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

line CFD first comes nearer to the
straight line AB, and then goes further from it, before it cuts
it, which is impossible. If FE be said 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 also impossible.
Therefore FE is not unequal to AC, that is, it is equal to it.

P R O P. 2. If two equal straight lines AC, BD be each at right angles to the same ftraight line AB; the straight line CD which joins their extremities makes right angles with AC and BD.

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 base BC is equal to the base AD: And in the triangles ACD, BDC, AC, CD are cqual to BD, DC, and the base AD is equal to the base

F D BC; therefore the angle ACD is e. C С 8. a. qual to the angle BDC: From any

IG 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 E B shewn, the angle ACF is equal to the angle EFC: In the same 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 angles c; wherefore also the angles ACD, BDC are right an. gles.

Cor. Hence, if two straight lines AB, CD be at right angles to the same straight line AC, and if berwixt them a straight line BD be drawn at right angles to either of them, as 10. AB; then BD is equal to AC, and BDC is a right angle.

If AC be nor equal to BD, take BG equal to AC, and join CG: Therefore, by this Proposition, the angle ACG is a right angle; but ACD is also a right angle; wherefore the an.


10. def.

Book I.

A corollary is added to this propofition, which is necessary to
Prop. I. b. i. and otherwise.

PROP. XX. and XXI. B. I.

Proclus, in his commentary, relates, that the Epicurcans de rided this propofition, as being manifest even to affes, and need ing no demonftration; and his answer is, that though 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 21st. and some other plain propofitions, is, that the number of axioms ought not to be increased without neceffiry, as it must be if these propofitions be not demonstrated. Moni. Clairault, in the preface to his elements 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 togetber 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 lides 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.

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Some authors blame Euclid because he does not demonAtrate, 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 so dull, tho' only beginning to learn the elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet FH betwixt F and HD M

H because FD is less than FH; and that, for the like reason, the circle described from the centre G, at the distance GH or GM, must meet DG betwixt D


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 instead
of Euclid's, in the 49th page of
his elements of geometry, that he D M F G H
may supply the omiffion 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 shews the circles must
meet one another, 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 his demonstration must be changed into another.

PROP. 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 D equal to BAC; because, without this restriction, there might be three different cases of the proposition, as Campa: nus and others make.

Mr Thomas Simpson, in p. 262. of thc 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, 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 centre 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.

The proposition which is usually called the gih postulate, or
I sth axiom, by some the 12th, on which this 2ych depends, has

the words ve' rubwas, that is, in a straight line, or in the fame Book I. direction, be plain, when two straight 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 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 segment, and what is said of the angle of a semicircle, and the angles of segments, in the 16. and 31. propofitions of book 3. are the additions of soma less skilful editor : On which account, especially fince they are quite useless, these definitions are distinguished from the rest by inverted double commas.

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The words, “ which also divides the circle into two cqual "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 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, elle the straight lines from the center to the circumference would Dot be all equal: The same thing is easily deduced from the 31. prop. of book 3. and the 24. of the fame ; from the first of which it follows that semicircles are similar segments of a circle : And from the other, that they are equal to one another,


This definition has one condition more than is neceffary ; because every quadrilateral figure which has its opposite sides equal to one another, has likewise its opposite angles equal; and on

the contrary

Let ABCD be a quadrilateral figure of which the oppolite fides AB, CD are equal to one an A other; as also AD and BC: Join BD; the two fides AD, DB are e qual to the two CB, BD, and the base AB is equal to the base CD; there. B

С fore by prop. 8. of book 1. the angle ADB is equal to the angle CBD; and by prop. 4. B. 1. the an. gle BAD is cqual to the angle DCB, and ABD to BDC; and therefore also ihe angle ADC is equal to the apgle ABC.


Book I. And if the angle BAD be equal to the oppofite 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, and the two angles
BAD, ADC are together equal to
the two angles BCD, ABC: Where.
tore BAD, ADC are the half of all B

the four angles; that is, BAD and
ADC are equal to two right angles : And therefore AB, CD
are parallels by prop. 28. B. 1. In the same manner AD, BC
are parallels: Therefore ABCD is a parallelogram, and its ope
posite Gides are equal by 34. prop. B. 1.

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There are two cases of this proposition, 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 nece ary to the demonstration of this case, can be of no use at all in the ele. ments, or any where else, but in this demonstration; because the second 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 some 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 cafe explicitly 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 enunciation of this 7. prop. so as to preserve the very fame meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners.


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