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NOTES, &c.

DEFINITION I. BOOK 1.

T is necessary to confider a folid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, line, and superficies; for these all arise from a solid, and exist in it: The boundary, or boundaries which contain a folid are called superficies, or the boundary which is common to two folids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies: Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other solid, is called a fuperficies, and has no thickness: For if it have any, this thickness must

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part of the thickness of the solid AG; because, if this be removed from the solid BM, the superficies BCGF, the boundary of the folid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth.

The boundary of a fuperficies is called a line, or a line is the common boundary of two fuperficies that are contiguous, or which divides one superficies into two contiguous parts: Thus, if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies, and of the superficies KBCL which is contiguous to it, this boundary BC is called a line, and has no breadth: For, if it have any, this must be part either of the breadth of the fuperficies ABCD, or of the superficies KBCL, or part of each of them. It is not part of the breadth of the fuperficies KBCL; tor, if this superficies be removed from the superficies ABCD,

the

Book I.

the line BC which is the boundary of the fuperficies ABCD remains the fame as it was: Nor can the breadth that BC is supposed to have, be a part of the breadth of the superficies ABCD; because, if this be removed from the superficies KBCL, the line BC which is the boundary of the superficies KBCL does nevertheless remain: Therefore the line BC has no breadth: And becaufe the line BC is in a fuperficies, and that a fuperficies has no thickness, as was shewn; therefore a line has neither breadth nor thickness, but only length.

The boundary of a line is called a point, or a point is the

common boundary or extremity

of two lines that are contiguous:

HG

M

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of the length of KB; for, if the line KB be removed from AB, the point B which is the extremity of the line AB remains the fame as it was: Nor is it part of the length of the line AB; for, if AB be removed from the line KB, the point B, which is the extremity of the line KB, does nevertheless remain : Therefore the point B has no length: And becouse a point is in a line, and a line has neither breadth nor thickness, therefore a point has no length, breadth, nor thickness. And in this manner the definitions of a point, line, and superficies are to be understood.

DEF. VII. B. I.

Instead of this definition as it is in the Greek copies, a more distinct one is given from a property of a plane fuperficies, which is manifestly supposed in the elements, viz. that a straight line drawn from any point in a plane to any other in it, is wholly in that plane.

DEF. VIII. B. I.

It seems that he who made this definition designed that it should comprehend not only a plane angle contained by two straight lines, but likewise the angle which some conceive to be made by a ftraight line and a curve, or by two curve lines, which meet one another in a plane: But, tho' the meaning of

2

the

gles ACD, ACG are equal to one another, which is impoffible. Book I. Therefore BD is equal to AC; and by this Propofition BDC is a right angle.

PROP. 3.

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 ftraight 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 AE to G, so that EG be equal to AE; and produce FE to H, and make EH equal to FE, and join HG. Because, in the triangles AEF, GEH, AE, EF are equal to GE, EH, each to each, and contain equal angles, the angle a 15.1. GHE is therefore equal to the angle AFE which is a rightb 4. t.

b

angle: Draw GK perpendicular to AB; and because the straight

lines FK, HG

BM

are at right an

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double of FE.

In the fame 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 of 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. 4.

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

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Bifect

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 fide of AC to that on which AG is, and join FH: Therefore, in the triangles AFG, CFH the fides FA, AG are equal to FC, CH, each to each, and the angle

a

FAG, that is, EAB is equal to the

E

a 15. r.

b 4. 1. angie FCH; wherefore the angle
AGF is equal to CHF, and AFG to
the angle CFH: To these last add the
common angle AFH; therefore the
two angles AFG, AFH are equal to
the two angles CFH, HFA, which
two last are equal together to two CH

GA

B

F

D

c 13. 1.

right angles, therefore also AFG,

d 14. 1.

AFH are equal to two right angles, and consequently d 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. 5.

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 two right angles.

a 23. I

At the point Cin the straight line CE make 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 an

other, and that the angles

E

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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 HK perpendicular to CF meeting Book I. AB in L: And because AB, CF contain equal angles with AC on the fame fide 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 confequently the point H is in KL produced. Wherefore the ftraight lines CDH drawn betwixt the points C, H which are on contrary fides of AL, must neceffarily cut the straight line AB.

this

PROP. XXXV. В. І.

The demonstration of this Proposition is changed, because, if the method which is used in it was followed, there would be three cafes to be separately demonstrated, as is done in the tranflation from the Arabic; for, in the Elements, no case of a Propofition that requires a different demonstration, ought to be omitted. On this account, we have chosen the method which Monf. 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 ufe of Prop. 26. B. 1. from which the equality of the two triangles does not immediately follow, because, to prove that, the 4. of B. 1. must likewise be made use of, as may be seen, in the very same case in the 34. Prop. B. 1. it was thought bet ter to make use only of the 4. of B. 1.

PROP. XLV. B. I.

The straight line KM is proved to be parallel to FL from the 33. 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 ufed.

IN

PROP. XIII. B. II.

N this Proposition only acute angled triangles are mention. Book II. ed, whereas it holds true of every triangle : And the de monstrations of the cases omitted are added; Commandine and Clavius have likewise given their demonstrations of these cafes.

PROP. XIV. B. II.

In the demonstration of this, some Greek editor has ignorantly inferted the words, " but if not, one of the two BE, "ED

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