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and the magnitude of it being likewise given, to find its sides.

And the 4th Problem is the same with this: to find a point N in the given straight line AB produced, so as to make the rectangle AN, NB, equal to a given space: or, which is the same thing, having given AB the difference of the sides of a rectangle, and the magnitude of it, to find the sides.

PROPOSITION XXXI.

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 24th Prop. of Book 5. as Clavius has done.

PROPOSITION XXXII.

The enunciation of the preceding 26th 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 similar 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 32d Prop. was needful: and the 32d may be otherwise, and something more briefly demonstrated, as follows:

PROPOSITION XXXII.

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

Let GAF, HFC be two triangles which have two sides AG, GF, proportional to the two sides 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 straight line.

*

A

G

F

E

H

31. 1.

B

Draw CK parallel to FH, and let it meet GF produced in K: because AG, KC are each of them parallel to FH, they are parallel to one another, and therefore the alternate angles AGF, FKC are equal: and AG is to GF, as (FH to HC, that is *) CK to KF; wherefore the triangles AGF, CKF are equiangular*, and the angle AFG equal to the angle CFK: but GFK is a straight line, therefore AF and FC are in a straight line*.

30.1.

34. 1.

⚫ 6.6.

14. 1.

The 26th Prop. is demonstrated from the 22d, 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.

Produce EF, GF, to H, K, and join FA, FC; then because the parallelograms ABCD, AEFG are similar, DA is to AB, as GA to AE: wherefore the remainder DG

* Cor. 19. 5. is to the remainder EB, as GA to AE: but DG is equal to FH, EB to HC, and AE to GF therefore as FH to HC, so is AG to GF; and FH, HC are parallel to AG, GF; and the triangles AGF, FHC are joined at one angle in the point F; wherefore AF, FC are in the same straight line*.

* 32. 6.

32.6.

A

E

B

F

H

K

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 *.

PROPOSITION XXXIII.

The words "because they are at the centre," are left out, as the addition of some unskilful hand.

In the Greek, as also in the Latin translation, the words, ἃ ἔτυχε, σε any whatever", are left out in the demonstrations of both parts of the proposition, and are now added as quite necessary; and in the demonstration of the second part, where the triangle BGC is proved to be equal to CGK, the illative particle aça, 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 Ptolemy's Mɛyáλn Zúrταξις, Ρ. 50.

PROPOSITIONS B. C. D.

These three propositions are added, because they are frequently made use of by geometers.

BOOK XI.

DEFINITIONS.

IX. and XI. The 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 not follow, except in the case when the figures are triangles: the similar position of the sides 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 solid 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 same number, and even of the same magnitude, that are neither similar nor equal, as shall be demonstrated after the notes on the 10th Def.: 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 10th Def. it is sufficiently plain how much the Elements have been spoiled by unskilful editors.

X. Since the meaning of the word "equal" is known and established before it comes to be used in this definition; therefore the proposition which is the 10th Def. of this Book, is a theorem, the truth or falsehood 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 10th Def. That figures are similar, ought to be proved from the definition of similar figures; that they are equal, ought to be demonstrated from the axiom, "Magnitudes that wholly coincide are equal to one another;" or from Prop. A. of Book 5. or the 9th Prop. or the 14th of the same Book, from one of which the equality of all kinds 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 he called the first Def. of the third Book, is really a theorem in which those circles are said to be

* 12. 11.

* 4. 1.

S. 1.

• 4. J.

equal, that have the straight lines from the centres 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, though it were true, that solid 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 would make a definition of this proposition, which ought to be demonstrated. But if this proposition be not true, must it not be confessed that geometers have, for these thirteen hundred years, been mistaken in this elementary matter? And this should teach us modesty, and to acknowledge how little, through the weakness of our minds, we are able to prevent mistakes, even in the principles of science 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:

EIA

Let there be any plane rectilineal figure, as the triangle
ABC, and from a point D within it, draw the straight line
DE at right angles to the planc
ABC: in DE, take DE, DF, equal
to one another, 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 B
with DA, DB, DC, which it meets
in that plane; and in the tri-
angles EDB, FDB, the sides ED

F

C

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 same manner EA is equal to FA, and EC to FC and in the triangles EBA, FBA, the sides 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 triangle EBA equal to the triangle FBA, and the other angles equal to the other angles; therefore these tri

1 Def. 6.

angles are similar*: in the same manner the triangle EBC is 4. 6. & 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 bases the same straight lines AB, BC, CA: now the three triangles GAB, GBC, GCA are common to both solids, and the three others EAB, EBC, ECA, of the first solid, have been shewn equal and similar to the three others FAB, FBC, FCA, of the other solid, 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 manifest, because the first of them is contained in the other: therefore it is not universally true that solids are equal which are contained by the same number of equal and similar planes.

COROLLARY. From this it appears that two unequal solid 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 con-tains 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 solid 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.

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