H TT is neceffary 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 arife from a folid, and exift in it: The boundary, or boundaries which contain a folid are called fuperficies, or the boundary which is common to two folids which are contiguous, or which divides one folid into two contiguous parts, is called a fuperficies: Thus, if BCGF be one of the boundaries which contain the folid ABCDEFGH, or which is the common boundary of this folid, and the folid BKLCFNMG, and is therefore in the one as well as the other folid, is called a fuperficies, and has no thickness: For if it have any, this thickness must either be a part of the thickness of the folid AG, or the folid BM, or a part of the thickness of each of them. It cannot be E part of the thickness of the folid BM; because, if this folid be removed from the folid AG, the fuperficies BCGF, the boundary of the folid AG, remains ftill the fame as it was. Nor can it be a part of the thickness of the folid AG; because, if this be removed from the folid BM, the fuperficies BCGF, the boundary of the folid BM, does nevertheless remain; therefore the fuperficies BCGF has no thickness, but only length and breadth. A D G M F N B K 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 fuperficies into two contiguous parts: Thus, if BC be one of the boundaries which contain the fuperficies ABCD, or which is the common boundary of this fuperficies, and of the fuperficies 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 fuperficies KBCL, or part of each of them. It is not part of the breadth of the fuperficies KBCL; for, if this fuperficies be removed from the fuperficies 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 fuppofed to have, be a part of the breadth of the fuperficies ABCD; becaufe, if this be removed from the fuperficies KBCL, the line BC which is the boundary of the fuperficies 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 fhewn; therefore a line has neither breadth nor thickness, but only length. E H G M C B K D The boundary of a line is called a point, or a point is the common boundary or extremity of two lines that are contiguous; Thus, if B be the extremity of the, line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length: For, if it have any, this length muft either be part of the length of the line AB, or of the line KB. It is not part 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 neverthelefs 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 thicknefs. And in this manner the definitions of a point, line, and fuperficies are to be understood. A DE F. VII. B. I. Inftead of this definition as it is in the Greek copies, a more diftinct one is given from a property of a plane fuperficies, which is manifeftly fuppofed in the elements, viz. that a ftraight line drawn from any point in a plane to any other in it, is wholly in that plane. DE F. VIII. B. I. It feems that he who made this definition defigned that it fhould comprehend not only a plane angle contained by two ftraight lines, but likewife the angle which fome 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 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 ftraight 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 fhall be greater than any given ftraight line. Let AB, AC be two ftraight lines which make an angle with one another, and let AD be the given ftraight line; a point may be found either in AB or AC, as in AC, from which the perpendicular drawn to the other AB fhall be greater than AD. In AC take any point E, and draw EF perpendicular to. AB; produce AE to G, fo 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. angle: Draw GK perpendicular to AB; and because the straight lines FK, HG are at right an A gles to FH, and N that is, to the double of FE. b B F K M In the fame manner, if AG be produced to L, fo that GL be equal to AG, and LM be drawn perpendicular to AB, then LM is double of GK, and fo on. In AD take AN equal to FE, and AO equal to KG, that is, to the double of FE, or AN; alfo 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 fo 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 ftraight 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 fome ftraight line. U Bifect Book I. a 15. I. Bifect AC in F, and draw FG perpendicular to AB; take CH in the ftraight 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 FAG, that is, EAB is equal to the b 4. 1. angie FCH; wherefore the angle AGF is equal to CHF, and AFG to the angle CFH: To thefe laft add the common angle AFH; therefore the two angles AFG, AFH are equal to the two angles CFH, HFA, which E GA B Ε two laft are equal together to two CH C13. 1. right angles, therefore allo AFG, D d14. 1. AFH are equal to two right angles, and confequently GF and FH are in one ftraight 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. a 23. I b13.1. PROP. 5. If two ftraight lines AB, CD be cut by a third ACE fo as to make the interior angles BAC, ACD, on the fame fide of it, together lefs than two right angles; AB and CD being produced fhall meet one another towards the parts on which are the two angles which are lefs than two right angles. E At the point C in the ftraight line CE make the angle ECF equal to the angle EAB, and draw to AB the ftraight line CG at right angles to CF: Then, becaufe the angles ECF, EAB are equal to one another, and that the angles ECF, FCA are together equal to two right angles. the angles EAB, FCA are equal to two right angles. But, by the hypothefis, the MC F K N D angles EAB, ACD are to L gether lefs than two right angles; therefore the angle A OG B H FCA's greater than ACD, and CD falls between CF and AB: And becaufe 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 fhall be greater than the ftraight line CG: Let this point be H, and draw HK perpendicular to CF meeting Book I. AB in L: And becaufe AB, CF contain equal angles with AC on the same fide of it, by Prop. 4. AB and CF are at right angles to the ftraight line MNO which bifects 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, muft neceffarily cut the ftraight line AB. PRO P. XXXV. B. I. The demonftration of this Propofition is changed, because, if the method which is used in it was followed, there would be three cafes to be feparately demonftrated, as is done in the tranflation from the Arabic; for, in the Elements, no cafe of a Propofition that requires a different demonstration, ought to be omitted. On this account, we have chofen 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 Simpfon 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, becaufe, to prove that, the 4. of B. 1. muft likewise be made use of, as may be seen, in the very fame cafe in the 34. Prop. B. 1. it was thought bet ter to make use only of the 4. of B. 1. The ftraight 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 demonftrated to be ftraight lines. A corollary is added from Commandine, as being often used. IN PRO P. XIII. B. II. N this Propofition only acute angled triangles are mention Book II. ed, whereas it holds true of every triangle: And the demonstrations of the cafes omitted are added; Commandine and Clavius have likewife given their demonftrations of these cases. PRO P. XIV. B. II. In the demonftration of this, fome Greek editor has ignorantly inferted the words, "but if not, one of the two BE, |