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Let ABCD be a quadrilateral figure, of which the opposite sides AB, CD, are equal to one another; as 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 base CD; therefore, by Prop. 8. of Book 1. the angle ADB is equal to the B angle ČBD; and, by Prop. 4. B. 1. the angle BAD is equal to the angle DCB, and ABD to BDC; and therefore also the angle ADC is equal to the angle ABC.

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. l. 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 half of all 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 opposite sides are equal, by 34th Prop. B. 1.

PROP. VII. B. I.

There are two cases of this proposition, one of which is not in the Greek text, but is as necessary as the other: and that the case 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 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 5th and 13th, but none of these stand in need of it except the 7th Proposition, on account of which it has been added : besides, the translation from the Arabic has this case 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 7th,” as if the case of the 7th, 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 5th and 7th Propositions; for it is not worth while to relate his trifles at full length.

It was thought proper to change the enunciation of this 7th Prop. so as to preserve the very 'same 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 proposition, which is necessary to Prop. 1. B. 11. and otherwise.

PROP. 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 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 propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated. Mons. Clairault in the Preface to his Elements of Geometry, published in French at Paris, anno 1741, says

66 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.

PROP. 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 so dull, though only
beginning to learn the
Elements, as not to per-

DM F G H Н
ceive that the circle de-
scribed from the centre F, at the distance FD, must
meet FH betwixt F and 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 and G; and that these circles must meet
one another, because FD and GH are together greater
than FG? And this determination is easier to be un-
derstood 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 may supply
the omission he blames Euclid for, which determi-
nation 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 case his demonstration must be changed into
another.

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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 equal to BAC;
because without this restriction

D
there might be three different
cases of the proposition, as
Campanus and others make.

Mr. Thomas Simpson. in p.
262, of the second edition of
his Elements of Geometry,
printed anno 1760, observes in E
his notes, that it ought to have

G
been 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 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.

PROP. XXIX. B. I.

The proposition which is usually called the 5th postulate or 11th axiom, by some the 12th, on which this 29th depends, has given a great deal to do, both to ancient and modern geometers: it seems not to be properly placed among the axioms, as indeed it is not self-evident; but it may be demonstrated thus :

DEFINITION I.

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

DEF. II.

One straight line is said to go nearer to, or further from, another straight line, when the distances 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.

AXIOM.

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

C cuts it; and, in like manner,

B

D a straight line cannot go fur

E ther from another straight F

H line, and then come nearer to it; nor can a straight line keep the same distance from another straight line, and then come nearer to it, or go further from it; for a straight line keeps always the same direction.

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

page.

A to the point B, and then, See the

A.

B figure in from the point B to the point

C

D. preceding C, go further from the same

E DE: and, in like manner,

G

H the straight line FGH cannot go further from DE, as from F to G, and then, from G to H, come nearer to the same DE: and so in the last case, as in fig. 2.

PROP. I.

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 equal to AC, or BD.

If EF be not equal to AC, one of them must be greater than the other; let AC be the greater : then, because FE is less 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 T: but because DB is greater

F than FE, the straight line

D CFD is further from AB at the point D than at F, that is, FD goes further from AB from F to D: therefore the straight line A #

B 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.

PROP. II.

If two equal straight lines AC, BD be each at right angles to the same straight line AB; the straight line CĎ 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

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