304

Book I. lefs than CA, the ftraight line CFD is nearer to the ftraight

I.

8. 1

I.

F

F

E

line AB at the point F than at the
point C, that is, CF comes nearer
to AB from the point C to F: But
because DB is greater than FE,
the ftraight line CFD is further
from AB at the point D than at F,
that is, FD goes further from AB
from Fto D: Therefore the ftraight
line CFD firft comes nearer to the
ftraight line AB, and then goes further from it, before it cuts
it, which is impoffible. If FE be faid to be greater than CA,
or DB, the straight line CFD first goes further from the straight
line AB, and then comes nearer to it, which is alfo impoffible.
Therefore FE is not unequal to AC, that is, it is equal to it.

A

PROP. 2.

B

If two equal ftraight lines AC, BD be each at right angles to the fame ftraight line AB; the ftraight line CD which joins their extremities makes right angles with AC and BD.

a

Join AD, BC; and because, in the triangles CAB, DBA, CA, AB are equal to DB, BA, and the angle CAB equal to the angle DBA; the bafe BC is equal to the bafe AD: And in the triangles ACD, BDC, AC, CD are equal to BD, DC, and the base AD is equal to the base

b

BC; therefore the angle ACD is e-
qual to the angle BDC: From any
point E in AB draw EF unto CD,
at right angles to AB; therefore, by
Prop. 1. EF is equal to AC, or BD;
wherefore, as has been just now A

fhewn, the angle ACF is equal to

F

D
G

E B

the angle EFC: In the fame manner, the angle BDF is equal to the angle EFD: but the angles ACD, BDC are equal, therefore the angles EFC and EFD are equal, and right 10. def. angles; wherefore also the angles ACD, BDC are right angles.

COR. Hence, if two ftraight lines AB, CD be at right angles to the fame ftraight line AC, and if betwixt them a ftraight line BD be drawn at right angles to either of them, as to AB; then BD is equal to AC, and BDC is a right angle.

If AC be not equal to BD, take BG equal to AC, and join CG: Therefore, by this Propofition, the angle ACG is a right angle; but ACD is alfo a right angle; wherefore the angles

L

PROP. XI. B. I.

A corollary is added to this propofition, which is neceffary to Prop. 1. b. 11. and otherwise.

PROP. XX. and XXI. B. I.

Proclus, in his commentary, relates, that the Epicureans derided this propofition, as being manifeft even to affes, and needing no demonftration; and his answer is, that though the truth of it be manifeft to our fenfes, yet it is fcience which muft give the reason why two fides of a triangle are greater than the third : But the right answer to this objection against this and the 21st and fome other plain propofitions, is, that the number of axioms ought not to be increased without neceffity, as it must be if these propofitions be not demonftrated. Monf. Clairault, in the preface to his elements of geometry, published in French at Paris, ann. 1741, fays, That Euclid has been at the pains to prove, that the two fides of a triangle which is included within another are together less than the two fides of the triangle which includes it; but he has forgot to add this condition, viz. that the triangles must be upon the fame bafe; because, unless this be added, the fides of the included triangle may be greater than the fides of the triangle which includes it, in any ratio which is less than that of two to one, as Pappus Alexandrinus has demonftrated in Prop. 3. b. 3. of his mathematical collections.

PROP. XXII. B. I.

Some authors blame Euclid because he does not demonftrate, that the two circles made use of in the construction of this problem must cut one another: But this is very plain from the determination he has given, viz. that any two of the straight lines DF, FG, GH must be great

er than the third: For who is fo dull, tho' only beginning to learn the elements, as not to perceive that the circle defcribed from the centre F, at the diftance FD,

Book I.

must meet FH betwixt F and H,D M F G because FD is lefs than FH; and

H

that, for the like reafon, the circle defcribed from the centre G, at the distance GH or GM, muft meet DG betwixt D

and

Book I. and G; and that these circles, must meet one another, because

FD and GH are together greater
than FG? And this determina-
tion is easier to be understood than
that which Mr Thomas Simpson
derives from it, and puts inftead
of Euclid's, in the 49th page of
his elements of geometry, that he DM
may fupply the omiffion he blames

F G

H

Euclid for; which determination is, that any of the three ftraight lines must be lefs than the fum, but greater than the difference of the other two: From this he fhews the circles must meet one another, in one cafe; and fays, that it may be proved after the fame manner in any other cafe: But the straight line GM which he bids take from GF may be greater than it, as in the figure here annexed; in which cafe his demonstration must be changed into another.

PROP. XXIV. B. I.

