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In a ftraight line, or in the fame direction, be plain, when two Book I. ftraight lines are faid to be in a straight line, it does not appear what ought to be understood by thefe words, when a ftraight line and a curve, or two curve lines, are faid to be in the fame direction; at least 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 faid of the angle of a femicircle, and the angles of fegments, in the 16. and 31. Propofitions of Book 3. are the ad ditions of fome lefs fkilful Editor. on which account, efpecially fince they are quite useless, these Definitions are distinguished from the reft by inverted double commas.

DE F. XVIÍ. B. Í.

The words" which alfo divides the circle into two equal parts" are added at the end of this Definition in all the copics, 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 firft 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 opposite sides AB, CD are equal to one another; as

alfo AD and BC. join BD; the two A

fides AD, DB are equal to the two CB, BD, and the bafe AB is equal to

the bafe CD; therefore by Prop. 8. of B

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Book 1. the angle 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.

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Book I.

And if the angle BAD be equal to the opposite 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 four right angles, and the two angles BAD, ADC are together equal to the two angles BCD, ABC. wherefore BAD, ADC are the

A

half of all the four angles; that is, BAD B

and 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 opposite fides are equal by 34. Prop. B. 1.

PROP. VII. B. I.

There are two cafes of this Propofition, one of which is not ia 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 ufe at all in the Elements, 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. Propofition, on account of which it has been added. befides the tranflation from the Arabic has this cafe explicitely demonstrated. and Proclus acknowledges that the fecond part of Prop. 5. was added upon account of Prop. 7. but gives a ridiculous reason 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 itself. 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 enuntiation of this 7. Prop. fo as to preferve the very fame meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners.

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 Demonstration; and his answer is, that tho' the truth of it be manifest to our fenfes, yet it is fcience which must give the reason why two fides of a triangle are greater than the third. but the right anfwer to this objection against this and the 21. and fome other plain Propofitions, is, that the number of Axioms ought not to be encreased without neceffity, as it must be if thefe 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 lefs 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 base; 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 ufe of in the construction of this Problem muft 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 greater 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 center F, at the diftance FD, must meet FH betwixt F and H,DM becaufe FD is lefs than FH; and

FG

H

that, for the like reafon, the circle defcribed from the center G, at the diftance GH or GM must meet DG betwixt D and G; and

Book I.

Book I. that thefe circles must meet one another, because FD and GH are

together greater than FG? and this
determination is eafier to be under-
ftood than that which Mr. Thomas
Simpfon derives from it, and puts
inftead of Euclid's, in the 49. page
of his Elements of Geometry, that he DM
may fupply the omiffion he blames

FG

H

Euclid for; which determination is, that any of the three straight lines must be less 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 demonftration 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 reftriction,
there might be three different cafes of
the Propofition, as Campanus and others
make.

Mr. Thomas Simpfon in p. 262. of
the fecond edition of his Elements of
Geometry printed Ann. 1760. obferves E
in his Notes, that it ought to have been
fhewn that the point F falls below the

line EG; this probably Euclid omitted, 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 center D at the distance DF, and must be in
that part of it which is above the straight line EF, because DG falls
above DF, the angle EDG being greater than the angle EDF.

PROP. XXIX. B. I.

The Propofition which is ufually called the 5. Poftulate, or 11. Axiom, by fome the 12. on which this 29. depends, has given a great deal to do both to antient and modern Geometers. it feems

not to be properly placed among the Axioms, as, indeed, it is not Book I. felf-evident; but it may be demonftrated thus.

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The distance of a point from a straight line, is the perpendicular drawn to it from the point.

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One ftraight line is faid to go nearer to, or further from another ftraight line, when the distance of the points of the first from the other straight line become less or greater than they were; and two ftraight lines are faid to keep the fame distance from one another, when the distance of the points of one of them from the other is always the fame.

AXIOM.

A ftraight line cannot first come nearer to another straight line, and then go further from it, before

it cuts it; and, in like manner, a ftraight line cannot go further from.

A

B

another straight line, and then come

nearer to it; nor can a ftraight line F

keep the fame diftance from ano

G

H

ther straight line, and then come nearer to it, or go further from

it; or a ftraight line keep always the fame direction.

For example, the straight line ABC cannot first come nearer to

the straight line DE, as from the point

point B to the point C, go further D

A to the point B, and then, from the A

from the fame LDE. and, in like man- F ner, the straight line FGH cannot go

B

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See the fi gure above.

further from DE, as from F to G, and then, from G to H, come nearer to the fame DE. and fo in the laft cafe as in fig. 2.

PROP. I.

If two equal straight lines AC, BD be each at right angles to the fame ftraight line AB; if the points C, D be joined by the ftraight line CD, the straight line EF drawn from any point E in AB unto CD, at right angles to AB, fhall be equal to AC, or BD. If EF be not equal to AC, one of them must be greater than the ether; let AC be the greater; then becaufe FE is lefs than CA, the

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