in magnitude. and because AD is parallel to FG, and GH equal to HA; DH is equal to HF, and AD equal to GF. and DH is b. 4.

given, therefore HF is given in magnitude; and it is alfo given in pofition, and the point H is given, therefore the point F is given. c. 36. Dati

c

And because the ftraight line EFG is drawn from a given point

F without or within the circle ABC given in pofition, therefore d. 95. of the rectangle EF, FG is given. and GF is equal to AD, wherefore 96.Dat the rectangle AD, EF is given.

PROP. C.

IF from a given point in a straight line given in pofition, a ftraight line be drawn to any point in the circumfe rence of a circle given in pofition; and from this point a ftraight line be drawn making with the firft an angle equal to the difference of a right angle and the angle contained by the ftraight line given in pofition, and the ftraight line which joins the given point and the center of the circle; and from the point in which the fecond line meets the circumference again, a third straight line be drawn making with the fecond an angle equal to that which the first makes with the second. the point in which this third line meets the ftraight line given in pofition is given; as alfo the rectangle contained by the firft ftraight line and the fegment of the third betwixt the circumference and the ftraight line given in pofition, is given.

Let the ftraight line CD be drawn from the given point C in the ftraight line AB given in pofition, to the circumference of the circle DEF given in pofition of which G is the center; join CG, and

Ffa

from the point D let DF be drawn making the angle CDF equal
to the difference of a right angle and the angle BCG, and from
the point Flet FE be drawn making the angle DFE equal to
CDF, meeting AB in H. the point H is given; as alfo the rec-
tangle CD, FH.

Let CD, FH meet one another in the
point K from which draw KL perpendi-
cular to DF; and let DC meet the cir-
cumference again in M, and let FH meet
the fame in E, and join MG, GF, CH. A

Because the angles MDF, DFE are M
equal to one another, the circumferences
16. 3. MF, DE are equal; and adding or
taking away the common part ME, the
circumference DM is equal to EF; there-
fore the ftraight line DM is equal to the
ftraight line EF, and the angle GMD to

B. 8. I

G. 32. E.

b

c

K

KY

M

AC

D

F

ㄧB

HB

the angle GFE; and the angles GMC, E, GFH are equal to one another, because they are either the fame with the angles GMD, GFE, or adjacent to them. and because the angles KDL, LKD are together equal to a right angle, that is, by the hypothefis, to the angles KDL, GCB; the angle GCB or GCH is equal to the angle (LKD, that is to the angle) LKF or CKH. therefore the points C, K, H, G are in the circumference of a circle; and the angle GCK is therefore equal to the angle GHF; and the angle GMC is equal to GFH, and the ftraight line GM to GF; therefore d CG is equal to GH, and CM to HF. and becaufe CG is equal to GH, the angle GCH is equal to GHC; but the angle GCII is given, therefore GHC is given; and confequently the angle CGH is given. and CG is given in pofition, and the point e. 32. Dat. G; therefore GH is given in pofition; and CB is alfo given in pofition, wherefore the point H is given.

¿. 26. 1.

And becaufe HF is equal to CM, the rectangle DC, FH is f.95.or 96. equal to DC, CM. but DC, CM is given f, becaufe the point C Dat. is given; therefore the rectangle DC, FH is given.

F I N I S.

O N

EUCLID'S DATA.

TH

DEFINITION II.

HIS is made more explicit than in the Greek text, to prevent a mistake which the Author of the fecond Lemonftration of the 24th Propofition in the Greek Edition has fallen into, of thinking that a ratio is given to which another ratio is fhewn to be equal, tho' this other be not exhibited in given magnitudes. See the Notes on that Propofition which is the 13th in this Edition. befides by this Definition, as it is now given, fome Propofitions are demonftrated, which in the Greek are not fo well done by help of Prop. 2.

DE F. IV.

