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right angle; but ACD is also a right angle: wherefore the Book f. angles ACD, ACG are equal to one another, which is impossible. Therefore BD is equal to AC: and by this proposition BDC is a right angle.

PROP. III. If two straight lines, which contain an angle, be produced, there may be found in either of them a point from which the perpendicular drawn to the other shall be greater than any given straight line.

Let AB, AC be two straight lines, which make an angle with one another, and let AD be the given straight line; a point may be found either in AB or AC, as in AC, from which the perpendicular drawn to the other AB shall be greater than AD.

In AC take any point E, and draw EF perpendicular to AB; produce A E to G, so that EG may be equal to AE; and produce FE to H, and make EH equal to FE, and join HG. Because, in the triangles A EF, GEH, AE, EF are equal to GE, EH, each to each, and contain equal a angles, a 15. 1. the angle GHE is therefore equal to the angle AFE which b 4. 1. is a right angle: Draw GK perpendicular to AB; and because the straight lines FK, HG are at

А. F К. BM right angles to FH, and

KG at N right angles to o

E FK, KG is equal

H

G to FH, by Cor. Pr. 2, that is, to

L the double of FE.

In the
same manner, if AG be produced to L, so that GL be equal
to AG, and LM be drawn perpendicular to AB, then LM is
double GK, and so on. In AD take AN equal to FE, and
AO equal to KG, that is, to the double of FE, or AN; also,
take AP, equal to LM, that is, to the double of KG or AO;
and let this be done till the straight line taken be greater than
AD: Let this straight line so taken be AP, and because AP
is equal to LM, therefore LM is greater than AD. Which
was to be done.

PROP. IV.
If two straight lines AB, CD make equal angles EAB,
ECD with another straight line EAC towards the same parts
of it; AB and CD are at right angles to some straight line.

С

b 4. 1.

Book I. Bisect AC in F, and draw FG perpendicular to AB; take

CH in the straight line CD equal to AG, and on the contrary side of AC to that on which AG is, and join FH: Therefore in the triangles AFG, CFH, the sides FA, AG are equal to

FC, CH, each to each, and the angle a 15. 1. FAG, that is a, EAB is equal to the

E
angle FCH; wherefore b the angle
AGF is equal to CHF, and AFG

GA B
to the angle CFH: To these last,
add the common angle AFH; there-
fore the two angles AFG, AFH, F
are equal to the two angles CFH,

HFA, which two last are equal to-
c 13. 1. gether to two right angles “, therefore C H. D

also AFG, AFH are equal to two d 14. 1. right angles, and consequently GF and FH are in one straight

line. And because AGF is a right angle, CHF, which is equal to it, is also a right angle: Therefore the straight lines AB, CD are at right angles to GH.

PROP. V. If two straight lines AB, CD, be cut by a third ACE, so as to make the interior angles BAC, ACD, on the same side of it together less than two right angles ; AB and CD being produced shall meet one another towards the parts on which are the

two angles which are less than tuo right angles. a 23. 1. At the point C, in the straight line CE, make a the angle

ECF equal to the angle EAB, and draw to AB the straight
line CG at right angles to CF: Then, because the angles
ECF, EAB are equal
to one another, and the

E
angles ECF, FCA, are
b 13. 1. together equal to two
right angles, the angles

МС, F K
EAB, FCA are equal
two right angles.

N.
But, by the hypothesis,
the angles EAB, ACD

L
are together less than AOG B
two right angles; there-

H fore the angle FCA is greater than ACD, and CD falls between CF and AB: And because CF and CD make an angle with one another, by Prop. 3, a point may be found in either of them CD, from which the perpendicular drawn to CF shall be greater than the straight line CG. Let this point be H, and draw

;

a

to

D

HK perpendicular to CF, meeting AB in L: And because Book I. AB, CF contain equal angles with AC on the same side of it, by Prop. 4, AB and CF are at right angles to the straight line MNO, which bisects AC in N, and is perpendicular to CF: Therefore by Cor. Prop. 2, CG and KL, which are at right angles to CF, are equal to one another: And HK is greater than CG; and therefore is greater than KL, and consequently the point H is in KL produced. Wherefore the straight line CDH, drawn betwixt the points C, H, which are on the contrary sides of AL, must necessarily cut the straight line AB.

