NOTE S, &c. DEFINITION I. BOOK I. 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 fuperficies; 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 solid 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 muft either be a part of the thickness of the folid AG, or the folid BM, or a part of the thicknefs of each of them. It cannot be a 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; becaufe, 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. H G M F N D C L A 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 superficies 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 fupposed to have, be a part of the breadth of the fuperficies ABCD; because, 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 because the line BC is in a fuperficies, and that a superficies has no thickness, as was fhewn; therefore a line has neither breadth nor thickness, but only length. H G M D C L 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 extre- E mity 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 nevertheless remain: Therefore the point B has no length : And because 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, lipe, and superficies are to be understood. ABK 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 straight line drawn from any point in a plane to any other in it, is wholly in that plane. DE F. VIII.' B. I. It seems 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 straight line and a curve, or by two curve lines, which meet one another in a plane: But, tho' the meaning of the the words 'sudes, that is, in a ftraight line, or in the fame Book I. direction, be plain, when two ftraight lines are faid to be in a ftraight 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 faid to be in the fame direction; at least it cannot be explained in this place; which makes it probable that this definition, and that of the angle of a fegment, and what is faid of the angle of a femicircle, and the angles of fegments, in the 16. and 31. propofitions of book 3. are the additions of fome less skilful editor: On which account, efpecially fince they are quite useless, these definitions are distinguished from the rest by inverted double commas. DE F. XVII. B. I. The words," which also divides the circle into two equal "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 demonftrates 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 muft coincide, elfe the ftraight lines from the centre to the circumference would not be all equal: The fame thing is eafily deduced from the 31. prop. of book 3. and the 24 of the fame; from the first of which it follows that femicircles are fimilar fegments of a circle: And from the other, that they are equal to one another. DE F. XXXIII. B. I. This definition has one condition more than is neceffary; because every quadrilateral figure which has its oppofite fides equal to one another, has likewife its oppofite angles equal; and on the contrary. Let ABCD be a quadrilateral figure of which the oppofite fides AB, CD are equal to one an- A other; as alfo AD and BC: Join BD; the two fides AD, DB are e qual to the two CB, BD, and the bafe AB is equal to the bafe CD; there- B fore by prop. 8. of book 1. the angle C ADB is equal to the angle CBD; 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 Book I. D C And if the angle BAD be equal to the oppofite angle BCD, and the angle ABC to ADC; the oppofite fides 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: 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 áre parallels by prop. 28. B. 1. In the fame manner AD, BC are parallels: Therefore ABCD is a parallelogram, and its oppofite fides are equal by 34. prop. B. 1. B PROP. VII. B. I. There are two cafes of this propofition, 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 second part of prop. 5. which is neceffary to the demonftration of this cafe, can be of no use at all in the elements, or any where elfe, but in this demonftration; because the second part of prop. 5. clearly follows from the first part, and prop. 13. B. 1. This part muft therefore have been added to prop. 5. upon account of fome propofition betwixt the 5. and 13. but none of thefe ftand in need of it except the 7. propofition, on account of which it has been added: Befides, the tranflation from the Arabic has this cafe explicitly demonftrated: And Proclus acknowledges that the fecond part of prop. 5. was added upon account of prop. 7. but gives a ridiculous reafon for it, "that it might afford an answer to objections made "against the 7." as if the cafe of the 7. which is left out, were, as he exprefsly makes it, an objection against the propofition itself. Whoever is curious may read what Proclus fays 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. fo as to preferve the very fame meaning; the literal tranf lation from the Greek being extremely harth, and difficult to be understood by beginners. |