from the equality of the angles, the fimilitude of the figures does Book XI. not follow, except in the cafe when the figures are triangles. the fimilar position of the fides, which contain the figures, to one another, depending partly upon each of thefe. and, for the fame rea son, those are similar solid figures which have all their folid angles equal, each to each, and are contained by the fame number of fimilar plane figures. for there are fome folid figures contained by fimilar plane figures, of the fame number, and even of the fame magnitude, that are neither fimilar nor equal, as fhall be demonftrated after the Notes on the 10. Definition. upon this account it was neceffary to amend the Definition of fimilar folid figures, and to place the Definition of a folid angle before it. and from this and the 10. Definition, it is fufficiently plain how much the Elements have been spoiled by unfkilful Editors.

DEF. X. B. XI.

Since the meaning of the word "equal" is known and established before it comes to be used in this Definition, therefore the Propofition which is the 10. Definition of this Book, is a Theorem the truth or falfhood of which ought to be demonftrated, not affumed; fo that Theon, or fome other Editor, has ignorantly turned a Theorem which ought to be demonftrated into this 10. Definition. that figures are fimilar, ought to be proved from the Definition of fimilar figures; that they are equal ought to be demonstrated from the Axiom, "Magnitudes that wholly coincide, are equal to one ano"ther;" or from Prop. A. of Book 5. or the 9. Prop. or the 14. of the fame Book, from one of which the equality of all kind of figures muft ultimately be deduced. In the preceding Books, Euclid has given no Definition of equal figures, and it is certain he did not give this. for what is called the 1. Def. of the 3. Book, is really a Theorem in which thefe circles are faid to be equal, that have the straight lines from their centers to the circumferences equal, which is plain from the Definition of a circle, and therefore has by fome Editor been improperly placed among the Definitions. The equality of figures ought not to be defined, but demonftrated. therefore tho' it were true that folid figures contained by the fame number of similar and equal plane figures are equal to one another, yet he would juftly deferve to be blamed who should make a Definition of this Propofition which ought to be demonstrated. But if this Propofition be not true, muft it not be confeffed that Geome

Book XI. ters have for these thirteen hundred years been mistaken in this Elementary matter? and this fhould teach us modesty, and to acknowledge how little, thro' the weakness of our minds, we are able to prevent mistakes even in the principles of sciences which are juftly reckoned amongst the most certain; for that the Propofition is not univerfally true can be fhewn by many examples; the following is fufficient.

b. 4. I.

b

Let there be any plane rectilineal figure, as the triangle ABC, 3. 12. II. and from a point D within it draw the straight line DE at right angles to the plane ABC; in DE take DE, DF equal to one another, upon the oppofite fides of the plane, and let G be any point in EF; join DA, DB, DC; EA, EB, EC; FA, FB, FC; GA, GB, GC. because the straight line EDF is at right angles to the plane ABC, it makes right angles with DA, DB, DC which it meets in that plane; and in the triangles EDB, FDB, ED and DB are equal to FD and DB, each to each, and they contain right angles, therefore the base EB is equal to the base FB; in the fame manner EA is equal to FA, and EC to FC. and in the triangles EBA, FBA, EB, BA are equal to FB, BA, and the bafe EA is equal to the bafe FA; wherefore the angle EBA is equal to the angle FBA, and the triangle EBA equal to the triangle FBA, and the other

c. 8. I.

b

C

angles equal to the other

angles; therefore these tri

1. Def. angles are fimilar d. in the

6.

fame manner the triangle

B

G

E

C

EBC is fimilar to the triangle FBC, and the triangle EAC to FAC, therefore there are two folid figures each of which is contained by fix triangles, one of them by three triangles the common vertex of which is the point G, and their bases the straight lines AB, BC, CA; and by three other triangles the common vertex of which is the point E, and their bases the fame lines AB, BC, CA. the other folid is contained by the fame three triangles the common vertex of which is G, and their bafes AB, BC, CA; and by three other triangles of which the common vertex is the point F, and their

bafes the fame straight lines AB, BC, CA. now the three triangles Book XI. GAB, GBC, GCA are common to both folids, and the three others EAB, EBC, ECA of the firft folid have been fhewn equal and fimilar to the three others FAB, FBC, FCA of the other folid, each to each. therefore these two solids are contained by the same number of equal and fimilar planes. but that they are not equal is manifeft, because the first of them is contained in the other. therefore it is not univerfally true that folids are equal which are contained by the fame number of equal and fimilar planes.

COR. From this it appears that two unequal folid angles may be contained by the fame number of equal plane angles.

For the folid angle at B which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the folid angle at the fame point B which is contained by the four plane angles FBA, FBC, GBA, GBC; for this laft contains the other. and each of them is contained by four plane angles, which are equal to one another, each to each, or are the self fame; as has been proved. and, indeed, there may be innumerable folid angles all unequal to one another, which are each of them contained, by plane angles that are equal to one another, each to each. it is likewise manifeft that the before-mentioned folids are not fimilar, fince their folid angles are not all equal.

And that there may be innumerable folid angles all unequal to one another, which are each of them contained by the fame plane angles difpofed in the fame order, will be plain from the three following Propofitions.

PROP. I. PROBLEM.

Three magnitudes A, B, C being given, to find a fourth fuch, that every three fhall be greater than the remaining one.

Let D be the fourth, therefore D must be less than A, B, C together. of the three A, B, C let A be that which is not less than either of the two B and C. and firft, let B and C together be not lefs than A; therefore B, C, D together are greater than A. and because A is not lefs than B; A, C, D together are greater than B. in the like manner A, B, D together are greater than C. wherefore in the case in which B and C together are not less than A, any magnitude D which is less than A, B, C together will answer the Problem.

But if B and C together be lefs than A, then because it is re

Book XI. quired that B, C, D together be greater than A, from each of these taking away B, C, the remaining one D must be greater than the excess of A above B and C. take therefore any magnitude D which is less than A, B, C together, but greater than 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 construction, A, B, C are together greater than D.

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

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 also that any two of the three C, D, E fhall be greater than the third. Let A be not lefs

than B, and C not lefs than D.

Firft, 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 less than the fum of C and D, but greater than the excess 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 lefs than C and D together, and C and D together are greater than E; therefore likewise 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, because, 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 lefs than C and D together, but greater than the excess of A above B. Let this magnitude be E, and because E is greater than the excefs of A above B, B together with E is greater than A. and, as in the preceding 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 in each of the cafes it has been fhewn that any two of Book XI. the three A, B, E are greater than the third.

And because in each of the cafes E is greater than the excess 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 conftruction, C and D together are greater than E. therefore any two of the three, C, D, E are greater than the third.

There may be innumerable folid angles all unequal to one another, each 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 less than either of the other two, and fuch, that A and B together are lefs 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 lefs 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

MA

H

greater than the third. and because A and B together are less than two right angles, the double of A and B together is less 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 less 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