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from the equality of the angles, the similitude of the figures does Book XI. not follow, except in the case when the figures are triangles. the similar position of the fides, which contain the figures, to one another, depending partly upon each of these. and, for the same reason, those are similar solid figures which have all their folid angles equal, each to each, and are contained by the same number of similar plane figures. for there are some solid figures contained by similar plane figures, of the fame number, and even of the same magnitude, that are neither similar nor equal, as fhall be demonstrated after the Notes on the 10. Definition. upon this account it was necessary to amend the Definition of similar solid figures, and to place the Definition of a solid angle before it. and from this and

Definition, it is sufficiently plain how much the Elements have been spoiled by unskilful Editors.

the 10.

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 demonstrated, not assumed; so that Theon, or some other Editor, has ignorantly turned a Theorem which ought to be demonstrated into this 10. Definition. that figures are similar, 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 must 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 these circles are said 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 some Editor been improperly placed among the Definitions. The equality of figures ought not to be defined, but demonstrated. therefore tho’ it were true that folid figures contained by the same number of similar and equal plane figures are equal to one another, yet he would justly deserve to be blamed who should make a Definition of this Proposition which ought to be demonstrated. But if this Proposition be not true, muft it not be confessed that Geome

Book XI. ters have for these thirteen hundred years been mistaken in this

Elementary matter? and this should 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 justly reckoned amongst the most certain ; for that the Proposition is not universally true can be shewn by many examples ; the following is sufficient.

Let there be any plane rectilineal figure, as the triangle ABC, 2. 12. II. and from a point D within it draw a the straight line DE at right

angles to the plane ABC; in DE take DE, DF equal to one another, upon the opposite 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, thereb. 4. I. fore the base EB is equal to the base FB; in the same 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
base EA is equal to the

E
base FA; wherefore the
C. 8. I.

angle EBA is equal < to the
angle FBA, and the tri-
angle EBA equal to the
triangle FBA, and the other
B

C angles equal to the other 54.6. angles; therefore these tri1. Def. angles are similar d. in the 6.

F fame manner the triangle EBC is similar to the triangle FBC, and the triangle EAC to FAC. therefore there are two solid figures each of which is contained by six 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 same lines AB, BC, CA. the other solid is contained by the same three triangles the common vertex of which is G, and their bases AB, BC, CA; and by three other triangles of which the common vertex is the point F, and their

d.

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

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

For the solid angle at B which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the solid angle at the same point B which is contained by the four plane angles FBA, FBC, GBA, GBC; for this last 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 same; as has been proved. and, indeed, there may be innumerable solid 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 manifest that the before-mentioned solids are not similar, since their solid 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 same plane angles disposed in the same order, will be plain from the three following Propositions.

PROP. I. PROBLEM. Three magnitudes A, B, C being given, to find a fourth such, that every three shall 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 first, let B and C together be not less than A; therefore B, C, D together are greater than A. and because A is not less 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 less 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 excess 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 less than C and D together, and such that any three of them whatever are greater than the fourth; it is required to find a fifth magnitude E such, that any two of the three A, B, E shall be greater than the third, and also that any two of the three C, D, E shall be greater than the third. Let A be not less than B, and C not less than D.

First, Let the excess of C above D be not less 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 likewise than the excefs of A above B; wherefore E and B together are greater than A; and A is not less than B, therefore A and E together are greater than B. and, by the Hypothesis, A and B together are not less 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 Hypothesis, the three B, C, D are together greater than the fourth A; C and D together are greater than the excess 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 because E is greater than the excess of A above B, B together with E is greater than A. and, as in the preceding case, it may be shewn that A together with E is greater than B, and that A together with B is greater than E.

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

And because in each of the cases 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 less 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, each of which is contained by the same four plane angles, placed in the same order.

Take three plane angles A, B, C, of which A is not less than either of the other two, and such, 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 such, that A and B together be not less than C and D together. and by Problem 2. find a fifth angle E such, 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 A E C

F

B

K

D
M
^

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 consequently 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

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