In the Greek text Def. 4. is thus "Points, lines, fpaces and "angles are faid to be given in pofition which have always the fame "fituation." but this is imperfect and ufclefs, becaufe there are innumerable cafes in which things may be given according to this Definition, and yet their pofition cannot be found. for inftance, let the triangle ABC be given in pofition, and let it be proposed to draw a ftraight line BD from the angle at B to the oppofite fide AC which shall cut off the angle DBC which fhall be the feventh part of the angle ABC. fuppofe this is done, therefore the ftraight line BD is invariable in its pofition, that is, has always the fame fituation; for any other straight line drawn from the point B on either fide of BD cuts off an angle greater or leffer than the feventh part of the angle ABC; therefore, according to this Definition, the fraight line BD

a

B

is given in pofition, as alfo the point D in which it meets the a. 23. Dat, traight line AC which is given in pofition. but from the things

here given, neither the ftraight line BD nor the point D can be

found by the help of Euclid's Elements only, by which every thing in his Data is fuppofed may be found. this Definition is therefore of no ufe, we have amended it by adding "and which are either actually exhibited or can be found;" for nothing is to be reckoned given, which cannot be found, or is not actually exhibited.

66

The Definition of an angle given by pofition is taken out of the 4th, and given more diftinctly by itfelf in the Definition marked A.

DE F. XI. XII. XIII. XIV. XV.

The 11th and 12th are omitted because they cannot be given in English fo as to have any tolerable fenfe, and therefore wherever the terms defined occur, the words which exprefs their meaning are made ufe of in their place.

The 13. 14. 15. are omitted as being of no ufe.

It is to be obferved in general of the Data in this book, that they ale to be understood to be given Geometrically, not always Arithmetically, that is, they cannot always be exhibited in numbers; for inftance, if the fide of a fquare be given, the ratio of it to its diab. 44. Dat. meter is given geometrically, but not in numbers; and the diameter is given, but tho' the number of any equal parts in the fide be given, for example 10, the number of them in the diameter cannot be given, and the like holds in many other cafes.

Dat.

a. 1. Dcf.

b. 2. Def.

PROPOSITION I.

In this it is fhewn that A is to B, as C to D, from this that A is to C, as B to D, and then by permutation; but it follows directly, without thefe two steps, from 7. 5:

PROP. II.

The limitation added at the end of this Propofition between the inverted commas is quite neceffary, becaufe without it the Propofi tion cannot always be demonftrated. for the Author having faid "b.caufe A is given, a magnitude equal to it can be found, let "this be C; and because the ratio of A to B is given, a ratio, "which is the fame to it can be found b" adds, "let it be found, "and let it be the ratio of C to A." Now from the fecond Definition nothing more follows than that fome ratio, fuppofe the ratio of E to Z, can be found, which is the fame with the ratio of A to B ; and when the Author fuppofes that the ratio of C to A, which is

See Dr. Gregory's Edition of the Data.

BA

alfo the fame with the ratio of A to B, can be found, he neceffarily fuppofes that to the three magnitudes E, Z, C a fourth proportional may be found; but this cannot always be done by the Elements of Euclid; from which it is plain Euclid must have understood the Propofition under the limitation which is now added to his text. An example will make this clear; let A be a given angle, and B another angle to which A has a given ratio, for inftance, the ratio of the given ftraight line E to the given one Z, then, having found an angle C equal to A, how

M

E

Z

can the angle A be found to which

C has the fame ratio that E has to

Z? certainly no way, until it be fhewn how to find an angle to which a given angle has a given ratio, which cannot be done by Euclid's Elements, nor probably by any Geometry known in his time. Therefore in all the Propofitions of this book which depend upen this fecond, the above-mentioned limitation must be understood, tho' it be not explicitly mentioned.

PROP. V.

The order of the Propofitions in the Greek text between Prop. 4. and Prop. 25. is now changed into another which is more natural, by placing those which are more fimple before those which are more complex; and by placing together those which are of the fame kind, fome of which were mixed among others of a different kind. thus Prop. 12. in the Greek is now made the 5. and those which were the 22. and 23. are made the 11. and 12. as they are more fimple than the Propofitions concerning magnitudes the excefs of one of which above a given magnitude has a given ratio to the other, after which these two were placed; and the 24. in the Greek text is, for the fame reafon, made the

3.

PROP. VI. VII.

Thefe are universally true, tho' in the Greek text they are de monftrated by Prop. 2. which has a limitation; they are there fore now fhewn without it.