I NOTE S, &c. DEFINITION I. BOOK I. T is neceffary to confider a solid, 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 exist 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 solid, it 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 of the folid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the folid BM, becaufe if this folid be removed from H G M E FN D 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. 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 line BC which is the boundary of the fuperficies ABCD T Book I. 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 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 fuperficies has no thicknefs, as was fhewn; 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. 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 must either be part of the 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, line and fuperficies are to be understood. DEF. 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 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 defigned 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 the words 'selas; that is, in a straight line, or in the fame direction, be plain, when two ftraight lines are faid to be in a straight 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 segment, 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 lefs fkilful Editor. on which account, efpecially fince they are quite useless, these Definitions are distinguished from the reft by inverted double commas. DEF. XVII. B. I. The words "which alfo 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 must coincide, else the straight lines from the center to the circumference would not be all equal. the fame thing is easily deduced from the 31. Prop. of Book 3. and the 24. of the fame; from the firft of which it follows that femicircles are fimilar fegments of a circle. and from the other, that they are equal to one another. DEF. XXXIII. B. I. This Definition has one condition more than is neceffary; be cause 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 another; as alfo AD and BC. join BD; the two fides AD, DB are equal to the two CB, BD, and the bafe AB is equal to the bafe CD; therefore by Prop. 8. of B C Book 1. the angle 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 alfo the angle ADC is equal to the angle ABC. Book I. Book I. And if the angle BAD be equal to the oppofite angle BCD, and the angle ABC to ADC; the opposite 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 A half of all the four angles; that is, BAD B C and ADC are equal to two right angles. and therefore AB, CD are 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. 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 necessary to the Demonftration of this cafe, can be of no use at all in the Elements, or any where else, but in this Demonftration; 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 fome Propofition betwixt the 5. and 13. but none of these stand in need of it, except the 7. Propofition, on account of which it has been added. befides the tranflation from the Arabic has this case explicitely demonstrated. and Proclus acknowledges that the fecond 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 7." as if the case of the 7. which is left out, were, as he expressly 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 enuntiation of this 7. Prop. fo as to preferve the very fame meaning; the literal tranflation from the Greek being extremely harsh, and difficult to be understood by beginners. |