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Book VI.

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

Let AB be the given ftraight 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 fquare.

Draw AE, BF at right angles to AB, on the contrary fides of it, and make AE equal to C, and BF equal to D: Join EF, and bifect it in G; and from the centre G, at the diftance GE, defcribe a circle meeting AE again in H; join HF, and draw GL parallel to AE; let the circle meet AB produced in M, N, and upon BN defcribe the fquare NBOP, and complete the rectangle ANPQ: because the angle EHF in a femicircle is equal to the right angle EAB.

E

C

D

G

OP

Q

L

B

AB and HF are parallels, and M
therefore AH and BF are e-

qual, and the rectangle EA,

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AH equal to the rectangle

a 35. 3.

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EA, BF, that is, to the rectangle C, D: And becaufe 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 gi ven straight line AB the rectangle AP has been applied equal to the given rectangle C, D, exceeding by the fquare BP. Which was to be done.

Willebrordus Snellius was the firft, as far as I know, who gave thefe conftructions of the 3d and 4th problems in his Appollonius Batavus: And afterwards the learned Dr Halley gave them in the Scholium of the 18th prop. of the 8th book of Apollonius's conics reftored by him.

The 3d problem is otherwife enunciated thus: To cut a given fhaight line AB in the point N, fo as to make the rect angle AN, NB equal to a given fpace: Or, which is the face thing, having given AB the fum of the fides of a rectangle, and the magnitude of it being likewife given, to find its

fides.

And the 4th problem is the fame with this, To find a point
N in

N in the given ftraight line AB produced, fo as to make the Book VI. rectangle AN, NB equal to a given fpace: 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 fides.

PROP. XXXI. B. VI.

In the demonftration of this, the inverfion of proportionals is twice neglected, and is now added, that the conclufion may be legitimately made by help of the 24th prop. of b. 5. as Clavius had done.

PROP. XXXII. B. VI.

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 fame diameter; but likewife two fimilar parallelograms that have vertically oppofite angles, have their diameters in the fame ftraight line: But there seems to have been another, and that a direct demonstration of these cafes, to which this 32d propofition was needful: And the 32d may be other wife and fomething more briefly demonstrated as follows.

PROP. XXXII. B. VI.

If two triangles which have two fides 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 paral

G

lel to FH, and GF to HC; AF A

D

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alternate angles AGF, FKC are e

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

The 26th prop. is demonftrated from the 32d, as follows. If two fimilar and fimilarly placed parallelograms have an angle common to both, or vertically oppofite angles; their diameters are in the fame ftraight line.

Y 2

First,

e 14. I.

Book VI.

First, Let the parallelograms ABCD, AEFG have the angle BAD common to both, and be fimilar, and fimilarly placed; ABCD, AEFG are about the fame diameter.

Produce EF, GF, to H, K, and join FA, FC: Then becaufe the parallelograms ABCD, AEFG are fimilar, DA is to AB, as GA to AE; where

a Cor. 19. fore the remainder DG is to the A

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G

D

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b 33.6. F; wherefore AF, FC are in the fame straight line Þ.

Next, Let the parallelograms KFHC, GFEA, which are fimilar and fimilarly placed, have, their angles KFH, GFE vertically oppofite; their diameters AF, FC are in the fame 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 fame ftraight line b

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

In the Greek, as alfo in the Latin tranflation, the words

ά ετυχε,

TUE," any whatever," are left out in the demonstration of both parts of the propofition, and are now added as quite neceffary; and, in the demonftration of the fecond part, where the triangle BGC is proved to be equal to CGK, the illative particle apa 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 Meydan Zurzáğı,

p. 50.

PROP. B. C. D. B. VI.

Thefe three propofitions are added, because they are frequently made ufe of by geometers.

DEF.

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

THE fimilitude of plane figures is defined from the equality of their angles, and the proportionality of the fides about the equal angles; for from the proportionality of the fides 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 fimilar pofition of the fides, which contain the figures, to one another, depending partly upon each of thefe: And, for the fame reafon, thofe are fimilar folid 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 10th 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 10th definition, it is fuflìciently plain, how much the elements have been spoiled by unfkilful editors.

DE F. 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 10th definition of this book, is a theorem, the truth or falfehood of which ought to be demonftrated, not affumed; fo that Theon, or fome other Editor, has ignorantly turned a theorem which ought to be demonstrated into this 10th 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, 66 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 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 ift def. of the 3d book, is really a theorem in which thefe 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.

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 folid figures contained by the fame number of fimilar and equal plane figures are equal to one another, yet he would juftly deferve to be blamed who fhould make a definition of this propofition which ought to be demonftrated. But if this propofition be not true, muft it not be confeffed, that geometers have, for thefe thirteen hundred years, been miftaken in this clementary matter? And this should teach us modefty, and to acknowledge how little, through the weakness of our minds, we are able to prevent mistakes even in the principles of fciences which are justly reckoned amongst the moft certain; for that the propofition is not univerfally true, can be fhewn by many examples: The following is fufficient.

Let there be any plane rectilineal figure, as the triangle à 12. 11. ABC, and from a point D within it draw a the ftraight 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 b 4. 1. the bafe EB is equal

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angle FBA, and the tri-B
angle EBA equal to
the triangle FBA, and
the other angles equal to
the other angles; there-
fore thefe triangles are

G

E

F

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fimilar: In the fame manner the triangle EBC is fimilar to

the

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