PROP. XXXV. B. I. The demonstration of this proposition is changed, because, if the method which was used in it were followed, there would be three cases to be separately demonstrated, as is done in the translation from the Arabic; for, in the Elements, no case of a Proposition that requires a different demonstration ought to be omitted. On this account, we have chosen the method which Mons. Clairault has given, the first of any, as far as I know, in his Elements, page 21, and which afterwards Mr Simpson gives in his page 32. But whereas Mr Simpson makes use of Prop. 26, B. I, from which the equality of the two triangles does not immediately follow, because, to prove that, the 4th of B. 1. must likewise be made use of, as may be seen in the very same case in the 34th Prop. B. 1, it was thought better to make use only of the 4th of B. l.

PROP. XLV. B. I. The straight line KM is proved to be parallel to FL from the 33d Prop.; whereas KH is parallel to FG, by construction, and KHM, FGL have been demonstrated to be straight lines. A corollary is added from Commandine, as being often used.

PROP. XIII. B. II. In this proposition only acute angled triangles are mentioned, Book 11. whereas it holds true of every triangle: And the demonstrations of the cases omitted are added. Commandine and Clavius have likewise given their demonstrations of these cases.

PROP. XIV. B. II. In the demonstration of this, some Greek editor has ignorantly inserted the words, “ But if not, one of the two BE “ ED), is the greater: Let BE be the greater, and produce it

Bouk II. "to F,” as if it were of any consequence whether the greater

or less be produced: Therefore, instead of these words, there ought to be read only, “ but if not, produce BE to F."

PROP. I. B. III.

Book 111. Several authors, especially among the modern mathemati

cians and logicians, inveigh too severely against indirect or apagogic demonstrations, and sometimes ignorantly enough; not being aware that there are some things that cannot be demonstrated any other way: Of this the present proposition is a very clear instance, as no direct demonstration can be given of it: Because, besides the definition of a circle, there is no principle or property relating to a circle antecedent to this problem, from which either a direct or indirect demonstration can be deduced: Wherefore it is necessary that the point found by the construction of the problem be proved to be the centre of the circle, by the help of this definition, and some of the preceding propositions : And because, in the demonstration, this proposition must be brought in, viz. straight lines from the centre of a circle to the circumference are equal, and that the point found by the construction cannot be assumed as the centre, for this is the thing to be demonstrated: it is manifest some other point must be assumed as the centre: and if from this assumption an absurdity follows, as Euclid demonstrates there must, then it is not true that the point assumed is the centre; and as any point whatever was assumed, it follows that no point, except that found by the construction, can be the centre, from which the necessity of an indirect demonstration in this case is evident.

PROP. XIII. B. III.

As it is much easier to imagine that two circles may touch one another within in more points than one, upon the same side, than upon opposite sides, the figure of that case ought not to have been omitted; but the construction in the Greek text would not have suited with this figure so well, because the centres of the circles must have been placed near to the circumferences: On wliich account another construction and demonstration is given, wbich is the same with the second part of that which Campanus has translated from the Arabic, where, without any reason, the demonstration is divided into two parts.

PROP. XV.

B. III.

Book III.

The converse of the second part of this Proposition is wanting, though in the preceding the converse is added, in a like case, both in the enunciation and demonstration; and it is now added in this. Besides, in the demonstration of the first part of this 15th, the diameter AD (see Commandine's figure) is proved to be greater than the straight line BC by means of another straight line MN; whereas it may be better done without it: On which accounts we have given a different demonstration, like to that which Euclid gives in the preceding 14th, and to that which Theodosius gives in Prop. 6, B. 1, of his Spherics, in this very affair.

PROP. XVI. B. III.

In this we have followed neither the Greek nor the Latin translation literally, but have given what is plainly the meaning of the proposition, without mentioning the angle of the semicircle, or that which some call the cornicular angle, which they conceive to be made by the circumference and the straight line which is at right angles to the diameter, at its extremity; which angles have furnished matter of great debate between some of the modern geometers, and given occasion of deducing strange consequences from them, which are quite avoided by the manner in which we bave expressed the proposition. And in like manner, we have given the true meaning of Prop. 31, B. 3, without mentioning the angles of the greater or less segments. These passages, Vieta, with good reason, suspects to be adulterated, in the 386th page of his Oper. Math.

PROP. XX. B. III.

The first words of the second part of this demonstration, “ rexnec in din nánov," are wrong translated by Mr Briggs and Dr Gregory “ Rursus inclinetur;" for the translation ought to be « Rursus inflectatur," as Commandine has it: A straight line is said to be inflected either to a straight, or curve line, when a straight line is drawn to this line from a point, and from the point in which it meets it, a straight line making an angle with the former is drawn to another point, as is evident from the 90th Prop. of Euclid's Data: For this the whole line betwixt the first and last points is inflected or broken at the point of inflection, where the two straight lines meet: And in the like

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