in a straight line, or in the fame direction, be plain, when two Book I. straight lines are said to be in a straight line, it does not appear what ought to be understood by these words, when a ftraight line and a curve, or two curve lines, are said to be in the same 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 said of the angle of a femicircle, and the angles of segments, in the 16. and 31. Propofitions of Book 3. are the ada ditions of some less skilful Editor. on which account, especially since they are quite useless, these Definitions are distinguished from the reft by inverted double commas.

DE F. XVII. B. 1. The words “ which also 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 Coi rollary from it. Proclus demonstrates 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, else the straight lines from the center to the circumference would not be all equal. the fame thing is easily deduced from the 31. Prop. of Book 3. and the 24. of the fame; from the firft of which it follows that femicircles are similar segments 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 necessary; because every quadrilateral figure which has its opposite sides equal to one another, has likewise 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

A also AD and BC. join BD; the two sides AD, DB are equal to the two CB, BD, and the base AB is equal to the bafe CD; therefore by Prop. 8. of B

C 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 allo the angle ADC is equal to the angle ABC.

Book I.

And if the angle BAD be equal to the opposite angle BCD, and the angle ABC to ADC; the opposite sides are equal. Because by Prop. 32. B. 1. all the angles of the quadrilateral figure ABCD are together equal to four right angles,

A
and the two angles BAD, ADC are to-
gether equal to the two angles BCD,
ABC. wherefore BAD, ADC are the
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
Gides are equal by 34. Prop. B. 1.

PROP. VII. B. I. There are two cases of this Proposition, one of which is not ia the Greek text, but is as necessary as the other, and that the cafe left out has been formerly in the text appears plainly from this, that the second part of Prop. 5. which is necessary to the Demonstration of this case, can be of no use at all in the Elements, or any where else, but in this Demonstration ; because the fecond part of Prop. 5. clearly follows from the first part, and Prop. 1 3. B. I. this part must therefore have been added to Prop. 5. upon account of fome Proposition betwixt the 5. and 13. but none of these stand in need of it, except the 7. Proposition, on account of which it has been added. besides the translation from the Arabic has this case explicitely demonstrated. and Proclus acknowledges that the second

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 case of the 7. which is left out, were, as he expressly makes it, an objection against the proposition itself. Whoever is curious may read what Proclus says 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 preserve the very same meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners.

Book 1.

PRO P. XI. B. I. A Corollary is added to this Proposition, which is necessary to Prop. 1. B. IV. and otherwise.

PRO P. XX. and XXI. B. I. Proclus in his Commentary relates that the Epicureans derided this Proposition, as being manifest even to asses, and needing no Demonstration; and his answer is, that tho' the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third. but the right answer to this objection against this and the 21. and some other plain Propositions, is, that the number of Axioms ought not to be encreased without necessity, as it must be if these Propositions be not demonstrated. Mons. Clairauit in the Preface to his Eleinents of Geometry published in French at Paris Ann. 1741. says that Euclid has been at the pains to prove that the two sides of a triangle which is included within another are together less than the two , sides of the triangle which includes it; but he has forgot to add this condition, viz. that the triangles must be upon the same base; because unless this be added, the sides of the included triangle may be greater than the sides of the triangle which includes it, in any ratio which is less than that of two to one. as Pappus Alexandrinus has demonstrated in Prop. 3. B. 3. of his Mathematical Collections.

PRO P. XXII. B. I. Some Authors blame Euclid because he does not demonstrate 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 greater than the third. for who is fo dull, tho' only beginning to learn the Elements, as not to perceive that the circle described from the center F, at the distance FD, must meet FH betwixt F and H,DM FG H because FD is less than FH; and that, for the like reason, the circle described from the center G, at the distance GH or GM must meet DG betwixt D and G; and

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

together greater than FG? and this
determination is easier to be under-
stood than that which Mr. Thomas
Simpson derives from it, and puts
instead of Euclid's, in the 49. page
of his Elements of Geometry, that he DM FG H
may supply the omission he blames
Euclid for; which determination is, that any of the three straight
lines must be less than the sum, but greater than the difference of
the other two. from this he fews the circles must meet one ano-
ther, in one case; and says that it may be proved after the same
manner in any other case. 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 bis demonstration must be changed into another.

PRO P. XXIV. B. I.
To this is added “ of the two sides DE, DF, let DE be that
” which is not greater than the other ;" that is, take that side of
the two DE, DF which is not greater than the other, in order to
make with it the angle EDG equal to

D
BAC. because without this restriction,
there might be three different cases of
the Propofition, as Campanus and others
make.

Mr. Thomas Simpson in p. 262. of
the second edition of his Elements of
Geometry printed Ann. 1760. observes
in his Notes, that it ought to have becu

G shewn that the point F falls below the line EG; this probably Euclid omitted, as it is very easy to perceive that DG being equal to DF, the point G is in the circumference of a circle described 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 Proposition which is usually 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 anticnt and modern Geometers. it seems

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

1.

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

DEF. 2. One straight line is said to go nearer to, or further from another straight line, when the distance of the points of the first from the other straight line become less or greater than they were; and two straight lines are said to keep the same distance from one another, when the distance of the points of one of them from the other is always the same.

A X I 0 M. A straight line cannot first come nearer to another straight line, and then go further from it, before it cuts it; and, in like manner, a

A

B с straight line cannot go further from

comp another straight line, and then come nearer to it; nor can a straight line 7

G

H keep the fame distance from another straight line, and then come nearer to it, or go further from it; or a straight line keep always the same direction.

For example, the straight line ABC cannot first come nearer to the straight line DE, as from the point

B A to the point B, and then, from the A point B to the point C, go further D

C from the same LE. and, in like man- F G H ner, the straight line FGH cannot go further from DE, as from F to G, and then, from G to H, come nearer to the fame DE. and so in the last case as in fig. 2.

PROP. If two equal straight lines AC, BD be each at right angles to the same straight line AB; if the points C, D be joined by the straight line CD, the straight line EF drawn from any point E in AB unto CD, at right angles to AB, shall be cqual 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 because FE is less than CA, the

See the fir

gure above.

C