Book VI. PRO P. XXXI. B. VI. In the Demonstration of this the inversion of proportionals is twice neglected, and is now added, that the conclusion may be legitimately made by help of the 24. Prop. of B. 5. as Clavius had done.

The Enuntiation of the preceding 26. Prop. is not general
enough; because not only two similar parallelograms that have an
angle common to both, are about the same diameter; but likewise
two fimilar parallelograms that have vertically opposite angles,
have their diameters in the same straight line. but there seems to
have been another, and that a direct Demonstration of these cases,
to which this 32. Proposition was needful, and the 32. may be
otherwise and something more briefly demonstrated as follows.


a. 31.

Cb. 30.5.

If two triangles which have two sides of the one, &c.

Let GAF, HFC be two triangles which have two fides AG,
GF proportional to the two fides FH, HC, viz. AG to GF, as FH
to HC ; and let AG be parallel to FH,
and GF to HC; AF and FC are in

a A

straight line.
Draw CK parallel * to FH, and let E

it meet GF produced in K. because AG,
KC are each of them parallel to FH,
they are parallel b to one another, and B
therefore the alternate angles AGF,

FKC are equal. and AG is to GF, as (FH to HC, that is ) CKC 34 1
to KF; wherefore the triangles AGF, CKF are equiangular d, and d. 6. 6.
the angle AFG equal to the angle CFK. but GFK is a straight
line, therefore AF and FC are in a straight line

The 26. Prop. is demonstrated from the 32. as follows.

If two similar and similarly placed parallelograms have an angle
common to both, or vertically opposite angles; their diameters are
in the same straight line.

First, Let the parallelograms ABCD, AEFG have the angle
BAD common to both, and be similar, and similarly placed ;
ABCD, AEFG are about the same diameter.

c. 14. I.


Book VI. Produce EF, GF, to H, K, and join FA, FC. then because the V parallelograms ABCD, AEFG are fi

G milar, DA is to AB, as GA to AE;

D a.Cor.19.5. wherefore the remainder DG is 'to

the remainder EB, as GA to A E. but I
DG is equal to FH, EB to HC, and
AE to GF. therefore as FH to HC,
fo is AG to GF; and FH, HC are pa- 1
rallel to AG, GF; and the triangles

K AGF, FHC are joined at one angle, in the point F; wherefore b. 31. 6. AF, FC are in the same straight line b.

Next, Let the parallelograms KFHC, GFEA which are similar and similarly placed, have their angles KFH, GFE vertically opposite; their diameters AF, FC are in the same straight line.

Because AG, GF are parallel to FH, HC; and that AG is to GF, as FH to HC; therefore AF, FC are in the same straight line 6.

PRO P. XXXIII. B. VI. The words “ because they are at the center,” are left out, as the addition of some un kilful hand.

In the Greek, as also in the Latin Translation, the words a' itu. X, any whatever,” are left out in the Demonstration of both parts of the Proposition, and are now added as quite necessary. and in the Demonftration of the second part, where the triangle BGC is proved to be equal to CGK, the illative particle äge in the Greek Text ought to be omitted.

The second part of the Proposition is an addition of Theon's, as he tells us in his Commentary on Ptolomy's Meyaan Eurtas, P: 50.

PROP. B, C, D, B. VI. These three Propositions are added, because they are frequently made use of by Geometers.

Book XI.

DE F. IX. and XI. B. XI.

HE similitude of plane figures is defined from the equality of

their angles, and the proportionality of the sides about the equal angles; for from the proportionality of the sides only, or only

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 fimilar position of the sides, which contain the figures, to one another, depending partly upon each of these. and, for the same reafon, those are similar folid figures which have all their folid angles equal, each to each, and are contained by the same number of fimilar plane figures. for there are some solid figures contained by similar plane figures, of the same number, and even.of the same magnitude, that are neither similar nor equal, as shall be demonstrated after the Notes on the jo. 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 the 10. Definition, it is sufficiently plain how much the Elements have been spoiled by unskilful 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 fallhood 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 limilar 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 ail 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 i. 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 gures fiare equal to one another, yet he would juftly deserve to be blamed who should make a Definition of this Proposition which ought to be demonstrated. But if this Proposition be not true, must it not be confefied that Geo

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

Elementary matter ? and this Mould 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. 11. 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 anuther, upon the opposite sides 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 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
base EA is equal to the

base FA; wherefore the
angle EBA is equal to the
angle FBA, and the tri-
angle EBA equal to the
triangle FBA, and the other B

angles equal to the other 4.6. angles; therefore these tri1.Def. angles are simiiar d. in the same manner the triangle

म F 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 same lines AB, BC, CA. the other folid is contained by the same 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

C. 8.



bases the same straight lines AB, BC, CA. Dow the three triangles Book XI. GAB, GBC, GCA are common to both folids, and the three others LAB, EBC, ECA of the first folid have been shewn 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 similar planes. but that they are not equal is inanifeft, because the first of them is contained in the other. therefore it is not univerfally true that solids are equal which are contained by the same ouniber of equal and similar 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, CBC 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 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 folid angles all unequal to one another, which are each of them contained by plane angles that are equal to one another, each to eacli. it is likewise manifest that the before-mentioned folids are not fimilar, since 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 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 lefs than A, then because it is rs.

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