and therefore the straight lines GF, FK cannot coincide with ON, NQ, nor consequently can the folid angles, which therefore are unequal.

And because from the three given plane angles A, B, C there can be found innumerable other angles that will serve the fame purpofe with the angle D, and again from D or any one of thefe others, and the angles A, B, C, there may be found innumerable angles, fuch as E or M; it is plain that innumerable other folid angles may be constituted which are each contained by the fame four plane angles, and all of them unequal to one another. Q. E. D.

And from this it appears that Clavius and other Authors are miftaken who affert that thofe folid angles are equal which are con tained by the fame number of plane angles that are equal to one ano. ther, each to each. alfo it is plain that the 26. Prop. of Book II. is by no means fufficiently demonftrated, because the equality of two folid angles, whereof each is contained by three plane angles which are equal to one another, each to each, is only affumed, and not demonstrated.

PROP. I. B. XI.

The words at the end of this, " for a straight line cannot meet a straight line in more than one point," are left out, as an addi tion by fome unskilful hand; for this is to be demonstrated, not af fumed.

Mr. Thomas Simpson in his notes at the end of the 2d Edition of his Elements of Geometry p. 262. after repeating the words of this note, adds "Now can it poffibly fhew any want of skill in an editor" (he means Euclid or Theon) "to refer to an Axiom which "Euclid himself had laid down Book 1. No. 14." (he means Earrow's Euclid, for it is the 10th in the Greek)" and not to have "demonftrated, what no man can demonftrate?" But all that in this cafe can follow from that Axiom is, that if two ftraight lines could meet each other in two points, the parts of them betwixt thefe points must coincide, and fo they would have a fegment betwixt

X 2

Book XI. betwixt these points common to both. Now, as it has not been fhown in Euclid that they cannot have a common segment, this does not prove that they cannot meet in two points, from which their not having a common fegment is deduced in the Greek Edition. but, on the contrary, because they cannot have a common fegment, ns is fhewn in Cor. of 11. Prop. B. 1. of 4to. Edition, it follows plainly that they cannot meet in two points, which the remarker fays no man can demonstrate.

Mr. Simpfon in the fame notes p. 265. juftly obferves that in the Corollary of Prop. 11. Book 1. 4to. Edit. the ftraight lines AB, BD, BC, are fuppofed to be all in the fame plane, which cannot be affumed in 1. Prop. B. 11. this, foon after the 4to. Edition was published, I obferved and corrected as it is now in this Edition. he is mistaken in thinking the 10th Axiom he mentions here, to be Euclid's; it is none of Euclid's, but is the 10th in Dr. Barrow's Edition, who had it from Herigon's Curfus Vol. I. and in place of it the Corollary of 11. Prop. Book 1. was added.

PROP. II. B. XI.

This Propofition feems to have been changed and vitiated by fome Editor; for all the figures defined in the 1. Book of the Elements, and among them triangles, are, by the Hypothefis, plane fi gures; that is, fuch as are defcribed in a plane; wherefore the fecond part of the Enuntiation needs no Demonftration. befides a convex fuperficies may be terminated by three ftraight lines meeting one another. the thing that fhould have been demonstrated is, that two, or three ftraight lines, that meet one another, are in one plane. and as this is not fufficiently done, the Enuntiation and De innftration are changed into those now put into the Text.

PROP. III. B. XI.

In this Propofition the following words near to the end of it are let out, viz. therefore DEB, DFB are not straight lines, in the "like manner it may be demonstrated that there can be no other "ftraight line between the points D, B." because from this that two lines include a fpace, it only follows that one of them is not a fhaight line. and the force of the argument lies in this, viz. if the common fection of the planes be not a straight line, then two straight lines could include a space, which is abfurd; therefore the common fection is a ftraight line.

PROP.