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Book VI.

4. To apply a rectangle to a given straight line that shall be equal to a given rectangle, exceeding by a square.

Let AB be the given straight line, and the rectangle C, D the given rectangle, it is required to apply a rectangle to AB equal to C, D, exceeding by a square.

Draw AE, BF at right angles to AB, on the contrary fides of it, and make ab equal to C, and BF equal to D: Join EF, and bisect it in G; and from the centre G, at the distance GE, describe a circle meeting AE again in H; join HF, and draw GL parallel to AE; let the circle meet AB

E

C
produced in M, N, and upon
BN describe the
the square

D
NBOP, and complete the
rectangle ANPQ: because the

G 0 P
angle EHF in a semicircle is
equal to the right angle EAB.

L

B
AB and HF are parallels, and M

N
therefore AH and BF are e-
qual, and the rectangle EA,

H

F AH equal to the rectangle EA, BF, that is, to the rectang! C, D: And because ML is equal to LN, and AL to LB, therefore MA is equal to BN, and the rectangle AN, NB to MA, AN, that is, to the rectangle EA, AH, or the rectangle C, D: Therefore the rectangle AN, NB, that is, AP, is equal to the rectangle C, D, and to the given straight line AB the rectangle AP has been applied equal to the given rectangle C, D, exceeding by the square BP. Which was to be done.

Willebrordus Snellius was the first, as far as I know, who gave there constructions of the 3d and 4th problems in his Appollonius Batavus: And afterwards the learned Dr Halley gave them in the Scholium of the 19th prop. of the 8th book of A. pollonius's conics reftored by him.

The 3d problem is otherwise enunciated thus : To cut a giren itzaight line AB in the point N, so as to make the rectangle AN, NB equal to a given space : Or, which is the face thing, having given AB the fum of the sides of a rectangle, and the magnitude of it being likewise given, to find its fides. And the 4th problem is the same with this, To find a point

N in

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N in the given straight line AB produced, so as to make the Book VI. rectangle AN, NB equal to a given space : Or, which is the fame thing, having given AB the difference of the fides of a rectangle, and the magnitude of it, to find the sides.

PROP. 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 24th prop. of b. 5. as Clavius

had done.

PROP. XXXIÍ. B. VÌ. The enunciation of the preceding 26th prop. is not general enough; because not only two fimilar 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 feems to have been another, and that a direct demonstration of these cases, to which this 32d propofition was needful: And the 32d may be otherwise and something more briefly demonitrated as follows.

PRO P. XXXII. B. VI.

a

31. 1.

b

If two triangles which have two sides of the onė, &c.
Let GAF, #FC be two triangles which have two fides AG,
GF, proporsional to the two sides FH, HC, viz. AG 10 GF, as
FH to HC; and let AG be paral.

G
lel to FH, and GF to HC; AF A
and FC are in a straight line.
Draw CK parallel to FH, and

E

H let it meet GF produced in K: Because AG, KC are each of them parallel to FH, they are parallel

b 30. 1, to one another, and therefore the B

K alternate angles AGF, FKC are equal : And AG is to GF, as (FH to HC, that is c) CK to KF;c 34. 1. wherefore the triangles AGF, CKF are equiangular 4, and thed 6. 6. angle AFG equal to he angle CFK: But GFK is a straight line, therefore AF and FC are in a straight line

The 26th prop. is demonstrated from the 32d, 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,

e 14. 1.

Y 2

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Book VI. First, Let the parallelograms ABCD, AEFG have the angle

BAD common to both, and be fimilar, 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 Gimilar, DA

is to AB, as GA to AE; wherea Cor. 19. fore the remainder DG is to the A,

G

D
remainder EB, as GA to AE: But
DG is equal to FH, EB to HC,

F
E

H
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 B

K joined at one angle, in the point b 32.6. F; wherefore AF, FC are in the same straight line b.

Next, Let the parallelograms KFHC, GFEA, which are fimi. lar and fimilariy placed, have their angles XFH, GFE vertically oppofite; 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 fams straight line 6 PROP. XXXIII.

B. VI. The words “because they are at the centre,” are left out, as the addition of some unskilful hand.

In the Greck, as also in the Latin tranflation, the words c

έτυχε, whatever,” are left out in the demonftration of both parts of the propofition, 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, ihe illative particle apce in the Greek text ought to be omitted. The fecond part of the propofition is an addition of Theon's

, as he tells us in his commentary on Ptolomy's Meyann Surrück, p. 50.

any

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

DEF

Book XI. DEF. IX. and XI. B. XI. THE fimilitude of plane figures is defined from the e

quality of their angles, and the proportionality of the fides about the equal angles ; for from the proportionality of the sides only, or only from the equality of the angles, the fimilitude of the figures does not follow, except in the cafe 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 fame reason, those are ii. milar 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 fimilar plane figures, of the fame number, and even of the fame magnitude, that are neither similar nor equal, as fhall be demonftrated after the notes on the roth definition: Upon this account it was necessary to amend the definition of fimi. lar folid figures, and to place the definition of a solid angle before it : And from this and the roth definition, it is fufliciently plain, how much the elements have been spoiled by unskilful editors,

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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 roth definition of this book, is a theorem, the truth or falsehood of which ought to be demonstrated, not affumed ; so that Theon, or some other Editor, has ignorantly turned a theorem which ought to be demonstrated into this roth definition : Thar figures are fimilar, 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 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 ift def, of the 3d book, is really a theorem in which these circles are faid to be equal, that have the straight lines from their centres to the circumferences equal, which is plain, from the definition of a circle ; and therefore has by

fome

Y 3

Book XI. fome 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 should 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 clementary matter? And this should teach us modefty, and to acknowledge how little, through the w.akness of our minds, we are able to prevent mistakes even in the principles of sciences which are juftly reckoned amongst the moft certain ; for that the propofition is not universally true, can be shewn by many examples: The following is sufficient.

Let there be any plane rectilineal figure, as the triangle 1 13. 11. ABC, and from a point D within it draw a the straight line

DE at right angles to the plane ABC; in DE fake 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 with DA, DE DC which it meets in ihat plane ; and in the triangles EL B, FDB, ED and DB are equal to FD and

DB, each to each, and they contain right angles; therefore b 4. 1. the base EB'is equal o

to the base FB ; in the
same manuer EA is e.
qua! 10 FA, and EC to
FC: And in the triangles
EBA, FBA, EB, BA

E
are equal to FB, BA
and the base EA is e-
qual to the base FA;
whiciefore the angle

D 6. 8. 2. EBA is equal to the

angle TBA, and the ti-B
angle EBA cqual b to
the triangle FBA, and
the other angles equal to
the other angles; there.

F
fire life triangles are
finiard: In the fame manner the triangle EBC is similar to

the

Idf. 6.

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