D

To this is added, "of the two fides DE, DF, let DE be
"that which is not greater than the other;" that is, take that
fide of the two DE, DF which is not greater than the other, in
order to make with it the angle EDG
equal to BAC; becaufe, without this
restriction, there might be three differ-
ent cafes of the propofition, as Campa-
nus and others make.

Mr Thomas Simpfon, in p. 262. of
the fecond edition of his elements of
geometry, printed ann. 1760, obferves,
in his notes, that it ought to have been E
fhewn, that the point F falls below the
line EG; this probably Euclid omitted,

G F

as it is very eafy to perceive, that DG being equal to DF, the point G is in the circumference of a circle defcribed from the centre D at the distance DF, and must be in that part of it which is above the ftraight line EF, becaufe DG falls above DF, the angle EDG being greater than the angle EDF.

The propofition which is ufually called the 5th poftulate, or ith axiom, by fome the 12th, on which this 29th depends, has

the words 'rubas, that is, in a straight line, or in the fame Book I direction, be plain, when two ftraight lines are faid to be in a straight line, it does not appear what ought to be understood by these words, when a straight line and a curve, or two curve lines, are faid to be in the fame direction: at leaft it cannot be explained in this place; which makes it probable that this definition, and that of the angle of a fegment, and what is said of the angle of a femicircle, and the angles of fegments, in the 16. and 31. propofitions of book 3. are the additions of fome lefs fkilful editor: On which account, especially fince they are quite ufelefs, thefe definitions are distinguished from the reft by inverted double commas.

DE F. XVII. B. I.

The words, "which alfo divides the circle into two equal parts," are added at the end of this definition in all the copies but are now left out as not belonging to the definition, being only a corollary from it. Proclus demonftrates it by conceiving one of the parts into which the diameter divides the circle, to be applied to the other; for it is plain they must coincide, elfe the ftraight lines from the center to the circumference would not be all equal: The fame thing is eafily deduced from the 31. prop. of book 3. and the 24. of the fame; from the first of which it follows that femicircles are fimilar fegments of a circle: And from the other, that they are equal to one another.

DE F. XXXIII. B. I.

This definition has one condition more than is neceffary; becaufe every quadrilateral figure which has its oppofite fides equal to one another, has likewife its oppofite angles equal; and on the contrary.

Let ABCD be a quadrilateral figure of which the oppofite fides AB, CD are equal to one an- A other; as alfo AD and BC: Join BD; the two fides AD, DB are Cqual to the two CB, BD, and the bafe AB is equal to the bafe CD; there- B

fore by prop. 8. of book 1. the angle

C

ADB is equal to the angle CBD; and by prop. 4. B. 1. the angle BAD is equal to the angle DCB, and ABD to BDC; and therefore alfo the angle ADC is equal to the angle ABC.

And

800

Book I.

And if the angle BAD be equal to the oppofite angle BCD, and the angle ABC to ADC; the oppofite fides are equal; Because, by prop. 32. B. 1. all the angles of the quadrilateral figure ABCD are together equal to A four right angles, and the two angles BAD, ADC are together equal to the two angles BCD, ABC: Wherefore BAD, ADC are the half of all B the four angles; that is, BAD and

D

C

ADC are equal to two right angles: And therefore AB, CD are parallels by prop. 28. B. 1. In the fame manner AD, BC are parallels: Therefore ABCD is a parallelogram, and its op pofite fides are equal by 34. prop. B. 1.

PROP. VII. B. I.

There are two cafes of this propofition, one of which is not in the Greek text, but is as neceffary as the other: And that the cafe left out has been formerly in the text appears plainly from this, that the fecond part of prop. 5, which is neceffary to the demonftration of this cafe, can be of no use at all in the ele ments, or any where elfe, but in this demonstration; because the fecond part of prop. 5. clearly follows from the first part, and prop. 13. B. 1. This part must therefore have been added to prop. 5. upon account of fome propofition betwixt the 5. and 13. but none of these stand in need of it except the 7. pro pofition, on account of which it has been added: Befides, the tranflation from the Arabic has this cafe explicitly demonftrated: And Proclus acknowledges that the fecond part of prop. 5. was added upon account of prop. 7. but gives a ridiculous reafon for it," that it might afford an answer to objections made "against the 7." as if the cafe of the 7. which is left out, were, as he exprefsly makes it, an objection against the propofition iffelf. Whoever is curious may read what Proclus fays of this in his commentary on the 5, and 7. propofitions; for it is not worth while to relate his trifles at full length.

It was thought proper to change the enunciation of this 7. prop. fo as to preferve the very fame meaning; the literal tranflation from the Greek being extremely harfh, and difficult to be understood by beginners.

PROP