NOTES, ETC. DEFINITION I. BOOK I. It is necessary to consider a solid, 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 solid are called superficies; or the boundary which is common to two solids 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 in the other solid, it is called a superficies, and has no thickness; for if it have any, this thickness must either be H G M a part of the thickness of E the solid AG, or of the solid N BM, or a part of the thickness of each of them. It D! Ic cannot be a part of the thickness of the solid BM; because if this solid be re A B К. moved from the solid AG, the superficies BCGF, the boundary of the solid AG, remains still the same as it was. Nor can it be a part of the thickness of the solid AG; because if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth. The boundary of a superficies is called a line, or a line is the common boundary of two superficies 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 superficies ABCD, or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL: for, if this superficies be removed from the superficies ABCD, the line BC, which is the boundary of the superficies ABCD, remains the same as it was: nor can the breadth thạt 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 because the line BC is in a superficies, and that a superficies 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 D IC any, this A B length must 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 same 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; line, and superficies, are to be understood. ex 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 superficies, 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 straight line and a curve, or by two curve lines which meet one another in a plane : but, though the meaning of the words ta' súdeias, that is, in a straight line, or in the same direction, be plain, when two straight lines are said 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 said to be in the same 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 said of the angle of a semicircle, and the angles of segments, in the 16th and 31st propositions of Book 3, are the additions of some less skilful editor : on which account, especially since they are quite useless, these definitions are distinguished from the rest by inverted double commas. DEF. XVII. B. I. The words 66 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 demonstrates 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 centre to the circumference would not be all equal : the same thing is easily deduced from the 31st Prop. of Book 3. and the 24th of the same; from the first of which it follows, that semicircles are similar segments 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 necessary ; because every quadrilateral figure which has its opposite sides equal to one another, has likewise its opposite angles equal; and on the contrary, A Let ABCD be a quadrilateral figure, of which the opposite sides AB, CD, are equal to one another; as also AD and BC: join BD; the two sides AD, DB are equal to D the two CB, BD, and the base AB is equal to the base CD; therefore, by Prop. 8. of Book 1. the angle ADB is equal to the B angle ČBD; 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 if the angle BAD be equal to the opposite angle BCD, and the angle ABC to ADC; the opposite sides 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 are parallels by Prop. 28. B. 1. In the same manner, AD, BC are parallels: therefore ABCD is a parallelogram, and its opposite sides are equal, by 34th Prop. B. I. PROP. VII. B. I. There are two cases of this proposition, one of which is not in the Greek text, but is as necessary as the other : and that the case left out has been formerly in the text, appears plainly from this, that the second part of Prop. 5. which is necessary to the demonstration of this case, can be of no use at all in the Elements, or any where else, but in this demonstration; 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 some proposition betwixt the 5th and 13th, but none of these stand in need of it except the 7th Proposition, on account of which it has been added : besides, the translation from the Arabic has this case explicitly demonstrated. And Proclus acknowledges that the second 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 7th," |